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Declining Volatility in the US Automobile Industry

By VALERIE A RAMEY AND DANIEL J VINE

The automobile industry is a highly volatilesector of the US economy Motor vehicle pro-duction has accounted for almost 25 percent ofthe variance of aggregate GDP growth over thepast 40 years even though gross motor vehicleoutput represented on average less than 5 per-cent of the level of aggregate GDP1 In themid-1980s however the variance of automo-bile production declined drastically falling bymore than 70 percent relative to its past levelWhile the variance of auto sales also recededthis decline was smaller than the decline inoutput volatility Moreover the covariance ofinventory investment with sales became nega-tive in the 1980s suggesting that inventorieshad begun to more actively insulate productionfrom sales shocks At the same time assemblyplants began to adjust average hours per workermuch more often than in preceding decadeswhen most output volatility stemmed fromchanges to the number of workers attached toeach plant Interestingly many of these changeswere observed outside the motor vehicle indus-try as well2

This paper documents these developments inthe US auto industry and shows how thechanges observed in sales inventories and pro-duction in the 1980s could have stemmed fromone underlying factormdasha decline in the per-sistence of motor vehicle sales We analyzeindustry-level data and micro data on produc-tion schedules from 103 assembly plants in theUnited States and Canada to document thesedevelopments Using the original version of thelinear-quadratic inventory model formulated byCharles C Holt Franco Modigliani John FMuth and Herbert A Simon (1960) we thenshow that a decline in the persistence of salesleads to all of the changes noted above even inthe absence of technological change

I Structural Change in the US AutomobileIndustry The Facts

A Production Sales and InventoryVariances

The levels of monthly US car and truckproduction measured in physical units at sea-sonally adjusted annual rates are shown in thepanels of Figure 1 from January 1967 throughDecember 2004 Sales of domestic vehicles foreach of these market segments are also shownwhere sales of domestic vehicles include vehi-cles assembled in the United States Canadaand Mexico3

As seen in Figure 1 the historical trends forproduction and sales in the car segment havebehaved differently than the trends in the trucksegment Production and sales of cars have de-clined over time while production and sales oftrucks have steadily increased As there are ob-vious differences in the conditional means ofthese two market segments we treat them sep-arately in most of the analysis below

Ramey Department of Economics University of Cal-ifornia San Diego 9500 Gilman Drive La Jolla CA92093-0508 (e-mail vrameyucsdedu) Vine Division ofResearch and Statistics Board of Governors of the FederalReserve System 20th and C Streets NW Washington DC20051 (e-mail DanielJVinefrbgov) We are indebted toEdward Cho and Jillian Medeiros for outstanding researchassistance and to George Hall for his Chrysler data Wehave benefited from helpful comments from two anony-mous referees Nir Jaimovich and Garey Ramey ValerieRamey gratefully acknowledges support from National Sci-ence Foundation grant 0213089 The analysis and conclu-sions set forth are those of the authors and do not indicateconcurrence by the Board of Governors or the staff of theFederal Reserve System

1 The Data Appendix gives details of all calculations andestimates

2 Chang-Jin Kim and Charles R Nelson (1999) andMargaret M McConnell and Gabriel Perez-Quiros (2000)document a 50-percent decline in the variance of GDPgrowth beginning in 1984 John E Golob (2000) shows thatthe covariance of inventory investment and sales for mostindustries switched from being positive before 1984 tobeing negative after 1984 James A Kahn et al (2002)showed that within durable goods manufacturing the vari-

ance of production fell much more than the variance ofsales Ron Hetrick (2000) documents an increase in the useof overtime hours for many industries during the 1990sexpansion

3 The truck market segment includes vans and SUVs

1876

The variances of both car and truck output(relative to their respective trends) droppedsharply in the mid-1980s According to struc-tural break tests on the variance of detrended carproduction there was a statistically significantbreak between February and March in 1984 Fortruck production the break occurred betweenJanuary and February in 19834 Thus the struc-

tural break in the variance of automobile pro-duction occurred at essentially the same time asthe structural break in GDP growth volatilitywhich many studies place in 1984

In order to quantify the changes in industryvolatility that took place around 1984 Table 1reports the variances of key variables in the autoindustry in two sample periodsmdash1967 through

4 One important difference between the time-series prop-erties of physical unit data and chained-dollar data used inother studies is that stationarity tests on the logarithm ofphysical unit variables reject a unit root in favor of adeterministic trend To search for the structural break in thevariance we used data detrended with an Hodrick-Prescott

(HP) filter rather than trend breaks so as not to bias theresults for a particular period In particular we used sea-sonally adjusted output divided by the exponential of the HPfilter trend applied to the log of output The p-values wereessentially zero

1970 1975 1980 1985 1990 1995 2000 20050

2

4

6

8

10

12

14Millions of units annual rate

Car salesCar production

A

B

1970 1975 1980 1985 1990 1995 2000 20050

2

4

6

8

10

12


Truck salesTruck production

FIGURE 1 US AUTOMOBILE PRODUCTION AND DOMESTIC SALES

(January 1967 through December 2004)

Notes Production and sales are measured in millions of units at a seasonally adjusted annualrate Domestic sales include US sales of vehicles built in the United States Canada and Mexico

1877VOL 96 NO 5 RAMEY AND VINE DECLINING VOLATILITY IN THE US AUTOMOBILE INDUSTRY

1983 and 1984 through 2004 The variables ofinterest are derived from the standard inventoryidentity Yt St It where Y is production Sis sales and I is the change in inventories Forstationary variables the following relationshipexists between the variance of production andthe variance of sales

(1) VarYt VarSt VarIt

2 CovSt It

The numbers in Table 1 report each termfrom this variance decomposition for cars andtrucks separately and because production andsales have trends within each sample all quan-tities are first normalized by their respectivetrends The variance of production for cars fellby 70 percent after 1984 for trucks it fell by 87percent Moreover the variance of productionfor both cars and trucks fell by a larger percent-age than did the variance of sales and thecovariance of inventory investment with final

sales either switched from being positive tobeing negative or became more negative

The variances and covariances do not form aperfect identity in part because domestic salesand inventories include imports from Canadaand Mexico while domestic production in-cludes only vehicles produced in the UnitedStates A small portion of US production isalso exported5 Changes in North American mo-tor vehicle trade behavior however do not ap-pear to have explained the structural change inindustry volatility The addendum to Table 1shows the trade-augmented variance decompo-sition for cars where US production is aug-mented with North American imports and USsales are augmented with exports The changesin the variances after 1984 are very similar tothose shown in Table 1 (Trade data for trucksare not readily available)

