Further Regression Topics

8/11/2019 Further Regression Topics

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EC114 Introduction to Quantitative Economics19. Further Regression Topics I

Marcus Chambers

Department of EconomicsUniversity of Essex

13/15 March 2012

EC114 Introduction to Quantitative Economics 19. Further Regression Topics I
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Outline

1 Dummy Variables

2 Chow Tests

Reference: R. L. Thomas, Using Statistics in Economics,McGraw-Hill, 2005, sections 14.1 and 14.2.

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Dummy Variables 3/35

Sometimes the variables we want to use cant be

measured in a precise quantitative way.We therefore have to introducequalitativefactors into theanalysis.

For example, suppose we want to study the demand forbeef over time.

We might consider the regression model

Qt=+Yt+t, t= 1, . . . , n,

whereQis per-capita demand for beef,Y is per-capitadisposable income, and thetsubscript denotes the time

period which indexes observations.

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Suppose that we suspect the nature of demand for beef to

change in certain periods because of a scare due to mad

cow disease (Creutzfeld-Jakob disease, or CJD).How can we deal with such a qualitative factor?

The approach we will follow introduces a dichotomousordummy variablesuch that, in period t,

Dt= 1 when consumers fear getting CJD,Dt= 0 when consumers do not fear getting CJD.

We shall see how such variables can be used in aregression model to allow the parameters of the model to

change in certain periods.

We shall also consider tests of whether the parameters are

constant over the entire sample or whether they change inthe way described above.

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Suppose we have monthly data on the demand for beef for

5 years(n= 60).

Furthermore suppose that we know that the CJD fear is

particularly relevant for the 12 months of the second yearand for the first 6 months of the third year.

We can then define a dummy variable for the relevant

period as:

Dt= 1 fort= 13, . . . , 30,Dt= 0 for all othert.

This variable enters the data set as a series of zeros and

ones and is treated like any other variable.Note that we can only constructDtif we have information

on the period affected by the CJD scare.


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We can allow for the CJD scare to affect the intercept, ,simply by includingDtas an additional explanatory

variable:

Qt=+Dt+Yt+t, t= 1, . . . , n.

WhenDt= 0(no CJD scare) the model reduces to

Qt=+Yt+t.

But during the CJD scare,Dt= 1and so

Qt= (+) +Yt+t,

the intercept becoming+(and we would probablyexpect < 0so that demand falls for givenYduring theCJD scare).


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Suppose we run this regression and obtain

Qt= 25 12Dt+0.02Yt.

This implies that during the CJD scare (setting Dt= 1)

Qt= 13+ 0.02Yt,

while when there is no CJD scare (setting Dt= 0)

Qt= 25+ 0.02Yt.

This effect is illustrated in the following diagram:


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The presence of the dummy variable enables the

estimated demand equation to shift downwards during theCJD scare.

Note, however, that the slope remains unaffected.


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The shift in the demand equation, allowing different

demand equations in the different periods, was obtained byestimating asingleequation.

It means that we can test whether the shift is significant bycarrying out at-test for the significance of the dummy

variables Dt.

So, to testH0:= 0againstHA:= 0we could use

TS=

s

tn3 under H0,

wheredenotes the estimated coefficient on Dtands isits standard error.


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However, it is nevertheless possible that the effect of the

CJD scare is on the marginal propensity to consume beef

out of income i.e. on the parameter.This situation might appear to be a bit more complicatedbut it is also straightforward to handle.

Instead of including the variableDtby itself in the

regression, we now include the product of DtwithYtin theregression i.e. we include the variable YtDt.

It is easy to construct a new variable in regression software

that is the product of two variables.

For example, in Stata, if Dand Yare the two variables, we

can use the command:

gen yd=y*d

to generate the product.


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The starting point is now given by the equation

Qt=+Yt+YtDt+t.

WhenDt= 0it follows thatYtDt= 0and we have theoriginal equation

Qt=+Yt+t,

while whenDt= 1it follows that YtDt= Ytand we obtain

Qt=+ (+)Yt+t.

