Chapter 4

The Measurement of

Quality Change:

Constructing an Hedonic

Price Index for Computers

Using Multiple Regression

Methods

“Ha poll were taken ofprofessional economists and starisriians in

yey wi ssiemate (and by a wide nrargt ecsrobabitity they would design 7 “ crgint) the

sail of the price indexes to takehdl account of quality ¢ Ranges as

“wag”

the most bnportant defect in theseinde

NATIONAL BUREAU OF ECON!

JOMIC RESEARCH (1961 p 35]

“Vey envnt port af view18 that what the hedonic approach ines toda

is estintate aspects of the budget constraint {acing COUSHNCE .

allowing therebytheestimationof ‘missitg prices whenquali)

maeZVUGRILICHES (1988. D 120)

1) the automobileandairplane businesses had developed like

: 2. wd

the computer business, a Rolls Rovee wouldcost $2.75 and ritfo

ptilfion miles on onegallonof gas ‘And a Boving 767 worldcost just

sian) and circle theglobein 20 munutes on fivegallons of gas. .

“ :TOM FORESTER (1985. pi

iar

In this chapter we develop an understanding of how price indexes areconstructed and interpreted, how quality and price are related, and howmul-tiple regression analysis can be used to account for the effects of qualitychange. We focus attention on the interpretation of estimated coellicients ina multiple regression, including the dummyvariable parameters.

Price indexes play a very important role in today’s economies. Forexample, contracts between employers and employees often contain specificcost-of-living provisions for the automatic adjustment of wagesorsalaries ducto changes in the Consumer Price Index (CPI). Similarly. governmentpiments to seniorcitizens and other recipients of Social Security depend explic-itly on changes over time in the CPL. Many long-term contracts betweensell-ers andbuyers include detailed provisions specifying how fulure transactionsare to be affected by inflation. depending on changes in some particular price

sspeople, and analysts often want to know howmeasures such as sales, output, and costs should be deflated by a price indexin order to obtain “rea! quantity measures that adjust for the effects ofinflation.

Because price indexes are so pervasive in today’s economic:portant that they be constructed with as much reliability and accuracyasis practically possible. Much has been written concerning the largenumber of important conceptual and statistical sampling issues encount-

ered in constructing and interpreting both individual and aggregate priceindexes.

Onevery importantissue concerns howprice indexes should be adjusted

to account properly for quality change. For products such as stylized rawmaterials, the physical specifications and characteristics typically remainunchanged over long periods of time: and for such products, quality changeusually docs not present a problem, However, for some products whose spec-ifications and characteristics evolve rapidly overlime, accounting properlyforquality change becomes a very significantissue. In this chapter we will focuson one aspect of this problem and examine how regression methods can be

to construct quality-adjusted price indexes for individual products, inparticular, we will construct an hedonic price index for computer-relatedproducts using multiple regression procedures.

The chapter proceeds as follows. In Section 4.! we briefly summarizetraditional procedures used in constructing price indexes that attempt toaccount for quality change. In Section 4,2 we examine the relationshipbetween price and quality for a product at a given point in time. We showhowthis price-quality relationship can be analyzed by using regression anal-ysis in which transaction prices are regressed on a numberofexplanatory vari-abies, each measuring an important quality characteristic of the product.Then Section 4.3 we extend the price-quatity analysis into the intertem-poral domain by showing howmultivariate regression analysis can be used tomeasure the extent to which prices have changed over time, adjusting forquality by holdingits level fixed. This procedure involves considerable use of”

103

104° The Measurement of Quatity Cluinge

4.

dichotomous independent variables, more ofien simply called dummy

variables,In Section 4.4 we apply this general frameworkofhedonic price analysis

to thespecific problem of constructing a price index for computers, Comput-

ets are a particularly interesting application, since quality change in this

industry has been very rapid and pervasive. Here we survey and review the

empirical literature to date, including the recent innovations adopted by the

U.S, Bureau of Economic Analysis. In Section 4.8 we discuss a number of

econametri¢ issues: the use of weighted least squares to adjust for heteraske-

dasticity, the choice of functional form, and the conditions under which

parameters can he directly interpreted as estimates of the marginal costs or

marginal consumer valuations of changes in the quality characteristic:

Finally. the exercises presentedin this chapter involve you in the repli-

cation and extension ofclassic results reported in the empirical literature to

date andare based on three very interesting datasets, The first is that gathered

nd employed by Frederick V. Waugh [1928] in his pioneering study of the

effects of quality on vegetable prices. The second data set was constructed hy

Gregory C. Chow [1967] in his seminal study of demand for computers over

the 1985-1965 era. and thethird is a portion of one involving computers and

computerperipheral products constructed by [BM in its recent study in con-

junction with the U.S, Bureau of Economic Analysis.’ The econometric pro-

cedures that you employ in these exercises include multiple regression anil.

ysis, the interpretation of R? and multiple correlation, dummy. variable

hypothesis testing. tests for parameter stability. weighted least squares and.

tests for homoskedasticity, and Box-Cox estimation.

TRADITIONAL PROCEDURES FOR INCORPORATING

QUALITY CHANGESINTO PRICE INDEXES

The traditional procedure for controlling for the effects of quality change on

price is commonly called the “matched model” method. In this procedure the

only prices used in constructing the index are those for models or varieties

that are unchanged in specification between two. adjacent lime periods. The

idea underlying this method is that matching the models ensures that any

differences between the pricescollected forthe two timeperiodssolely reflect

price change, rather than a change in what in fact was purchased. In the

United States the Producer Price Indexes (PPI), used for deflating many com-

ponents of producers’ durable equipment are constructed by using the

imatehed model method.

Twoproblems can arise with the matched madel method(hat ma

vent it from properly accounting for quality change.’ One error can arise:

when the price changes observed for matched models do not accurately rep-

resent the price movements taking place for all models. For example, if new

models of some product are introduced that embady an improved technol.

pres

4.2

4.2. Anatyzing the Price-Quality Relationship 105

ogy. and ifthe price indexis based only on information gained trom fotlowi:prices of the incumbent models until they disappear from the mark Litepossible that asubstantial portion of the overall price change w ld theuncovered by using matched mode) methods, Be would not be

A second error can occur when modelsthatare in fact natidentical arenevertheless matched. This can arise when information on someof the onifications andcharacteristics of the modelsis unavailable or is not takenmioaccount,so that some models that appear to be matches are in feet differ .Alternatively, in some cases the statistical agency may know that the womadels are nat truly identical but may cone¢lude that if the differences .small, it is preferable to make the matchrather than dropthe price i tor naGon from the index entirely, pagenornae

Dealing with cachofthese twopossible errars i es facing a tradeThe stricter the criteria for acceptingtwo modelanet the, in mennumber of models that will be excluded from the price index This i mpliesthat. with the matched mode! method, the more one guards against thesec.ond Possible error (matching not completely identical models), the torelikelyit is that the index will contain thefirst error (inferring incorrect information for unmatched models based on that observed for matched models),

Regression analysis can help considerably in reducingthe severit: of thistradeoft. Afteryears of encouragementfrom a number ofdistinguished economists, in 1986 the U.S. Bureau of Economic Analysis published and incor 7porated intoits official statistics the results of a study on computer prices inwhieh the mittched model method was augmented by a particular t roe ofregression analysis knownas hedonicprice analysis. In one type of edo fcprice analysis the matched model methadis employed whenever the a pro.priate data are available, and then hedonic regression methods are usedtoimpute missing prices for newly introduced or just discontinued models,thereby accounting more fulty for the price changes associated with turnoverof modelsavailable in the market. In another type of hedonic price anal sisthe orice index is estimated directly from a regression equation." In eitherei lonic price analysis forms the basis of the measurement of quality

ANALYZING THE PRICE-QUALITY RELATIGIVEN POINTIN TIME ONSHP ATA

tn this se ‘tion we discuss the relationship between price and qualityata give!point in timehy reviewing some ofthe carly empiricalliterature in terms oftoday’s standards for econometric practice, this pioneering literature le Memuch to be desired, Interestingly, while much recent hedonic price rm marchhas focused on quality-adjusted prices for computers, thefirst hedonie priceequations were estimated without the benefit of any computers whatsoever.Therefore in assessing this carly literature. it is worth bearing jin mind that

106 The Measurement of Quality Change

today's computational hardware and software tools were simply not availablein the 1920s. As weshall sce, however, an important contribution ofthe earlystudiesconsisted ofestablishing the notion thatprice variationsreflect qualitydifferentials.

The early empirical research on measuringtheeffects ofquality on pricewasfor the most part not concerned with implications for price indexes, buthad other purposes in mind, Apparently, the first empirical studyrelatingprice and quality wasthat of Frederick Waugh, an agricultural economist whoin 1927 wrote a paper entitled “Quality Factors Influencing VegetablePrices.” The goal of Waugh's research was, using statistical analysis, “to dis-coverthe important qualityfactors which cause high ar low prices” (1928. p.186). In bis paper, Waugh reported results. based on multiple correlationanalysis. in which he considered the effects of physical characteristics—size.shape. color, maturity, uniformity. and other factors—on theprices ofaspar-agus, tomatoes, and hothouse cucumbers as recorded in daily sales transac-

ons at the Faneuil Hall wholesale market in downtown Boston,Massachusetts.

Waugh had rather practical motivations for his research. Noting thatprofit-maximizing farmers can to some extent adjust both the quantity andquality of commodities produced 1o mect market demandconditions, Waughemphasized the usefulness of his research by stating that (1928. p. 187]

It can be demonstratedthat there is a premiumforctypesof products, and if that premium is more than large enoughto paythe increased cost of growing a superior product, the individual andwill adapt his production and marketing poficies to market deo

Fo eliminate the effects of seasonal and day-to-day changes in prices,

Waugh used as a left-hand or dependentvariable in his regression equationstheratio of the actual price recorded for a wholesale marketlot transaction ofa particular commodity (denoted ?,,) to the average market quotationfor thatcommoditythat day (PA), Call this relative price of the »th lot transaction.for the ith commodityp,,. and note that it is defined as

Pn = PiafPM, (4.1)

Waugh regressed p,, on the physical characteristics of vegetables that hebelieved were related to their actual or perceived quality variations. Althoughhe reported separate results for asparagus. tomatocs, and hothouse cucum-

bers, here we confine our attention to his analysis of asparagus.

Waugh inspected 200 individual lots of asparagus in Boston over thelime period of May6 to July 2, 1927, Apparently, great deal of quality varieation existed amonglots of asparagus, since even on the single day ofJuly 2,prices per bushel box varied from a low of $4.50 to a high of $12.00. Waughcharacterized qualitative differences amonglots as follows:

‘One lot was extra fancy. It was green fromtips to butts, The stalks werelarge. straight and ofuniform size. The bunches were compact

4.2 Anal:

ing the Price-Qualily Relationship 107

io!ws classified

a

x

“junk.” I was white, The stalkswere small. craokedand uneven in size. The bunches were loose. Th vere ja iandanes sc. The butts were jagged and

To quantify the effecis of qualit; i i iily variables on relative prices, W:chose three measures that he believed were reiated to perceived quality, aefirst was the numberof inches of Breen color on the asparagus; hame thisvariable GREEN,The second variable was designed to capture the averagesize of stalks and was measured bythe numberof stalks in he bunches: callthis variable NOSTALKS.Since the standard size of bunches was about 18ounces, the numberofstalks in a bunch increased as their size or diameterbecame smaller. Finally, to capturetheeffect of uniformity. a record was keptof the actual diameter of cach stalk, and then the variation insured

h

y

using th artile coellici is ion; i iant ig the quartile cocflicient of dispersion; denote this variable

Waughis not completely clear on precisel i, t L ly how heperformed his regresstonanalysis. mt it appears 10 have had someproblems, at least by ‘odas'sstandards. Waugh's aim was to esti i iple re 0

standard S to estimate parameters in the multiple regression

Pa = th + 8,» GREEN, + #; » NOSTALKS, 4+ Gy DISPERSE, + u, (2)for # = 1... 200, where the least squares estimatesof 5 :interpreted as representing the partialeffect ofa change onenaafeacteristic onprice, all other quality characteristics being held fixed. Althou, chWaughassumed thai «, was a random disturbance lerm with meanzero, hdid notgive it a specific interpretation, vee. Using what he called the method of mukiple cosrelation analysis,w ‘augh (192! PB. 144]reports the followingestimates of theregression qua-tion parameters in Eq. (4.2), where ¢, is the equation residual: : cae

Pa = Bq + 0.13826 - GREEN, — 1.53394 NOST,le

>

1.53304 - ‘ALKS,~ 0.27553 - DISPERSE, + ¢, (3)

Noestimateis provided for the interceyestimated standard errors and #-statistics.

Waugh commented on his empirical findin, i. Waugi d eS as follows, First,reported that green color is byfar the mast important qualityfactor infersmg asparagus prices in Boston: “Boston wants Breen asparagus” [1928, p,188), Waugh noted that the predicted price per dozenbunches of asparnaesthal were green throughout a length ofsix inches trom the lip was 38.5 centsmore than asparagus having only five inchesofgreen color. The coefficient ofdetermination for the GREEN variable 0.40837, which led Waughtostate that (1928,p, 188) “the iniluence ofthis one factor explained 41 percentof theween in prices foundin the 200 inspections,”

ith respect to size of the asparagus, Waugh (1929.; cl ize of PD. 144] commenterthat on the basis ofhis estimated regression coefficient, for each audition

term fy, nor a i :x Bu, NOF are values given for the

108 ‘The Measurement of Quatity Change

stalk in the bunchtheprice waspredicted to decrease by about4.6 cents per

dozen bunches. In termsofexplanatory power, Waugh reported that the cocf-

ficient of determination for the NOSTALKSvariable was 0.14554, Finally,

concerning uniformity, Waughs analysis revealed that the premium paid for

uniformityin size was small; further, the coefficient of determination for the

DISPERSEvariable wasalso rather small, equaling only 0.02133.

Thesize ofthe relative coefficients of determinationfor these three sep-

arate regressors led Waugh to conclude that the uniformity variable

was about one-seventh as important as size of stalk and one-twenticthas

important as green color in its ¢flect on prices. The three togetherexplained $8 percent of the price variation. The cociticient of multiple

correlation between these three quality factors and prices reccived was.

01.76,"

‘To interpret Waugh's regression results, we focus on two questions.1, were his estimates of the multiple regression coeflicients correctly com-

puted? Apparently not. This issue wil! be dealt with in Exercises 1 and 2at

the end ofthis chapter. Second, did he measure goodnessoffit properly? Here

the answeris clearly no, at Jeast by today’s standards, Waughrelied heav:

on the notion of the coefficient of determination in interpreting which vari-

ables were most important and howthey contributed to explaining variation

in prices.In the multiple regression context it may be tempting to decompose the

R? from an estimated equation into additive components, indicating what

proportion of the variance in the dependent variable is distinctly attributable

tovariations in cachof the right-hand variables.

