IN 1961, Arrow, Chenery, Minhas, and Solow (ACMS) (1961) introduced their now familiar production function V y[8 K-P + (1-8)L-P]-1/P (1.) where V is value added per man-year, K is capital, L is man-years of labor, and y, 8, and p are the efficiency, distribution, and substitution parameters, respectively. It is well known that the elasticity of substitution, 1/(1 + p), can be estimated by estimating b in the profit maximizing conditi'on log (V/L) _log a + b log w + u (2) where w is the annual wage rate of production workers. fact, ACMS obtained very good results by using international data from 19 countries for various census years between 1949 and 1955. These data represented up to 24 ISIC industries at the three-digit level. 1963, C. E. Ferguson (1963) used U.S. Census of Manfactures data to fit the regression equation (2). Whereas ACMS obtained good results, Ferguson was disappointed in his: In the entire list of 129 items, R2 is significant at P < .05 in only 50%o of the cases. The bcoefficient is significant 70% of the time . . . But in more than half of these, b was not found to be significantly different from one (1963, p. 306). Ferguson recognized a possible reason for such results. The requires different relative factor prices for different observations. With only a little variation in the wage rate, the regression coefficients will have large standard errors. Unfortunately, when Ferguson's paper appeared there was no way to correct or improve the sample. Now, it is possible to pool time-series and cross-sectional data and to recognize the possibility of cross-sectional heteroscedasticity and time-wise autoregression of the disturbance terms. Jan Kmenta has termed this a cross-sectionally heteroscedastic and time-wise autoregressive model (1971, p. 509). We shall use this to estimate the elasticity of substitution, b, in a modification of regression equation (2). We expect that the increased variability of the independent variable will improve the results. This system of production functions for various indtustries provides a classic example of a case where the method of seemingly unrelated regressions may be applied. Thus, we also obtain two-stage Aitken estimates of the elasticity of substitution' using the pooled data. Estimation of the elasticity of substitution by pooling time-series and cross-section data requires a modification of regression equation (2). As the regression stands, there is an implicit assumption of no technological progress over time. This assumption is removed by specifying the as log (V/L)_ log a + b logw + c2T2 + C3T3 + C4T4 + u (3) where T2, T3, and T4 are dummy variables representing the years 1958, 1963, and 1967, respectively.' The introduction of the dummy variables into equation (3) allows for the possibility of technological progress in each of the cross-section years of 1958, 1963, and 1967. Although this results in a loss of three degrees of freedom, it does allow our to capture the influence of technological progress. The Received for publication August 8, 1973. Revision received for publication October 12, 1973. * We have benefited greatly from the comments offered by our colleagues,. David Denslow and Frank Sloan. The encouragement and helpful suggestions of the late C. E. Ferguson, Jan Kmenta, John Moroney, and an anonymous referee are gratefully acknowledged. Jerry R. Jackson provided invaluable assistance in the computations. Of course, we must exonerate everyone but ourselves of all blame for what follows. An earlier version of this paper was presented at the annual meetings of the Econometric Society in December 197 1. 1 We are indebted to an anonymous referee for suggesting this means of accounting for technological progress.

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

LMOST all discussions concerning the imA portance of money in affecting economic activity make reference to the liquidity-trap hypothesis. This important hypothesis states that the elasticity of the demand for money with respect to the rate of becomes infinite at low rates. As J. M. Keynes himself expresses it, after the rate of has fallen to a certain level, liquidity preference may become virtually absolute in the sense that almost anyone prefers cash to holding a debt which yields so low a rate of interest (1936, p. 207). Studies by Bronfenbrenner and Mayer (1960), Konstas and Khouja (1969), Laidler (1966), Meltzer (1963) and White (1972), among others, have attempted to confirm or disconfirm this hypothesis by testing whether the elasticity of the demand for money increases as the rate of falls, on the basis that this is the only way it can pass from a finite to an infinite value. Thus far, the evidence mainly disconfirms the liquidity-trap hypothesis. This evidence, however, has generally been obtained by employing ordinary least squares regression methods. Yet, as David Laidler points out, it is not possible to fit directly by regression analysis a function which has a negative slope over part of its range and no slope at all over another part . (1969, p. 97). Past studies, therefore, have not been directly able to determine whether the elasticity becomes infinite at low rates. The purpose of this paper is to test the liquidity-trap hypothesis by employing spline functions. Briefly, these functions represent a special class of approximating functions which allow the dependent variable in a regression to take on different functional relationships with respect to the independent variable in various subintervals of the domain of the independent variable in a continuous fashion. In this way, the problem inherent in previous studies using ordinary least squares techniques can be avoided, permitting a more direct test of the liquidity-trap hypothesis. In short, this paper will provide new and more direct evidence bearing on the issue of an infinitely elastic demand for money function as well as the way in which the important but relatively unknown spline functions may be used to capture various empirical economic relationships. The plan of the remainder of the paper is as follows. The next section contains a discussion of spline theory, followed by a section containing the empirical results obtained by using spline functions. The summary and conclusions are then reported in the last section.