5 The variance identity also does not hold because thevariables are seasonally adjusted and detrended separately

ADDENDUM COMPARISON OF PRODUCTION AND SALES INCLUDING IMPORTS AND EXPORTS

Cars

19672ndash198312 19842ndash200412

Var(US Production N American Imports) 294 094Var(US Sales Exports) 240 096Var(I) 020 015Cov(S I) 001 018VarY

VarS123 097

TABLE 1mdashDECOMPOSITION OF MOTOR VEHICLE OUTPUT VOLATILITY

Cars Trucks

19672ndash198312 19842ndash200412 19672ndash198312 19842ndash200412

Var(Y) 313 092 916 115Var(S) 245 101 689 094Var(I) 020 015 029 009Cov(S I) 003 020 015 008VarY

VarS128 091 133 122

Notes Monthly data are measured in seasonally adjusted physical units Cars and trucks are measured separately Y production S sales and I change in inventories Production and sales were normalized by the exponential of a fittedlinear trend to the log of the variable estimated separately over each period Inventory investment was normalized by the fittedtrend in the log level of inventories The variances and covariances in the table are 100 times the actual ones (See DataAppendix for data sources and details)

1878 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006

Golob (2000) and Kahn et al (2002) uncov-ered a similar decline in the volatility of aggre-gate chain-weighted durable goods outputSome researchers have linked this decrease involatility to a decline in the inventory-salesratio As shown in Figure 2 however the in-ventory-sales ratio (or ldquodays-supplyrdquo) for lightvehicles shows no evidence of change after19846 Thus the changes we have documentedin the auto industry occurred for some otherreason7

B Intensive and Extensive Labor Margins

High volatility of output in the motor vehicleindustry is often associated with the ways auto-makers adjust work schedules and the rates of

production at assembly plants8 Adjustments tothe workweek are sometimes temporary suchas when a plant schedules overtime hours orcloses for a week (called an ldquoinventory adjust-mentrdquo) At other times changes to the work-week are more permanent such as when asecond shift is added Automakers adjust therate of production at a plant by raising or low-ering the line speed which has historically re-quired a change in the number of workers oneach shift

While fluctuations in total worker hours areroughly proportional to the number of vehiclesassembled changes in employment and averagehours depend on the margins of adjustment thatare being used Overtime hours and inventory shut-downs are intensive adjustments to hours workedwhile changes to the number of shifts and to the linespeed are largely extensive adjustments to the num-ber of workers Although inventory adjustments en-tail temporary layoffs the spell typically lasts only aweek or two and involves negligible adjustmentcosts Inventory adjustments essentially allow theplant to reduce average hours per worker measuredon a monthly basis

6 Inventory data for light trucks are not available before1972 The numbers prior to 1972 are for cars only Wheninventory data for both market segments exist in the 1970sthe inventory-sales ratios for cars and light trucks were verysimilar

7 This does not imply that improvements in inventorymanagement had no impact on the auto industry In fact theauto industry pioneered just-in-time inventories in the1980s and the ratio of materials and work-in-progress in-ventories to shipments for the entire automobile industry(SIC code 371) did fall after 1984

8 Plant entry and exit historically are responsible for thesecular trend in output (See Timothy F Bresnahan andRamey 1994)

FIGURE 2 INVENTORY-TO-SALES RATIO FOR DOMESTIC CARS AND LIGHT TRUCKS


Notes Domestic unit inventories and sales include vehicles built in the United States Canadaand Mexico Quantities are seasonally adjusted Seasonally adjusted daysrsquo supply is based on307 selling days per year Daysrsquo supply prior to January 1972 excludes light trucks


The costs associated with adjusting each pro-duction margin have interesting implications forthe volatility of output in the auto industry (seeAna M Aizcorbe 1992 Bresnahan and Ramey1994 George J Hall 2000) To summarizeseveral key results changes to extensive mar-gins entail large adjustments costs and are there-fore used when shifts in demand are perceivedto be persistent Transitory shocks to sales onthe other hand are best accommodated throughintensive margins While overtime hours andinventory adjustments do affect marginal coststheir use incurs no adjustment costs

Using a dataset that tracks production sched-ules at US and Canadian assembly plantsoperated by the Big Three automakers (Gen-eral Motors Ford and the Chrysler portion ofDaimlerChrysler) we calculate the contribu-tions of intensive and extensive adjustments tothe variance of monthly output and the resultsof this exercise are recorded in Table 2 Sepa-rate measurements are made for the periodsbefore and after 19849 Contributions to vari-ance are measured as in Bresnahan and Ramey

(1994) the variance of actual output is com-pared to the variance of an artificial outputmeasure that holds each margin in turn con-stant at each plant The difference in the variancesof actual and constructed output determines theimpact of each margin on the variance of plant-level output after holidays supply disruptionsand the summer shutdowns are removed Thenumbers in Table 2 are weighted averagesacross all plants and do not sum to 100 becauseof nonlinearities and covariance terms

The contribution of extensive adjustments tothe variance of output declined from 65 percentin the first period to 37 percent in the secondperiod A change in the way shifts were addedand pared between the two periods accountedfor most of this change The contribution tovariance of adding and cutting shifts fell from33 percent of the monthly production variancein the 1970s and 1980s to only 5 percent duringthe 1990s

Adjustments to intensive margins accord-ingly became more important in the latter pe-riod The contribution of overtime hours andinventory adjustments rose from about 40 per-cent of plant-level variance in the early periodto 51 percent in the latter period In particularthe contribution of overtime hours to total out-put variance stepped up significantly from acontribution of less than 10 percent in the earlyperiod to more than 24 percent in the secondperiod The contribution from temporary plantclosures increased by a smaller amount

C The Persistence of Sales

The decline in the variance of auto salesshown in Table 1 though smaller than the de-cline in the variance of output could havestemmed from a reduction in the size of salesshocks or a reduction in their persistence Tosee this assume that sales are represented by asimple AR(1) process with a first-order autocor-relation of and an error-term variance of 2Variance is then given by the expression 2(1 2)