This temptation should be resisted. The problem with such a decom-

positionis that in general it is impossible to compute unambiguously the indi-vidual contribution of cach right-hand variable to the explanation ofvaria-

tions in the dependent variable. The only exception to th when each of

the right-hand variables is arthogonat to cachofthe other regressors,Coelticients of determination atlempt to measure contribution

ables, but since there is some ambiguity about the definition of coefficient of

determination,it will be useful for us first todistinguish partial from separate

coefficients of determination.Considerfrst the partial coefficient of determination, which attempts to

measure the marginal proportional contribution of a regressor to the expta-

nation of variance in the dependent variable, given afl the other regresser

The partial coefficient of determination for variable X,is computed as the RX

froma regression equationin whichtwosets ofresiduals are regressed agaeachother, one from a regression of ¥ onall X°s except 4, and the other from

ssion of X, on all other X°s.'' The major problem with this measureisthat the sum of the marginal contributions over the variousright-hand vari-ables docs not in general equal the X? from ihe original multiple regressionequation, in part because each ofthe partial coefficients of determinationsiscomputedgiven a diferent set ofother regressors,

4.2 Awatyzing the Price-Quality Relationship tog

buti An alternative measure used by sometoattemptto measure the contri-bution of an individual regressoris called the separate cosfficient ofdeter-mination. Suppose we had a multiple regression equation of the form

Yim Bot BiXy + BXn + BXu b+ + Beky toyfel..ya “4

and denoteleast squares estimates of the A's with b'ste least § s s with 4's. The separate couiliciof determination for the jth regressor is computed as . oelicient

Tih HK -HBy = bh; zSw (4.5)

It may be tempting to sum the separate coefficien f determinatiover all the regressors and interpret the result as the wolalpopestanofvaration “explained” by the regressors, thatis, to interpret this sum as e atingthe Conventional R* goodness-of-fit measure. Unfortunately, however th cis no clear relationship between the conventional R? measure from me tiplerepression equation analysis and the DM?) », mee, hich notion of the coefficient of determination di ‘i ave imind? We doknow that the 58% totalcoefficient of determinerhareported by him equals the sum ofthe coefficient of determination a isfrom the three Separate regressors—0.$8 = 0.41 + 0.15 + 0,02 Moreover,aswehal see in Exercise 2, Waugh apparently computed coefficients ‘ofseermination using Eq. (4.5), that is, he computed scparate coefficients of

It is important to notethat unless the regressors GREEand DISPERSEare pairwise uncorrelated ‘and this in inetvos novtheweethe 0.58 figure calculated as this sum is not equal to the traditional 2? fi mecomputed in today's typical regression programs,"* Hence nat ont was itincorrect for Waugh to attribute the relative importance of contributio }from the regressors GREEN, NOSTALKS,and DISPERSEas equaling 0. aL0.15, and 0.02, respectively, butit was also incorrectfor him to Slate, ‘s wasquoted above, “Thethre [regressors} together explained 58 percent of theprice variation. * An implication ofthis is that without being able to exami °further the original data, it is not clear what stati tical significance canbeattributed wy Waugh's empi ) results. Fortunately, Waugh published hisoriginal data in appendices at the end ofhis 1929 book, and this will cl ableus to analyze them further in Exercises { and 2 at the end of this chapter, .

By today's computational andstatistical standards, therefore, Waugh’analysis was somewhat flawed. However, for our purposes it should be bor :in mind that Woaugh’s particularly valuable and enduring insight was to sethe Dassibility of using statistical methods for assessing the relationshipbetween price and quality at a given point in time, having eliminated ihe

1d

43

The Measurement of Quality Change

effects of seasonal and day-to-day price variations by using relative prices, In

the next section we extend Waugh’s reasoning by examining how quality vari-

ations amongvarieties and over time can be dealt with simultaneously by

using multiple regression analysis,

MEASURING PRICE-QUALITY RELATIONSHIPS

OVER TIME

Although one often observes quality diflerentials among goods at a given

pointin time, quality changes also occur over time, owingin part to the forces

oftechnological change. As we shall nowsee. regressionanalysis, particularly

involving dummyvariables, can help considerably in dealing with quality

adjustment of price indexes aver time, This type of regression analysis has

become known as hedonic price analysis. A briefhistorical background on

hedonic prices may be ofparticular interest here,

In the late $930s in the United States, owing in part to the relatively

Sarge size of General Motors (GM) and the existence of substantial cyclical

unemployment, public policy debates arose in Congress and elsewhere over

whether GM should be required to vary prices ofits automobile models in

order to stabilize production volumes and employmentlevels. Same critics

argued that GM had been using its monopoly power ruthlessly and ¢hat it

could be employed more constructively. As evidence, they noted that over the

1925-1938 timeperiod. the U.S. Bureau of LaborStatistics (BLS) official new

car price index for the CM brands had increased 45%moreover, the index

af GM new earsales demonstrated considerable variability from year to year,

Stung bythis criticism and alarmed at the prospect of such government

intervention, in (938 GM fundeda study by Andrew T. Court af the Auto-

mohile Manufacturers’ Association to assess the effects of auto price changes

onthe total volumeofauto sates,"*Oneofthefirst issues that Court faced was the choice ofthe price vari-

able to be used as an explanatory variable in the demand for automobile equa-

tion. Should it be average list price? If it were, it would not properly account

for the quality improvements over the 1925-1935 time period in the

increased horsepower, comfort, speed, and reliability of the GM models.

Court could have used the BLSofficial new carprice index, but at that time

this price index was based simply an average list prices by brands, and no

account was taken ofclanging specifications and quality improvements." In.

particular, within a brand, no distinction was made between standard, fully

equippedcars andspecial, stripped economy models thal were offered with-

outstarter, battery. generator, and sparetire. This sole reliance on brand cone

trasted with the BLS procedures that were operative at that time for farm

tractors and trucks, the official price indexes of which were computed by using

matched model methods involving certain physical specifications. Appar-

ently, the BLSbelieved that use of matched model methods for automobiles

would have been inadequate because of the complexity ofthe problem.

ys

4.3 Measuring Price-Quatity Relationships 111

__ Court devised an alternative procedure, which he called the hedonicpricing method. Invokingutilitarian philosophies that promoted hedonisticthinking—secking the greatest happiness of the community as a whole—Court [1939, p. 107] defined hedonicprice comparisonsas “those which ree-ognize the potential contribution of any commodity, a motor car in thisinstance, to the welfare and happiness ofits purchasers and the community.”Automobiles, he noted, produce a numberofservices that consumers enjoy.It would be desirable to measure directly the amount of happiness andincreased welfare provided by automobile services, but such quantificationwould, of course, he impossible. However, it might be reasonable ta relate theenjoyment consumers receive from automobiles to physical design and oper-ing characteristics, such as power, 5] i a a ike.oereachasD er, speed, internal room, safety, and the like.

{Inthe case af passenger cars, if the relative importanceto the customer afhorsepower, braking capacity, window arca, seat width, tire size, ete.coutd be established, the data reflecting these characteristics could becom.bined into an index of usefulness and desirability. Prices per vehicledivided by this index of Hedonic content would yield valid comparisonsin the fice of changing specifications.” |

; Following a suggestion made to him by Sidney W. Wil eChief Statistician of the U.S. Bureau of Labor Statistics, Countmaceeaeatnsmeasure the relative importance of automobile characteristics 10 customersbyestimating parameters in a multiple regression equation, based on histar-ical data, by model and by year, on list prices and measures of relevant auto-mobile characteristic specifications.

As an example of Court's thinking, consider the cuse ariautomobile models introduced in three,model years, say, 153sone and1927, Suppose, as did Court. that three characteristics relate to ‘automobilequality: the dry weight ofthe car in pounds (WT), the lengthof the wheelbasein inches (LH), and the advertised horsepower (HP), Denote the price ofmodeljas ?,, and in addition to the intercept term, define two dummy vari-ables, Diosa variable that takes on the value of | if model / is from themade!year 1926 and zero otherwise, and Djoz,,. a variable that takes on thevalue of | if model / is from model year 1927 and zero otherwise. :

Within this context, Court proposed estimati iN s . s stimating the following | ieregression equation: ue redone

In Pe ao to: Diy, + ay Dis, + Be WT,

+ By LH, + ay HP, + oy, (4.6)

where the a's and f’s are unknown parameters to he estimated and w, isassumed to be an identically and independently normally distributed distur-bance term with mean zero and variance o’, Incidentally, the intercept terma refers {o the 1925 year, and so it is necessary to have only 1wo dumm:variables, Dix, and Dysy. even though data from three years are being

112) The Measurement of Quality Change

employed, Data necessary to estimate parameters in Eg. (4.6) consisted of a

numberofobservations on different new car models sald in made! years 1925,

1926, and 1927,The importanceof this hedonic regression equation to the construction

ofprice indexes that account for quality change emerges from the interpre-

tation ofthe regression coefficients, particularly «, and «;, Suppose that two

automobile models were introduced,alike in all quality respects except that

one was introduced in model year 1925 and the other in model year 1926:

denote their commonweight, length, and horsepower as WT", LH*, and HP*,

respectively, and their distinct prices as Pizs and Pi. Since quality-related

characteristics are identical, any price change between the twoyears is a pure

price change, unrelated to quality variations. Theleast squarespredictedprice

for the madelintroduced in 1925 is

th Prone Ay + Ay WT + fy LY + By HPS (4.7)

while the predicted price for the model introduced in 1926is

In Pay = dn + &) + A WT + A: LHS + a HPP (4.8)

where the 4's and A's referto least squaresestimates of « and 3 (obtained, say.froma larger sample), and the ?implies the predicted or fitted log-price.

Since in this example any price change between years 1925 and (926isa puse price change unrelated to quality variations, we can compute the pure

price change simply by taking differences in the predicted priecs af 1926 and

$925, Subtracting Eq. (4.7) fram Eq, (4.8) yields

Th Poy, = Ln Page = & 49)

Hence the estimate of a, represents the estimated change in the natural log-

arithm ofthe price index for new car models from 1925 to 1926, quality beingheld fixed.

If an identical model were also sold in 1927, similar reasoning wouldyield an interpretation ofthe estimateof cr; as representingthe estimatedtotalchange in the natural logarithm ofthe quality-adjusted price index for new

car models over the twoyears from 1925 to 1927, The change in the natural

logarithm ofthe estimated quality-adjusted price index for new cass over the

single year from 1926 to 1927 would therefore be equal to the diflerence

between the estimates of x, and cy.While the above interpretation of «, and «, is based on an example in

which quality tevels have been constrained to be identical in 1925 and $926,this interpretation holds much more generally. Recall that in a multipleregression equation the least squares estimate of a parameter refers to achange in oneright-hand variable, all others being held fixed, In this partic-ular case we can therefore interpret the estimate of «, as reflecting the esti-mated changeinin ?, due 10 the passage of time, holding all other variables—

in this case, includingall quality variables—constant, Thusin a very natural

i

4,3. Measuring Price-Quality Relationships 113

waythe estimated coefficients on the time dummy variables represent changesn the natural logarithm ofthe price index for new cars, quality being heldXC,

Let us now arbitrarily normalize thelevel of the quality. adjusted newcar price index to 1.00 in 1925. To create estimates ofthis price index for1926 and 1927, we take appropriate antilogarithms, that is, we exponentiateas follows;'*

Quality-adjusted price index for 1926: ¢*! (4.19)Quulity-adjusted price index for 1927; e

Incidentally, had Court specified a regression equation in which thedependent variable was P rather than In ?, then the quality-adjusted priceindexes would have been computed as 1.00, 1 + &,, and | + a, for 1925.1924, and 1927, respectively, The reason Court chose the semilogarithmicequation specification overthe linear one was groundedin goadness-of-fit cri-teria; on the basis of a preliminary analysis, Court reportedthat th semilog-arithmic form was foundto give “higher simple correlations,"””

Within the context of Court's hedonic price equationit is possible tolest statistically a numberofinteresting hypotheses. One hypothesis is of theform that “quality does not matter,” thatis, that price variations in no wayreflect quality differences. Notice thatif in Eq, (4.6) #) = A: = 8. = 0, thenvariations in WT. LH, and HP woutd not be associated with changes in In P.Hence the “quality does not matter” hypothesis could be implementedempirically by testing the joint null hypothesis that 6, =f; = 9, = O againstthe alternative hypothesis that 8, * 0, 8; # 0, or 8) # 0,

In addition to this joint test involving three coefficients, one could ofcourse also test thal one particular quality characteristic is notsignificant. Inthe case of Eq. (4.6), for example, one could test the hypothesis that wheelbaselength (LH) does not matter, simply by testing the null hypothesis that #; =O against the alternative hypothesis that 8: # 0, .

A different hypothesis is thatall price variations fram 1925 to 1927 cor-respond with qualityvariations and therefore that during this time span therehas been no change in the quality-adjusted price index, that is, once adjustedfor quality change, there has been no inflation. In the context of Eq. (4.6),such a null hypothesis is a, = a, = 0, while the alternative hypothesis is thatoy * 0 and a, # 0. The null hypothesis that there has been no quality-adjusted price change between only two years, say, 1926 and 1927, could beimplemented by testing e; -- a, = 0 against the alternative a) — a, #0,

Finally, suppose one wanted to measure how much quality change perautomobile had in fact taken place over the 1925-1927timeinterval. Recallthat while prices per automobite had risen considerably over the 1925-1927time period, according to Court,at least someofthis price increase should beattributed ta quality improvements. The quality levels embodied in 1927

14 The Measurement of Quality Change

models as compared to 1925 models could be computed in twodifferent, but

numerically equivalent, ways. This can be seen by subtracting estimates of

fey! Digs + 2 + Dyoy,) from both sides of the estimated version of Eq.(4.6)

asfollows:

InP — f+ Dow — A Dore . ,= dy +B, WT, + ay LH, + Ay BP, 4.01)

To imerpret this equation, let us suppose that wo models were being

compared, model } = A from 1925 and model i = 8 from 1927. Given the

values of the characteristics of these two models and the estimated parame-

ters, ane could focus on and compute eitherside of Eq. (4.11). If one caleu-

lated theleft-handside.in essence the logarithm ofthe quality-adjusted price

index (the logarithmof Eq.(4.10)) would be subtracted from the least squares.

fitted quality-unadjusted price in logarithms. This is equivalent to taking the

logarithm ofthe ratioof the fited quality-unadjusted price to the estimated

quality-adjusted price. The difference between these twois the logarithm of

the estimated quality difference between models A and B.

