 1 ]

In writing out the expansions of the entries of Ο referred to in falsifying the KLM version of the model given in the Appendix, the above notation is used.

The resulting estimated KLM behavioral equations are presented in Table 2.

Table 1. Klein model I behavioral equations.

Note: t-ratios in parenthesis; *p < 0.05; **p < 0.01.

Table 2. KLM model behavioral equations.

Note: t-ratios in parenthesis; *p < 0.05; **p < 0.01.

The Sargan overidentification test statistics are no longer significant: consumption (p = 0.0782), investment (p = 0.0993), private wages (p = 0.1117). Thus, whatever else might be said about this model, it appears to be a valid reduction from the reduced form. The resulting matrix representation of the KLM model is as follows.

$\begin{array}{l}\left[\begin{array}{ccccccc}\beta 1& 0& 0.810& 0& 0.047& 0& 0\\ 0& \beta 1& 0& 0& 0.461& 0& 0\\ 0& 0& \beta 1& 0& 0& 0& 0.478\\ 1& 1& 0& \beta 1& 0& 0& 0\\ 0& 0& 0& 1& \beta 1& \beta 1& 0\\ 0& 0& 1& 0& 0& \beta 1& 0\\ 0& 0& 0& 1& 0& 0& \beta 1\end{array}\right]\left[\begin{array}{c}C\\ I\\ {W}_{1}\\ Y\\ P\\ W\\ E\end{array}\right]\\ =\left[\begin{array}{cccccccc}\beta 0.810& \beta 0.192& 0& 0& 0& 0& 0& 0\\ 0& \beta 0.455& 0.072& 0& 0& 0& 0& 0.113\\ 0& 0& 0& \beta 0.106& \beta 0.330& 0.695& 0& 0\\ 0& 0& 0& 0& 0& 1& \beta 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ \beta 1& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& \beta 1& 0& 0\end{array}\right]\left[\begin{array}{c}{W}_{2}\\ {P}_{\beta 1}\\ {K}_{\beta 1}\\ {E}_{\beta 1}\\ Year\\ TX\\ G\\ {W}_{\beta 1}\end{array}\right]\end{array}$

$\mathrm{sgn}\mathrm{\Xi ²}=\left[\begin{array}{ccccccc}\beta & 0& +& 0& +& 0& 0\\ 0& \beta & 0& 0& +& 0& 0\\ 0& 0& \beta & 0& 0& 0& +\\ +& +& 0& \beta & 0& 0& 0\\ 0& 0& 0& +& \beta & \beta & 0\\ 0& 0& +& 0& 0& \beta & 0\\ 0& 0& 0& +& 0& 0& \beta \end{array}\right]$ , $\mathrm{sgn}\mathrm{\Xi ³}=\left[\begin{array}{cccccccc}\beta & \beta & 0& 0& 0& 0& 0& 0\\ 0& \beta & +& 0& 0& 0& 0& +\\ 0& 0& 0& \beta & \beta & +& 0& 0\\ 0& 0& 0& 0& 0& +& \beta & 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ \beta & 0& 0& 0& 0& 0& 0& 0\\ +& 0& 0& 0& 0& \beta & 0& 0\end{array}\right]$

Does this solve the impossibility issue? We use the same Monte Carlo methodology reported on in Buck and Lady [10] to determine whether this sign pattern is possible. A version of the computer application that implements the Monte Carlo that we used is available at https://astro.temple.edu/~gmlady/RF_Finder/Finder_Page.htm which also contains the Stata programs and data used to estimate the models presented herein.

The reduced form estimated for the KLM model is given below in Table 3.

The Monte Carlo method takes quantitative samples of the structural arrays with the sign patterns given above, computes $\mathrm{Ο}={\mathrm{\Xi ²}}^{\beta \text{1}}\mathrm{\Xi ³}$ , and then checks to see if the resulting sign pattern of Ο as computed from the sample corresponds to the sign pattern of the estimated reduced form given above. The sample is constructed as follows.