In auto industry data most of the reduction inthe variance of sales that occurred in the 1980scame from a weakening in the persistence ofsales as opposed to a reduction in the varianceof shocks to sales This distinction is importantin the production-smoothing model because the

9 Our data for the more recent sample start in 1990 andnot in 1984 because we did not have access to AutomotiveNews for 1984ndash1989

TABLE 2mdashIMPORTANCE OF INTENSIVE AND

EXTENSIVE MARGINS OF ADJUSTMENT FOR THE

VARIANCE OF MONTHLY MOTOR VEHICLE OUTPUT

(Percent of average plant-level varianceattributed to use of each margin)

Margin of adjustment 1972ndash1983 1990ndash2001

Changes in extensive margins 646 373Shifts 330 54Line speeds 211 178

Changes in intensive margins 395 510Temporary closures

(Inventory adjustments) 323 331Overtime hours 96 243

Notes Plant-level variance is calculated after holidays sup-ply disruptions model changeovers and extended closuresare removed Percent impact of each margin on outputvariance is calculated by comparing variance of actual pro-duction with the variance of hypothetical production if eachmargin (in turn) were held fixed Contributions to varianceare the weighted average among all plants operating in eachperiod Contributions of extensive and intensive margins donot sum to 100 because of covariance terms The same istrue for individual margins within each category See DataAppendix for data sources


variance of production relative to sales dependsa great deal on but is invariant to 2

To see the change in persistence for autosales consider the univariate model shown inequation (2)

(2) St 0 1 t 2 St 1 Dt

0 1 t 2 St 1 t

St is the logarithm of seasonally adjusted salesand t is a normally distributed error term withmean zero and variance equal to 2 3 DtDt is an indicator variable that equals zero fromJanuary 1967 through December 1983 and oneover the rest of the sample

The model is estimated with monthly domes-tic auto sales (seasonally adjusted) from January1967 to December 2004 and it allows all pa-rameters of the sales process to change in 1984including the coefficient on lagged sales theconstant the slope of the trend and the varianceof the residual We estimate this model viamaximum likelihood for cars alone light trucksalone and for the sum of cars and light truckscalled ldquolight vehiclesrdquo

The estimates are summarized in Table 3 andit appears that the process governing sales haschanged significantly between the two periodsThe constant and the trend differ across the twoperiods for all three aggregates reflecting themix shift in sales between cars and light trucksthat was shown in Figure 1 The parameter 3 which measures the change in the variance ofthe sales shocks shows a significant decline in1984 for light trucks but is unchanged for carsand the light-vehicle aggregate The first-orderautocorrelation of sales on the other hand fellin the late period for all three aggregates asevidenced by the negative and significant pointestimates of 2 For cars the first-order auto-correlation fell from 085 to 056 and for trucksit fell from 093 to 069 When all light vehiclesare grouped together this estimate declinedfrom almost 09 to about 06

The changes in the estimates imply that salesin the post-1984 period returned to the meanmuch more quickly following a surprise thanwas the case in earlier decades Also most ofthe change in the unconditional variance ofsales described in Table 1 appears to have comefrom a change in the propagation of sales

shocks rather than a change in the variance ofsales shocks

The persistence of sales could have declinedfor a number of reasons One possibility is thatthe automakers began responding to shocksmore aggressively with their pricing policiesAnother possibility is that the types of shockshitting the industry such as oil price shocks ormonetary shocks became less persistent Theproduction-scheduling model we present in thenext section does not depend on the source ofthe change in persistence but only on the factthat it occurred We take this change in persis-tence as given and examine the implications forthe behavior of production inventories averagehours and employment

II Production Scheduling Model withInventories and Workforce

The changes in the sales process described inthe preceding section have large effects on therelationship between production inventoriesand sales The channel through which the per-sistence of sales affects production volatility inthe auto industry was in fact described byOlivier J Blanchard (1983) before the changes

TABLE 3mdashESTIMATES OF AGGREGATE AUTOMOBILE

SALES PROCESS

Coefficient CarsLighttrucks

Lightvehicles

0 (constant) 0367 00045 0279(0136) (0042) (0117)

0 ( constant) 0628 0287 0654(0215) (0086) (0234)

1 (trend) 00002 000017 00005(000013) (000016) (000011)

1 ( trend) 00003 00011 000050(000017) (000035) (000020)

2 (AR(1)) 0851 0934 0886(0049) (0030) (0043)

2 (AR(1)) 0293 0243 0271(0108) (0076) (0102)

2 (innovVariance)

00070 00089 00066(000116) (00012) (00011)

3 ( innovVariance)

000046 00040 00013(00016) (00014) (00014)

Log likelihood 4903 5023 5232

Notes Estimated coefficients and standard errors (in paren-theses) from equation (2) Standard errors were computedusing Eicker-White methods The sample is from February1967 through December 2004 (455 observations)

Significantly different from zero at the 5-percent level


uncovered here had even occurred Blanchardconcluded that the structure of costs in the autoindustry was such that inventories would eitherstabilize or destabilize production depending onthe persistence of sales Fluctuations in autosales were very persistent during Blanchardrsquossample period 1966 through 1979 so invento-ries destabilized production

In this section we analyze the original Holt etal (1960) model of industry costs which distin-guishes between intensive and extensive labor ad-justments and show how the persistence of salesaffects the decision rules for inventories and work-force in a production-smoothing model In thenext section we show how the persistence of saleshas an impact on the variances of the decision rules

A The Structure of Costs

Consider a plant that faces a stochastic salesprocess and must choose the size of its work-force Nt and the level of output Yt in order tominimize the discounted present value of pro-duction workforce-adjustment and inventory-holding costs The level of workforce determinesthe minimum efficient scale of production in eachperiod denoted Nt The cost-minimizationproblem is shown in the following expression

(3) minNt j Yt j

Ct EtlimJ3

j 0

J

t j 1

2[1Yt j

2(Yt j Nt j)2 3Nt j

4(Nt j Nt j1)2

1(It j1 2 St j)2]

where 0 1 i 0 for i 1 through 41 13 0 and 2 0 It is the stock of inventoriesat the end of period t and St is sales in period tThe minimization is subject to the inventoryidentity Yt It It1 St and the processgoverning sales St c St1 t where tis an iid shock to sales with mean zero andvariance equal to