Alternatively, one could focus on the right-hand side of Bq. (4.11) and

simply substitute into it the values of the WT, LH, and HP characteristics for

models A and B along with the corresponding parameter estimates. The dif

ference between the two model values would then be directly interpreted as

the logarithm oftheir quality ratio.

Note that in either case the hedonic technique converts the “quality

problem” into a quantity measure, Court's methodological contribution to

the construction of quality-adjusted price indexes was therefore a most impor-

tant and significant one, As we shail sec in the next section, much recent

work—including that on computers—has been based on the framework

established by Coust in 1939.

‘The curious reader might wonder what ultimately became of Court's

hedonic price research. In terms of contributing to the public policy debate

involving more cyclical pricing behavior by GM,Court's findings were dra-

matic and significant. While the BLS official new car price index rose 45%

over the 1925-1935 time period, Court's proposed quality-adjusted new car

price index decreased approximately 55%,'* GM officials used these empirical

findings along with other data in developing their argumentthat automobile

manufacturers had already been reducing quality-adjusted prices and thal any

further price decreases designed to stabilize employment would likely lead the

auto manufacturers to the “brink of insolvency.” since the required ‘break-

even volume would be much larger than the price-induced increase in

demandfor new cars,"In termsofgenerating new empirical research, however, it is somewhat

surprising that Court's methodological contributions were not immediately

influential. Indeed, for more than two decades the only subsequent research

4.3 MeasuringPrice-QualityRelationships

ZVI GRILICHES

Father ofModern Hedonic Price Analysis

he ongoing profes-sional interest in

.. lhe use of regres-sion analysis to adjustprice indexes for qualitychangeis due in large part

to the work of Zvi Gril-iches. who in 1961 pre-

sented before hearings inthe U.S. Congress a semi-nal analysis entitled “He-donic Price Indexes forAutomobiles: An Econo-metric Analysis of Quality Change."Griliches’ research is widely recog-nized as having formed the basis ofmodern hedonic price analysis.Zi Griliches was born in Lithua-

nia of Jewish parents in 1930. At ageV1. Griliches was moved, along with

his parents, into the German-occu-pied Kaunas Ghetto, where he re-mained until 5944, when he and hisfather were separated from hismother and were shipped to a work-ing camp of the Dachau concentra-tion system. Orphaned at the age of14, Griliches was liberated by Pat-ton’s 3rd Army in May 1945 and ar-rived in Palestine via the “under-ground railroad”(actually a boat) inSeptember 1947 after a seven-monthinternment in a British camp on Cy-prus, He then spent several years invarious kibbutzim,focusing much ofhis spare time on learning Englishand mathematics, teaching himselfenough to pass an external high

uel equivalence examination in

After studying historyfor a year at the HebrewUniversity. Griliches ap-plied to and was acceptedby the University of Cati-fornia at Berkeley. Gril-iches chose agriculturaleconomics as his majorfield of study because toleave Isracl at that time,one had to be going tostudy something “essen-tial.” Griliches completed

Berkeley's undergraduate degree re-quirements in two years, receiving aB.S. degree with highest honors in19$3 and an M.S. in 1954, andlearned his first econometrics framGeorge Kuznets (Simon's brother).Hethen transferred to the Universityof Chicago, where he studied addi-lional econometrics with Roy Rad-ner, Henri Theil, Trygve Haavelmo,and Carl Christ, His 1957 Ph.D. dis-sertation, “Hybrid Corn: An Explo-tation of the Economics of Techno-logical Change“ (published inEconometrica) earned him the bestpublished research award from theAmerican Farm Economic Associa-tion.

Although Griliches is widely rec-ognized for his work on hedonic priceanalysis, his research has rangedwidely, and today he is equally wellknown for major contributions to theanalysis of productivity growth andtechnical change (see Chapter 9 ofthis book), estimating rates of returnto education (Chapter 5), analyzing

1s

16 The Measurementof Quality Change

patentstatistics, and the econometric

specification and interpretation of

distributed lags (Chapters 6. 7. and

8).Zvi Griliches’initial academic ap-

pointment was at the University of

Chicago. In 1969 he moved to Har-

vard University, where he now is the

Paul M. Warburg Professor of Eco-

nomics. He icaches graduate courses

in applied econometrics and ¢cono-

metric methods and conducts two

seminars, one in econometrics and

the other in labor, He is a Fellow of

the Econometric Society, served asits

President in 1975, was coeditor ofEconometrica for a decade.

recipient of the distinguished John

Bates Clark Medal from the Ameri-

can Economic Association.Heis also

an elected Member of the NationalAcademyofSciences.

To Zvi Griliches. doing empirical

econometrics involves one in a won-

derful research venture that can in-volve numerousfrustrations and sur-

Econometrica article he described

this process, quoting from A. A. |

Mitne’s The House at Pooh Corner as |

follows:

prisingly circuitous paths. In a 1977 |

“How would it be.” said Poohif, as soan as we're out of

ight of this Pit, we try tofindit

again?”What's the good afthat?” said

ailybit.“Well.”said Pooh, “we keep: 1

Jovking for Home andnotfinding.

it, so T thought fF we looked for

this Pit, we'd be sure nat to findit,which would be a Good Thing,

because then we might finding that we weren't looking

chich might be just what wewere looking for. really.”

“Pdon’t sce aiuch sense in tbat”

said Rabbit.“No,” said Pooh humbly, “there

isn'L, But there was going to bewhen I began it, 1'sjust thatsomething happened to it on the

pubtish

ed on hedonic pricing was relatively obscure. con:

ing of several

theoretical notes and empirical illustrations, including one by Richard

Staage:

: on computing a “price per unit of inebriatian” for alcoholic bever-

Court's hedonic multiple regression approach to the construction of

price indexes was finallyrevived in 1961 by Zvi Griliches.” Unlike Court's.

Griliches’ work immediately stimulated a substantial and very influential

body of new research, both theoretical and empirical, that continues to this

day.” Although Court's notion of hedonic prices focused on the demand side,

the post-Griliches research typically envisages hedonic prices as ihe outcome

of shifling supply and demand curves for characteristics. Further. as was

notedin the introduction to this chapter, Griliches’ revival of the hedonic

method has now attained an official status; in 1986 the U.S. Bureau of Eco-

nomic Analysis released official price indexes for computers based in part on

the results of hedonic multiple regressions.””

4.4 Applicationsof the Hedonic Method 117

4.4 APPLICATIONS OF THE HEDONIC METHODTO PRICEINDEXES FOR COMPUTERS

Although Court's hedonic price framework has been applied in a host ofical studies analyzing quality change in various goods and commodities,

it is computers and automobiles that have received the greatest share of alicn-n from applied econometricians. In this section we survey and review var-

idus applications of the hedonic price procedure to the estimation of qual-ity-adjusted price indexes for computers. Computers provide a particularlyinteresting application, since quality improvement in the computer industryhas been rapid and pervasive. Before we begin this survey, however, we firstmake some more general comments on the hedonic pricing procedure.

Implicit in the hedonic price framework is the assumption that thenumerous models and varieties of a particular commodity can he viewed as.consisting of various combinations, bundles, or composites of a smaller num-ber of characteristics or basic attributes. In brief, the hedonic hypothesis is“that_heterogencous goods are aggregationsofcha istics. Moreover,implicit marginalpricesforthe characteristics can be calculated as derivativesof the hedonic price equation with respect to levels of the characteristics

Once one views heterogencous goods as aggregates of individual char-acteristics, it becomes clear that the relationship between the overall bundleprice and the level or quantity oflhe various characteristics need not be con-slant over time, that implicit prices may change over lime. Firms can beviewed as supplying various combinationsof characteristics. and consumerscan be envisaged as demanding them. When supply or demand curves forcharacteristics shift, the implicit price relationships between the overall priceofthe bundle and the individual characteristics might also change.

For example, recall the motivation of Waugh’s research on quantifyingtheeffects of consumers’ valuations of vegetable quality on price. Supposeproducers responded 10 a particularly perceived premium and increased thesupply of, say, green asparagus, Would this changethesize of the market pre-mium for green asparagus? More generally,

Will the premium paid for products ofhighqualitytend to rematacon-5 wc stant, or will it change withvariationsin the proportion ofthe different

gradesof qualities on the market?”

The answerto this question depends, of course, on the underlying supply unddemand relationships for the characteristics. As we shall sce later on, while ingeneralit ficult to recover consumers’utility functions and/or producers’cost Cunctions from the hedonic regression equation, viewing observations of

characteristic combinations as representing the outcomes ofan under-lying and potentially changing supply-demand framework suggests that inempirical implementations, particular care should be given to checkingwhether the parameters of hedonic price equationsare stable over time.

In this context it is useful 10 note very briefly the major changes that

138 The Measurement of Quality Change

have occurred since the 1950s in the computer industry.”Itis traditional 10

characterize the commercial history of mainframe computers as consisting of

three generations. By most accounts thefirst generation of computer history

began with theearliest models sold commercially in the United States, begin-

ning in 1953 and 1954, and ended in about 1959-1960, when models were

introduced in which solid-state transistors replaced vacuum tubes. This sec-

ond generation of computers reigned until about 1964 or 1965, when inte-

grated circuit technology was introduced. The third-generation computers

embodying the integrated circuit technology. exemplified by the IBM 360

series, displayed a considerably improved price-performanceratio relative 10

ils predecessors.In addition to these changes in computer hardware. the marketing of

software, previously bundled with hardware, was altered considerably begin-

ning in the mid-to late 1960s. In particular, by the late 1960s, computer hard-

ware began to become unbundled from sofware and associated maintenance.

Moreover, whereas in the past, bundled computers were primarily rented on

a monthly basis, by around 1968 or so the outright purchase of unbundled

computers became the norm.‘Oneimplication of these computer hardware and marketing changes for

the construction ofprice indexes is the following: If price indexes are to adjust

properlyfor quality change, then it would scem to be importantthat the con-

struction of indexes over the 1965-1972 time period should incorporate the

effects of quality reductions associated with unbundling and that rental prices

should be properly distinguished from purchase prices.

With this as background,let us now examine the pioncering study of

demand for computers by Gregory Chow [1967]. The purpose of Chow's.

study was to explain the growth in demand for computerservices in the

United States over the 1955-1965 time period. in particular to separate out

the effects on demand that would have occurred had technological change

(quality improvements) not occurred from demand changes induced by tech-

nological change. To dothis, it was necessary for Chow to construct a quality-

adjusted price index for computerservices, using hedonic price analysis and

multiple regression techniques. Incidentally, since Chow's data spanned the

time in which computers were rented, his measureofprice is the rental price

per month in thousands ofdollars. Moreover,ail his data come from an era

in which computer software and hardware were bundled.

Recognizing that it would be impossible to obtain measuresofall rele-

vant quality variables, Chow chose but three characteristics and hoped that

any omitted variables would be very highly correlated with the variables that

he included.Thefirst variable chosen was multiplication lime, computed as the aver-

age time required to obtain and complete the multiplication instruction, in

microseconds. Chow expected this multiplication time variable, named

MULT,to have a negative effect on computer rental price. Presumably, the

MULTvariable measures one aspectofthe speed of the computer.

4.4 Applications of the Hedonic Method

GREGORY C. CHOW

A Pioneerin Econometrics

\though GregoryC. Chow is un-doubtedly —best-

known to econometricsstudents as developer ofthe widely used “Chowtest” for parameter cqual-ity in differing data sam-

ples, his contributionshave spanned a widerange of other topics aswell, Chow's 1967 Amteri-can Economic Reviewarelicle on the demand for computers,discussed in detail in this chapter, isrecognized as the first hedonic priceanalysis of quality changes embodiedin computers: his work on using op-limal control theory in dynamic eco-nomic models and his research oncomputational algorithms for econo-metric estimators are also regarded asseminal,

Born in Macau. South China, in1929, Chowemigrated to the UnitedStates in the late 1940s, He receiveda B.A, degree from Cornell Univer-sity in 1951 and M.A. (1952) andPh.D. (1955) degrees from the Uni-versity Chicago, all in economics.Afterinitiating an academic career atthe Massachusetts Institute of Tech-nology (1955-1959) and Cornell

University (1959-1962),Chow moved in 1962 tothe IBM Thomas J. Wat-son Research Center inYorktown Heights, NewYork, where he served asStaff Member and Man-ager of Economic Models.In 1970, Chow returnedto academia, becomingProfessor of Economicsand Director of the Econ-ometric Research Pro-

gram at Princeton University.In the last decade, much of Chow's

attention has focused on economicissues in the People’s Republic ofChina (PRC). He has served as Chair-man of the American Economic As-socii n Commitice on Exchanges

in Economics with the PRC. hastaught in the PRC on numerous oc-casions, holds honorary professor-ships and an honorary degree fromChinese universities, and has writteneconometric theory and economicstextbooks that have been widely readby university students in China.Chowis a Fellow of the EconometricSociety and has served on the edito-rial boards of numerous professionateconomics journals.

19

However, since computers are able to execute a wide variety ofinstruc-

lions, multiplication time might not necessarily be representative of the lyp-ical tasks executed on the computer. Other single-instruction speed measurescould include addition time, but a more preferable measure might be aweighted composite ofall of the instruction execution rates for a typical job

Change 44. Applications af the Hedonic Method 121

The Measurementof Qu:

mix (such as MIPS—millionsofinstructionsper second). Asis nated in the

survey by Triplett [1989], there is some controversy among computer indus-

try experts concerning the choice of speed measure, in part because it is

thought that some measures favor one computer architecture (and therefore

the manufacturer adoptingit) over another.”

The second characteristic chosen by Chow as a measure of quality was

memory size, called MEM,which was computed as the product of the num-

ber of words in the main memory (in thousands) and the number of binary

digits per word, with allowances made for different types ofdigits.” Chow

expected MEM(tobe positively related to quality and so expected the esti-

mated coefficient of MEMin the hedonic price equation to he positive. |

The third characteristic used by Chowwas also related to speed and

involved the average time required by the computer to retrieve information

from the memory. This access time variable is named ACCESS. Chow pre-

dicted that the effect of ACCESSon rental price would be negative.

Chowobtained data for his study from a number ofsources cluding

a published U.S. governmentsurvey of computer rentals and characteristics

and issues of the monthly magazine Computers and Automation. For cach

yearfrom 1955 to 1959, Chow obtained data on nine to eleven models rented

(but not necessarily newly introduced)that year. while from [960 to 1965 his

data included only newly introduced models, which varied from 3 sample of

10 new models in 1960 to 18 in 1964.

In termsofempirical results, Chow reported that addition time was con-

sidered as a measure of speed instead of MULT but thatresults with it were

slightly inferios, The estimated coefficients obtained by Chowbased on sep-

arate, year-by-year regressionsare reproduced in Table 4.1."

