If we apply least squares to the transformed data we obtain. which yields the generalized least squares estimator of β, i.e.,

Transcription

1 Econ 388 R. Butler 04 revsons lecture 6 WLS I. The Matrx Verson of Heteroskedastcty To llustrate ths n general, consder an error term wth varance-covarance matrx a n-by-n, nxn, matrx denoted as, nstead of σ I nxn. Then we wll fnd a transformaton, T, that rotates ε nto a sphercal dstrbuton. It works lke ths: y = β + ε ε ~ N0,. Transform the model and data by premultplyng by the matrx T [TY] = [T]β + [Tε] So ths T s one of the C -type transformatons dscussed n lecture 4. Revew t f you haven t already commtted t to memory. If we can construct a matrx T such that Tε ~ N0, T T = σ I, t follows that T T = σ I Transformed error terms Tε, satsfy the classcal error assumptons, so that OLS on the transformed better yeld desrable estmators. From the last result above we see that Σ = σ T T Σ = σ T T If we apply least squares to the transformed data we obtan β T = [T T] T TY β T = T T T TY = y y whch yelds the generalzed least squares estmator of β,.e., β T = Σ Σ Y. What s the covarance? Substtute y = β + ε, then T, Whch you transpose both sdes, multply, and take expected values to get V T you should prove ths to yourself. HETEROSKEDASTIC ROBUST COVARIANCE MATRI: Suppose you do OLS, but the errors have covarance matrx. In ths case you have so that now take the expected value of both sdes to get V Ths s the robust covarance estmator. If the OLS assumpton that I V holds, then the robust varance estmator reduces to the usual OLS V. Otherwse, t s estmated by the general matrx formula V, Or n summaton notaton N N N V Where I have employed an overly general notaton one that apples to systems of equatons as well as sngle equatons for the terms n partcular, f t s just a sngle equaton model than s the th row observaton of the data matrx, and =.

2 II. Consder the model from the ndvdual observaton perspectve to get a better feel for what the transformaton does and what t looks lke: y = β + ε = β + β x βkxk + where ~ N 0, σ We transform ths model to one characterzed by homoskedastcty by premultplyng the orgnal formulaton by σ σ,.e., where σ s an unknown constant y x = x k k + Note that the varance of the transformed random dsturbance s gven by Var σ σ ε = σ σ Varε = = and σ σ satsfes assumptons A.-A.4. The transformaton matrx n ths the heteroskedastcty case s gven by / / T = σ 0 0 / / n Note that: T T / 0 0 / / /... n = σ I and the transformed data matrces are gven by: Y / Y* = Y / TY... Y n / n......

3 * = / x / x k / / n x n / n x nk / n = T An applcaton of least squares to the transformed data wll yeld BLUE of β. It can be verfed that TT = σ -. Note: In the GLS estmator the multplcatve constant n the transformaton matrx s arbtrary and wll cancel out. Denote T by σd so that T T = D σ D = σ Σ or D D = Σ If the model s y = + ε we obtan = ~ - = TT TTy = σ - - σ - y = y = ~ - = D D D Dy = y Thus when choosng a T matrx for data transformaton the unknown constant σ need not be specfed, as just the D part of the matrx s necessary. III. Correctng for Heteroskedastcty by Data Transformatons A. Redefne the varables: sometmes log transformatons help, sometmes puttng the data nto per capta terms see the example n chapter 8. B. Weghted Least Squares: Use the tests n the last lecture to detect the presence of heteroskedastcty, then use the [w=whatever] opton n Stata whch wll be the elements of the dagonal matrx above, the same for the weght=whatever; opton n SAS..A couple of examples: a. f you happen to know that Z where terms out to be about two, then the factor of proportonalty s Z. b. So create a / Z varable call t nv_zsq, and then use the followng type command the square root of these elements form the dagonal elements of the T-matrx above, STATA CODE: regress wklywg educ age female whte experen [w=nv_zsq];

4 If you assume that V = σ,.e., the varance for each ndvdual s proportonal to ther value of. Then the weghted least squares soluton n STATA CODE s: gen wt=/x; regress y x x [w=wt]; Recall we dd somethng smlar to ths when we dscussed the lnear probablty model n a prevous lecture.. When you don t know the factor of proportonalty, but need to estmate t, there are some flexble alternatves to dong FEASIBLE GENERALIZED LEAST SQUARES estmaton. Two useful approaches mentoned n our text nvolve regressons of the log of the squared resduals on ether a all the ndependent varables, or b the predcted value of y yhat and ts square. For approach a for example, the weghts used would be / ĥ where the ĥ = exp k k where the estmated alpha parameters come from the regresson of the log of the squared resduals on the ndependent varables,,,, k, as follows log = k k For approach b, do the same but n the frst step regress log of the squared resduals on yhat and yhat squared log = y y 0 A STATA example follows the demand for cgarettes example from chapter 8 n the Wooldrdge text: # delmt ; *accessng wooldrdge data ; nfle educ cgprc whte age ncome cgs restaurn lncome agesq lcgprc usng "g:\classrm_data\wooldrdge\smoke.raw"; * frst, test for heteroskastcty; regress cgs lncome lcgprc educ age agesq restaurn; predct resds, resduals; hettest; hettest, rhs; mtest, preserve whte; gen resdsq = resds*resds; gen lnres_sq = logresdsq; regress lnres_sq lncome lcgprc educ age agesq restaurn; predct res_hatd; gen nv_wt = expres_hat;

