12.4. Suppose you want to test the hypothesis that the real interest rate is constant, so that all changes in the nominal interest rate reflect changes in expected inflation. Thus your hypothesis is 1 = r + Emit+1. (a) Consider a regression of it on a constant and at+1. Does the hypothesis that the real interest rate is constant make a general prediction about the coefficient on at+1? Explain. (Hint: For a univariate OLS regression, the coefficient on the right-hand-side variable equals the covariance between the right-hand-side and left-hand-side variables divided by the variance of the right-hand-side variable.) (b) Consider a regression of +1 on a constant and i. Does the hypothesis that the real interest rate is constant make a general prediction about the coefficient on it ? Explain. (c ) Some argue that the hypothesis that the real interest rate is constant implies that nominal interest rates move one-for-one with actual inflation in the long run-that is, that the hypothesis implies that in a regression of i on a constant and the current and many lagged values of a, the sum of the coefficients on the inflation variables will be 1. Is this claim correct? (Hint: Suppose that the behavior of actual inflation is given by at = pit-1 + et: where e is white noise.)