Chow comments onhis results as follows:

Numberof

Observations

t

0.0461

O.1476

0.0972,

0.2414

0.0794

V.0978

0.2518

0.1476

KR

0.947

U.R9

U.

0.941

0,976

0.99

3

0.943

O44

0.89

5

OR77

0.908

3

zE&

2 %g

2= (0

.078

3)

(0.1

228)

{0,0565) =

O.1617

>

(0.0

395)

O42

OR21)

nputere”

draeit

canEsemarn

Reston

Vol

S7.Ni

0.4297

(0.1797)

0.5507

(0.1078)

(0

.073

2)

0.6867

(0.0

754)

0.5778

M4495

(0.0

697)

Memory

Judging from the I cross section regressions for the individual years. |

memorysize has a larger coefficient (in absolute value) than either accesstime of multiplication time. Three of the coefficients of multiplicationtime, andone coefficient ofaccess time, have wrong signs, though they aresmall fractions of their standard errors, While the standard errors are largefor manycoefficients, as a result of the high correlations among the threeexplanatoryvariables and ofthe small sample sizes. the orders of magni- i

tude of the three coefficients do not appear to have changed drastically 1through time. Note also that the intercent tends to be smallerfor later :

years, but its decline is far from being uniform.”

ine

0.0108

0.0549

(0.1

596)

0.0171

—4891-Dy

g&

€ (0.0729)

0.0786 z

Ss(0

.0690)

(0,0284)

{0.0

891}

(0.0525

-0.0654

2

4¢1 0

.0675

=2Ss

t

Uechno

logica

lCha

ngea

ndthe

Dematf

zs

O48Ele

e

Sete

Pau,

Das,

Oay

Usa.

andPx

y atedu

mmy

vanabl

esor

theyea

rs19

61-1

465

‘Sourc

e:Da

tafr

omGre

gor

aB

= 0.140

S

0.542

Chon

Because his pre-1960 data was scarce, and to avoid mixing first- and

second-generation computer models, Chow pooled his dataforthe years 1960

to 1965, constraining the slope coefficients on the three quality variables to

he equal across all years but specifying dummy variable intercept termsfor |

cach year 1961 to 1965. Thetest statistic corresponding to the null hypothesis

that these slope coellicients were equal across the

six

years suggests nonrejec-

tion ofthe null hypothesis; Chow reports a test-statistic of0.74, which is con-

siderablyless than the critical value of the F-statistic at any reasonable level

4 l a

ARY

x

0,003

2404

—0.801

~1590

+£354

EstimatedStandard

Errors

inParentheses(All

variables

innaturallo

gari

thms

)

Year T

able4.1

Chow'sEstimatedHedonic

Pric

eEquations

forComputerServices,

1955-1965:

1953

1956

1937

1958

1959

1960

1961

1962

1963

1964

1965,

Pooled

Run.

1960-1963

ts S The Measurement of Quality Change

of significance, Results fromthis pooted regression are presented in the bot-tom rows of Table 4.1.

A numberofpoints are worth noting. First, in spite of strong correla-

tions amongthe explanatory variables, cachofthe coefficients on the three

quality variables, MULT, MEM,and ACCESS.is statistically significant, the

MEM variable having a particularly targe implied t-statistic (0.5793/0.0354

= 16.36). Second,since the regression invalves logarithms ofboththe depen-

dent and explanatory variables, the estimated coefficients can be interpreted

directly as elasticities. The three elasticities are estimated to be 0.58 for MEM,

—0,14 for ACC and —0.07 for MULT.Third,coeflicients on the dummyvariables decrease uniformly as time increases, reflecting sustained price

declines for computers, adjusted for quality change. An cxaminationofthese

coeflicients reveals that the price decline is particularly large from 1961 to

1962 (corresponding with the substantial market penetration of second-gen-

eration solid-state computers); anot! considerable decline in quality-

adjusted price occurred in 1964, coineiding with price reductions on oxisting

models following the announced introduction of the new IBM Series 360

models,Quality-adjusted price indexcs for computers could be computed in a

number of ways. Chow suggests choosing a base year. say. 1960, and com-

puting the base period quantity of computerservicesas thefitted value of the

regression equation (with the —0,1045 intercept for 1960), using the arith-

metic mean of the model characteristics for that year as measures of MULT,

MEM,and ACCESS. Following the procedure completely analogousto that

described for automobiles in the previous section ofthis chapter beneath Eq.

(4.11), the average quantities of (quality-adjusted) computerservices for other

years are then computed, using the same 1960intercept but replacing 1960valuesofthe characteristics with values for cach of the models intraducedineach year, 1961 to 196S, The quality-adjusted price index for each model

could then be computed simply as the rental price divided by the above

model-specifte quantity-quality index. To calculate an aggregate price index

for the year overall models, Chowsimply employed an arithmetic average

over each of the models introduced that year.An alternative procedure is much more direct and simply invofves tak-

ing antilogarithmsofthe estimated coefficients of the dummyvariables in the

bottom row of Table 4.1. Normalizing the buse year value to unity in 1960,

one obtains a timeseries of quality-adjusted price indexes over the 1960~1965

time period, presented in the middle columnofTable 4.2, This hedonic price

index conyputed directly from the regression coefficients can be comparedwith that obtained using Chow'sfirst suggested procedure, whichis presentedin the final columnofTable 4,2. A comparisonofthese svo columnsindicatesthat in this case the two alternative methods outlined above for computing,quality-adjusted price indexes give very similarresuits,

The above quality-adjusted price indexes can be used 10 compute theiraverage annual growth rate (AAGR}, When this is donefor the directly com-

4.4 Applicntions of the Hedonic Meihod 123

Toble 4,2. Quality-Adjusted Price Indexas for Computers, 1960-1965. Basedon Chow's Dummy Variable Coeticlent Estimates and on Chow’sArithmetic-Weighted Procedure

Estimated Price Index Price Index*Year Coefficient (Antilogarithm) (Arithmetic-Weighted)

1960 1.0000 1.00001961 —0.1398 0.8695 O84 381962 — 0.4891 0,632 0.64141963 0.5938 0.8522 0.53301964 —0,9248 0.3966, 0.3906,1968 . 1,163 O.3125 0.3188,“Enimies in the hos column are taken fiom Chow [1H0.Fante 3p W124) normahoed wr anny an 1860, ~~

puted hedonic price index, one finds that the AAGR is approximately— 20.8%per year, which implies that the quality-adjusted price index for com-puters declined approximately 21%per year aver the 1960-1965 time periad.Such a result is. of course, dramatically different from that implicitly assumedbythe U.S. Bureau of Economic Analysis (BEA) until 1986, namely, that theprice index for computers was constant at 1.000 over the entire 1953-1985time period."

Chow's research on the demand for computers marked the beginning ofa substantial numberofstudies estimating quality-adjusted price indexes forcomputers. These studies have recently been surveyed by Triplett [1989],

Triplett divides the various hedonic price studies on computers intothose using data before 1972 and those using post-1972 data. With respect tothe pre-1972 studies, Triplett notes that none reports anyattemptto deal withthe consequences of unbundling, which began to occurin the late 1960s. Asa result, Triplett argues that the price indexes for the late 1960s and early1970s estimated by the hedonic method quite likely overstated quatityimprovements,althoughthesizeof the bias is unknown.Triplett conjecturesthat it is conceivable that computer pri quality-adjusted, actually roseslightly in the late 1960sor early 1970s,

; Afler critically surveying a number of empirical studies, Triplett com-bines their hedonic price index estimates judgmentally, using weights thatreflect what he calls “best practice” researchcriteria, such as quatity of theunderlying data sources, choice of index computation method used, andinvestigatoreffects, This yields a most interesting set ofresults, reproducedhere in Table 4.3.

UA numberofcommentsare worth noting, First, a striking result in Table4.3 is that the quality-adjusted price of computers declined every year from1953 lo 1972. Second, and even morestriking, by 1972 a computer's quality-adjusted price was approximately only 1%of what it cost when computersfirst came on the market in 1953. Third, the advent of second-generationcomputers in 1958-1959 set off an accelerated rate ofprice decreasesofabout

- The Measurement of Quality Chunge

Table 4.3 Triptett’s “Best Practice” Research Price Index for Computers.

1953-1972 -i ice Index Annual %Price tndex Anmual % . Price i

Year (1965 100) Change Year (1965 = 100) Change| - 963 183.0 -2i61983 1320 19631954 1139 -137 1964 139.0 -42

1955 1010 -3 1968 100.0 -2781986 R62 -147 1966 38.0 6151987 71 18 1967 29 -30.1

1988 689 -94 1968 M3 -971989 59) = 182 1969 M2 -041960 4as = 2d 1970 23.31961 332 -247 1971 IRA1962 239 9 1972 14.8

s

25%per year, which increased dramatically to over 60% when third-genera-n computers were delivered in 1966, Over the t 967-1970 time period these

price decreases slowed considerably: they then picked up again in $975- 1972with the introduction of the IBM Madel 360series, Note. however, that fail-ure {o account for the unbundling of computers implies that the post-1968price indexes maybe biased downward. Fourth and finally, the AAGRofthequality-adjusted price index for computers over the entire 1953-1972 timeperiod is about — 27%, implying that quality-adjusted Price decreases in com-puters over this time period have occurred at a Prodigious rate,

Turning now to the post-1972 time period,we begin by noting that asubstantial number of empirical studies examining computerprice move-ments have been published over this time period, Although Triplett surveysthese in considerable detail. here we confine our allentianto just one hedonicprice study, the IBM analysis published by Cole ct al. [1986]. which inturnformed part of the basis of the official computer price index later publishedbythe U.S. Department of Commerce. Bureau of Economic Analysis.“One aspectof the Cole et al. studythat is worthy of special note was its

focus on the individual components or “boxes” of a computer system, ratherthan on the system as a whole. In particular, Cole et scparatel examined

computerprocessors, intermediate andlarge (hard) disk drives, printers, andgeneral-purpose displays (keyboards and monitors). ; |

Although theyexperimented with a number of alternative functionalforms (e.g., log-log, semilog, and linear), Cole et al. obtained preferred resultswith the log-log specification, For the computer processors the quality char-

acteristics chosen were speed (MIPS. or millions of IBM 370-cquivalentinstructions per second) and memory{in megabytes); for disk drives the char-

4.4 Applicationsof the Hedonic Method 125

acteristics used were speed and capacity. While capacity was relativelystraightforward 10 measure (the numberof Megabytes that could be stored onthe disk drive), speed (in units of kilobytes per second) was computed as |over the sum ofthree clements: average seck time {the average time for theread/write head to locate and arrive atthe correct track on the disk) plus aver-age rotation delay (the time required by the disk to rotate so that the read/write head is'lined upat the correct point on the track) plus the transfer rate(the time required to transfer the data between the drive and the main mem-oryoncethe correctposition on the disk has been located)." Data were takenfrom a numberofsources, including IBM sales manuals and trade and generalPress sources such as Datamation and Computerworld, Prices for the variousmodelsare list prices for a quantity of one and so do not necessarily corre-spond with transactions prices, particularly in the case of large-volumediscounting,

Since they employed a log-log specification, Cole et al. were able tointerpretthecoefficient estimates on the characteristic variables as elasticitiesofprice with respectto qualitycharacteristics. One hypothesis that they exam-ined in detail was what they called homogeneity, by which they meant thehypothesis that the sum oftheelasticities (coefficient estimates) on the qualitycharacteristic variables equaled 1, Underthis type of homogeneity, a doublingin the values of each of the characteristic variables would double the price.Significantly, for each of the four lypes of computer equipment thal Cole etal. examined, the null hypothesis that the characteristics coefficients summedfo | could not he rejected.

Onthebasis oftheir hedonic regression results, Cole ct al. constructeda numberofalternative price indexes. Here we confine our discussion to theirestimates for processors and intermediate-large disk drives, estimates that areteproduced in cotumns1-4 of Table 4.4,

The most striking result that is observed in Table 4.4 is that the tradi-tional matched model procedure for accounting for quality change appears tobe woefully inadequate. For example, while the matched model proceduregenerates an AAGR of —8.5% for computer processors, the hedonic regres.sion procedure yields a much larger(in absolute value) AAGR of — 19.2%:for disk drives, the matched model and hedonic regression procedures yieldprice indexes that grow at an AAGR of —6.9% and — 16.845, respectively,

Finally, in column5 of Table 4.4 we reproduce from Cartwright [1986]the new official U.S. Department of Commerce. Bureau of Economic Anal-ysis, price index for the commodity aggregate of computers and computer-peripheral equipmenttypes, based largely on hedonic regression procedures.”Notethatthis price index continues the steady decline over time first revealedearlier for the 1953~1972 time period in Table 4.3. Over the 1972-1984 era,the AAGRofthis quality-adjusted price index is ~ 13.8%.

Ione combinesthe resuits from Table 4.3 for the 1954- 1972 timeinter-val with those of Table 4.4 for the 1972-1984 epoch, one finds th |. Qualityadjusted, computers that cost $531.88 in 1953 cost only $1.00 in 1982; in

126

45

The Measurement of Quality Change

Toble 4.4 Price indexesfor Processors and Disk Drives Based on the Cole et

al. Study* and New Offclal Hedonic-BasedBEA Price indexesfor

Computers’ * (1982 = 100)

Processors Disk Drives Computers

gy @Q) Q) @ OoMatched Hedonic Matched Hedonic New Official

Year Model Regression Model Regression BEAPrice Index

1972 2141 990.1 2087 4274 408.11973 214.6 1047.5 200.9 429.5 369.3

1974 219.9 R148 134.5 21

1975 2228.9 721 143.4 265.11976 THR (4.0 2311977 499.0 199.7

1978 147.3 2624 BLA 147.0 169,31979 136.4 242.6 107.7 HO 146.21980 115.4 177.2 910 96.2 117.51981 wha 129 92.9 96.6 107.41982 100.0 100.0 100.0 100.0 100.01983 89.7 90.1 86.5 34.3 Al1984 737 712 R31 46.9 68.5

AAGR1972-1984 —85 = 19.2 -6.9 7 16.8 — 13.8

Cole etal. [(986, Fable 7. p. 29]ree Canceright {1986, Tah 1. 0B)

other words, what would have cost more than halfa million dollars in 1953cost only $1000 in 1984. Moreover. since the calculations in Colect al. andCartwright involve only mainframe and minicomputers and exclude personal(micro) computers,it is possible that this price index understates the amountof quality improvement. .

This concludes our survey of empirical hedonic research on priceindexes for computers. We now change ourfocus and considerseveral ccon-ometric issucs that arise in the estimation of hedonic regression equations.