5 gen wt = /nv_wt; regress cgs lncome lcgprc educ age agesq restaurn [w=wt]; wth some of the output as follows:. nfle educ cgprc whte age ncome cgs restaurn lncome > agesq lcgprc usng "C:\Documents and Settngs\rjb99\My Documents\BYU_Classes > \econ388\classrm_data\wooldrdge\smoke.raw"; 807 observatons read. * frst, test for heteroskastcty;. regress cgs lncome lcgprc educ age agesq restaurn; Source SS df MS Number of obs = F 6, 800 = 7.4 Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = cgs Coef. Std. Err. t P> t [95% Conf. Interval] lncome lcgprc educ age agesq restaurn _cons predct resds, resduals;. hettest; Breusch-Pagan / Cook-Wesberg test for heteroskedastcty Ho: Constant varance Varables: ftted values of cgs. hettest, rhs; ch = 56.7 Prob > ch = Breusch-Pagan / Cook-Wesberg test for heteroskedastcty Ho: Constant varance Varables: lncome lcgprc educ age agesq restaurn ch6 = 69.6 Prob > ch = mtest, preserve whte; Whtes test for Ho: homoskedastcty aganst Ha: unrestrcted heteroskedastcty ch5 = 5.7 Prob > ch = 0.00 Cameron & Trveds decomposton of IM-test Source ch df p Heteroskedastcty Skewness

6 Kurtoss Total gen resdsq = resds*resds;. gen lnres_sq = logresdsq;. regress lnres_sq lncome lcgprc educ age agesq restaurn; Source SS df MS Number of obs = F 6, 800 = 43.8 Model Prob > F = Resdual R-squared = Adj R-squared = 0.47 Total Root MSE =.469 lnres_sq Coef. Std. Err. t P> t [95% Conf. Interval] lncome lcgprc educ age agesq restaurn _cons predct res_hat; opton xb assumed; ftted values. gen nv_wt = expres_hat;. gen wt = /nv_wt;. regress cgs lncome lcgprc educ age agesq restaurn [w=wt]; analytc weghts assumed sum of wgt s.9977e+0 Source SS df MS Number of obs = F 6, 800 = 7.06 Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = cgs Coef. Std. Err. t P> t [95% Conf. Interval] lncome lcgprc educ age agesq restaurn _cons FINAL NOTE ON PROJECTION WITH GLS: Intuton 0: Whle orthogonal projectons are always possble mechancally, they cease to be the best BLUE statstcal technques when model assumptons fal: heteroskedastcty, autocorrelaton, and errors correlated wth the s. These all change

7 the nce sphercal shape of the errors dstrbuton of the multvarate normal errors, centered as they are, by assumpton, n the regresson plane nto non-sphercal dstrbutons. Therefore, what our transformatons wll do s to project Y onto the - plan, not orthogonally whch are for spheres but oblquely whch are for the nonsphercal. Snce GLS y, then the GLS projecton s y P y wth the error space projecton beng GLS I M You can readly show that these projectons are dempotent, and that M P 0. But the projectons are not symmetrc. Ths means they are not orthogonal. If they were orthogonal, then we would have M y P y, or that M y P y 0 or that y M P y 0 But you can check and show that M P 0 t would be zero f the operators were symmetrc. Thus, n general, GLS resduals are not orthogonal to GLS predcted values. Ths s because the GLS predcted ftted or P y values le n the - plane though P does not project orthogonally onto, t projects onto to so that P but P, but the GLS resduals le n the space orthogonal to. Hence the GLS projecton s oblque not at a rght angle, so the resdual vector must necessarly be longer than the OLS resdual vector by constructon, OLS s the shortest possble. See Davdson and McKnnon, ESTIMATION AND INFERENCE IN ECONOMETRICS, 993, for more on ths p [[[[Do you Want a Whole Hershey Bar?. Correcton for heteroskedastcty nvolves transformng the data wth a T matrx such that a. and Y are pre-multlpled by T b. T s a dagonal matrx c. the generalzed least squares estmator follows the usual - Y formula wth T and TY replacng and Y d. all of the above *. The Generalzed Least Squares approach to heteroskedastcty a. s the orthogonal projecton of TY onto T b. s the oblque projecton of Y onto * c. both a and b d. nether a nor b 3. The robust opton n regress Stata: a. corrects for all forms of heteroskedastcty [[not mpure heteroskedastcty]]

8 b. corrects only for that specfc form of heteroskedastcty specfed n a prevous weght statement c. corrects for all types of falures of the V I assumpton d. none of the above *