If < 0and + > 0the intercept remains unchanged butthe slope falls while remaining positive.


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Suppose we run this regression and obtain

Qt= 25+ 0.02Yt 0.01YtDt.

This implies that during the CJD scare (setting Dt= 1)

Qt= 25+ 0.01Yt,

while when there is no CJD scare (setting Dt= 0)

Qt= 25+ 0.02Yt.

This effect is illustrated in the following diagram:


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y

The presence of the product dummy YtDtresults in theslope of the estimated equation changing.

Note that the intercept remains unchanged.




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y

We can, of course, allow both the intercept and the slope

to change at the same time.

This requires adding bothDtandYtDtto the regression:

Qt=+1Dt+Yt+2YtDt+t.

WhenDt= 0it follows thatYtDt= 0and we have the

original equation

Qt=+Yt+t,

while whenDt= 1it follows that YtDt= Ytand we obtain

Qt= (+1) + (+2)Yt+t.

We can generalise this to any multiple linear regression.


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Example. Consider the logarithmic money demand model

ln(M) =1+2ln(G) +

whereMdenotes the money stock andGdenotes GDP.

Data set 9.1 in Thomas contains observations for 30countries in 1985 and we shall split the sample into those

countries with GDP$4,000.We can define an appropriate dummy variable:

D= 1 if GDP$4,000.

We shall run regressions to see if the intercept and/orslope are different for countries in these two ranges ofGDP per head.

Including the dummy variable yields:


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. regress lm lg d

Source | SS df MS Number of obs = 30

-------------+------------------------------ F( 2, 27) = 119.10Model | 57.8613015 2 28.9306507 Prob > F = 0.0000

Residual | 6.5585244 27 .242908311 R-squared = 0.8982

-------------+------------------------------ Adj R-squared = 0.8906

Total | 64.4198259 29 2.22137331 Root MSE = .49286

------------------------------------------------------------------------------

lm | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

lg | .9030628 .132545 6.81 0.000 .6311029 1.175023d | -.4534076 .3644604 -1.24 0.224 -1.201219 .2944033

_cons | -1.519285 .3323406 -4.57 0.000 -2.201191 -.8373782

------------------------------------------------------------------------------

The dummy variable is statistically insignificant and so

there does not appear to be any difference in the interceptbetween these two groups.

Including the variableD ln(G)yields:


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. regress lm lg dlg


-------------+------------------------------ F( 2, 27) = 116.31Model | 57.7200283 2 28.8600142 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.8883

Total | 64.4198259 29 2.22137331 Root MSE = .49814

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

lg | 1.089832 .0828707 13.15 0.000 .9197953 1.259869dlg | -.1650309 .1697023 -0.97 0.339 -.5132313 .1831695

_cons | -1.958399 .1146874 -17.08 0.000 -2.193718 -1.72308

------------------------------------------------------------------------------

The product variable is also statistically insignificant

suggesting no difference in the slope parameter (incomeelasticity of money demand) between the two groups ofcountries.


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Dummy variables are widely used in Econometrics.

In cross-sections they can be used to represent things

such as:gender: D= 1if female,D= 0if male;employment status: D= 1if employed,

D= 0if unemployed;

marital status: D=1

if married,D= 0if unmarried.

In time series dummies can be used to represent thingssuch as:

season: Dj= 1if quarterj,

Dj = 0otherwise (j= 1, . . . , 4);particular event: D= 1during wartime,

D= 0otherwise.


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It is also possible to test whether coefficients in aregression remain constant over two pre-specified

sub-samples using theChow test for parameter stability.Suppose we split the sample of nobservations into two

sub-samples, the first containing n1observations, thesecond containingn2 observations, where n1+ n2=n.

Suppose, in the first sub-sample, we have the populationregression

E(Y) =1+2X2+. . .+kXk,

while in the second sub-sample we have the population

regression

E(Y) =1+2X2+. . .+kXk.