ECONOMETRIC ISSUESIN THE ESTIMATION OFHEDONIC PRICE EQUATIONS

Oursurvey of empirical hedonic research on price indexes for computers hasalready raised a numberofeconometric issues. In this section we briefly con-sider some other important topics.

4.5 Econometric Issues 127

4.5.4 Heteroskedasticity

One econometric issue that has received someattention in the hedonicpriceliterature concerns heteroskeda: y of the disturbance terms. Suppose, forexample, that data consist of average prices of cach computer model and thatthe various models have rather different sales volumes. Let the disturbanceterm wu, in the regression equation

Y= Bot Bo Xt Ay Xy be the Xt Fah

originally involving NV individual computer observations be distributed nor-mally with mean zero and variance a”, butlet the A¢ observations consist ofaverage prices for cach of the Af models, where Af < N. Denote the volumeof sates for the nth madel as S.,. When Af model-specific averaged data abser-vationsare used rather than the original data for cach of the N computers, thevariance ofthe disturbance term wu? is no longer «? but instead equals 4°/S.,.This implies a hetcroskedastic disturbance term,large-volume models havingsmaller disturbance variances than models with low-volumesales.

In orderthat estimates of the parameters be efficient and thatestimatedstandard errors be consistent,

it

is necessary to use generalized or weightedleast squares rather than ordinary least squares. Intuitively, in this case, geneeralized least squares weights the large-volume, low-disturbance-variancemodels mare and the low-volume, high-disturbance-variance modelsless thandoes ordinary least squares. Note, however, that in this particular example,ifonefirst transforms each of the variables(including the vector of ones for theintercept term) by multiplying them by VS,, and then performsordinaryleastsquares on the transformed data, the results obtained will be numericallyequivalent to employing generalized or weighted least squares on the untrans-formed data. For further details on generalized least squares, see your econ-ometric theory textbook.

4.5.8 Choice of Functional Form

Several timesin the previous sections we noted that, on the basis of economicor other theory, fewif any restrictions were placed on the functional form ofthe hedonic price equation. While Waugh considered only the linear form.Court compared the semilog form with the linear one and noted that on thebasis of goodness-of-fil criteria, with his automobile data the semilog formwas preferable, Similarly, Cole et al. compared results based on thelinear,semilog, and log-log specifications and, on the basis of their data for four dif-ferent types of computer equipment. found thatin general the log-log speci-fication was preferred. It should be noted here that a comparison of goodnessof fit from the various functional forms is not necessaril straightforward,since explaining variations in the natural logarithm of price is not the sameas explaining variationsin price.

128 ‘The Measurement of Quality Change

Althougha full discussionis beyond the scope of this chapter, itis worth

mentioninghere bricfly that there is a statistical basis that can be used to com-

parealternative functional forms, and notjust the log-log, semilog. and linear

specifications. This procedure is known as the Box»Cox or Box-Tidwell trans-

formation,"’ The Box-Cox procedure involves transforming a variable from

NX, 10 Xt as follows:

(4.12)Vee Xia' eN

where \ is. a parameterto be estimated. When A = { for all variables (¥ and

the X°s), the functional form is linear, and in the limit as \ — 0 for each

variable, ¥% — In ¥,and the A— In X,, implyinga log-log, specification. One

canalsospecify a different \ for Y; thanforall of the 4, right-hand variubles,

in which case ane would have a mixed specification; if the for Y were

restricted to 0 and thatfor all the A°s to 1,semilogarithmic functional form

woutd result. In the Box-Tidwell procedure the individual 1s are permitted

to have differing A's as well. ;

An important point here is that A need not be restricted to values of 0

‘or | but can take on a wide variety of other values, including those outside

the 0=1 interval.” In fact, in his survey, Triplett [1989] notes that the log-log

specification used in many econometric studies is inconsistent with certain a

priori knowledge ofthe computerindustry. He suggests that valuesof d other

than 0 or ( should alsa be investigated, including negative values,

4.5.C Choice of Variables, MakeEffects, andEffects of Omitted Variables

The hedonic hypathesis essentially involvestreating varicties of products as

alternative bundles of a smaller numberof characteristics, These characteris-

tics should of coursereflect measuresof quality. But how should measures of

quality be chosen? In the case of computers, Cole at al. Feport that the [BM

study group spent considerable effort in understanding the views of computer

industry engincering design and marketing personnel and that on the basis of

such discussionstheysettled on a numberof quality variables. This approach

is attractive, since knowledge of the particular industry is most useful in

choosing measures of quality. ;

As Chow, Cole, and others have freely acknowledged, however, it is

impossible tospecifyall the relevant quality variables entering an hedonic

price equation for computers. In general, unicss the omitted variables are

cither perfectly carrelated or perfectly uncorrelated with cach of the included

variables, the omission of significant quality variables canresult in biased esti-

mates of the parameters, .

Some have argued that one particularly important quality variable that

4.5 Econometric [ssucs 129

is typically omitted from computer hedonic price equations concernsthe reli-ability of the model, or the service and maintenanceif mayrequire, Presum-ably.-this variable is omitted because ofdifficulty in obtaining an accuratemeasure ofit. However, if such a quality were associated with the manufac-(urer of the model, andif it were assumed that this quality effect is constan'over time, then one could incorporateit into the hedonic equation byspeci-fying dummyvariables for various manufacturers. Whenthis is done, esti-matesof the resulting parameters are often called “makeeffects,"

Moregenerally,it is often the case that the quality variables employedin hedonic regression equationsare notin themselves measures ofquality butare presumed to be highly correlated with consumers’ perceptionsofqualities.Inthe automobile study by Court, for example, the weight of the auto modelin pounds was uscd as a measure of quality, Consumers surely do not judgethe quality of an auto by its weight, yet in fact weight has often been highlycorrelated with other quality attributes, such as safety and power,

This raises the issue of how one can properly estimate parameters of thehedonic price equation when the quality attributes are unobserved or aremeasured with error. There is a great deal ofliterature elsewhere in econo-metrics on coping with measurement error or unobserved variables, but thatliterature is only now finally beginning to be applied to the estimation ofhedonic price equations." We can expect much more research on this topicin the future.

4,5.D Identification of Underlying Supply andDemand Functions

Earlier in this chapter it was noted that since price-quality observations canbe viewed as the market outcome of underlying supply-demandrelations forcharacteristics, shifts in either function could result in volatile parameteresti-mates in the estimated hedonic price equation. To be specific, recall thatimplicit prices of the characteristics can be computed as partial derivatives ofthe price of the product with respect to the level of the characteristic. With alinear functional form, these derivatives are simply the parameters, and inthat case the parameters provide direct estimates of the implicit prices. Withother functional forms, such as the semilog or log-log specifications, thesederivatives can depend on levels of the characteristics and the productpriceas well as the parameterestimates. This therefore suggests that tests for param-

ity in the hedonic price equation are closely related to tests for shifisderlying supply-demandrelationships for characteristics.”type of reasoning has led a numberofresearchers to extend the

hedonic price framework and to attempt to estimate, in addition to thehedonic price equation, the underlying supply and demand equations forcharacteristics, However, this more ambitious analysis of hedonic marketsraises a number of important econometric issues.

“The Measurement of Quality Change

Note first that the estimated hedonic price equation contains three

pieces ofinformution: the price of the praduct, the quantities of the charac-

teristics, and the implicit prices of the characteristics. This information makes

it possible to define the budget constraint facing consumers; for example, how

muchofcharacteristic / must be given up in order to obtain more ofcharac-

teristic 4, productprice being held constant. Second, since with many func-

tional formsthe implicit prices dependonthelevelsof the characteristics, the

slope ofthe budget constraint is not necessarily constant, and so the budget

constraintis nontinear. This contrasts with the usual case dealt with in eca-

nomicstexts, in which the budget constraint for productsis linear.

Owingin large part to nonlinearity of the budget constraint, it turns out

that in generalit is very difficult, andat times even impossible, to obtain con-

sistent estimatesofthe supply and demandfunctions underlying the hedonic

price equation. This has been shown in important papers by Sherwin Rosen

[1974] and Dennis Epple [1987]; also see Timothy J. Bartik [1987].

There are, however, some special cases in which one can interpret (hese

implicit prices as reflecting directly the marginal costs of production orthe

valuations of consumers. Suppose first that the supply of characteristics

fixed, that is, supply is perfectly price inelastic. In such a case, demandcurve:

for the various characteristics will intersect with vertical supply curvesat the

implicit prices estimated by the hedonic price equation. Hence ins aUSUS,

implicit prices will reflect the market's valuations ofthe character

addition it is assumed that all consumersare alike, then the implicit prices

retlect valuations of the representative consumer. fn practice. empirical

hedonic studies based on perfectly inclastic supply assumptions have often

dealt with secondhand markets such as those for automobiles, trucks, or

housing.”’Alternatively, suppasethat supply curves for characteristics are perfectly

flat andelastic.firmsare identical, and markets are competitive. In such cases

the demand curves for the various characteristics will always intersect with

the supply curves at an implicit price that is equal to the average cd iter

ginal) costs of production,since with competitive markets, price equals aver-

age and marginal costs.Incidentally, the perfect competition assumption

can berelaxed to that of a constant markupsituation; for an empirical imple-

mentation, sce Makoto Ohta [1975).

Jn general, however,the strong results reported by Rosen, andespecially

by Epple, indicate that while hedonic prices reflect the shape of budget con-

straints, identification of the underlying supply and demand functions for

characteristics is very difficult and that a great dealofcareis requiredin both

the specification and the estimation of equations in order to obtain a valid

inference. The subtleties invalved become even more complexifone specifies

that markets for new products can be in temporary disequilibrium in the

sense that supply prices are not necessarily instantancously adjustedto equal-

ity with consumers’ valuations.”

46

4.6 Hands On with Hedonie Regressions 131

4.5.E Final Comments

In this section we have briefly raised a number of important econometricissue’ that are involved in the estimation and interpretation af parameters inhedonic price equations. Exciting features of the hedonic price framework arethat its practical importance is obvious, it can be implemented relatively eas-ily, and yet it raises a host ofvery significant and difficult econometric issues.Therefore the analysis of hedonic prices provides both beginning andadvanced econometricians with opportunities for challenging research. Withthat as background we now take a hands-on approach andinvolve youin theestimation and interpretation of hedonic price equations.

HANDSON WITH HEDONIC REGRESSIONS,PRICEINDEXES, AND COMPUTERS

The Purpose ofthe following exercisesis to help you gain an empirical under-standing ofhedonic regression equations, of price indexes constructed fromes mated hedonic regression equations, and of the importance ofselectedcharacteristics on prices of computers. Although you might not experiencehedonistic pleasures fromthese exercises involving hedonic regressions, infet they are very useful, and you might even enjoy them!

Exerel se | is particularly enlightening; in it you attempt to replicateWaugh’s pioneering study and find that replication is not possible. This is ausefut (and, unfortunately, not uncommon) experience in applied economel-rics. This leads you to examine his data more carefully. In Exercise 2 youperform a number of simple and multiple regression estimations usingWaugh’s data, and then you examine relationships among their R? measures.as well as coefficients of determination and correlations. ‘

In Exercise 3 you compute a numberofalternative price indexes forcomputers, using Chow's data set, multiple regression techniques, anddummyvariables. You also test an hypothesis involving the specification ofthe MEM variable, Again, you will experience some problemsin attainingcomplete replication of Chow'soriginal findings. Then in Exercise 4 you rep-Heate (with success!) the findings reported by Cole et al. in their analysis ofhard disk drives of computers, findings (hat were later adopted in part by theU.S, Department of Commereeforits official computerprice index. You willtest for homogeneity of the characteristics and will also cxamine allernativeways of modeling price changes over time.; Whether the hedoni¢ coefficients are stable over time is an importantissue; therefore in Exercise 5$ you have the opportunityto test this hypothesis,using a variety of parameter equality tests, including the well-known Chowtest. The application here invalves examining parameterstability over the firstand second generations of computer modets and employs Chow's data, Thenin Exercise 6 you investigate a different way of constructing price indexes

132 The Measurement of Quality Change

based on hedonic equations. namely, using adjacent year regression proce-dures and Chow’s data.

Finally, for students with more advanced backgrounds. in Exercise 7you engage in an examination of choice of functional form, using the Box-Cox estimation method and your choice of either the Waugh asparagus dataor the Chow computerdata.

SaeveeceesIn the data diskette provided (have you made a backup?}, you will find

a subdirectorycalled CHAP4.DAT with data files. In these data files, threedata series are provided. including the asparagus data from the originalWaugh study(in a file named WAUGH).the data series collected and usedby Chow in his 1967 study of demand for computer services (CHOW), and aportion ofthe large data set used by Cole et al. in the IBM studyfor the U.S.Department of Commerce. The Cole et al. data here are those for interme:diate and large (hard) disk drives and are foundin the file named COLE.

Note that hefore the data diskette can be used, the data files must beproperly edited and formatted, depending on the requirements of your par-ticular computer software. For further information, refer back to Chapter 1,

ion 1.3, MAKE SURE THAT ALL DATAFILES TO BE USED IN THEERCISES BELOW ARE PROPERLY FORMATTED.

EXERCISES

EXERCISE 1: Exarnining Wough's 1927 Asparagus Dota

Ina numberofexercises in this book you will be asked lo replicate a rescarch-er's reported empirical findings. The appropriate data will be provided te you.and unless yoursoftware is awed, you should be able to replicate successfullypreviously reported results. In some cases, however, it will not be possible toachieve a complete replication or reconciliation of findings, and this willrequire you to dig further and examine the underlying data more closely, Thatis what we ask you to do in this exercise. The purposeofthis exercise, there-fore, is to involve you in an importantpart ofthe scientific method, namely,to attempttoreplicate others’ empirical findings.

In your data diskette subdirectory CHAPS.DAT is a file calledWAUGH,which contains 200 data points on four variables: (1) the relativeprice per bunchof asparagus, named PRICE:(2) the numberofinches ofgreen color on the asparagus (in hundredths of inches). called GREEN:(3) the numberofstalks of asparagus per bunch, denoted NOSTALKS:and(4) the variation in size (the interquartile coefficient) of the stalks, denotedDISPERSE.“

Exercises 133

{a} Using these data, estimate the parameters of the multiple regressionequation in which PRICEis regressed on a constant term, GREEN,-NOSTALKS,and DISPERSE. Compare the parameter estimates thatyou obtain with those reported by Waugh, reproduced in Eq.(4.3) ofthis chapter. Which parameter estimates differ the most from those ofWaugh?

