Imposing and Testing Equality Constraints in Models

Transcription

1 Impoing and Teting Eqality Contraint in Model Richard William, Univerity of Notre Dame, Lat revied ebrary 15, 015 Overview. We have previoly diced how to impoe and tet vario retriction on model. In thi ection we will extend thi dicion by explaining how to tet whether two or more coefficient within a model are eqal; we ll alo how how to tet more complicated ort of eqality contraint. Tet for eqality of parameter within a model. Sppoe yo wih to tet H0: β1 β HA: β1 β It may be helpfl to note that thi i the ame a teting H0: β1 - β 0 HA: β1 - β 0 That i, yo want to tet whether two variable have eqal effect. or example, in a model of family deciion-maing, yo might hypotheize that wive have the ame amont of inflence a their hband. Or, yo might want to tet whether time pent in one type of activity ha the ame effect a time pent in another activity. There are at leat way of doing thi. Option 1. Wald Tet. Wald tet are compted ing the etimated coefficient and the variance/covariance of the etimate from the ncontrained model. A nice featre of Wald tet i that they only reqire the etimation of one model. Thi i the approach ed by Stata tet command, where it i qite eay and imple to e. Here i the rationale for thi approach: Recall that teting βedc βjobexp i eqivalent to teting βedc - βjobexp 0. Alo recall that V(X ± Y) V(X) + V(Y) ± COV(X,Y) σ X ± Y Thi implie that V(bEdc - bjobexp) V(bEdc) + V(bJobexp) - COV(bEdc, bjobexp) Hence, an appropriate tet tatitic for thi problem i 1, N 1 ( b Edc b bedc Job exp + ) ( β Edc β Job exp ) 0 ( b Edc b Job exp b Job exp b b +, exp Edc bjob exp Edc bjob ) bedc, bjob exp We have een thi idea many time before: Oberved Vale Vale Predicted by the Nll (the predicted difference in thi cae being zero) divided by the etimated tandard error of the etimator. Since an tet i being reported, all of thi i qared. Impoing and Teting Eqality Contraint in Model Page 1

2 Stata Example. Here i a modified verion of the income/edcation/job experience example we have been ing. I have rewored the data o that it i now a ample of 100 blac and for hndred white. We want to tet whether a year of job experience (JOBEXP) ha the ame effect on income a a year of edcation (EDUC).. e clear. reg income edc jobexp Sorce SS df MS Nmber of ob (, 497) Model Prob > Reidal R-qared Adj R-qared Total Root MSE edc jobexp _con tet edc jobexp ( 1) edc - jobexp 0 ( 1, 497) Prob > The tatitic i highly ignificant, which mean we reject the hypothei that the two effect are eqal. To mae re that Stata did thing correctly, yo can e the vce command after rnning regre to get the variance and covariance (etat vce alo wor). vce edc jobexp _con edc jobexp _con Hence, 1, N 1 ( ) bedc bjob exp ( ) b * Edc + bjobexp b Edc, bjobexp Incidentally, in Stata, if yo want to ee what the contrained parameter etimate loo lie, add the coef parameter to the tet command, e.g. Impoing and Teting Eqality Contraint in Model Page

3 . tet edc jobexp, coef ( 1) edc - jobexp 0 ( 1, 497) Prob > Contrained coefficient Coef. Std. Err. z P> z [95% Conf. Interval] edc jobexp _con Epecially if yo have tried to tet a fairly complicated hypothei, it i good to loo at the contrained coefficient to mae re yo pecified thing correctly, e.g. if for ome reaon yo hypotheized that βedc 3 * βjobexp 1,. tet edc 3*jobexp - 1, coef ( 1) edc - 3 jobexp -1 ( 1, 497) Prob > Contrained coefficient Coef. Std. Err. z P> z [95% Conf. Interval] edc jobexp _con tet alo mae it eay to tet impler hypothee, e.g.. tet edc ( 1) edc ( 1, 497) 1.58 Prob > Converely, Stata tetnl command let yo tet complicated nonlinear ort of relationhip among coefficient, e.g. if for ome reaon yo hypotheize Sqrt(βEdc) βjobexp,. tetnl qrt(_b[edc]) _b[jobexp] (1) qrt(_b[edc]) _b[jobexp] (1, 497) Prob > Impoing and Teting Eqality Contraint in Model Page 3

