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Econometrics I, Lecture 10

Thu, 15 May 2014

Yasushi Kondo

Heteroskedasticity is known up to a multiplicativeconstant

(8.21)

Example: Savings function (8.22), (8.23)

Example: Group data

Per capitaV(u|x) inv. proportional to population

TotalV(u|x) proportional to population

8.4 Weighted least squares estimation

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Var| 2

0+ 1+ Var | 2

8.4 Weighted least squares estimation

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Eqs. (8.24)(8.26)

Scaled error terms are homoskedastic;WLS (regressing on s w/o intercept) is a GLS

should be estimated, Feasible GLS

Pages 276278

0+ 11+ + + ,Var| 2 0 1+ 1

1+ + +

00 + 11 + + + ,Var| 2

WLS as Feasible GLS (Pages 276278)

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should be estimated, Feasible GLS

Regression of

log is preferred because it always provides positive

estimates of variances.

2 on 1, 2, , to get log(2) on 1, 2, , to get , = exp()log(2) on , 2 to get , = exp()

Example 8.7 Demand for Cigarettes

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. use smoke.dta, clear

. regress cigs lincome lcigpric educ age agesq restaurn

Source | SS df MS Number of obs = 807

-------------+------------------------------ F( 6, 800) = 7.42

Model | 8003.02506 6 1333.83751 Prob > F = 0.0000

Residual | 143750.658 800 179.688322 R-squared = 0.0527

-------------+------------------------------ Adj R-squared = 0.0456

Total | 151753.683 806 188.280003 Root MSE = 13.405

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

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

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

lincome | .8802682 .7277832 1.21 0.227 -.548322 2.308858

lcigpric | -.7508586 5.773343 -0.13 0.897 -12.08355 10.58183

educ | -.5014982 .1670772 -3.00 0.003 -.8294597 -.1735368

age | .7706936 .1601223 4.81 0.000 .456384 1.085003

agesq | -.0090228 .001743 -5.18 0.000 -.0124443 -.0056013

restaurn | -2.825085 1.111794 -2.54 0.011 -5.007462 -.6427078

_cons | -3.639841 24.07866 -0.15 0.880 -50.90466 43.62497------------------------------------------------------------------------------


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. predict uhat, residuals

. generate log_uhatsq = log(uhat 2)

. regress log_uhatsq lincome lcigpric educ age agesq restaurn,notable noheader

. predict ghat, xb

. generate hhat = exp(ghat)


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. regress cigs lincome lcigpric educ age agesq restaurn[aweight=1/hhat]

(sum of wgt is 1.9977e+01)

Source | SS df MS Number of obs = 807-------------+------------------------------ F( 6, 800) = 17.06

Model | 10302.646 6 1717.10767 Prob > F = 0.0000

Residual | 80542.159 800 100.677699 R-squared = 0.1134

-------------+------------------------------ Adj R-squared = 0.1068

Total | 90844.805 806 112.710676 Root MSE = 10.034

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

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

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

lincome | 1.29524 .4370118 2.96 0.003 .4374148 2.153065

lcigpric | -2.940312 4.460144 -0.66 0.510 -11.69528 5.814656

educ | -.4634463 .1201587 -3.86 0.000 -.6993099 -.2275828

age | .4819479 .0968082 4.98 0.000 .2919197 .671976

agesq | -.0056272 .0009395 -5.99 0.000 -.0074713 -.0037831

restaurn | -3.461064 .795505 -4.35 0.000 -5.022588 -1.899541

_cons | 5.635463 17.80314 0.32 0.752 -29.31092 40.58184

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

Weighted least squares in matrix form

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Assumed structure of heteroskedasticity

Var = Var =

( is diagonal)

WLS as GLS

=

=

0+ 11+ + + ,Var| 2 0 1+ 1

1+ + +

Weighted least squares in matrix form

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True structure of heteroskedasticity

Var = Var =

( is diagonal)

Variance matrix of WLSE

WLS + 11 1VarWLS 2 11 11 11 2 11 if

Use an estimate of this form because the assumption of heteroskedasticity

may not be true.

The last command in the Stata example should be followed by robust optionregress cigs lincome lcigpric educ age agesq restaurn [aweight=1/hhat], robust

WLS vs OLS

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The WLS and OLS estimates may be substantially

different, although both estimators are consistent

May indicate that E(y|x) is misspecified

Recommended:

Draw Residual-Fitted value plot

Perform BP and/or White tests

OLS/WLS with Robust SE

OLS with Robust SE is much more popular than WLS with Robust SE.

However,WLS (w/ Robust SE) is preferred in the case of strong

heteroskedasticity.

WLS is expected to be more efficient than OLS.

8.5 The Linear Probability Model Revisited

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Binary response, 0,1

Linear probability model

= + + + + = +

= P = 1 = E = +

Heteroskedasticity

Var = 1

Estimation of skedastic function

= , = 1 Some modifications may be necessary if 0,1

OLS w/ robust SE, WLS w/o (or w/) robust SE

Ch. 9. More on specification and data issues

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9.1 Functional form misspecification

9.2 Using proxy variables for unobserved explanatory

variables

9.3 Models with random slopes

9.4 Properties of OLS under measurement error

9.5 Missing data, nonrandom samples, and outlying

observations

9.6 Least absolute deviations estimation


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9.4 Properties of OLS under measurement

error

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Measurement error in dependent variable

The measurement error is uncorrelated with independent

variables.OLSE is consistent, but its variance is larger

under the (additional) assumption

0+ 11+ + 0 0+ 11+ , + 0E0|1 0


error

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Measurement error in an independent variable

Classical errors-in-variables: OLSE is inconsistent even

when the measurement error () is uncorrelated withunobserved variable (). (9.33)

0+ 11+ , 1 1+ 1 0+ 11+ , 11Cov1, 1 0 Cov1 , 112

Unobserved variable

Measurement errorObserved variable w/ error


error

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Results of the classical errors-in-variables

(9.33), p. 311

Estimated OLS effect is weakened

Attenuation bias

plim1 1+ Cov1, Var1 1 11212 + 12

1 12

12

+ 12

Homework assignments

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Computer Exercise C1, Chapter 9

Computer Exercise C1: (i) Apply RESET from equation (9.3) to

the model estimated in Computer Exercise C5 in Chapter 7

This is C9.1 in the 4th intl ed.

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