(b) Since the resultsdiffer, further investigation appears to be warranted. Inhis Appendix, Waugh [1929, Table 4, p. 144] reports summary statisticsof his underlying data. In particular, he reports the arithmetic means ofthe variables PRICE, GREEN. NOSTALKS, and DISPERSE to be90.095, 5.8875, 19.555, and 14.875, respectively. Compute means ofthese variables. Are his statistics consistent with those based on the datain your file WAUGH? Do you have any hunches yel on the source ofthe inability to replicate Waugh's findings?

{c) Waugh’s Appendix also provides stati ‘on the product moments

(variances and covariances) ofthe four variables. as follows:

PRICE GREEN NOSTALKS DISPERSE

1063.64 3430.89 — 100.92 “R245

ABI719 -17M

6hARDISPERSE

Using your computer software and the data provided in the fileWAUGH,compute the moment matrix and compareit 10 Waugh’s, asreproduced above. Notice that the sample variances for the variablesGREENand DISPERSEare very similar to those reported by Waugh."they are not quite as close for NOSTALKS, and are very different forPRICE,Are all your covarianceslarger than those reported by Waugh.or does the relative size vary? Does there appearto be any pattern to thedifferences that might help to reconcile the findings?

(d)} Even thoughit does not appearto be possible to reconcile Waughs datawith his reported estimates ofregression coefficients, are anyofhis prin-cipal qualitative findings concerning the effects of variations in GREEN,NOSTALKS. and DISPERSEaffected? Howdifferent are your findingsfrom his concerning the quantitative effects of one-unit changes in eachofthe regressors on the absolute price per bunch ofasparagus? (To dothese calculations, you will need to refer back to Section 4.2 and willalso need to know that the average market quotation PAM, was $2,782.)Commentalso onthestatisticalsignificanceofthe parameterestimates.

(e) Do you have anyfinal thoughts on why results differ? (Hint: Computeleast squares estimates using his estimated variances and covariances,reproduced above.)

134 The Measurementof Quality Change

ISE 2: Exploring Relationships among F?, Coeficients of

EXERCISE Determination, and Correlation Coeficients

Earlier in this chapter we noted the extensive use Waugh made ofhis esti-

mated coefficients of determination and how he apparently erred in inter~

preting his results, The purpose ofthis exercise is to gain an understanding of

relationships among the various coefficients of determination. R and cor

relation coefficients, as well as to comprehend better the implications of the

extent of correlation among regressors.

a file irect P4,.DATyou have obser-a) Inthe data file WAUGH of subdirectory CHAP >

‘ ions on the variables PRICE, GREEN, NOSTALKS, and DIS-

PERSE. Using your computer software, compute simple correlations

hetweeneachofthese variables, The correlation matrix that you obtain

should be the following:

- PRICE GREEN NOSTALKS DISPERSE

1.00000, 0.74834 0.40656 -O334ad

1.00000 014% 12605

4.00000 0.35003

Which variables are most highly correlated? Which variables are almost

orthogonal? .

(b) Run tee simple regressions, PRICE on GREEN. PRICE on

NOSTALKS, and PRICE on DISPERSE, where cach regression also

includes a constant term. Take the R? fromeack of these three simple

regressions, and compute its square root, Then compare its value with

the appropriate correlation coefficient reported in the first row of the

abovetable, Whyare they equal (except for sign)? Now Supposeya

ct and had inadvertently run the “reverse” regressions, E

orPRICE,NOSTALKS on PRICE. and DISPERSE on PRICE, What

R? measures would you have obtained? Why do they equal those from

‘carrect” regressions? |te) Noice the valueofthe Ré measure from the simple regression of PRICE

on GREEN, computed in part (6). What do you expect We happen lo

this value of R? if you now add the regressor NOSTALKS,that is, run

a multiple regression equation with PRICE on constant, GREEN and

NOSTALKS? Why? Given the correlation between the GREEN and

NOSTALKSvariables shown in the above table, do you expect the

change in R? to be large or small? Why? Run this regression equation,

(d)

(e)

Exercises 135

. and check to see whether your intuition is validated. Then comment onthe change in the R? value from the simple regression of PRICE onGREENor PRICE on DISPERSE when PRICEis regressed on bothGREEN and DISPERSE;is this change consistent with the samplecorrelation between GREEN and DISPERSE? Similarly, what is thechange in the R? value from the simple regression of PRICE onNOSTALKSor PRICE on DISPERSE when PRICE is regressed onboth NOSTALKS and DISPERSE?Is this change consistent with thesample correlation coefficient between NOSTALKS and DISPERSE?Why?In all three cases considered in part (c), the R? from the multiple regres-sion (with two regressorsin addition to the constant) is less than the sumof the R”s from the corresponding two simple regressions, Is the R?fromthe multiple regression equation withall three tegressors (GREEN,NOSTALKS,and DISPERSE)greater thanorless than the sum oftheR? from the three simple regressions? Nore; tt might be tempting to con-clude from this that the R? from a multiple regression with a constantterm and X regressorsis a/wayss tess than or equal to the sum of the R?values from the K simple regressions. However, this is not always.the case, as has been shown in an interesting theoretical counterexample by Harold Watts (1965) and validated empirically byDavid Hamilton [1987]. (See Exercise 3 in Chapter3 in this hook for fur-ther details.)Using Eq, (4,5) in this chapter to compute separate coefficients ofdeter:tuination, based on the regression coefficient estimates reported byWaugh and reproduced in Eq. (4.3), see whether you can replicateWaugh’s reported cocflicient of determination values for he GREEN,NOSTALKS, and DISPERSE variables as 0.40837, 0.14554, and9.02133, respectively. You should be able to replicate Waugh forNOSTALKS and DISPERSE but not for GREEN, Waugh [1929, p.(13] states: "The sumt of the coefficients of determination is7524,indicating that 57.524 percentofthe squared variability in the percent-age prices is accounted for by the three factors studied.” Is this correct?Whatshould Waugh havestated instead? Why?Asin part (a) of Exercise 1, estimate parameters in the multiple regres-sion equation of PRICE on a constant, GREEN, NOSTALKS,and DIS-PERSE.Note the value of the R? from this regression, and then com-pute and retrieve the fitted or predicted values, Now run a simpleregression cquation in which the dependent variable is PRICE and thefegressors include a constam and the fitted value from the previousregression equation, Compare the R? from this regression to ghat fromthefirst regression. Why doesthis result occur? Whyis the valueoftheestimated intercept term zero and the estimated slope coefficient unityin this regression?

136 The Measurement of Quality Change

EXERCISE 3; Constructing Alternative Price Indexes for Computers

Based on Chow's Data

‘The purpose ofthis exercise is to construct a price index for computers using

hedonic regression techniques and then to assess the sensitivity of the Price

index to changes in the specification of the underlying hedonic regression

equation,In the data file CHOW in the subdirectory CHAP4.DAT youwill find

data serics consisting of 137 observations on 11 variables, These variables (not

necessarily in the same orderas they appearin the data file) include the num-

her of newinstallations of that computer model in that year (VOLUMF). the

monthly rental of computers (RENT), the numberofwordsin main memory,

in thousands (WORDS), the numberof binary digits per word (BINARY).

the number of equivalent binary digits (DIGITS), the average time required

to obtain and complete multiplication or addition instructions (denoted as

MULT and ADD,respectively), the average time required to access infor-

mation from memory(ACCESS). the year in which the model wasintroduced

(YEAR), a dummyvariable taking on the valuc of | if the model was man-

ufaetured bythe IBMcorporation (BMDUM), and the number of the obser-

vation (ORDER), These data were provided by Gregory Chow,

(a) Construct the appropriate variables. In particular, take natural loga-

rithms of the variables RENT. MULT, ACCESS, and ADD, and

rename them using the prefix LN (¢.g., LNRENT). Form the variable

for memory space, MEM,as the product WORDS*BINARY*DIGITS,

and then take the natural logarithm of this new MEM variable and call

it LNMEM.Finally, construct the dummyvariables Ds, Dx: Dan Daas

and ys that take on the value of unity if the model was introduced in

the year 1961. 1962. 1963, 1964, or 1965, respectively. and otherwise

are zero. Note thatif a model was introduced in, for example. 1962, the

valueofthe variable YEARforthat observation would be 62. Using the

data from 1960 through 1965, and then from 1954 to 1959, compute

and examine the correlation matrices measuring simple correlations

among the above variables. Are the correlations among variables.

roughlythe same over the two time intervals? Was Chow's concern over

collinearity amongregressors justified?

(hb) Having constructed these data, replicate Chow byestimating parameters

of the multiple regression equation of LNRENTon a constant term,

Day Dax Des Dour Doss LNMULT, LNMEM, and LNACCESS,usingobservations fram 1960 through 1965 (observations 56 to 137 in the

variable ORDER). Untess your software (or data construction proce-

dures} is flawed, you should be ableto obtain the sameresults as Chow

reported, reproduced in the bottom rows of Table 4.1 in this chapter.

Finally. normalize the price index for computers to 1.000 in 1960, take

tc)

(d)

Exercises 137

‘ antilogarithmsof the estimated coefficients on the dummyvariables,and then form a price index covering the 1961~1965 time period. Com-pare this price index series to that reported in Table 4.2.Arguments can be made that for purposes of computational accuracythe number of equivalent binary digits per ward (the productBINARY*DIGITS)might be more importantthan the numberofwordsin memory (WORDS). Show that in Chow's logarithmic specification ofthe variable MEM in the hedonic price equation,the logarithmsofthesetwo variables (BINARY*DIGITS and WORDS)are implicitly assumedto have equalslope coefficient estimates. Now construct a new separatevariable measuring word length, LENGTH = BINARY*DIGITS: thenform the two separate logarithmic variables LNLENGTH andLNWORDS,Next, estimate parameters in the multiple regression equa-tion of LNRENTona constantterm, Dey, 242 Dox Dos Doss LNMULT,LNLENGTH, LNWORDS,and LNACCESS,using observations from1960 through 1965 (observations 56 to 137 in the variable ORDER).Test the null hypothesis that coefficient on LNLENGTHcquals that onLNWORDS,using a reasonablelevel of significance. Is this null hypoth-esis rejected of not rejected? Which specification do you prefer, andwhy?Next construct dummyvariables Du. Du. Do, Dae Doo and Dio thattake on the value | if the appropriate year is 1955, 1956, 1957, 1958,1959, or 1960, respectively. Using the entire sample of 137 observations,estimate parameters in the multiple regression equation ofLNRENT onthe dummy variables Ds through Dy, LNMULT, LNMEM,andLNACCESS. On the basis of values of the estimated dummy variablecoefficients, construct a price index for computers covering the 1954~1965 time period, normalized to unity in 1954. Compare this priceindex with the “best practice” index computed by Triplett, reproducedin Table 4.3. (You might want to renormalize one of the lwoseries sothat the base years are common.)

In Section 4.5.4 of this chapter it was noted that if the differing com-Puter models range considerably in their volumesofsales, and if thepure data reflected arithmetic means over varying amounts ofsales bymodel, then the disturbance terms may be heteroskedastic. Chow col-lected data on the number of computers installed by year for cachmodel: in the data file CHOW this variable is named VOLUME. Usingthe pooled data over the 1960-1965 timeperiod, follow the procedures.outlined in Section 4.5.A, transform the variables appropriately (includ-ing the intercept and dummy variables), and then estimate parametersofthe transformed model data by ordinary least squares. Are anyof theparameter estimates affected significantly? Whatis the effect on theesti-mated standard errors of adjusting for this type of hetcraskedasticity? Isthe rationale for this type of heteroskedasticity correct, since LNRENTrather than RENTis the dependent variable? Why or whynot?

138 The Measurement of Quality Change

EXERCISE 4: Price Indexesfor Disk Drives: A Closer Look at theIBM Study

The purpose of this exercise is to replicate, interpret. and extend selectedresults reported by Coleet al. in the IBM studyofprice indexes for hard diskdrives. Using data consisting of 91 observations on 30 devices marketed by10 vendors over the 1972-1984 timeperiad in the United States, Coleetal.obtained the following results. in which LN refers to the natural logarithm,PRICEis thelist price. SPEEDis 1 aver the sum of average seek time plusaverage rotation delay plus transfer rate. CAPis the capacityof the disk drivein megabytes. the numbers in parentheses are fstalistics, and s is the esti-mated standarderror of the residuals:

LNPRICE, = Dummies + 0.41 LNSPEED, + G461L.NCAP, + ¢,) (5.8)

ot Ri = 0.844 y= 0.051

The data used by Cole et al. are found in the data file COLE within the sub-directory CHAP4.DAT.Thisfile also contains a number of other variables,discussed in the README.DOCfile of the CHAP4.DATsubdirectory, Fur-ther information can also be obtained byreading the article by Cole al. inthe January1986 Survey of Current Business.

Veryimportant note: Aller publishingtheir article, Cole et al. discovered

several data errors, Three data corrections have been madesince the originalresearch was published. These data revisions affect parameterestimates. Thedata nowappearing at the beginning of the COLEfile reflect these corrections,while original values for these data are found in the last part of the COLEdata file. Further informationis available in the README.DOCdatafile ofthe CHAP4.DATsubdirectory.

(a) Using the original data, construct the variables LNPRICE, LNSPEED,and LNCAP, as well as dummy variables by year for cach year from1973 to 1984, Estimate parameters of the multiple regression model ofELNPRICE on a constant, the 12 yearly dummies, LNSPEED. andLNCAP. You should be able toreplicate successfully the Cole ct al.results reproduced above.

(b) Nowconstruct the samevariables using the corrected data, and estimateparameters ofthe same regression equation, Compare the results. Areany of the parameter estimates affected in a substantial manner?

fc) Cole et al. report that in many cases they were unable toreject the nullhypothesis of homogencity, which they interpreted to imply that thesum ofthe estimated regression coefficients on LNSPEED and LNCAPequals unity. Since most computerregression software programs permityou lo print outofretrieve the estimated variance-covariance matrix ofthe estimated coefficients, you should be able to oblain estimates of the

Exercises 139

variances and covariances of these two regression coefficients. Givenyour parameter and variance-covariance estimates [rom part (b), con-struct a 95% confidenceintervalfor the sum ofthe regression coefficientson the LNSPEED and LNCAP variables. (You will nced to use the vari-ance of a sum rule—see anystatistics text to refresh your memory.) Isthe null hypothesis of homogeneity rejected? Why or why not? Com-Ment on yourtest results.

(d) The model specification employed in parts (a) and (b) allows the annualrate of price decrease, quality adjusted, 10 vary over time. A morerestrictive assumption is that this annual rate of price decrease is con-stant. Show that if one replaces the 12 dummyvariables in part (a) or(b) with the single variable TIME, where TIME takes on the value | in1972, 2 in 1973, .... and 12 in 1984, then such a model specificationimposes therestriction that the annualrate ofprice decrease is constant,Generate the TIME variable, and then, using the corrected data and thevariable TIMEin place of the annual dummyvariables, estimate such amodel. Test the null hypothesis of constant rate of price decreasesagainst the alternative hypothesis of differing rates of price decrease foreach year. Do the data support the constani rate of price decreaseassumption? Why or why not?

