# 2 30 points suppose that y t iid n µ σ 2 with µ 6

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2.) (30 points) Suppose that y t i.i.d. N ( µ, σ 2 ) with µ 6 = 0 and let q T = T/ P T t =1 y t . Prove that T ( q T µ 1 ) L N (0 , v ) and calculate v. Does v = lim T →∞ E ( q T µ 1 ) 2 ?

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–2– Econ 220B, Winter 2010 3.) (100 points) Suppose y t = x 0 t β + ε t where x t is a ( k × 1) vector of explanatory variables and E [ ε t x t | ε t 1 x t 1 , ε t 2 x t 2 , ..., ε 1 x 1 ] = 0 with { ε t , x t } stationary and ergodic and E ( ε 2 t x t x 0 t ) = σ 2 Q . You can further assume that T 1 P T t =1 x t x 0 t p Q which has rank k . The OLS estimates are given by b = ³ P T t =1 x t x 0 t ´ 1 ³ P T t =1 x t y t ´ and s 2 = ( T k ) 1 P T t =1 ( y t x 0 t b ) . a.) Derive the asymptotic distribution of b under the stated assumptions. b.) Write down the formula for the standard OLS F test for testing the null hypothesis that β 1 = β 2 , where β 1 denotes the fi rst element of the vector β and β 2 denotes the second element. c.) Derive the asymptotic distribution of the test you proposed in part (b) under the stated assumptions. Be sure your derivation accurately describes this particular example
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