PSC 405, Spring 2017
Problem Set #2
Problem set is due Tuesday, Mar. 7th in class.
1. Suppose we estimate the model yi = µ + ui , where ui ∼ N (0, σi2 ).
(a) What does the OLS estimator simplify to in this case?
(b) Directly obtain a consistent estimate of the variance of µ̂. Show
that this equals White’s heteroskedastic consistent estimate of
the variance.
2. Suppose the DGP is yi = β0 xi + ui , where ui = xi i , and xi and i are
both standard normal. Assume that the data are independent over i
and that xi is independent of i . (The first four central moments of
N [0, σ 2 ] are 0, σ 2 , 0, and 3σ 4 .)
(a) Show that the error term ui is heteroskedastic.
(b) Calculate plim N −1 (X0 X)−1 . (Get E[x2i ] and apply a law of large
numbers.)
(c) Calculate σ02 = Var[ui ], where the expectation is with respect to
all stochastic variables in the model.
(d) Calculate plim N −1 X0 Ω0 X = lim N −1 E[X0 Ω0 X], where Ω0 =
Diag[Var(ui |xi )]
(e) Using the above, give the default OLS
√ result for the variance
matrix in the limit distribution of N (β̂ − β 0 ), ignoring heterskedasticity.
(f) Using
√ the above, give the variance matrix in the limit distribution
of N (β̂ − β 0 ) taking account of heteroskedasticity.
(g) What do you notice about your answers to (e) and (f)?
3. Consider the linear regression model y = Xβ + u.
(a) Calculate the formula for β̂ that minimizes u0 Wu, where W is
full rank. (Use the chain rule for matrix differentiation.)
(b) Show that this simplifies to the OLS estimator if W = I.
(c) Show that this gives the GLS estimator of W = Ω−1 .
4. Consider least squares estimation of the model y = Xβ + u, where
Σ = σ 2 (I+AA0 ), where A is an N ×m matrix with k < m < N and for
simplicity, assume that σ 2 and A are known, and X is nonstochastic.
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PSC 405, Spring 2017
Problem Set #2
(a) Calculate the variance of the OLS estimator using
a
β̂ ∼ N (β, (X0 X)−1 X0 ΩX(X0 X)−1 ).
(b) Compare the above to the default variance estimate σ 2 (X0 X)−1 .
(c) Give the variance of the GLS estimator of β, using the result that
(I + AA0 )−1 = IN + A(Im + A0 A)−1 A0 .
(d) In general, GLS is more efficient than OLS. But what if we compare the default variance of OLS with the true variance of GLS?
Comment.
5. King (1986) points out that one can calculate standardized regression
coefficients by multiplying the unstandardized regression coefficient
by “the ratio of the standard deviation of the respective independent
variable to the standard deviation of the dependent variable.” If you
do this for simple regression (one independent variable), what is the
resulting value? What would be the effect of increasing the variation
in X on this value?
6. Using the accompanying data set, run a regression of the proportion
of votes cast for the Democratic candidate on the state unemployment
rate and west.
(a) Check the regression for influential observations and reestimate
if necessary.
(b) Substantively interpret all estimated coefficients, standard errors,
and the residual standard error.
(c) Demonstrate how and actually calculate every number in the regression summary (25 numbers).
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