Testing for Omitted Variables

Transcription

1 Testng for Omtted Varables Jeroen Weese Department of Socology Unversty of Utrecht The Netherlands emal tel fax Prepared for North Amercan Stata users meetng Boston, March 2001 testomt 1/8 jeroen weese March 21, 2001

2 A short recap: Classc testng methods The three classc lkelhood-based approaches to test smooth hypotheses about parameters H : g( ) = 0, LR test: estmate model wth and wthout constrant g ( ) = 0. A large derence between t statstcs (e.g., devance) s evdence aganst H. Wald test: estmate the model wthout the constrant. Test whether the parameters satsfy a lnearzed verson of the constrant. (ecent) Score/Lagrange Multpler test: estmate the restrcted model. If the t crteron (log-lk) sharply ncreases n drectons away from the constrant, ths s evdence aganst the constrant. testomt 2/8 jeroen weese March 21, 2001

3 How to choose? Methods are often asymptotcally equvalent (under the null). Lkely, the hgher order asymptotc propertes of LR are better. Lttle s known n general about small sample propertes. Computatons may vary wdely It may be hard to estmate the restrcted model (e.g., non-lnear constrants g) It may be hard to estmate the unrestrcted model (e.g., n random eects/coef models, n whch the restrcton eectvely elmnates the random eects/coefs) testomt 3/8 jeroen weese March 21, 2001

4 Omtted varables n lnear form models Dd I use the rght set of predctor varables? non-lnear transformatons of an ncluded x-var (e.g., a squared term) s t rght to treat a varable as an nterval varable, or should t be treated as a categorcal varable (e.g., level of educaton) nteractons between x-varables What about some of the varables that I dd not enter n the model? (To hell wth theory!) Sometmes ths may nvolve ancllary parameters, e.g. the scale parameter n regresson-type models the between-equaton correlaton n selecton models the cutponts n an ordnal regresson model We typcally assume that these parameters are constant between subjects, but there s consderable attenton to heteroscedastcty ssues n regresson-style models, not so n other regresson-type models. testomt 4/8 jeroen weese March 21, 2001

5 Score testng for omtted varables n lf models Parameters a parameter-vector parttoned as = (, ), 1 2 and ˆ =(ˆ, 0). 0 1 x x11 x are the parameters of the restrcted model are assocated wth the omtted varables. Lnear predctor: lp = = + l s log-lkelhood contrbuton of -th observatons The score statstc l() l() U()= = x = sx lp Stata calls s a score varable. Let U ( ) = U( ) = sx It depends on the estmator only va s testomt 5/8 jeroen weese March 21, 2001

6 score tests Score tests are based on the large sample dstrbuton under H of the quadratc form 1 U (ˆ ) var( U) U (ˆ ) l() The score varable has to be evaluated under ˆ lp 0. And so t s computed f a score() estmatngthe restrcted model. 0 2 U E l k opton s speced whle How to estmate var( U)? The classc model-based estmator uses the fact under under regularty condtons, var( ) = () = I ( ) and so ths requres addtonal nformaton about the model that was estmated, namely the (expected) Fsher nformaton. An alternatve based on the hessan / observed nformaton s feasble. Yet another alternatve s the outer-product of gradents estmator, U (ˆ ) U (ˆ )= 2 s xx Ths requres only the score varable s. The modcaton of the OPG estmator to the case of clustered observatons and complex survey data s straghtforward. testomt 6/8 jeroen weese March 21, 2001

7 Desgn consderatons for a Stata command Language to specfy potentally omtted varables varables not yet n model (lp), transformatons of varables already n model (lp) factoral versons of vars n model (lp) Quadratc extenson of the current model (lp) Derent types of tests Lkelhood rato test Wald Score test, wth three estmator of the varance of the scores Unvarate as well as smultaneous tests. Adjusted P-values (Bonferron, Holm, Sdak,... ) testomt 7/8 jeroen weese March 21, 2001

8 Contnuaton The presentaton contnues wth the presenatons of the command (boston.do) testomt 8/8 jeroen weese March 21, 2001