Heteroskedasticity Example

Transcription

1 ECON 761: Heteroskedasticity Example L Magee November, 2007 This example uses the fertility data set from assignment 2 The observations are based on the responses of 4361 women in Botswana s 1988 Demographic and Health Survey The variables are: NAME DESCRIPTION 1 age woman s age in years 2 educ her years of education 3 ceb # of children ever born 4 urban =1 if she lives in an urban area; =0 if rural After generating some regressor variables, the model from question 2 of assignment 2 is estimated by OLS The hettest command results in a Cook-Weisberg test This is one of many tests that reject H 0 : no heteroskedasticity if there is evidence of a relation between the error variance and the regressors This one tests H 0 : τ = 0 in the variance formula σi 2 = σ2eŷiτ, where ŷ i is the usual predicted value It very strongly rejects H 0 This is not surprising The younger women will have had less children (lower ŷ i ) and a smaller variance in the distribution of number of children (lower σi 2) hettest age is a slight variation of this test, that checks for a relation between the error variance and the variable age, or H 0 : τ = 0 in the variance formula σi 2 = σ 2 e (age) iτ Again, the evidence of heteroskedasticity is very strong Next, OLS is run again, but with the robust option in order to produce asymptotically valid standard errors and t statistics, using an HCCME The coefficient estimates are unchanged As is often, but not always, the case, this adjustment results in larger standard errors, and consequently smaller t statistics and wider confidence intervals, for most of the estimates This makes the results look a bit less accurate than they did before, but that is only because the original results reported incorrect standard errors The next two lincom post-estimation commands are not directly related to the heteroskedasticity issue I included them so there would be something more tangible coming from this OLS estimation to compare with the GLS results that follow lincom produces an estimate of a linear combination of the regression coefficients, along with t-stats, etc These linear combinations were chosen to produce the predicted mean number of children of two rural women, each having 5 years of education, one age 20 and the other age 40 The predicted means are in the Coef column The predict post-estimation commands save the residuals and y-hats for use in generating weights which will be used later for weighted least squares (WLS) First, σi 2 is modelled as a linear 1

2 function of the y-hats in the reg res2 yhat command The predicted values from this regression (res2hat1) are the predicted variances The summarize command reveals that some of these are negative This is undesirable for at least two reasons First, Stata will drop observations with negative weights when running WLS, but since they probably are the small-variance observations, we want to give them more weight than they would get in OLS Second, if there are some negative estimated variances, there also may be estimated variances that are positive, but very very close to zero much closer than are the true variances Since WLS weights by the reciprocal of the estimated variance, those observations may be given an enormously high weight, which would distort the estimates A different model of res2 is estimated next This time it is modelled as a function of eyhat, which is exp(yhat) This results in predicted variances that are a concave function of the yhats, and as the summarize command at the bottom of p3 shows, the predicted variances (res2hat2) are all comfortably positive The WLS estimation is done at the top of p4 The coefficient estimates differ a bit from the OLS estimates, and the WLS standard errors are smaller This is to be expected since the standard errors are estimating the standard deviations of the coefficient estimates, and the goal of WLS is to make these smaller than the OLS ones Finally, the same lincom commands are executed as were after the OLS command Like the coefficients, these linear combinations, or predictions, are similar to the OLS ones, but with slightly smaller standard errors, and narrower, or tighter confidence intervals Overall, it looks like WLS has improved the results, but not by very much, considering the enormous test statistics when testing for het on pp 1 and 2 Some reasons for this small reduction in standard errors: (1) The large number of observations (n = 4361) As n gets large, the power of a test increases A test might then strongly reject, even if H 0 is only slightly false (2) Having a lot of het means OLS is not BLUE, but it doesn t necessarily mean that OLS is way worse than WLS (3) The model that I used to estimate the variances (in the command reg res2 eyhat) might not have captured as much of the het as was found in the original tests for het on pp 1 and 2 There may be variance models that produce better WLS estimators in this estimation problem 2