EXERCISE 5: Assessing the Stability of the Hedonic Price Equation forFirst- and Second-Generation Computers

In this exercise we assess the stability of the hedonic price equation for com-puters over time, One implicit hypothesis underlying the hedonic method isthat goods such as computers can be viewed as the aggregate of a number ofcharacteristics, Since firms supply various computer models embodyingalter-native combinationsofcharacteristics and consumers demand them,the rela-tionship between price and characteristics reflects the outcome of a marketprocess. When dramatic technological changes occur, factorprices vary,or ifconsumerpreferences change,therelationship between the overallprice of thebundle and the individual characteristics might also change. The dita to beused are in the file CHOW within the subdirectory CHAP4.DAT: names ofvariables are described at the beginning of Exercise 3 above.

{a} Conventional wisdom in the computer industry dates the first genera-lionofcomputers as occurring from 1954 to about [959and the secondgeneration as taking place between 1960 and about 1965, Chow[1967,Pp. 1123] reports thalhetested the null hypothesis that the three “slope”coefficients were equal over the 1960-1965 time period and could notreject the nuit hypothesis; his F-test statistic was 0.74, muchless thanthe critical value at any reasonablelevel ofsignificance. Construct theappropriate variables as in part (a) of Exercise 3, and then estimate

140. The Measurement of Quality Change

(by

{c)

(a)

parameters in two models, one in which the slope coefficients are con-Strained to be the sameinall years 1960-1965(a pooled regression) and

the other in which these coelficients are allowedto differ (separate, year-

by-year regressions). You should be able 10 replicate Chow's results,reproduced in Table 4.1 of this chapter. Based on the sums of squaredresiduals from these regressions, test the null hypothesis that the slopecoefficients are equa! over the 1960-1965 time period. Be particularlycarefulin calculating the appropriate degrees of freedom for the test.Form appropriate dummyvariables for cach of the years from 1955 to1959, and thenrepeat part (a) and test the nul! hypothesis that the slopecoefficients are equal over the 1954-1959 cra, first by running a pooledtegression over the 1934-1959 data and then by doing ycar-by-yearregressions. 1954 to 1959. Incidentally, the year-by-year results that youwill obtain for 1955-1958 will differ somewhat from those reported by(Chowandreproduced in Table 4.1; the reason for these discrepanciesisunclear, although for years 1957 and 1958 the CHOWdata set has onlynine observations, whereas in Chow's [1967} originalarticle he indicatesthe use of ten observations for both years.In essence, parts (a) and (b)tested for slope parameter stability withiathe first and the second generations of computers, respectively. To testwhether the hedonic relationship changed between the first and second

generations,it will be uscful to run one additional regression coveringthe entire 1954-1965 time period, namely, a specification in whichLNRENTisregressed on a constant, ycar-specific dummyvariables for

1955 through [965, LYMEM, LNMULT, and LNACCESS. Having

tun this regression, and initially assuming equality of the slope para-

meters within thefirst (1954-1959) and the second (1960-1965) genera-

lions,test the null hypothesis that the slope coefficients ofthe first gen-

eration equal those of the second gencration. Does this result surpriseyou? Why or why not? Next, relax the assumption of slope parameterequality within cach generation, and test the null hypothesis that slopeparameters are equal over the entire 1954-1965 time span against thealternative hypothesis that these slope coefficients varied from year 10year. Note thatcalculation of the appropriate F-statistic requires com-paring the sumsof squared residuais from the 12 separate year-by-yearregressions with that (rom the pooled 1954-1965 regression and thenadjusting by the appropriate degrees offreedom.tnterpret yourresults.

Are the twotest results of part (c) mutually consistent? Why or why not?

The abovetests for parameterstability are in essence a variant of thewell-known Chowtest (yes, the same Gregory Chow who wrote the com-puterarticle). described in most econometrics textbooks. In order thatthis test statistic be valid, it is necessary to assume that disturbances aredistributed independently among models and over time and. perhapsmore important, that their vasiance is constant over time. Chow's esti-mates of the disturbance variance, denoted s, are reproduced in Table

Exercises 141

4.1. Using anyone ofthetests for homoskedasticity described in youreconometrics textbook, test whether the homoskedasticity assumptionis valid within the two generations and between the two generations.What do your results imply concerning the validity of the tests per-formedinparts (a) through (c)?

EXERCISE 6: Using Time-Varying Hedonic Price Equations toConstruct Chained Price Indexes for Computers

The proceduresfor constructing quality-adjusted price indexesfor contputers,based on estimated hedonic price equations discussed in this chapter assuinedthat the slope coefficients were constant over time. In this exercise we relaxthe assumption of constant parameters over the entire data sample andinstead employadjacentyearregression procedures 10 construct chained priceindexes, The data used in this exercise are in the file named CHOW, withinthe subdirectory CHAP4.DAT.andare described atthe beginning of Exercise

(ay Consider the following regression equation, based on data from twoadjacent years, for example, 1954 and 1955:

LNRENT, = a, + 3, DUM, + 3, LNMEM,+ @; LNMULT, + 8, LNACCESS,

where DUM,is a dummyvariable taking on the value of | if modeliwas introduced in the currentyear(say, 1955) and 0 if it was introducedin the adjacent previous year (1934), The estimate of , indicates thechangein the natural logarithm ofthe price from 1954 to 1955, holdingqualityfixed. Sucha regression equation couldbe specified for each pairof adjacent years, such as 1934-1955, 1955-1956, 1956-1957, .,.1964-1965. An attractive feature of the adjacent year regressionapproachis that the slope coefficients are allowed to vary over time.Using the data in the file CHOW, construct the appropriate variables,estimate the I} adjacent year regression equations by ordinaryleastsquares, and then retrieve the [1 estimates of 6, denoted as Biase Brosea sor see Bins. Next, using data covering the entire 1984-1965 timeperiod. estimate the more Uaditional hedonic regression equation inwhich LNRENTis regressed on a constant, 1) dummy variables Dy.10 Diss, LNMEM, LNMULT,and LNACCESS. Compare lo-yearchanges in the estimated coefficients of these 1} dummy variables withthe devels of the Hd, estimates. Whyis it appropriate to compare year-to-ycar changesin the estimated dummyvariable coefficients with levelsof the estimated 3,2? Comment on and interpret anydifferences thatappear to be substantial,

142 The Measurement of Quality Change

{b) Calculate a traditional hedonic price index for computers over the1954-1965 time period, normalized to unity in 1954, by simply expo-nentiating values ofthe estimated coefficients on the 11 dummyvari-ables, Dyas 10 Dias Then construct a chained price index, using thefollowing sequential procedure: For 1955, exponentiate Ares; for 1956,exponentiate the sum (81s; + Brsa); for 1957, exponentiate the sum

(Buss + Biss + Bros). Continue this for each year, until for 1965 thequality-adjusted price index is computed as the antilogarithm ofthe sum(Bross + Brose + Biysr + * + Biaes). Why is such an index called achained price index? Empirically compare this chained price index withthetraditional hedonic price index. Do they differ in any substantial orsystematic manner? Which index do you prefer, and why?

EXERCISE 7: Exploring Alternative Functional Forms for the HedonicPrice Equation Using Box-Cox Procedures

In Section 4.5 it was noted thalthe choice ofa functional form forthe hedonicprice equationis often one made with a priori information having only lim-ited influence, and goodness-of-fit criteria instead play a major role. The pur-pose ofthis exercise is to use the Box-Cox procedure to choose amongalter-native functional forms for the hedonic price equation. To dothis exercise,you will need 10 have access to software programsthat are capable of doingBox-Cox transformations.” The following nomenclature will be adopted: ABox-Cox transformation of the dependentvariable involving a \ variable asin Eq. (4.12) will be called a A, transformation, while a Box-Cox transfor-mation ofthe independentvariables will be called a 4, transformation. Thereare four commion special cases of the Box-Cox transformation:

1.4, = 4. la linear equation (jon x),2.4, = A, = 0: log-log equation (in yon Inv).

RA FOAMS semilog equation (In yon .4.4, = 1,4, = 0: another semilog equation (1 on In_x).

I is importantto note, however, that A, and A, can take on values other than0 or |. This implies that the Box-Cox procedure is not limited to the set oflinear, log-log. and semilogarithmic functional forms.

Forthis exercise, choose and work through either part (a), dealing withthe Waughasparagus data, or part (b), based on the [BMstudyofdisk drives.(Note: If you choose to work with the Waugh data. you will need to deletethe 10 observationsfor which DISPERSE = 0.)

(a) Using the data on asparagus PRICE, GREEN, NOSTALKS,and DIS-PERSEin the data file WAUGHofthe subdirectory CHAP4.DAT.esti-matethe four above special cases of the Box-Cox (ransformation, wherein eachcase the same A, transformation is applied to cachofthe regres-

Notes 143

sors GREEN, NOSTALKS,and DISPERSE.Onthebasis of the samplevalue of the log-likelihood function, which of these four functionalformsis preferred? Now estimate the parameters A, and d,, where >, isstill constrained to be the same forall three regressors but differs from4, Are the estimated values ofA, and X,close to anyofthe fourspecialcases outlined above? Comment onthe signs of the estimated cocffi-cients and the implied shape of the hedonic price function. Using thelikelihood ratio testing procedure,test cach of these four special cases asa null hypothesis, where in each case the alternative hypothesis leavesA, and 4, unconstrained. Finally, test the null hypothesis that A, = A,against the alternative hypothesis that A, # A,.

(b) Data on PRICE, SPEED, and capacity (CAP)for hard disk drives in theUnited States over the 1972-1984 time period are found in the fileCOLEwithin the subdirectory CHAP4.DAT. Form set of (1 dummyvariables, D+, Du... . Dsa. that take on the value of unity if the diskdrive model was introduced in that particular year andis zero otherwise,In this exercise, Box-Cox transformations are performed on PRICE,SPEED,and CAP, but not on theintercept term or the dummy vari-ables, Estimate the four above commonspecial cases of the Box-Coxtransformation, where in each case the same , transformation isapplied to each of the regressors SPEED and CAP. Onthebasis of thesample value of the log-likelihood function, which of these four func-tional formsis preferred? Doesthis result concur with that reported byColeet al., discussed in Section 4.47 Now estimate separately the param-eters A, and A,, where A,is still constrained to be the samefor both sloperegressors. Are the estimated values of A, and , close to any ofthe fourspecial cascs outlined above? Commenton thesigns ofthe estimatedcoefficients and the implied shapeofthe hedonicprice function. Do val-ues of the transformation parameters all lie in the 0-1 domain, or aresome,as Triplett [1989] conjectured,in the negative domain? Using thelikelihoodratio testing procedure,test cach ofthese fourspecial cases asa null hypothesis, where in each case the alternative hypothesis leaves4, and A, unconstrained. Then test the null hypothesis that 4, = A,against the alternative hypothesis that 4, # d,. Finally, briefly outlinehow you would construct a price index series for disk drives based onyour preferred functional form.

CHAPTER NOTES

t. A similar quoteis atiributed by Robert J. Gordon (1989. fn. 2} to the December22, 19K0issue of Forbes magazine, which in urn atiributes it to an unspecifiedissue of Computerworld.

. For a discussion of such issucs and appropriate references, see U.S. Deparimerof Lahor, Bureau of LaborStatistics (1982, Vol. 1]. Franklin M. Fisher and KarlShel [1983]. and Robert A. Pollak [1989].

144 The Measurement of Quality Change

34.

ry

”

10,VW.

13.1

17.tk.

. For a discussion of R?rel

. See, for example, the interpretation ofCourt's findings by

. Sve. for example, Hendrik §, Howthakker [1952], RichardSta

See Rosanne Cole et al, [1986].

For a more detailed discussion ofthese issues, sec Jack E. Triplett [1986] and thereferences cited therein,See Rosanne Cole et at. (1986) and David W. Cariwright [ 1986).

Yet anotherapproach also involves regression analysis hut focuses on changes intheprices of characteristics.

per was part ofWaugh’s Ph.D.dissenation at Columbia University. It built

on carlier work dating back to 1923 and was subsequently published as FrederickV. Waugh [1928]Waugh (1928, p. 18].Since Waugh (1929. p, 144] provides means for the dependent and independentwaniables. it is possible to work backwards and, using his parameter estimates forthe slope coefficients and the fict that the least squares ting always passes throughthe point of means, calculate the estimateof 4. In this case this calculation turnsont to be: 90,095 ~ 0.13826 + $8B,75 4 1.53394 - 19,855 + 0.27853» 1L.R7S

42.789,Waugh (£928, p. 189).A discussion of partial and separate coctlivients of determination is found in,amongothers, Arthur S. Goldberger [1964, pp. 197-200]

nships in multivariate and multiple simple regressionequations, see Arthur S. Goldberger [1963], especially Chapter4.This study was published in Andrew T. Court [1939].

it appears that the BLS price index was based primarilyon data about the FordModel T. a model that wasrelatively unchanged from year toyear. andfor whichthe matched model method worked quite well until the aveclerationof technicalchange in the early 1930s,

$. Courd [1939, p. 107]

If one exponentiates: (4,6) and then takes expected values, the expected valueofthe exponentiated disturbanceterm is nolonger zero but instead equals 0.5a°:

for ussion, sce John Aitchison and James A. C. Brown [1966] or Dale M.Heien [1968]. An implication of the Aitchison-Brownresult is that theof 0.Sa' should be added to each predicted price index before exponent(4.6). This adjustmentis seldom done by empirical researchers, however, perhaps

because it is typically quantitatively insignificant. Furthermore. if one is interestedonlyin examiningdifferences in predicted price indexes hetween for prac-tical purposesthis term will drop out.Court [£939p. 110).

Although a reduction in the growth rate ofprice indexes is common when aneattemptstoadjust for quality change. it is not abways the case that price indexesbasedon hedonic regressionsyield lower growth rates in times of technical prageress than these obtained using traditional matched model techniqnes. For inter-esting countcrexamptes, see Meyer L. Burstein {1961} and Jack E. Triples [1974].

GMofficial in Stephen

M. Dubrul (1939).

(1956), JanTin-hergen [1956], and an obscure paper by William M. Gorman [1957].

+ Sce Zvi Griliches (1971] andIrma Adelman and Zvi Griliches [ [961].» Some of this rescarchin the 1961-1970 decadeis summarized and referencedin

w &

Mu.38

6.|. A decimal digit counted as four binary digits, while an octal digit counted as tree

2

NM.