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The null hypothesis of interest is

H0:1=1, 2=2, . . . , k=k,

i.e. that the coefficients are the same in the two

sub-samples.

The alternative hypothesis is

HA:j =j for at least one j.

We need to conduct three regressions and obtain the sumof squared residuals, SSR, from each regression.


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The required regressions are as follows:Regression 1: use then1observations, obtain SSR1;

Regression 2: use then2observations, obtain SSR2;Regression 3: use allnobservations, obtain SSRp.

Regression 3 is sometimes known as a pooled regressionbecause all the observations have been pooled together.

There are two versions of the Chow test.

The first version compares all three values of SSR, while

the second compares either SSR1or SSR2with SSRp.


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The first test statistic is

TS=

(SSRp SSR1 SSR2)/k

(SSR1+SSR2)/(n1+ n2 2k) Fk,n1+n22k

underH0.

The usual decision rule applies:

ifTS> F0.05rejectH0in favour of HA;ifTS< F0.05do not rejectH0,

whereF0.05is the 5% critical value from the Fk,n1+n22kdistribution.

This test assesses whether the gain from estimating twoseparate regressions, rather than the pooled regression, isstatistically significant (as measured by comparing SSRpwith SSR1+SSR2).


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The second test statistic is

TS=

(SSRp SSR1)/n2

SSR1/(n1 k) Fn2,n1k

underH0.

The usual decision rule applies:

ifTS> F0.05rejectH0in favour of HA;ifTS< F0.05do not rejectH0,

whereF0.05is the 5% critical value from the Fn2,n1kdistribution.

This test is sometimes known as a predictive failure test,because it is assessing whether the additional n2observations included in the pooled regression result in astatistically significant proportionate change in SSR.


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In time series applications the second sub-sample of n2observations follows chronologically from the first

sub-sample ofn1observations, so there is a clear orderingof the observations.

In cross-sections, however, there is often no uniqueordering, and so the roles of the two sub-samples can be

reversed, resulting in the test statistic

TS=(SSRp SSR2)/n1

SSR2/(n2 k) Fn1,n2k

underH0.

In this case then2observations are taken as the reference

point, which wouldnt make sense with time series.


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Another way to think of the Chow test is in terms of dummyvariables.

LetDdenote the following dummy variable:

D= 1 if sub-sample 1 (n1observations),D= 0 if sub-sample 2 (n2observations).

We can then define the population regression

E(Y) = 1+2X2+. . .+kXk+ 1D +2(DX2) +. . .+k(DXk).

In sub-sample 1, we have

E(Y) = (1+1) + (2+2)X2+. . .+ (k+k)Xk,

while in sub-sample 2 we have

E(Y) =1+2X2+. . .+kXk.


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The differences in the coefficients between the two

sub-samples are captured by thej parameters.

If thej are all zero then there are no differences.

We can therefore consider our test of parameter stability as

being a test of

H0:1= 0, 2= 0, . . . , k= 0

againstHA:at least onej = 0.

We can carry out an F-test of thesekrestrictions by

running just two regressions, as follows:


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Regression 1 is the unrestricted regression

Y = 1+2X2+. . .+kXk

+ 1D +2(DX2) +. . .+k(DXk) +;

from this we need the SSR, denoted SSRU.

Regression 2 is the restricted regression

Y=1+2X2+. . .+kXk+;

from this we need SSRR.

We then construct the test statistic

TS=(SSRR SSRU)/k

SSRU/(n 2k) Fk,n2k under H0

and apply the usual decision rule.