2The plant observes St before it chooses em-

ployment and production in period t While theplant does take sales as given in this cost min-imization problem this does not imply that

sales are exogenous to the firm Rather we usea standard micro result that allows one to focuson only the cost minimization part of the overallprofit maximization problem

The terms in equation (3) summarize severalkey features of plant-level cost associated withscheduling production and holding inventoriesThe second term captures the cost per worker ofscheduling overtime or short weeks which is anintensive adjustment To see this consider theproduct of a normal ldquofull-timerdquo workweek (suchas 40 hours) and let Yt ht Nt where htrepresents average hours per worker The sec-ond term in equation (3) can thus be rewritten as2((ht 40)Nt)

2The fourth term in equation (3) is the cost of

adjusting the number of workers attached to theplantmdashthe extensive margin Adjustments to theworkforce shift the static marginal cost curve hor-izontally and redefine the minimum efficient scaleof production In contrast to varying the work-week increasing the number of workers does notlead to increasing static marginal costs but this movedoes incur dynamic adjustment costs

The last term in equation (3) captures the trade-offbetween inventory-holding costs and stock-out costswhich depends on the level of sales For industriesthat produce to stock such as motor vehicles this isa standard way to obtain an industry equilibrium inwhich inventories are nonzero

The modern production-smoothing model ofinventory behavior is a simplified version of thisoriginal Holt et al model The models used byBlanchard (1983) and those surveyed by Rameyand Kenneth D West (1999) for example donot distinguish between the intensive and exten-sive margins of labor input and thus are specialcases of the Holt et al model in which 0and output equals (or is in fixed proportion to)workforce All increases in production implyrising marginal costs in these models and thereis no distinction between boosting the work-week and hiring more workers This distinctionis very important in the auto industry however

The main reason that the special case of theoriginal inventory model became dominant isthat it has only one endogenous state variablemdashthe level of lagged inventories In contrast theHolt et al model has an additional endogenousstate variablemdashthe level of lagged workforceGoing from one state variable to two state vari-ables makes the system significantly more dif-ficult to solve analytically


B Solution Optimal Production Scheduling

For simplicity we set 1 3 and c equal tozero since these linear terms affect only themeans of the variables in the solution and not thedynamics The first-order conditions with respectto workforce and inventories in the current periodare written respectively as equations (4) and (5)

(4) 4 Nt Nt 1 2 Yt Nt

Et4 Nt 1 Nt

(5) Et2 Yt Nt 1 It 2 St 1

Et2 Yt 1 Nt 1

First-order condition (4) states that employ-ment given some level of output is optimizedwhen the cost of adding one more worker thisperiod less the savings in current-period produc-tion costs equals the discounted cost of adjustingworkforce by one less worker next period First-order condition (5) is analogous to the first-ordercondition obtained in the simple model withoutworkforce and it states that the cost of producingone more unit in the current period and storing itin inventory equals the discounted savings of pro-ducing one less unit next period When workforceand output decisions are optimized simulta-neously the solution path is homogeneous of de-gree zero in 1 2 and 4

The decision rules for Nt and It assuming ra-tional expectations depend on two state variablesmdashNt1 It1mdashand on St as shown in equation 610

(6) Nt

It C Nt 1

It 1 d St

The persistence of sales affects only thecoefficients in d and not those in C

III The Effects of Sales Persistence on theVariances of Output Inventories

and Workforce

Expressions for the variances of output in-ventories and workforce and the covariance

between inventory investment and sales arereadily derived from the decision rules Becausethe decision rules from the general model arerather complicated functions of the parametersit is first useful to analyze two special casesThe results are then shown to hold up in thegeneral model as well

A Special Case 1 Simplified ModernProduction-Smoothing Model

Consider first a special case of the Holt et almodel that sets 0 and 4 0 This is asimpler version of the model used by Blanchard(1983) in which 0 and Y N For a particularparameterization of his model he showed that theproduction response to a sales shock was greater ifthe persistence of sales was greater

The optimal decision rule for production inthis simplified model assuming rational expec-tations is given by

(7) Yt 1 It 1 13St

where 1

2 1

1

1

2

1

1

1

22

4

and 13

1 12

2

1

As long as 1 2 and 2 are nonnegative will lie between zero and unity and 13 will bepositive While depends on neither 2 nor 13 is increasing in both of these parameters

The relative variances of production and salesand the covariance of sales and inventory in-vestment in the simplified model are given inequations (8) and (9)

(8)Y

2

S2 1

21 13 1

1

213 121 2

1 1

10 See the Mathematical Appendix (available at httpwwwe-aerorgdatadec0620040244_apppdf) for the fullsolution


(9) SI 13 1S

2

1 1

The value of 13 relative to unity is an importantdeterminant of the relationship between produc-tion sales and inventories and 13 is an increas-ing function in If 13 13 1 which is more likelywhen shocks to sales are persistent the covari-ance is positive and the variance of productionis greater than the variance of sales The intu-ition is as follows when is high the firmanticipates that sales will remain elevated for alonger time following a positive shock to salesand it should raise production more in order toprevent its inventory-sales ratio from dippingtoo low for an extended period If 13 1 on theother hand the covariance of sales and inven-tory investment is negative and the variance ofproduction is potentially but not necessarilyless than the variance of sales

A decline in alone thus could turn the signof the covariance between sales and inventoryinvestment from positive to negative and itcould also reduce the variance of output belowthat of sales Neither measure however ismonotonically increasing in for all possibleparameter values In order to determine suffi-cient conditions under which these measures arestrictly increasing in we numerically investi-gated the parameter space for these functionswith preset to 0997 (annual discount rate of 4percent) We searched over values of 2 from 0to 5 months values of 12 from 0001 to 5and values of from 001 to 099

The covariance of sales and inventory in-vestment is monotonically increasing in forvirtually all parameter values An increase in always leads to an increase in the covari-ance of sales and inventory investment aslong as the penalty for deviating from desiredinventories is not too small relative to themarginal cost of production or as long as12 13 005 For 2 25 which coincideswith the average inventory-sales ratio inthe automobile industry (stated in months)12 13 0027 guarantees that the covarianceis increasing in

Most parameter values also imply that an in-crease in leads to a rise in the variance ofproduction relative to the variance of sales When 06 Y

2S2 is monotonically increasing in for

all values of 2 and 12 within the ranges ex-plored For 2 25 as shown in Figure 3 the

ranges of values for 12 and over which thederivative with respect to is not positive is con-fined to high values of when 12 exceedsabout 01 a threshold that is higher than the levelconsistent with the historical volatilities of USproduction and sales of autos as we show below