The Monte Carlo sampling procedure takes the sign patterns proposed for the arrays beta and gamma and quantifies the arrays consistent with the given sign patterns. The resulting quantifications are then used to compute pi, the reduced

Table 3. KLM estimated reduced form.

form, the sign pattern of which is then compared to the sign pattern of the estimated reduced form. Call the sign patterns proposed for beta and gamma the text-files and the particular quantification chosen the sample-files. The steps in selecting the sample-files from the text-files are as follows.

1) The text files for Ξ² and Ξ³ are read into their corresponding sample arrays.

2) All entries in the sample-files are set equal to zero if the corresponding entries in the text-files are equal to zero, preserving the zero restrictions.

3) The value of each remaining (not set to zero) entry of the sample-files are chosen individually and randomly from the open interval (0, 100) using a uniform distribution.

4) The values in the sample-files chosen in step 2) above that correspond to negative entries in the text-files are set negative.

5) The value of Ξ³(1, 1) in the sample file is set equal to the negative of Ξ²(1, 3) in the sample-file, a restriction of both the Klein Model I and the KLM structural form.

6) All entries other than Ξ²(1, 1) that are equal to β1β or ββ1β in the structural form are set equal to the absolute value of the value with chosen for beta (1, 1) in the sample file, keeping their appropriate sign as given in the text-files.1

1This produces a sample of coefficients that are consistent with the input text files. However, the values of 1 and β1 (indicating normalization and identities) have all been changed to the random value chosen for Ξ²(1, 1). To restore these values to unit values, one could multiply all rows by abs (Ξ² (1/(1, 1)), chosen in step 2) above. This sets all entries in the sample-files that are equal to β1β or ββ1β in the structural form equal to the same value, scaling all other entries in the sample files as appropriate to maintain the ratios of all entries in the sample files. However, since this is an elementary row operation it does not change the sign pattern of the reduced form achieved by using the sample-files that result by step 6). As a result, the actual sampling algorithm used does not utilize this step.

The sample-files are then used to compute the implied reduced form, the sign pattern of which is compared to the sign pattern of the estimated reduced form. Subject to the precision of the real variables utilized by the computer software being used (Visual Basic Version 6, service pack 6), the sample-files would in principle allow any possible quantification of the text-files with the given sign patterns and structural form restrictions.

This issue is entirely resolved, as shown below, by documenting why a reduced form sign pattern is not found by examination of the expansions of the arrays used in calculating entries of the reduced form.

The sign pattern of the KLM estimated reduced form was never found after investigating millions of possibilities. The software reported that the sign patterns of the sixth column and the seventh row in the estimated reduced form were never found, independent of the rest. Further analysis showed that in the sixth column, the signs of the triplet of entries: $\mathrm{Ο}\left(1,6\right)<0$ , $\mathrm{Ο}\left(\text{2},\text{6}\right)<0$ and $\mathrm{Ο}\left(\text{7},\text{6}\right)>0$ , were never found, independent of the rest. Also, the signs of the triplet of entries $\mathrm{Ο}\left(\text{7},\text{5}\right)>0$ , $\mathrm{Ο}\left(\text{7},\text{6}\right)>0$ and $\mathrm{Ο}\left(7,8\right)<0$ , where never found, independent of the rest.

The chance that the Monte Carlo sampling procedure βmissedβ a sample which would have led to the estimated reduced form sign pattern is vanishingly small for the large number of samples generated; however, the probability of this happening is nonzero. Accordingly, for the first triplet above we wrote out the expansions the cofactors of Ξ² times entries of Ξ³ appropriate to the computation of these entries of $\mathrm{Ο}={\mathrm{\Xi ²}}^{\beta \text{1}}\mathrm{\Xi ³}$ . Having done this, we found that,

$\mathrm{Ο}\left(7,6\right)=\mathrm{Ο}\left(1,6\right)+\mathrm{Ο}\left(2,6\right).$

