We can report with confidence that the land input caused considerable headaches in all the computations and may have been the cause of the extraordinary number of iterations required to achieve an optimal solution, perhaps because its quantity index is rather flat and exhibits very little variability. The estimates of the expected quantities and prices obtained from the phase I estimation problem are neither unbiased nor consistent. This is due to our ignorance of the true € λ ratios that weigh the objective function . We have already suggested that a Bayesian approach to the errors-in-variables problem may produce consistent estimates, albeit with a much more complex estimator. Hence, we are willing to accept some level of non-consistency of the estimates in exchange for a manageable estimator that can be implemented by normal practitioners. The problem, of course, is how to gauge what is an acceptable level of inconsistency. We do not have an analytical answer to this question. We only suggest that a small residual error may be an indication of the smallness of inconsistency. We proceed under this conjecture. A measure of the estimates obtained from the phase I model can be viewed in Figure 1 and Figure 2 that report a comparison between the sample and the estimated quantities and prices.In general, the estimated expected quantities and relative prices track the measured counterparts pretty closely. An exception is represented by the land input quantity index that has fluctuated around the value of 1—in a suspicious saw-tooth pattern—during the sample period. Another synthetic view of the phase I results can be gleaned by the trend of the expected and measured input shares as reported in figure 3.
Overall, the estimated expected series track the measured series rather closely.With the estimates of the expected quantities and relative prices from phase I, a NSUR model such as described by equations – was estimated using the NL option of the SHAZAM package. Unfortunately,growing blueberries in containers this SHAZAM option does not allow the imposition of parameter constraints that cannot be directly incorporated into the definition of the various equations. Hence, we were not able to test the negative definiteness of the S matrix using the SHAZAM package. An autocorrelation scheme of order 1 was implemented in this phase II of the estimation procedure. In order to gauge the validity of the PITP model, a translog model of the traditional theory was estimated using the same primal-dual approach and using the same estimated expected relative quantities and prices. This traditional model, therefore, is nested into the PITP model and the difference in the level of the two loglikelihood functions could determine whether or not the PITP conjecture ought to be rejected. The PITP model has 89 parameters versus 55 of the traditional model. The results of this comparison are reported in Table 1.The difference between the values of the logarithm of the two likelihood functions is equal to 322.84 for a number of restrictions equal to 34. Hence, the likelihood ratio test, which gives a chi-squared variable constructed as twice the difference of the logarithm of the two likelihood functions, is equal to 645.68, well above any imaginable critical value. This preliminary test, therefore, does not reject the null hypothesis that the PITP model is suitable for interpreting 81 years of technical progress in US agriculture. The relevant test, however, is given by the negative semi-definiteness of the expanded Slutsky matrix S defined in equation . Three of the four eigenvalues of the S matrix corresponding to the estimated PITP model of Table 1 are negative and one is positive indicating that the matrix is indefinite.
We were not able to test whether the PITP model, subject to the restriction that the S matrix in equation be negative semi-definite, is rejected by the sample data. In order to pursue this objective from a different angle, however, we coded the NSUR problem in GAMS achieving a level of the log-likelihood function that is close to, but not exactly equal to the value achieved with the SHAZAM package. This event is undoubtedly due to the highly nonlinear and non-convex problem at hand, and to the different optimizationalgorithms used in the two packages . Another shortcoming of this approach is that we did not compute the standard errors of the estimates, as their programming in GAMS is beyond our limited ability. In any event, the value of the unrestricted concentrated loglikelihood function obtained with GAMS was 1831.820 versus 1840.040 of SHAZAM. The determinant of the MSR matrix was computed internally to the maximization program by the LU decomposition, with the determinant defined as the product of the diagonal terms of the U matrix. When the negative semi-definiteness condition of the modified Slutsky matrix given in equation was imposed on the problem , the value of the log-likelihood function was 1829.505. A chi-squared test of the negative semidefinite condition, constructed as twice the difference between the values of the two log-likelihood functions , gives a measure of 4.630 with 46 degrees of freedom , indicating that the null hypothesis is not rejected even at a very small level of significance. The Cholesky values of the S matrix estimated under constraint are and the rank condition is satisfied. On the strength of this result and of the likelihood ratio test reported above, we will continue the discussion of the empirical results assuming that the PITP model presented in Table 1 was not rejected by the sample data.
