[SOLVED] Econometrics ii homework 2 spring 2025 p0

1. Frisch-Waugh with 2 regressors. Assume you regresss
yi = β1x1i + β2x2i + error .
In vector notation:
Y = β1X1 + β2X2 + error .
Now
′ X1X1 X1′X2 ! ′
(X X) = X1′X2 X2′X2 .
Find the vector of parameters β = (β1,β2)′, writing in terms of the inner products (like etc.). It is not hard to invert X′X, because the inner products are scalars. If you find it easier, you can assume without loss of generality that the units are chosen such that = 1 and (“Without loss of generality,” because we can change the units of the regressors without changing the results.) Now regress
X2 = X1ξ + error ,
and find the fitted value P1X2 = X1 ∗ ξˆ (remember that P1X2 is proportional to X1) and the residual M1X2 = X2− P1X2.
Finally, regress
Y = (M1X2)β2 ,
and verify that the βˆ2 that you get from this second regression is the same as the βˆ2 from the original regression.
2. Computer question (continuation of homework 1). In Matlab, regress real per capita U.S. data consumption growth on income growth and the interest rate using the posted dataset. (This is the what you did in homework 1.)
a) Regress income growth on the interest rate and take the residual which we can call MrY .
b) Regress consumption growth on MrY and compare the estimated coefficient to the one from the first regression.
c) Regress consumption growth on the interest rate and take the residual which we can call MrC
d) Regress MrC on MrY and compare the estimated coefficient to the one from the first and second regressions.
e) Draw a series of iid mean zero terms and add to income growth, getting Y ∗ and regress consumption on Y ∗ (income with measurement error) and the interest rate. Verify that the estimated coefficient to income gets closer to zero when there is measurement error. What happens to the coefficient to the interest rate? Repeat the regression with a much higher variance of the measurement error (“larger” measurement errors). What happens?