Testing and Dummy Variables

• Let

y = X + Z + u;

and consider the restriction

= 0:

Show that the Wald statistic and the Likelihood Ratio statistic are the same.

• Consider the model

y = X + u

where u N 0;²I. Consider testing the null hypothesis,

H₀ : g () = 0 vs H_A : g () 6= 0

where g() is a nonlinear function of g(.);

1. Find a consistent estimator of
2. Find the asymptotic distribution of your estimator; and
3. Use it to construct a Wald test statistic.

• Consider the models logW_i = ₀ + ₁Educ_i + ₂Female_i + ₃Black_i + ₄Asian_i + u_1i; logW_i = ₀ + ₁Educ_i + ₂Male_i + ₃White_i + ₄Asian_i + u_2i: a) Write each term as a linear function of the terms,

;

.

1. Show that

where and are OLS estimates.

1. Show that theb b for both equations are identically the same.