Si,

Ales. Calculations were done under the assumption that the averuge amount of data

. 0 pri

Notes 145

Zvi Griliches [1971]. This volumealso contains a number of other importantPapers commissioned bythe Price Statistics Committee of the Federal Reserve

Board. A more recentreview is given in Griliches (1988, Pant I).

. It might be noted, however, that for about 20 years previously the BLS had usedhedonic procedures to construct a new housing price index. In private correspon-dence, Jack Triplett has indicated that “the new house price indexis a price index

for characteristics—it is the ratio of current characteristics prices to those prevail-ing in 1982, weighted by the total quantities of characteristics sold.”Waugh (1929. p. 95].For more detailed historical accaunts, see, among others, Stan Augarten [1984].Histarica i ues. particularly those involving mainframe computersand the International Business Machines Corporation, are discussed in FranklinM.Fisher, John J. McGowan, and Joen Greenwood [1983]. Also sce Jack E.Triplett [1989] and Robert J. Gordon [1989].Onthis, alsa see Cote et al. [1986].

binary

. Table 4.1 corrects two typographical errors in Chow[1967, Table 1}. namely, thesigns on the estimated coefficients of the MEMand ACCESSvariables in Chow[1967]are incorrect for the year 1961.

29, Chow [1967, pp. 1121-1123].. Chow reports that results were virtuallyidentical when geometric means replacedthe arithmetic means.

~ As of 1990, the official BEA price index for computers aver the 1953-1969is stillconstant; it is only from 1970 onward that it declines.

¢ Robert J. Gordon(1989).

transferred at one time is 2 kilobytes. Further. if multiple read/write headsappeared on a device, the speed of the device was measured as the speed per setlimes the numberofsets.“The Cartwright deflatoris in effect a Paasche price index with a movingreferenceyear. Fora deflator with fixed 1972 weights bascd on Laspeyres procedures. sceTriplett (1989, Tabie 4.14, p. (96),For further discussion, sce your econometric theory textbook, A compact text

hooktreatmentis given in George G. Judge ct al. (1985, Chapter 20]. Theoriginalarticles are George E. P. Box and David R. Cox (1964) and Cicorge E. P. Box andPaul W. Tidwell [1962], Issues concerning estimation, inference. and computa-tional algorithmsare discussed by. amongothers, John J. Spitzer (1982. 1984].

We correspondence. Ellen R. Dulberger has indicated that in her ownresearch in Dulberger [1986] she employed Box-Cox transformations varying >from010 1 in increments of0.1,

A relatedissue concernstheset of qualityvariables that should be included in thehedonicregression equation.Sincealternative variables are afienhighly correlatedwith one another, parameter estimates frequently vary considerably with thechoice of included variables. One possible procedure, used by Phocbus Dhrymes

[1971] butlittle since then, employs factor analysis or principal components. Dis-cussion offactor analytic methods, however, is also beyondthe scope ofthis chap-ter. on this, see Harry H, Harman[1976].

146 The Measurement of Quality Change

38, For furtherdiscussion, see Fisher, McGowan, and Greenwood (1983). It is worthnoting that makeeffects might also capture manufacturer-specifie pricing policies,

such as failing (o keep up with competitive price reductions.See Dean F. Amel and Ernst 8. Bernt[1986] and the references cited therein.For an interesting application involving the effects ofchanges in gasoline priceson the vahrations offuel-efficiency characteristics of used automobiles, see Ma-koto Ohta and Zvi Griliches [1986].

41. See. for example, Robert E. Hall [1971] as well as the biGriliches (1971, 1988).

42. Fora related application.see Franklin M. Fisher, Zvi Griliches. and Carl Kaysen(1962).Onthis, sce, for example,Duiberger [1986].

a4, The data are taken from Waugh (1929, Append:45, Your computedsample variances may be identic

on whether your computer software computes sample variances bydividing thesquared deviations around the mean by# = 200 or by 2 — t= 199,

Information on modifying conventional noalinear least squares programsto per=form Boa-Cox estima as well as other computationalissues, are discussedin.John J. Spitzer [1982, 1984],

#0,40,

jographics at the end of

43

her. MeGowa

and Cireenwood [1983] and Ellen R,

- Table 1, pp.

46,

CHAPTER REFERENCES

Adetman, Irma and Z+i Ciriliches (1961, "On anIndex of Qu:of the American Statistical Association, 86 ember, 535-548,

Aitchison, John and James A.C. Brown [1966], The Lognormal Distribution, Cam-bridge, England: Cambridge University Press,

Amel, Dean F. and Ernst R. Berndt (1986), “Depreciation the Swedish AutomobileMarket: An Integratian of Hedonic and Latent Variable Approaches.” Cam-bridge, Mass.; Massachusetts Institute of Technology, Center for Energy PolicyResearch. Working Paper MIT El. 86-007WP, March.

Angarten, Stan [1984], Hit fy Bit: An Hlustrated Historyof ComputerTickoor & Fields.

Bartik. Timothy J. [1987], imation of Demand Parameters in HedamicPriceMadels.” Junrnad af Political Ecanomy, 93:1, January, 81-88,

orge E. P. and David R. Cox (1964). “An Analysis of Transtarmations,”* Jour.nalof the Reval Statistical Sucicty, Series B, 26:2, April. 2tt-243.

P. and Paul W. Tidwell [1962], “Transformation of the Independent* Terhnometric November, 531-550,

Meyer | [1961], "Measurement ofQuatity Change in Consumer Durables.”The Manchester School of Econonucs aud Soctal Studies, 29:4, September, 267-279,

Cartwright, David W. {4986}. “Improved Deflation of Purchases of Computers,” Surhaf Current desiness, 66:3, Match, 79.GregoryC. [1967]. “Technological Change and the Demand {or Computers.”

Anwrican nomic Review, 57:5, December, 1117=1130,Cole, Rosanne, ¥.C. Chen, Joan A. Barquin-Stolleman, Ellen Dulberger. Nurhan Hel-

” fonrnad

New York:

Box,

References 147

vacian, and James H. Hodge [1986], “Quality-Adjusted Price Indexes for Com-puter Processors and Sclected Peripheral Equipment.” Surveyof Current Busi-ness, 66:1, January, 41-50.

Court, Andrew T, [1939]. “Hedonic Price Indexes with Automotive Examples.” inThe Dynamics ofAutomobile Demand, New York: The General Motors Corpo-fation, 99-117,

Dhrymes, Phoebus J. [1971], “Price and Quality Changesin Consumer Capital Gaods:‘AnEmpirical Study." Chapter 4 in Zvi Griliches, ed., Price Indexes and QualityChange: StudiesiNewSethods of Measurement, Cambridge, Mass: Harvatd

Dubrul, Stephen M. [1939], “Significance of the Findings.” The Dynamics of Auto-mobile Demand, NewYork: The General Motors Corporation, 123-139.

Dulberger, Ellen R. {1986}, The Application of an Hedonic Model to a QualityAdjusted Price Index far Computer Processors, unpublished Ph.D, dissertation,City University of New York, Department of Economics, Portions of thistation were published as “The Application of a Hedonic Model to a Quality-Adjusted Price Index for ComputerProcessors,” Chapter 2 in Dafe W. Jorgensonand Ralph Landau, eds,. Technologyand Capital Formation, Cambridge, Mass.:MITPress, 1989, pp. 37-75.

Epple, Dennis [1987], “Hedonic Prices and Implicit Markets: Estimating Demand andSupply Functions for Differentiated Products.” Journal of Political Economy,98:1, January, $980,

anklin M. and Karl Shell 1983}, Te Economic Theoryof Prive Indexes:Tuo Essays on the Effects of Taste. Qualityand Technological Change, Canebridge, Mass; MIT Press.

Fisher, Franklin M,, Zvi Giiliches,

Model Changessince 1949," Jotrnal of Political

451,Fisher, Franklin M., John J. McGowan, and Joen E. Greenwood[1983], Falded. Spin-

dled and Mutilated: Economie Analysis and US, vs. LBM, Cambridge, Mass.:MITPress,

Forester, Tom, ed. (1985), The fuformation Technology Revolution, Cambridge.Mass.: MIT Press.

Goldberger. Arthur S. [1964], Keonometric Theory, New York: John WileyGoldberger, Arthur 5, (1968), Topics in RegressionAnalysis, NowYorGordon, Robert J. [1989], “The Postwar Evolution of Computer Prices,

in Dale W.Jorgenson and Ralph Landau, eds. Technology and Capital Forma.tion, Cambridge. Mass: MIT Press. pp. 77-525.

Gorman,William M. [1957]. "A Possible Procedure for Analyzing Quality Differeails in the Egg Market,” lowaAgricultural Experiment Station, Working Paper.

revised, February. Reissued in January 1976 as Discussion Paper No. BA, Lon-don; London Schogt of Economn

Griliches, Zvi [1961], “Hedonic Price Indeses for Automobiles: An EconometricAnalysis of Quality Change,” in ThePrice Statistics ofthe Federal Government,Gieneral Series No, 73, New York: Columbia University Press for the NationalBureau of Economic Research, pp. 137-196. Reprinted in Zvi Griliches, od., Priceindexes and Quality Change: Studies in NewMethnds of Measurement, Cambridge, Mass.: Harvard University Press, 1971, pp. 55-87.

ndCarl Kaysen [1962]. “The Casts of AutomobileEconomy, 10:5, October, 433-

nd Sans.

148 The Measurement of Quality Change

Griliches, Zvi [1971], “Introduction: Hedonic Price Indexes Revisited.” in Zvi Grit.iches, ed., Price Indexes and Quality Change: Shudies in New Methads of Mea-surement, Cambridge, Mass.: Harvard University Press, pp, 3-1

Griliches. Zvi [1988]. Technology, Education, and ProductivityBlackwell.

Hall, Robert E. [1971], “The Measurement of Quality Change (rom Vintage PriceData.”Chapter in Zvi Griliches, ed.. PriceIndexes and QualityChange: Studiesin New Methods of Measurement, Cambridge, Mass: Harvard University Press,pp. 640-271.

Hamilton, David {1987}Always Redundan

Harman, Harry H. [1976],sityof Chie

Hern, Dale M, [1968]

ew York: Basit

"Sometimes Ay + Ayn Correlated Variables Are Notmerican Statistician, 40.2, May, 129-132,

fodern Factor Analysis, Third Ei

Iniver-

Journal of the American

“A Note on Log-Linear RegressiotStanstical Association, 6. September, 1034- (038,

Houthakker, Hendrik §. [1952]. “Compensated Changes in Quantities and QualitiesConsumed.” Reviewof EconomicStudies, 193), 185-164,

Jenlge, George G.. William E. Grilfiths, Ro Carter Hill, Helmut Lutkepohl, andTsoung-Chao Lee [1985]. The Theoryand Practice of Econometrics, Second Edi-tion, New York: John Wiley and Sons,

National Bureau of Economic Research (1961), The Price Statistics of the FederalGovernment, General Series No, 73, New York: Columbia University Press forthe National Bureau of Economie Research,

Ohta, Makoto [1975], “Product Technotogies ofthe US.Industries and Hedonic Price Indexes for Their Products:Approach,” Journal ofPolitical Feommny, 83:1, February, 1

Ohta, Makoto and Zvi Grifiches (1986). “Automobile Prices and Quality: Did theoline Price Increases Change Consumer Tastesin the U,8.2" Journal of Business

4:2, April, 187-198,Pollak, Robert A. [1989]. The Theoryof the Cost-of- Livuig Index, New York: Oxford

University Press.Rosen, Sherwin M. [1974], “Hedonic Prices and Implicit Markets:

tiation in Pure Competition,” fournal of Political Eeonamy, 82:1.tuary, 34-55,

Spitzer, John J. [t982), “A Primer on Box-Cox Estimation,” ReviewofEconomies andMatisties, 64:2, May, 307-313.

Spitzer, John J. (1984), “Variance Estimatemation; Implicationsfor Estimation and Hypothesis Testing.” Reviies andStatistics, 66:4, November, 645-652,

Stone, Richard 11956), Quality and Price Indexes in National Accounts,nization for European Ecanomic Cooperation.

Tinberge (1956), “On the Theoryof Income Distribution.” Welnwirtscharthiches. 155-173, Reprinted in L.H, Klaassen, |..M. Koyck, and HJ. Witte:

veen, eds. Selected Papers afJan Tinhergen, Amsterdam: North-Holland, 1959,pp. 249-263.

Triplett, Jack E. [1971]. “Quality Bias in Price Indexes and New Methods of QualityMeasurement.” Chapter 6 in Zvi Griliches. ed.. Price indexes and QualityChange, Cambridge, Mass,: Harvard UniversityPress, pp. 180-214.

in Models with the Box-Cox Transfor-rafEconomn.

P; Onw-

4

Further Readings 149

Triplett, Jack E. (1986), “The Economic Interpretation af Hedonic Methods,” Survey

5 ofCurrent Busin 1, January, 36-40,Triplett, Jack E. {1989]. “Price and Technological Change in a Capital Good: A Survey

ofResearch on Computers,”Chapter 4 in Date W.Jorgenson and Ralph Landau.eds., Technologyand Capital Formation, Cambridge, Mass.: MIT Press, pp. 127=23

U.S. Departmentof Labor, Bureau of Labor Statistics [1982]. ALS Handbook ofMeth-ods, Vols. LandI, Washington, D.C.: U.S. GovernmentPrinting Office.

Watts, Harold W. {1965}. “The Test-o-Gram: A Pedagogical and PresDevice.” American Statistician, 19:4, October, 25-28.

Waugh, Frederick V. {1928}. “Quality Factors Influencing Vegetable Priceof FarEconomics, 10:2, April, (85-196,

Waugh, Frederick V. [1929], Qualityas a Determinant of Vegetable Prices: A Statis-tical Snadyof Quality Factors Influencing Vegetable Prices in the Boston Wholesale Market, New York: Columbia UniversityPress. Reprinted by the AMSPress,Ine., New York, 1968,

FURTHER READINGS

Gordon, Rober J. {1990}, The Measurement of Durable Guods Prices, Chicago: Uni«versity of Chicago Press.See ¢

Griliches, Zvi ed. [197], Price Indexof Measurement, Cambridge, Miclassic arti

Citiliches, 7:

BlackweLancaster. Kelvin [1971], Consumer Demand: 4 NewApproach, New York: Columbia

University Press. A theoretical framework emphasizing the role of characteristicsandattributes in demand analysi:

phen M, [1986], The Historyof Statistics. The Measurementof Uncertaintybefore 1900, Cambridge, Mass.: Harvard University Press. An enjoyable historicalaccountofstatistical measurementissues.