B Special Case 2 Model without Inventories

Consider next a special case in which plantscannot hold inventories but are able to chooseworkforce (ie Yt St and 1 2 0) Thestochastic Euler equation for this problem is given by

(10) Et4 Nt 1 1 4 22Nt

4 Nt 1 2St

The rational expectations solution to equation(10) that satisfies the transversality condition is

(11) Nt Nt 1 2

4

1 St

where 1

2 1

1

22

4

1

1

22

42

4

FIGURE 3 PARAMETER REGION FOR WHICH

VarYtVarSt

0

Notes Parameters other than and 12 are held fixed is set to 0997 (4-percent annual discount rate) and 2 is setto 25 months


As long as all parameters are positive willbe between zero and unity The variance ofworkforce relative to the variance of sales(which in this simple case is equal to the vari-ance of production) is given by

(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1

This expression is always increasing in andthe intuition is as follows Because changes toworkforce entail adjustment costs while changesto average hours do not workforce will accountfor a larger part of output volatility when theadjustment costs pay offmdashwhen sales are per-sistent and the new workforce is expected to yieldlower static marginal costs well into the future

C General Model

In the general model plants choose both out-put and workforce in each period While closed-form solutions to the model do exist the effectsof individual parameters on the decision rulesare difficult to see in these complicated expres-sions Fortunately the intuition developed inthe special cases is largely unchanged wheninventories and workforce are optimized jointly

The variance of output relative to sales thecovariance between sales and inventory invest-ment and the variance of workforce relative tooutput are plotted against values of in the toprow of Figure 4 The solid lines represent thesemeasures of volatility for a baseline set of pa-rameters chosen so that the variance of outputrelative to sales and the variance of workforcerelative to output in the model are consistentwith the empirical counterparts in industry dataprior to 198411 We do not attempt to estimatethe parameters from this model because while

having the advantage of being simple and intu-itive the model does not capture the importantnonconvexities in the automobile industry12

The other (not solid) lines plot these measuresfor alternative parameterizations of the modeland will be discussed below

The relationship between the variance of pro-duction and is pictured in the left-most graphProduction is less volatile than sales for lowvalues of though the ratio is not monotoni-cally increasing in for all values of Asshown in the middle graph the covariance be-tween sales and inventory investment increasesmonotonically in with this set of parametervalues The variance of workforce relative tooutput shown in the right-most graph is gen-erally increasing in but again not universallyThe ratio increases in as long as is suffi-ciently large

The impact of on the variances is consistentwith the magnitude of the changes observed inthe data For monthly sales of domestic carsrecall that declined from 085 to 056 Accord-ing to the baseline parameterization of themodel this would lead the ratio of the output-to-sales variance to decline from well over unityto around 07 even greater than the changeshown in Table 1 The persistence of truck salesdeclined from 093 to 069 and the model pre-dicts a smaller decline in the variance ratiowhich also matches the pattern in the data

The decline in the covariance between inven-tory investment and sales that occurs in themodel when the persistence of sales is reducedis qualitatively consistent with the change mea-sured in industry data This simple convex ap-proximation to the industry cost function doesnot however replicate the exact magnitude ofthe change very well Covariance declines from028 to 020 in the model while in the indus-try data it declined from 003 in the earlyperiod to 020 in the late period

Similarly the ratio of the variance of work-force to output in the model declines from about065 to 050 when falls In the plant-level datathe ratio fell from 065 in the early period to 0411 Specifically 2 is set to 25 which is the average

inventory-sales ratio (in months) is normalized to 1 isestimated from the first-order autocorrelation of sales to be085 and is preset to 0997 (4-percent annual rate) Withthese parameters in place 12 0085 and 4 2 165together yield decision rules in which Y

2 S2 128 and

N2 Y

2 0647 values which match their empirical coun-terparts for cars in Tables 1 and 2

12 Ramey and Vine (2004) analyzed a model of industrycosts that accounted for these nonconvexities A furthercomplication for estimation is that the worker-output ratio() is not constant in the data creating time-varyingcoefficients


0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ

FIGURE 4 VARIANCE OF SALES OUTPUT AND EMPLOYMENT AS A FUNCTION OF

Notes Parameters not subject to variation in each graph are fixed at their benchmark level12 009 42 165 1 2 25 0997 The variance of sales innovationsis fixed so that the variance of sales equals 245 when 085 This implies that covarianceis measured on the same scale as cars in Table 1


in the late period a decline that was morepronounced than the model predicts

D The Effects of Other Model Parameterson Variance

Could changes to parameters other than also be consistent with the changes measured inthe auto industry data around 1984 To inves-tigate this the effects of various parameter val-ues for 12 2 42 and on the keyvolatility measures (as a function of ) areshown in the panels of Figure 4 Alternativeparameterizations appear as the dashed and dot-ted lines in each graph

The effects of raising 12 on the volatilitymeasures are shown in panel A When the pen-alty for deviating from the desired inventory-sales ratio is higher the contribution ofworkforce adjustments to output variancemoves down The reason is that average hoursper worker can adjust quickly following a salesshock so this margin is used more intensivelywhen maintaining inventories becomes rela-tively more important The variance of outputrelative to the variance of sales howevermoves up when 12 is higher and the corre-spondence between the covariance of sales withinventory investment and also shifts up

Panel B describes the effects of lowering thetarget inventory-sales ratio 2 a change thathas been observed in some industries (thoughnot in inventories of finished autos) The vari-ance of output declines relative to the varianceof sales and inventory investment becomes lessprocyclical The contribution of workforce tooutput variance however increases for all val-ues of between zero and one

Increases in the cost of adjusting workforce42 shown in panel C do move the keyvolatility measures in the desired directionsHowever the sensitivities of changes in thevariance of output relative to sales and the co-variance of inventory investment with sales tochanges in 42 are very low Even if 42 israised to 3000 the effects on the variance ofoutput relative to sales and on the covarianceare small

Lastly the graphs in panel D show that ahigher value for boosts the variance of outputrelative to sales and the covariance of inventory

investment with sales This change also raisesthe variance contribution of workforce substan-tially for all values of less than one becauseevery dollar paid in workforce adjustment costsmoves the minimum efficient scale of produc-tion by a larger magnitude This in turn makesworkforce a more cost-effective margin of ad-justment

