Find the variance of the OLS estimator for the set up in (a) and use this to show that this estimator is

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1. Suppose that the variables  yt, wt, zt, are observed over T time periods

(t = 1,…,T) and it is thought that E[yt] depends linearly on wt (without a constant) but not on zt for the first T1 periods, and E[yt] depends linearly on zt (without a constant) but not on wt for the last T2 periods, with T1 + T2 = T. Write down a linear regression model that could be used to test this hypothesis, and specify the nature of the hypothesis in terms of the coefficients of the model.  [Hint: try using four regressors.]

2. For the simple "location" model yt =  + ut, t = 1,...,T

(a) Suppose that ut = t + z where the t are T i.i.d. (0, 2) random variables and z is a random variable, independent of each of the t and with E[z] = 0, var[z] = 1. Find the T  T covariance matrix of the ut. Show by means of Kruskal's theorem or otherwise that the OLS estimator of  the parameter  is BLU.

(b) Find the variance of the OLS estimator for the set up in (a) and use this to show that this estimator is consistent if  = 0 but inconsistent if   0. (To demonstrate consistency here show that the variance of the estimator tends to zero as T  , and for inconsistency show that this does not happen.) Does this make sense intuitively?

(c) Suppose that ut = t + t-1, where the t are i.i.d. (0, 2). Find the covariance matrix of the ut, and use this to show that the condition of Kruskals's theorem is satisfied, except for the first and last rows of the matrices appearing in the condition. What can we deduce from this ?

3. Consider the autoregessive model yt = yt-1 + ut, t = 1,...,T

ut = t + t-1

Where the t are independently and identically distributed with E[t] = 0, E[t2] = 2, and y0 = 0 = 0.

(a) Suppose that  = 0. Is the OLS estimator, b = i=2,T ytyt-1 / i=2,T yt-12, of  consistent? You may confine attention to the case where  = 0. 

(b) Suppose that   0. Find the probability limit of b in the case where  = 0, and use this to conclude that the OLS estimator is inconsistent in this case. What is the intuition behind this result?

(c) In order to get around the inconsistency in part (b), a researcher proposes using yt-2  as an instrumental variable to obtain the estimator

bIV = i=3,T ytyt-2 / i=3,T yt-1yt-2.

What is a justification for this suggestion? Find the probability limit of this estimator when

  0. You may confine attention to the case where  = 0.

4. Consider the Seemingly Unrelated Regressions model y1 = X1β1 + u1 y2 = X2β2 + u2

in which X1, X2 are n  k non-stochastic matrices with X1'X2 = 0, X1'X1 = X2'X2 = Ik , and E[uj] = 0, j = 1,2  E[ujuh'] = σjhIn for  j, h  with σjh = 1 for  j = h  and σjh = r for  j  h  (with -1 < r < 1), j, h = 1,2. Assume r  is known.

(a) Find the covariance matrix of the GLS estimator of β1.

(b) If you knew that 1 = 2, show how you can make use of this information in constructing a restricted GLS estimator of 1 (and of 2). Compare the covariance matrix of the estimator you propose with that of the unrestricted GLS estimator of β1 in part (a).

(c) Someone notes that when it is known that β1 = β2, we can write, for any choice of the scalar λ,

y1 + λy2 = (X1 + λX2)β + u1 + λu2,

or

y* = X*β + u*

where y* = y1 + λy2, X* = X1 + λX2, u* = u1 + λu2 . Find the covariance matrix of u* and hence the covariance matrix of the OLS estimator of β in the regression of y* on X*.

Suppose that λ is chosen so that λ = 1 if r ≤ 0 and λ = -1 if r > 0: compare the covariance matrix of this estimator with the covariance matrix of the restricted GLS estimator in part (b).

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