Testing and Dummy Variables
1) Let
y = X¯ + Z° + u;
and consider the restriction
° = 0:
Show that the Wald statistic and the Likelihood Ratio statistic are the same.
2) Consider the model
y = X¯ + u
where u » N ¡0;¾2I¢. Consider testing the null hypothesis,
H0 : g (¯) = 0 vs HA : g (¯) 6= 0
where g(¢) is a nonlinear function of g¯(.¯);
a) Find a consistent estimator of
b) Find the asymptotic distribution of your estimator; and
c) Use it to construct a Wald test statistic.
3) Consider the models logWi = ¯0 + ¯1Educi + ¯2Femalei + ¯3Blacki + ¯4Asiani + u1i; logWi = ®0 + ®1Educi + ®2Malei + ®3Whitei + ®4Asiani + u2i: a) Write each ® term as a linear function of the ¯ terms,
;
.
b) Show that
¯
where ® and are OLS estimates.
c) Show that theb b for both equations are identically the same.