To summarize changes in only two param-eters 42 and are capable of simulta-neously moving all three key variancemeasures in the right direction The effects ofchanges in 42 on two of the key varianceshowever are much smaller than the effects ofchanges in on the variances Thus amongthese parameters only changes in are likelyto reproduce the changes in the data that weredocumented earlier

IV Conclusion

This paper has documented significant changesin the behavior of production sales and inven-tories in the automobile industry The varianceof production has fallen more than 70 percentsince 1984 whereas the variance of sales hasfallen by less The covariance of inventory in-vestment and sales has also become more neg-ative Moreover plants are now more likely tovary average hours per worker than the numberof workers employed

These changes in production scheduling oc-curred together with a significant decline in thepersistence of sales shocks Our theoreticalanalysis of the original Holt et al (1960) pro-duction-smoothing model showed that a reduc-tion in sales persistence all else equal lowersthe volatility of output relative to sales lowersthe covariance of inventory investment andsales and reduces the portion of output volatil-ity that stems from employment changes Changesin the other model parameters cannot replicateall three of these changes

Many of these developments in productionvolatility have been documented in other in-dustries as well Our analysis suggests that itwould be fruitful to determine whether thebehavior of sales changed in these industriesas well Determining the source of the declinein persistence is also an important area forfuture research


DATA APPENDIX

The contribution of the variance of motorvehicle production to GDP is calculated bycomparing the variances of the growth rates ofthe following two variables chain-weighted to-tal GDP and chain-weighted GDP less motorvehicles Both of these variables are availablefrom Table 123 from the National Income andProduct Accounts From 1967 through 2004the variance of GDP growth is 1130 and thevariance of the growth of GDP less motor ve-hicles 865 We used nominal GDP figures tocompare levels

Figure 1 and Table 1mdashCar and truck salesare seasonally adjusted by the Bureau of Eco-nomic Analysis (BEA) and are in millions ofunits at an annual rate We would have pre-ferred to limit our analysis to light vehicles butlight-truck production is not distinguishablefrom total truck production prior to 1977 anddata for light-truck inventories are not availablebefore 1972 Thus Figure 1 and Table 1 includeall trucks All production data are seasonallyadjusted by the Federal Reserve Board Carinventories are seasonally adjusted by the BEAthough we had to use our own seasonal adjust-ment method for trucks For truck inventoryinvestment we regressed the unadjusted inven-tory investment on the difference between sea-sonally adjusted and unadjusted production andsales of trucks

Figure 2mdashCar sales car inventories andlight-truck sales are available on a seasonallyadjusted basis from the BEA The level of lighttruck inventories was seasonally adjusted withX12ARIMA

Table 2mdashThe dataset was constructed fromindustry trade publications in part by Bresnahanand Ramey (1994) who collected the data cov-ering the 50 domestic car assembly plants op-erating during the 1972ndash1983 period and byRamey and Vine (2004) who extended it toinclude all 103 car and light truck assemblyplants operating in the 1972ndash1983 and 1990ndash2001 periods The data were collected by read-ing the weekly production articles in AutomotiveNews which report the following variables forall North American assembly plants (a) the

number of regular hours the plant works (b) thenumber of scheduled overtime hours (c) thenumber of shifts operating and (d) the numberof days per week the plant is closed for (i) unionholidays (ii) inventory adjustments (iii) supplydisruptions and (iv) model changeovers Ob-servations on the line speed posted on eachassembly line were collected from the WardsAutomotive Yearbook

REFERENCES

Aizcorbe Ana M 1992 ldquoProcyclical LabourProductivity Increasing Returns to Labourand Labour Hoarding in Car Assembly PlantEmploymentrdquo Economic Journal 102(413)860ndash73

Automotive News Various issues 1972ndash2001 De-troit Crain Automotive Group

Blanchard Olivier J 1983 ldquoThe Production andInventory Behavior of the American Auto-mobile Industryrdquo Journal of Political Econ-omy 91(3) 365ndash400

Bresnahan Timothy F and Valerie A Ramey1994 ldquoOutput Fluctuations at the Plant LevelrdquoQuarterly Journal of Economics 109(3) 593ndash624

Golob John E 2000 ldquoPost-1984 InventoriesRevitalize the Production-Smoothing ModelrdquoUnpublished

Hall George J 2000 ldquoNon-Convex Costs andCapital Utilization A Study of ProductionScheduling at Automobile Assembly PlantsrdquoJournal of Monetary Economics 45(3) 681ndash716

Hetrick Ron L 2000 ldquoAnalyzing the RecentUpward Surge in Overtime Hoursrdquo MonthlyLabor Review 123(2) 30ndash33

Holt Charles C Franco Modigliani John FMuth and Herbert A Simon 1960 PlanningProduction Inventories and Workforce Engle-wood Cliffs NJ Prentice-Hall

Kahn James A Margaret M McConnelland Gabriel Perez-Quiros 2002 ldquoOn theCauses of the Increased Stability of theUS Economyrdquo Federal Reserve Bank ofNew York Economic Policy Review 8(1)183ndash202

Kim Chang-Jin and Charles R Nelson 1999ldquoHas the US Economy Become More Sta-ble A Bayesian Approach Based on a


Markov-Switching Model of the BusinessCyclerdquo Review of Economics and Statistics81(4) 608ndash16

McConnell Margaret M and Gabriel Perez-Quiros 2000 ldquoOutput Fluctuations in theUnited States What Has Changed since theEarly 1980rsquosrdquo American Economic Review90(5) 1464ndash76

Ramey Valerie A and Daniel J Vine 2004

ldquoTracking the Source of the Decline in GDPVolatility An Analysis of the AutomobileIndustryrdquo National Bureau of Economic Re-search Working Paper 10384

Ramey Valerie A and Kenneth D West 1999ldquoInventoriesrdquo In Handbook of Macroeco-nomics Vol 1B ed John B Taylor andMichael Woodford 863ndash923 AmsterdamElsevier Science North-Holland


The variances of both car and truck output(relative to their respective trends) droppedsharply in the mid-1980s According to struc-tural break tests on the variance of detrended carproduction there was a statistically significantbreak between February and March in 1984 Fortruck production the break occurred betweenJanuary and February in 19834 Thus the struc-

tural break in the variance of automobile pro-duction occurred at essentially the same time asthe structural break in GDP growth volatilitywhich many studies place in 1984

