Spline Estimation of the Liquidity Trap: A Reply

Abstract

In his comment on our paper (1976) McCulloch makes two main points. First, he argues that we have set up our spline function in the wrong way. Second, he argues that our computations are seriously in error. McCulloch also makes one minor point about the placing of knot points. In this reply, we will show that none of these comments in way affect our findings. The first point concerns the way in which we set up our spline As McCulloch correctly points out, we treat M/ Y as spline function of r. The reason for this particular setup, of course, is that we do indeed regard M/ Y as being causally function of r. McCulloch argues, however, that to detect trap one should treat r as spline function of M/ Y. The reasonableness of this argument depends in part on the definition of liquidity According to McCulloch, a trap consists of horizontal section of the demand for money function, or at least horizontal asymptote under the demand for money function, when the interest rate r is placed on the vertical axis and money (or money divided by income, M/ Y) on the horizontal axis. Clearly, by treating M/ Y as spline function of r, we were able to test whether or not the interest elasticity of the money demand function approached infinity at some low interest rate. Furthermore, we were able to test this definition of trap without entailing bias in the coefficients. McCulloch's point, then, concerns only the other definition of trap. More specifically, he argues that our setup did not permit us to determine whether the interest elasticity of the money demand function was infinite at some low interest rate. The reason, according to McCulloch, is that while spline can fit horizontal segment, it cannot fit vertical segment, since it is piecewise polynomial. Apparently, McCulloch is arguing that spline can estimate zero slope coefficient but not an infinite slope coefficient. We have no quarrel with this argument. However, if one were to obtain zero slope coefficient by treating r as spline function of M/ Y, it seems reasonable to assume that one would obtain very large slope coefficient (if the spline program generated output at all) by treating M/ Y as spline function of r. Since we obtained relatively small slope coefficient, we felt justified in concluding that our empirical results did not provide any evidence of horizontal segment to the money demand function. In event, we have re-run our equations treating r as spline function of M/ Y as McCulloch suggests. As we suspected, the results indicate that our finding that there is no evidence of horizontal segment to the money demand function remains intact. Moreover, these results are consistent with our finding that the interest elasticity of the demand for money tended to decline as the interest rate became small, finding that is not unique to our study, as we reported in our paper. The second point concerns the relationship between the estimated parameters and continuity. McCulloch calculates the functional value at the second knot forj= 0 using our estimated parameters for the 1920-1970 period. He finds that for j=0 the value is 2.482, whereas the value for j =1 is + 0.322. McCulloch considers this to be discontinuity and therefore questions the elasticities calculated from these parameters. We agree with McCulloch that this is considerable discontinuity. However, the computations are not seriously in error. The problem is that the decimal point was misplaced. Unfortunately, when our figures were copied from the printouts, the symbol D and the accompanying numbers were completely ignored. The D and the numbers, however, indicate the correct placement of the decimal point. After correctly placing the decimal point, one finds that using the parameters for j=0 for the 1920-1970 period to calculate the functional value for the second knot (XI = 2.286) gives SJ(2.286 -) = .391 -.0477h