4 Option : Incremental Tet. The incremental tet i another approach. Appendix A review incremental tet in general, and Appendix B how the math involved for teting eqality contraint; in thi ection we will imply otline the logic. Some ey advantage of thi approach are that (a) yo can e it with mot tatitical oftware, and (b) even thogh thi in t alway the eaiet approach, it i important to ndertand it becae the trategy ed here i imilar to the trategy that i optimal for other tatitical techniqe lie logitic regreion. Yo can proceed a follow: Regre Y on X1, X, and any other IV in the model. Store the relt. We refer to thi a the ncontrained model, becae the effect of X1 and X are not contrained to be eqal. That i, yo are etimating the model y α + β X + β X + β X + ε 1 1 Compte a new variable that i eqal to the m of the two variable yo hypotheize to have eqal effect, e.g. gen m1 x1 + x Rn a econd regreion in which yo regre Y on SUM1 and any other IV in the model. (Do NOT inclde X1 and X thogh.) Store the relt. We refer to thi a the contrained model, becae, by adding X1 and X together, yo are forcing their etimated effect to be eqal (i.e. only one beta i being etimated for both variable). That i, yo are etimating the model In practice, we etimate thi via 3 y α + β ( X + X ) + β X + ε y α + βm1 + β X + ε Do an incremental tet. A Appendice mae clear, yo are baically teting whether the R from the ncontrained model (where coefficient are not eqal) ignificantly differ from the R from the contrained model (where coefficient have been contrained to be eqal). If they do ignificantly differ then yo reject the hypothei that the coefficient are eqal. Thi procedre can be eaily modified to tet imilar hypothee. or example, if yo hypotheize that X1 and X have eqal bt oppoite effect, compte a variable lie DI1 X1 - X. If yo thin that the effect of X1 i twice that of X, compte omething lie WSUM1 X1 + X. If yo hypotheize that 3 variable have eqal effect, compte SUM13 X1 + X + X3. (Note that J in thi cae.) EXAMPLE. irt, we etimate the ncontrained model. INCOME i regreed on EDUC and JOBEXP, yielding the following: Impoing and Teting Eqality Contraint in Model Page 4

5 . e " clear. reg income edc jobexp Sorce SS df MS Nmber of ob (, 497) Model Prob > Reidal R-qared Adj R-qared Total Root MSE edc jobexp _con et tore ncontrained Now, we etimate the contrained model. irt, we compte JOBED EDUC + JOBEXP. Then, we regre INCOME on JOBED.. gen jobed jobexp + edc. reg income jobed Sorce SS df MS Nmber of ob ( 1, 498) Model Prob > Reidal R-qared Adj R-qared Total Root MSE jobed _con et tore contrained We can now e Maarten Bi ftet command, which can be downloaded from SSC. ftet let yo do tet of neted model. It i particlarly efl when model are neted (a in thi cae) bt cannot be etimated via the netreg command.. ftet ncontrained contrained Amption: contrained neted in ncontrained ( 1, 497) prob > Conclion. If yo are doing OLS regreion and yo are ing Stata, Option 1 (Wald tet) i probably the eaiet way to go. HOWEVER, if yo are ing a maximm lielihood techniqe lie logitic regreion, a modified verion of Option (ing a chi-qare tatitic intead of ) tend to be optimal. Alo, while we have primarily taled abot teting the eqality of coefficient, e.g. β 1 β, we have alo een that mch more complicated ort of tet are poible. Yo can alo do impler tet, lie β 1 3. Of core, any tet yo do hold have a rationale behind it. Yo don t do tet jt becae they are poible, yo do them becae there are btantive reaon that motivate them. Impoing and Teting Eqality Contraint in Model Page 5

6 Appendix A: Incremental Tet abot a bet of coefficient. We ometime wih to tet hypothee concerning a bet of the variable in a model. or example, ppoe a model inclde 3 demographic variable (X1, X, and X3) and peronality meare (X4 and X5). We may want to determine whether the peronality meare actally add anything to the model, i.e. we want to tet H0: β4 β5 0 HA: β4 and/or β5 0 One way to proceed i a follow. 1. Etimate the model with all 5 IV inclded. Thi i nown a the ncontrained model. Retrieve the vale for SSE and/or R (hereafter referred to a SSE and R.) [NOTE: If ing the R vale, copy them to everal decimal place o yor calclation will be accrate.]. Etimate the model ing only the 3 demographic variable. We refer to thi a the contrained model, becae the coefficient for the exclded variable are, in effect, contrained to be 0. Retrieve the vale for SSE and/or R (hereafter referred to a SSEc and R c). 3. Compte the following: ( R Rc) * ( N 1) ( SSEc SSE) / J ( SSEc SSE) * ( N 1) J, N 1 (1 R) * J SSE / ( N 1) SSE* J where J the nmber of contraint impoed (in thi cae, ) and the nmber of variable in the ncontrained model (in thi cae, 5). Pt another way, J the error d.f. for the contrained model min the error d.f. for the ncontrained model. If J 1, thi procedre will lead yo to the ame conclion a two-tailed T tet wold (the above will eqal the T from the ncontrained model. ) If J, i.e. all the IV are exclded in the contrained model, the incremental and the Global become one and the ame; that i, the global i a pecial cae of the incremental, where in the contrained model all variable are contrained to have zero effect. Yo can ee thi by noting that, if there are no variable in the model, R 0. When yo can e incremental. In order to e the incremental tet, it mt be the cae that The ample i the ame for each model etimated. Thi amption might be violated if, ay, miing data in variable ed in the ncontrained model caed the ncontrained ample to be maller than the contrained ample. Yo hold be carefl how miing data i getting handled in yor tatitical rotine One model mt be neted within the other; that i, one model mt be a contrained, or pecial cae, of the other. or example, if one model contain IV X1-X5, and another model contain X1-X3, the latter i a pecial cae of the former, where the contraint impoed are Impoing and Teting Eqality Contraint in Model Page 6

