2. Consider the least-squares regression estimates
b , the 95% asymptotic con…dence interval for , in terms of b , V bb , R, and z = 1:96 (a) Write down C (the 97.5% quantile of N (0; 1)). b ” is an asymptotic 5% test of H0 : = 0 . (b) Show that the decision “Reject H0 if 0 2 =C yi = x1i b 1 + x2i b 2 + e bi yi = e 2 x2i + e e2i
and the “one regressor at a time” regression estimates yi = e 1 x1i + e e1i
3. Take a regression model with i.i.d. observations (yi ; xi ) and scalar xi yi = xi + ei E (ei j xi ) = 0
with resቤተ መጻሕፍቲ ባይዱduals e bi = yi b e =
1X 2 2 x e n i=1 i i 1X 2 2 x e b n i=1 i i
n
n
xi b . Consider the estimates of
= n e n b
(a) Find the asymptotic distribution of (b) Find the asymptotic distribution of
p p
as n ! 1. as n ! 1.
(c) How do you use the regression assumption E (ei j xi ) = 0 in your answer to (b)?
Under what condition does e 1 = b 1 and e 2 = b 2 ?
The parameter of interest is
=
2
bb for Vb : (b) Suggest an (approximate) biased-corrected estimator b using an estimate V (c) For b to be potentially unbiased, which estimate of Vb is most appropriate? Under which conditions is b unbiased? 4. Take a regression model with i.i.d. observations (yi ; xi ) and scalar xi yi = xi + ei E (ei j xi ) = 0
2 = E x2 i ei
(a) Find E (bjX ) using our knowledge of E ( b jX ) and Vb = var( b jX ): Is b biased for ?
2 . Consider the OLS estimates b and b = b
Let b be the OLS estimate of
Econometrics 710 Midterm Exam March 16, 2017 1. Take the model yi E (xi ei ) with parameter of interest variance estimate. = R0 with R k = x0 i + ei = 0 bb its 1. Let b be the least-squares estimate and V