The derivation of this result is given in the Appendix. Further study revealed that this result could be readily derived from the system of equations. For , the result found can be expressed by the condition,

$\text{d}E=\text{d}C+\text{d}I.$

From Equation (7),

$\text{d}E=\text{d}Y+\text{d}TX;$

and from Equation (4),

$\text{d}Y=\text{d}C+\text{d}I\beta \text{d}TX.$

Substituting this into the relationship from Equation (7) yields,

$\text{d}E=\text{d}C+\text{d}I.$

the result derived from the expansions involved in the computation of $\mathrm{Ο}={\mathrm{\Xi ²}}^{\beta \text{1}}\mathrm{\Xi ³}$ . Accordingly, this condition can only be satisfied if $\mathrm{sgn}\mathrm{Ο}\left(7,6\right)$ equals (at least) one of ( $\mathrm{sgn}\mathrm{Ο}\left(1,6\right)$ , $\mathrm{sgn}\mathrm{Ο}\left(2,6\right)$ ). Thus, the signs of this triplet of entries in the estimated reduced form falsifies the model, i.e., are impossible. Further, this result is based entirely upon the accounting relationships (4) and (7) and is therefore independent of the signs and values of the behavioral coefficients. Thus, although our amendments to Kleinβs original specification solved the overidentification problem, the model is nevertheless falsified and is falsified independent of the signs and values of the behavioral coefficients.

4. Conclusions

The theme of this paper is that qualitative analysis and its potential to falsify a structural form based upon its sign pattern and the sign pattern of the estimated reduced form is a tool for model development as well as model dismissal. The demonstration presented here was to react to the falsification of the original Klein Model I, as presented in Buck and Lady [10] , and test the model for overidentification, which we found was present. Given this, we introduced some modest amendments to the original model, yielding our version, the KLM model. These amendments resolved the overidentification problem, but further analysis revealed that, nevertheless, the KLM model was also falsified by the estimated reduced form sign pattern. Significantly, the model was falsified based upon the zero and other restrictions, with the finding independent of either the signs or magnitudes of the behavioral coefficients of the model, i.e., the falsification was based upon accounting relationships embodied in the modelβs structural forms.

Our point is then that such a finding should prompt additional analysis in the further development of the modelβs structural form and the underlying data upon which the model is based. The important point is that, among other tests, a qualitative analysis could provide a useful analytical tool to support development. The Monte Carlo technique utilized for this analysis is easily programmed into a spread sheet and can readily reveal what reduced form sign patterns (appear) possible, among other things based only upon the structural formβs restrictions. In our opinion, when these restrictions are falsified by the estimated reduced form sign pattern, further development is appropriate and initiating the second stage of two stage least squares is not.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Lady, G.M. and Moody, C.E. (2019) Econometric Modeling and Model Falsification. Advances in Pure Mathematics, 9, 762-776. https://doi.org/10.4236/apm.2019.99036

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Appendix: Derivation of Falsification of the KLM Model

In Section III above it was reported that the triplet of signs in the estimated reduced form of the KLM version of Kleinβs model I,

$\mathrm{Ο}\left(1,6\right)<0$ , $\mathrm{Ο}\left(2,6\right)<0$ , and $\mathrm{Ο}\left(7,6\right)>0$ ,

was not found by the Monte Carlo methodology in millions of samples. The expansions of the underlying computation of these entries of the reduced form calculated as $\mathrm{Ο}={\mathrm{\Xi ²}}^{\beta \text{1}}\mathrm{\Xi ³}$ revealed that,

$\mathrm{Ο}\left(1,6\right)+\mathrm{Ο}\left(2,6\right)=\mathrm{Ο}\left(7,6\right).$

As a result, the sign pattern found in the estimated reduced form was not possible, since $\mathrm{sgn}\mathrm{Ο}\left(7,6\right)$ must equal (at least) one of ( $\mathrm{sgn}\mathrm{Ο}\left(1,6\right)$ , $\mathrm{sgn}\mathrm{Ο}\left(2,6\right)$ ) for this equality to hold. This Appendix presents the derivation of that result, using the notation for the KLM structural form given in Section 3.