3 do "C:\DOCUME~1\magee\LOCALS~1\Temp\STD tmp" ****************************************************************************** ** het07do : November 2007 ** ****************************************************************************** clear capture log using "C:\Documents and Settings\courses\761 and 762\f07\heterosk\het07log", replace * load fertility data used in assignment 2 use "C:\Documents and Settings\courses\761 and 762\f07\heterosk\a2dta" * generate regressor variables gen age2=age^2 gen age3=age^3 gen educ2=educ^2 gen urban_age=urban*age gen urban_age2=urban*age2 gen urban_age3=urban*age3 * run OLS and test for presence of het reg ceb age age2 age3 educ educ2 urban urban_age urban_age2 urban_age F( 9, 4351) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = 1515 ceb Coef Std Err t P> t [95% Conf Interval] age age age educ educ urban urban_age urban_age urban_age _cons hettest Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance Variables: fitted values of ceb chi2(1) = Prob > chi2 =

4 hettest age Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance Variables: age chi2(1) = Prob > chi2 = * re-run OLS, with HCCME-based standard errors using "robust" option reg ceb age age2 age3 educ educ2 urban urban_age urban_age2 urban_age3, robust Linear regression Number of obs = 4361 F( 9, 4351) = Prob > F = R-squared = Root MSE = 1515 Robust ceb Coef Std Err t P> t [95% Conf Interval] age age age educ educ urban urban_age urban_age urban_age _cons * predict mean fertility at age 20 and age40 for rural women with 5 years of education lincom _cons+20*age+(20*20)*age2+(20*20*20)*age3+5*educ+(5*5)*educ2 ( 1) 20 age age age3 + 5 educ + 25 educ2 + _cons = 0 ceb Coef Std Err t P> t [95% Conf Interval] (1) lincom _cons+40*age+(40*40)*age2+(40*40*40)*age3+5*educ+(5*5)*educ2 ( 1) 40 age age age3 + 5 educ + 25 educ2 + _cons = 0 ceb Coef Std Err t P> t [95% Conf Interval] (1) * save residuals and predictions ("y-hat"s) for modelling the variance for FGLS predict resid,residuals predict yhat,xb gen res2=resid*resid 2

5 * predict variance as a linear function of yhat reg res2 yhat F( 1, 4359) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = 415 res2 Coef Std Err t P> t [95% Conf Interval] yhat _cons predict res2hat1,xb * notice that some of the predicted variances ("res2hat"s) are negative summarize res2hat1 Variable Obs Mean Std Dev Min Max res2hat * predict variance using exp(yhat) instead gen eyhat=exp(yhat) reg res2 eyhat F( 1, 4359) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = res2 Coef Std Err t P> t [95% Conf Interval] eyhat _cons predict res2hat2,xb * now all of the predicted variances are positive and not too close to zero summarize res2hat2 Variable Obs Mean Std Dev Min Max res2hat

6 * proceed with FGLS estimation using "aweight" option and the "res2hat" variance estimates reg ceb age age2 age3 educ educ2 urban urban_age urban_age2 urban_age3 [aweight=res2hat2^(-1)] (sum of wgt is 30293e+03) F( 9, 4351) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = ceb Coef Std Err t P> t [95% Conf Interval] age age age educ educ urban urban_age urban_age urban_age _cons * predict mean fertility at age 20 and age40 for rural women with 5 years of education, but now using FGLS lincom _cons+20*age+(20*20)*age2+(20*20*20)*age3+5*educ+(5*5)*educ2 ( 1) 20 age age age3 + 5 educ + 25 educ2 + _cons = 0 ceb Coef Std Err t P> t [95% Conf Interval] (1) lincom _cons+40*age+(40*40)*age2+(40*40*40)*age3+5*educ+(5*5)*educ2 ( 1) 40 age age age3 + 5 educ + 25 educ2 + _cons = 0 ceb Coef Std Err t P> t [95% Conf Interval] (1) * results are similar to OLS results, but now standard errors are smaller log close log: C:\Documents and Settings\courses\761 and 762\f07\heterosk\het07log log type: text closed on: 23 Nov 2007, 11:56: