[SOLVED] Econometrics ii homework 3 spring 2025 p0

1. More Frisch-Waugh! Assume that your run a regression with 2 regressors (think of demeanedregressors). Assume the fitted value is
Yˆ = X1 + 4X2 .
Assume that you instead run the regression
(∗) Y = γ1 X1 + γ2 M1X2 + error ,
where M1 is the residual maker from regressing on X1. If P1 is the projection matrix on X1 and
P1X2 = 1.5X1 ,
what would be the estimated values of γ1 and γ2 in the regression (*)?
2. For the bivariate Normal distribution, derive the formula for the conditional density f(X2|X1).
,
where all the “OLS-assumptions” – including normality of ϵt – hold. The regressors Dkt are quarterly
dummy variables, such that
D1t = 1 in the 2nd quarter ; 0 otherwise
D2t = 1 in the 3rd quarter ; 0 otherwise
D3t = 1 in the 4th quarter ; 0 otherwise
Now assume that ¯y = 5 and if we let ¯yj ; j = 2,3,4 denote the average of the y-values in the kth quarter, assume that
y¯2 = 4 ,
y¯3 = 2 ,
y¯4 = 0 .
Also assume that ¯x = 0 and that xt is orthogonal to Dk ; k = 1,2,3.
Based on the given information, find the values of the OLS-estimates βˆ0,βˆ1,βˆ2 and βˆ3.
3. Computer question (continuation of homeworks 1 and 2). 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) Calculate the residual maker M and (using Matlab) calculate and display the eigenvalues and eigenvectors of M.
b) Generate the C matrix and the diagonal matrix of eigenvalues Λ (in the notation of class) and verify that CΛC′ = M. Display the values of C and Λ.
4. Computer question.
a) Generate two vectors of standard normally distributed variables e1 and e2 of length N = 100.
b) Generate X1 = e1 and let X2 = e1 + e2 and calculate the variance-covariance matrix Σ for X =
(X1,X2). (You can do that by hand, of course, but you will need to use it in the next question.)
c) Find a square root Σ1/2 of Σ using Matlab.
d) Calculate Y = (Y1,Y2) as Y = Σ−1/2X.
e) Calculate the covariance between Y1 and Y2 and verify that it is (close to) zero.