It is interesting to notice that the conventional S matrix for the traditional primal-dual model of Table 1 is indeed negative definite without imposing such a condition, with eigenvalues . In this case, however, the rank condition is not satisfied. The biases induced on input quantities by a price-induced technical progress of the type described in this paper were computed according to equation and are reported in Figure 4.The spikes are due to peculiar combinations of parameters and logarithmic values in the complex formula of equation . For example, by changing the level of constant input prices, it is possible to reduce those spikes, while maintaining the general pattern of the diagrams. Abstracting from the spikes, the common characteristic of three out of four inputs biases is a trend toward a zero level, with a substantial amount of PITP bias at the beginning of the last century. The bias of machinery input is negative until soon after WWII, indicating an input-saving PITP, and then becomes slightly positive. The bias of the labor input has the opposite trend, remaining an input-using PITP until 1960 for, then, becoming an input-saving PITP. The bias of the fertilizer input was negative prior to 1950, indicating an input-saving TP,square pots and then became slightly positive after that date. The land bias indicates a rapidly diminishing input-using PITP.A second version of the PITP model includes the public and private R&D and extension expenditures as explanatory variables of the portion of inputs attributable to the PITP hypothesis. A synthetic representation of this specification is given in equations. Before reporting on the empirical results, we present the series of public and private R&D and extension expenditures in Figure 5. All three series show a very similar trend, a fact that may lead to multi-collinearity and/or to non-significant estimates. As anticipated in a previous section, we took inspiration from the empirical results of Thirtle, Schimmelpfennig and Townsend who attributed the explanation of the non-substitution portion of their input ratios to a distributed lag specification of relative prices, along with private and public R&D. More accurately, in their machinery/labor factor ratio, they reported that only a series of annual private R&D expenditures was significant, together with the lagged machinery/labor price ratio. In their fertilizer/land factor ratio ), the lagged public R&D series was significant.
While the price ratios were specified with a maximum lag of order 2, the private and public R&D series took on lags of 15 and 25 periods, respectively. Version 2 of the model stated in equations – specifies a lagged relationship between the portions of expected inputs attributed to the PITP hypothesis and expected relative prices, R&D and extension expenditures as explanatory variables. This relationship, then, feeds into the production function and the input price equations in the joint determination of the parameters of interest. In figure 6, we present the decomposition of the estimated expected inputs into their complementary portions attributable to a substitution effect and a PITP effect, as they resulted from version 1.The machinery diagram shows almost similar trends of the substitution and PITP components suggesting that, throughout the last century, the machinery substitution effect had about the same strength than the price induced TP effect. The labor diagram, on the contrary, shows that the PITP component of labor was rather minimal throughout the sample period except during the two war periods . The fertilizer diagram indicates that the PITP component is similar to the machinery pattern, with a substantial effect from the early part of the century. Finally, the substitution and the PITP components in the land diagram have an almost mirror-symmetric trend because the total land is roughly constant , as already pointed out. It is intriguing to notice that the most pronounced substitution effect of the land input took place in a period that begins with world war II and catches up with the general trend by the middle of the fifties. At this stage, the problem is to specify the type and the length of the distributed lag series that can plausibly explain the variation of the PITP component of the estimated expected input quantities. As there is no theory that can guide the choice of explanatory variables and the length of their lags, some data mining is inevitable. In Table 2 we present the variables and their lags that were selected in the explanation of the PITP component of the estimated expected input quantities. The information of Table 2 refers to OLS estimates. The symbols for the variables should be read as: Exp = expected, MA = machinery, LB = labor, FR = fertilizer, LA = land, P = price. The lag is indicated explicitly and was restricted to a maximum of 6 periods for the expected input prices and of 7 periods for the R&D and extension variables. These cut-off periods were selected arbitrarily but with the goal of limiting the loss of degrees of freedom in a sample of only 81 observations.The machinery and the fertilizer equations, with all variables in natural units, fit the respective PITP components fairly well, with R-square measures of 0.96 and 0.92, respectively. The labor and the land equations, in semi-log specification, fit the respective PITP component less well, with an R-square measure of 0.75 and 0.85, respectively. The a-priori selection of the maximum lags may be responsible, at least in part, for the relatively low fit of these equations. The extension-expenditures variable enters only the machinery equation; no R&D and extension expenditure variables enter the labor equation; both private and public R&D expenditures enter the fertilizer equation jointly. In spite of the imperfect fit of the PITP equations, the overall information gleaned from the results of Table 2 suggests that a proper combination of lagged expected prices, R&D and extension expenditures may indeed explain the PITP components of the input quantities. A traditional model was also estimated and reported in Table 3. The difference between the logarithmic value of the two likelihood functions is equal to 204.404, which translates into a likelihood ratio test of 408.808, well above any imaginable critical value for a chi-square statistics with 34 degrees of freedom. This preliminary test, therefore, does not reject the hypothesis that a price induced technical progress prevailed during 80 years of US agriculture. As for the previous version 1 of the PITP model, we used the GAMS package to impose the comparative statics condition represented by equation . The implementation of the NSUR program gives a value of the unrestricted and concentrated log-likelihood function equal to 1864.147, while the restricted value is equal to 1858.630. The likelihood ratio test corresponds to a chi-square variable of 11.034 for 46 degrees of freedom, well below the critical value for any reasonable significance level.