1. Consider the following data set: to make numbers manageable we will use an unrealistically small sample size of 12. The variable to be 'explained' is Y, the growth rate of nominal output (%) across a sample of twelve countries, and the 12 observations are:
Y = [2.7 1.2 4.2 0.9 3.5 5.1 2.8 3.8 1.7 2.9 4.3 3.7] (12x1 matrix)
while the inflation rates of the twelve countries are π = [4.6 7.8 2.1 12.0 4.4 4.5 9.0 2.5 3.8 3.3 5.9 3.6 ] (12x1 matrix)
a) Estimate the regression parameters β in the model yi = β0 + β1πi + εi , where β = (β0, β1)' . Compute the standard errors of these estimates and obtain the t−type statistic ^β1/ se^β1 , which is a test of H0 : β1 = 0. Why would you expect to have difficulty in rejecting this null in this sample, even if it is false?
b) Consider the matrices Mx = X(X'X)-1X' and Px= I − X(X'X)-1X' , where X is the 12 x 2 the matrix of regressors, i.e. the constant and π. Show algebraically that these matrices must be idempotent, give their dimensions, and use one of them to compute the regression residuals. Show that the residuals sum to zero.
c) Another economist estimates the model yi = γ0 + γ1πi + γ2Zi + εi , where Zi is a measure of aggregate investment in the economy over the previous five years. If this person is right to include Zi , and if Zi is positively correlated with πi , what does this imply about the results from your model?