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Example. Lets return to the money demand example,

where we divided the countries into those withGDP>$4000 per head and those with GDP F = 0.0000


-------------+------------------------------ Adj R-squared = 0.8885

Total | 64.4198259 29 2.22137331 Root MSE = .49765

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

lg | 1.04467 .068569 15.24 0.000 .9042126 1.185127

_cons | -1.912253 .104309 -18.33 0.000 -2.12592 -1.698586

------------------------------------------------------------------------------


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The two sub-sample regressions are:

. regress lm lg if g F = 0.0000Residual | 4.6831173 17 .275477489 R-squared = 0.6938

-------------+------------------------------ Adj R-squared = 0.6758

Total | 15.2933819 18 .849632328 Root MSE = .52486

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

lg | .9226822 .148673 6.21 0.000 .6090096 1.236355

_cons | -1.970365 .1216961 -16.19 0.000 -2.227121 -1.713609------------------------------------------------------------------------------

. regress lm lg if g>4


-------------+------------------------------ F( 1, 9) = 3.52

Model | .714288023 1 .714288023 Prob > F = 0.0934


-------------+------------------------------ Adj R-squared = 0.2012Total | 2.54105312 10 .254105312 Root MSE = .45053

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

lg | .7237502 .3858088 1.88 0.093 -.1490099 1.59651

_cons | -1.117129 .8758755 -1.28 0.234 -3.098497 .864239

------------------------------------------------------------------------------


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For the first test we find that

SSRp= 6.9345, SSR1= 4.6831, SSR2= 1.8268

withn1= 19,n2= 11and k= 2.

The test statistic is

TS = (SSRp SSR1 SSR2)/k

(SSR1+SSR2)/(n1+ n2 2k)

= (6.9345 4.6831 1.8268)/2

(4.6831+ 1.8268)/(19+ 11 4)= 0.8479.

UnderH0(that the coefficients are the same in the two

sub-samples)TShas anF2,26distribution, and soF0.05= 3.3690.

AsTS= 0.8026< 3.3690we are unable to reject H0.


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For the second test we have

SSRp= 6.9345, SSR1= 4.6831, SSR2= 1.8268

withn1= 19,n2= 11and k= 2.The test statistic is

TS = (SSRp SSR1)/n2

SSR1/(n1 k)

= (6.9345 4.6831)/11

(4.6831)/(19 2) = 0.7430.

UnderH0(that the coefficients are the same in the twosub-samples)TShas anF11,17distribution, and so

F0.05 2.4153(the values forF10,17andF12,17are 2.4499and 2.3807, respectively).

AsTS= 0.7430< 2.4153we are unable to reject H0.


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We can also conduct the second test by reversing the roles

of the two sub-samples, by computing

TS = (SSRp SSR2)/n1

SSR2/(n2 k)

= (6.9345 1.8268)/19

(1.8268)/(11 2) = 1.3244.

UnderH0(that the coefficients are the same in the twosub-samples)TShas anF19,9distribution, and so

F0.05 2.9365(this is actually the value for F20,9).

AsTS= 1.3244< 2.9365we are once more unable to rejectH0.


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Finally, lets compute the alternative version of the first testusing dummy variables, definingD= 1for countries with

GDP F = 0.0000


-------------+------------------------------ Adj R-squared = 0.8873

Total | 64.4198259 29 2.22137331 Root MSE = .50038

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

lg | .7237502 .4285011 1.69 0.103 -.1570464 1.604547

d | - .8532358 .979691 -0.87 0.392 -2.867019 1.160548

dlg | .198932 .4513347 0.44 0.663 -.7287999 1.126664

_cons | -1.117129 .9727969 -1.15 0.261 -3.116742 .8824836

------------------------------------------------------------------------------


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We therefore have SSRR= 6.5099while our earlier resultsgive SSRU= 6.9345; also,k= 2and n= 30.

The test statistic is

TS=(6.9345 6.5099)/2

6.5099/(30 4) = 0.8479;

this is exactly the same value as the first Chow statistic we

computed!In fact, it is possible to prove that the two statistics areidentical.

BecauseTShas anF2,26distributions under H0(as before),

we know the 5% critical value is 3.3690 and the test resultis the same (do not rejectH0).

We have actually carried out exactly the same test but by aslightly different approach.


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Summary
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Su a y

Dummy variables

Chow tests

Next week:

Heteroskedasticity, autocorrelation and dynamic models

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