The details of the computation of these three entries of the reduced form are given by,

$\mathrm{det}\left(\mathrm{\Xi ²}\right)\mathrm{Ο}\left(1,6\right)=\beta {c}_{4}a\left(1,3\right)+a\left(1,4\right)\beta a\left(1,7\right);$

$\mathrm{det}\left(\mathrm{\Xi ²}\right)\mathrm{Ο}\left(2,6\right)=\beta {c}_{4}a\left(2,3\right)+a\left(2,4\right)\beta a\left(2,7\right);$ and,

$\mathrm{det}\left(\mathrm{\Xi ²}\right)\mathrm{Ο}\left(7,6\right)=\beta {c}_{4}a\left(7,3\right)+a\left(7,4\right)\beta a\left(7,7\right);$

where the $a\left(i,j\right)$ are the appropriate entries in the adjoint of Ξ².

The expansions of the cofactors that correspond to these entries in the adjoint of Ξ² are given by:

$a\left(1,3\right)={a}_{3}\left(1\beta {b}_{1}\right)\beta {a}_{1};$

$a\left(1,4\right)={a}_{1}+{c}_{1}\left({a}_{3}\beta {a}_{1}\right);$

and,

$a\left(1,7\right)={c}_{1}{a}_{3}\left(1\beta {b}_{1}\right)\beta {c}_{1}{a}_{1};$

so that,

$\mathrm{det}\left(\mathrm{\Xi ²}\right)\mathrm{Ο}\left(1,6\right)={c}_{4}{a}_{1}\beta {c}_{4}{a}_{3}+{c}_{4}{a}_{3}{b}_{1}+{a}_{1}+{c}_{1}{a}_{3}{b}_{1}.$

$a\left(2,3\right)={a}_{3}{b}_{1}\beta {b}_{1};$

$a\left(2,4\right)={b}_{1}\beta {b}_{1}{c}_{1};$

and,

$a\left(2,7\right)={c}_{1}{b}_{1}{a}_{3}\beta {c}_{1}{b}_{1};$

so that,

$\mathrm{det}\left(\mathrm{\Xi ²}\right)\mathrm{Ο}\left(2,6\right)=\beta {c}_{4}{a}_{3}{b}_{1}+{c}_{4}{b}_{1}+{b}_{1}\beta {c}_{1}{b}_{1}\beta {c}_{1}{b}_{1}{a}_{3}+{c}_{1}{b}_{1}.$

$a\left(7,3\right)={a}_{3}\beta {b}_{1}\beta {a}_{1};$

$a\left(7,4\right)=1;$

and,

$a\left(7,7\right)=1\beta {a}_{1}\beta {b}_{1};$

so that,

$\mathrm{det}\left(\mathrm{\Xi ²}\right)\mathrm{Ο}\left(7,6\right)={c}_{4}{b}_{1}+{c}_{4}{a}_{1}\beta {c}_{4}{a}_{3}+{a}_{1}+{b}_{1}.$

Based on these expansions, the reader can quickly verify that,

$\mathrm{det}\left(\mathrm{\Xi ²}\right)\mathrm{Ο}\left(1,6\right)+\mathrm{det}\left(\mathrm{\Xi ²}\right)\mathrm{Ο}\left(2,6\right)=\mathrm{det}\left(\mathrm{\Xi ²}\right)\mathrm{Ο}\left(7,6\right)={c}_{4}{b}_{1}+{c}_{4}{a}_{1}\beta {c}_{4}{a}_{3}+{a}_{1}+{b}_{1}.$

Significantly, this result is independent of the signs and magnitudes of the behavioral coefficients of the KLM model.