In order to quantify the changes in industryvolatility that took place around 1984 Table 1reports the variances of key variables in the autoindustry in two sample periodsmdash1967 through

4 One important difference between the time-series prop-erties of physical unit data and chained-dollar data used inother studies is that stationarity tests on the logarithm ofphysical unit variables reject a unit root in favor of adeterministic trend To search for the structural break in thevariance we used data detrended with an Hodrick-Prescott

(HP) filter rather than trend breaks so as not to bias theresults for a particular period In particular we used sea-sonally adjusted output divided by the exponential of the HPfilter trend applied to the log of output The p-values wereessentially zero

1970 1975 1980 1985 1990 1995 2000 20050

2

4

6

8

10

12


Car salesCar production

A

B

1970 1975 1980 1985 1990 1995 2000 20050

2

4

6

8

10

12


Truck salesTruck production

FIGURE 1 US AUTOMOBILE PRODUCTION AND DOMESTIC SALES


Notes Production and sales are measured in millions of units at a seasonally adjusted annualrate Domestic sales include US sales of vehicles built in the United States Canada and Mexico




2 CovSt It






Cars

19672ndash198312 19842ndash200412


VarS123 097


Cars Trucks



VarS128 091 133 122




































(2) St 0 1 t 2 St 1 Dt

0 1 t 2 St 1 t










SALES PROCESS


Lightvehicles

0 (constant) 0367 00045 0279(0136) (0042) (0117)

0 ( constant) 0628 0287 0654(0215) (0086) (0234)

1 (trend) 00002 000017 00005(000013) (000016) (000011)

1 ( trend) 00003 00011 000050(000017) (000035) (000020)

2 (AR(1)) 0851 0934 0886(0049) (0030) (0043)

2 (AR(1)) 0293 0243 0271(0108) (0076) (0102)

2 (innovVariance)

00070 00089 00066(000116) (00012) (00011)

3 ( innovVariance)

000046 00040 00013(00016) (00014) (00014)









(3) minNt j Yt j

Ct EtlimJ3

j 0

J

t j 1

2[1Yt j

2(Yt j Nt j)2 3Nt j

4(Nt j Nt j1)2

1(It j1 2 St j)2]














(4) 4 Nt Nt 1 2 Yt Nt

Et4 Nt 1 Nt


Et2 Yt 1 Nt 1



(6) Nt

It C Nt 1

It 1 d St



and Workforce






(7) Yt 1 It 1 13St

where 1

2 1

1

1

2

1

1

1

22

4

and 13

1 12

2

1



(8)Y

2

S2 1

21 13 1

1

213 121 2

1 1



(9) SI 13 1S

2

1 1










(10) Et4 Nt 1 1 4 22Nt

4 Nt 1 2St


(11) Nt Nt 1 2

4

1 St

where 1

2 1

1

22

4

1

1

22

42

4


VarYtVarSt

0




(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1


C General Model










2 S2 128 and

N2 Y




0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES




















2 CovSt It






Cars

19672ndash198312 19842ndash200412


VarS123 097


Cars Trucks



VarS128 091 133 122




































(2) St 0 1 t 2 St 1 Dt

0 1 t 2 St 1 t










SALES PROCESS


Lightvehicles

0 (constant) 0367 00045 0279(0136) (0042) (0117)

0 ( constant) 0628 0287 0654(0215) (0086) (0234)

1 (trend) 00002 000017 00005(000013) (000016) (000011)

1 ( trend) 00003 00011 000050(000017) (000035) (000020)

2 (AR(1)) 0851 0934 0886(0049) (0030) (0043)

2 (AR(1)) 0293 0243 0271(0108) (0076) (0102)

2 (innovVariance)

00070 00089 00066(000116) (00012) (00011)

3 ( innovVariance)

000046 00040 00013(00016) (00014) (00014)









(3) minNt j Yt j

Ct EtlimJ3

j 0

J

t j 1

2[1Yt j

2(Yt j Nt j)2 3Nt j

4(Nt j Nt j1)2

1(It j1 2 St j)2]














(4) 4 Nt Nt 1 2 Yt Nt

Et4 Nt 1 Nt


Et2 Yt 1 Nt 1



(6) Nt

It C Nt 1

It 1 d St



and Workforce






(7) Yt 1 It 1 13St

where 1

2 1

1

1

2

1

1

1

22

4

and 13

1 12

2

1



(8)Y

2

S2 1

21 13 1

1

213 121 2

1 1



(9) SI 13 1S

2

1 1










(10) Et4 Nt 1 1 4 22Nt

4 Nt 1 2St


(11) Nt Nt 1 2

4

1 St

where 1

2 1

1

22

4

1

1

22

42

4


VarYtVarSt

0




(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1


C General Model










2 S2 128 and

N2 Y




0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES



















































(2) St 0 1 t 2 St 1 Dt

0 1 t 2 St 1 t










SALES PROCESS


Lightvehicles

0 (constant) 0367 00045 0279(0136) (0042) (0117)

0 ( constant) 0628 0287 0654(0215) (0086) (0234)

1 (trend) 00002 000017 00005(000013) (000016) (000011)

1 ( trend) 00003 00011 000050(000017) (000035) (000020)

2 (AR(1)) 0851 0934 0886(0049) (0030) (0043)

2 (AR(1)) 0293 0243 0271(0108) (0076) (0102)

2 (innovVariance)

00070 00089 00066(000116) (00012) (00011)

3 ( innovVariance)

000046 00040 00013(00016) (00014) (00014)









(3) minNt j Yt j

Ct EtlimJ3

j 0

J

t j 1

2[1Yt j

2(Yt j Nt j)2 3Nt j

4(Nt j Nt j1)2

1(It j1 2 St j)2]














(4) 4 Nt Nt 1 2 Yt Nt

Et4 Nt 1 Nt


Et2 Yt 1 Nt 1



(6) Nt

It C Nt 1

It 1 d St



and Workforce






(7) Yt 1 It 1 13St

where 1

2 1

1

1

2

1

1

1

22

4

and 13

1 12

2

1



(8)Y

2

S2 1

21 13 1

1

213 121 2

1 1



(9) SI 13 1S

2

1 1










(10) Et4 Nt 1 1 4 22Nt

4 Nt 1 2St


(11) Nt Nt 1 2

4

1 St

where 1

2 1

1

22

4

1

1

22

42

4


VarYtVarSt

0




(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1


C General Model










2 S2 128 and

N2 Y




0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES







































(2) St 0 1 t 2 St 1 Dt

0 1 t 2 St 1 t










SALES PROCESS


Lightvehicles

0 (constant) 0367 00045 0279(0136) (0042) (0117)

