R = X3
■ 1 _
How would you find out from the correlation matrix whether (a) there is perfect collinearity, (b) there is less than perfect collinearity, and (c) the X's are uncorrelated.
Hint: You may use |R| to answer these questions, where |R| denotes the determinant of R. *10.18. Orthogonal explanatory variables. Suppose in the model
X2 to Xk are all uncorrelated. Such variables are called orthogonal variables. If this is the case:
a. What will be the structure of the (X'X) matrix?
c. What will be the nature of the var-cov matrix of (0?
d. Suppose you have run the regression and afterward you want to introduce another orthogonal variable, say, Xk+1 into the model. Do you have to recompute all the previous coefficients ft to ft ? Why or why not?
10.19. Consider the following model:
GNPt = + ftMt + ftMt_1 + ft(Mt - Mt_1) + Ut where GNPt = GNP at time t, Mt = money supply at time t, Mt-1 = money supply at time (t — 1), and (Mt — Mt-1) = change in the money supply between time t and time (t - 1). This model thus postulates that the level of GNP at time t is a function of the money supply at time t and time (t - 1) as well as the change in the money supply between these time periods.
a. Assuming you have the data to estimate the preceding model, would you succeed in estimating all the coefficients of this model? Why or why not?
b. If not, what coefficients can be estimated?
380 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
c. Suppose that the ftMt-1 term were absent from the model. Would your answer to (a) be the same?
d. Repeat (c), assuming that the term ftMt were absent from the model. 10.20. Show that (7.4.7) and (7.4.8) can also be expressed as
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