7 β4 β5 0. If, however, the econd model inclded X1-X3 and X6, it wold not be neted within the firt model and an incremental tet wold not be appropriate. Other type of contraint can alo be teted with an incremental tet. or example, we might want to tet the hypothei that β1 β, i.e. two variable have eqal effect. We ll dic ch poibilitie later. Other comment Contrained and ncontrained are relative term. An ncontrained model in one analyi can be the contrained model in another. In reality, every model i contrained in the ene that more variable cold alway be added to it. Wald tet, which are eaily done in Stata, are an alternative to incremental tet. We are alo often intereted in doing ch tet when etimating eqence of neted model. So, for example, Model 1 may inclde X1, X and X3; Model may add X4 and X5; Model 3 add X6 and X7; and o on. With each model we may want to tet whether the variable added in that model have effect that ignificantly differ from 0. Impoing and Teting Eqality Contraint in Model Page 7

8 Appendix B: Incremental Tet for eqality contraint Sppoe we wih to tet hypothee lie H0: β4 β5 HA: β4 β5 To tet for eqality contraint ing an incremental tet, the procedre wor a follow: Compte a new variable that i eqal to the m of the two variable yo hypotheize to have eqal effect, e.g. gen m1 x1 + x Regre Y on X1, X, and any other IV in the model. Retrieve the Uncontrained Error Sm of Sqare (SSE) or ele the ncontrained R (e everal decimal place). We refer to thi a the ncontrained model, becae the effect of X1 and X are not contrained to be eqal. That i, yo are etimating the model y α + β X + β X + β X + ε 1 1 Rn a econd regreion in which yo regre Y on SUM1 and any other IV in the model. (Do NOT inclde X1 and X thogh.) Retrieve the contrained error m of qare (SSEc) or ele the contrained R (e everal decimal place). We refer to thi a the contrained model, becae, by adding X1 and X together, yo are forcing their etimated effect to be eqal (i.e. only one beta i being etimated for both variable). That i, yo are etimating the model In practice, we etimate thi via 3 y α + β ( X + X ) + β X + ε y α + β1sm1 + 3 Do an incremental tet. In thi cae, J 1, o we get β X + ε ( SSEc SSE) / 1 ( SSEc SSE) *( N 1) SSE / ( N 1) SSE * 1 1, N 1 ( R Rc ) *( N 1) ( 1 R ) * 1 Impoing and Teting Eqality Contraint in Model Page 8

9 EXAMPLE. Uing the ame example a before, etimate the ncontrained model firt.. e " clear. reg income edc jobexp Sorce SS df MS Nmber of ob (, 497) Model Prob > Reidal R-qared Adj R-qared Total Root MSE edc jobexp _con Hence, SSE , R.8163, N 500,. Now, we etimate the contrained model. irt, we compte JOBED EDUC + JOBEXP. Then, we regre INCOME on JOBED.. gen jobed jobexp + edc. reg income jobed Sorce SS df MS Nmber of ob ( 1, 498) Model Prob > Reidal R-qared Adj R-qared Total Root MSE jobed _con Th, we get SSEc , R c.6074, J 1. The incremental tet i then 1, N 1 ( SSEc SSE) * ( N 1) SSE * 1 ( R Rc ) * ( N 1) ( 1 R ) * 1 ( ) * 497 ( ) * Thi vale i highly ignificant. Ergo, we reject the nll hypothei that edcation and job experience have eqal effect. Impoing and Teting Eqality Contraint in Model Page 9