0 ( constant) 0628 0287 0654(0215) (0086) (0234)

1 (trend) 00002 000017 00005(000013) (000016) (000011)

1 ( trend) 00003 00011 000050(000017) (000035) (000020)

2 (AR(1)) 0851 0934 0886(0049) (0030) (0043)

2 (AR(1)) 0293 0243 0271(0108) (0076) (0102)

2 (innovVariance)

00070 00089 00066(000116) (00012) (00011)

3 ( innovVariance)

000046 00040 00013(00016) (00014) (00014)









(3) minNt j Yt j

Ct EtlimJ3

j 0

J

t j 1

2[1Yt j

2(Yt j Nt j)2 3Nt j

4(Nt j Nt j1)2

1(It j1 2 St j)2]














(4) 4 Nt Nt 1 2 Yt Nt

Et4 Nt 1 Nt


Et2 Yt 1 Nt 1



(6) Nt

It C Nt 1

It 1 d St



and Workforce






(7) Yt 1 It 1 13St

where 1

2 1

1

1

2

1

1

1

22

4

and 13

1 12

2

1



(8)Y

2

S2 1

21 13 1

1

213 121 2

1 1



(9) SI 13 1S

2

1 1










(10) Et4 Nt 1 1 4 22Nt

4 Nt 1 2St


(11) Nt Nt 1 2

4

1 St

where 1

2 1

1

22

4

1

1

22

42

4


VarYtVarSt

0




(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1


C General Model










2 S2 128 and

N2 Y




0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES




















(2) St 0 1 t 2 St 1 Dt

0 1 t 2 St 1 t










SALES PROCESS


Lightvehicles

0 (constant) 0367 00045 0279(0136) (0042) (0117)

0 ( constant) 0628 0287 0654(0215) (0086) (0234)

1 (trend) 00002 000017 00005(000013) (000016) (000011)

1 ( trend) 00003 00011 000050(000017) (000035) (000020)

2 (AR(1)) 0851 0934 0886(0049) (0030) (0043)

2 (AR(1)) 0293 0243 0271(0108) (0076) (0102)

2 (innovVariance)

00070 00089 00066(000116) (00012) (00011)

3 ( innovVariance)

000046 00040 00013(00016) (00014) (00014)









(3) minNt j Yt j

Ct EtlimJ3

j 0

J

t j 1

2[1Yt j

2(Yt j Nt j)2 3Nt j

4(Nt j Nt j1)2

1(It j1 2 St j)2]














(4) 4 Nt Nt 1 2 Yt Nt

Et4 Nt 1 Nt


Et2 Yt 1 Nt 1



(6) Nt

It C Nt 1

It 1 d St



and Workforce






(7) Yt 1 It 1 13St

where 1

2 1

1

1

2

1

1

1

22

4

and 13

1 12

2

1



(8)Y

2

S2 1

21 13 1

1

213 121 2

1 1



(9) SI 13 1S

2

1 1










(10) Et4 Nt 1 1 4 22Nt

4 Nt 1 2St


(11) Nt Nt 1 2

4

1 St

where 1

2 1

1

22

4

1

1

22

42

4


VarYtVarSt

0




(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1


C General Model










2 S2 128 and

N2 Y




0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES






















(3) minNt j Yt j

Ct EtlimJ3

j 0

J

t j 1

2[1Yt j

2(Yt j Nt j)2 3Nt j

4(Nt j Nt j1)2

1(It j1 2 St j)2]














(4) 4 Nt Nt 1 2 Yt Nt

Et4 Nt 1 Nt


Et2 Yt 1 Nt 1



(6) Nt

It C Nt 1

It 1 d St



and Workforce






(7) Yt 1 It 1 13St

where 1

2 1

1

1

2

1

1

1

22

4

and 13

1 12

2

1



(8)Y

2

S2 1

21 13 1

1

213 121 2

1 1



(9) SI 13 1S

2

1 1










(10) Et4 Nt 1 1 4 22Nt

4 Nt 1 2St


(11) Nt Nt 1 2

4

1 St

where 1

2 1

1

22

4

1

1

22

42

4


VarYtVarSt

0




(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1


C General Model










2 S2 128 and

N2 Y




0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES




















(4) 4 Nt Nt 1 2 Yt Nt

Et4 Nt 1 Nt


Et2 Yt 1 Nt 1



(6) Nt

It C Nt 1

It 1 d St



and Workforce






(7) Yt 1 It 1 13St

where 1

2 1

1

1

2

1

1

1

22

4

and 13

1 12

2

1



(8)Y

2

S2 1

21 13 1

1

213 121 2

1 1



(9) SI 13 1S

2

1 1










(10) Et4 Nt 1 1 4 22Nt

4 Nt 1 2St


(11) Nt Nt 1 2

4

1 St

where 1

2 1

1

22

4

1

1

22

42

4


VarYtVarSt

0




(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1


C General Model










2 S2 128 and

N2 Y




0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES


















(9) SI 13 1S

2

1 1










(10) Et4 Nt 1 1 4 22Nt

4 Nt 1 2St


(11) Nt Nt 1 2

4

1 St

where 1

2 1

1

22

4

1

1

22

42

4


VarYtVarSt

0




(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1


C General Model










2 S2 128 and

N2 Y




0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES



















(12)N

2

S2

N2

Y2

1

1 2 2

4

1 21

2

1


C General Model










2 S2 128 and

N2 Y




0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES


















0 02 04 06 08 1

minus05

0

05

1

ρ

Cov

aria

nce

ρ













IV Conclusion





DATA APPENDIX






REFERENCES



























IV Conclusion





DATA APPENDIX






REFERENCES


















DATA APPENDIX






REFERENCES
























Documents

Declining Volatility in the U.S. Automobile Industryeconweb.ucsd.edu/~vramey/research/Ramey_Vine_AER_published.pdf · Declining Volatility in the U.S. Automobile Industry ... assistance