A Theory-Based Formal-Econometric Interpretation of an Econometric Model

Abstract

The references of most of the observations that econometricians have are ill defined. To use such data in an empirical analysis, the econometrician in charge must find a way to give them economic meaning. In this paper, I have data and an econometric model, and I set out to show how economic theory can be used to interpret the variables and parameters of my econometric model. According to Ragnar Frisch, that is a difficult task. Economic theories reside in a Model World and the econometrician’s data reside in the Real World; the rational laws in the model world are fundamentally different from the empirical laws in the real world; and between the two worlds there is a gap that can never be bridged To accomplish my task, I build a bridge between Frisch’s two worlds with applied formal-econometric arguments, invent a pertinent model-world economic theory, walk the bridge with the invented theory, and use it to give economic meaning to the variables and parameters of my econometric model. At the end I demonstrate that the invented theory and the bridge I use in my analysis are empirically relevant in the empirical context of my econometric model.

1. Introduction

In empirical studies that aim at a unification of the theoretical-quantitative and the empirical-quantitative approaches to economic problems, economic theory can be used in different ways. One way is to give economic meanings to the variables and parameters of an econometric model. On pp. 32–38 in his 1930 Yale lectures Ragnar Frisch describes how such an interpretation is to be carried out (Bjerkholt & Qin, 2011).

In his lectures, Frisch assumed that an economic theory resides in a Model World, and that the data reside in the Real World. In addition, he claimed that the rational laws in the model world are fundamentally different from the empirical laws in the real world. Between the two worlds there is a gap that can never be bridged. In this paper I add axioms for a theory universe, axioms for a data universe, bridge principles that link variables in the theory universe with variables in the data universe, and statistical arguments to Frisch’s analysis. With these additions, I use his ideas to search for an economic theory that can provide economic meanings to the variables and parameters of a given econometric model.

The given econometric model has three variables, $y_1$, $y_2$, and $y_3$, and a name, Commodity-exchange. I have 400 observations of each variable, and I believe that the observations describe the equilibrium configurations of a perfectly competitive market for one commodity. To see if that makes sense, I will search for an economic theory in Frisch’s Model World that I can use to underpin such an interpretation of Commodity-exchange.

A theory-based interpretation of an econometric model is a description of a theory-based model of the data generating process. The data generating process has one true model, which I denote by the acronym TPD, with T short for true, P for probability, and D for distribution. The data generating process has many theory-based models, each one of which I denote by the acronym MPD, where M is short for marginal, P for probability, and D for distribution. An MPD is a probability distribution of the data variables that is induced by the probability distribution of the theory variables and a model of the bridge principles. Such an MPD may be very different from the TPD.

An MPD has many interesting properties. For example, in a given empirical context, a model of the MPD is data admissible if its parameters lie in a 95% confidence band around an estimate of MPD parameters. Moreover, a data admissible MPD is congruent if it encompasses the true data generating process, the TPD. The congruence of an MPD implies that the MPD can account for the relevant statistical results obtained by the TPD (Hendry & Richard, 1989; Mizon, 1995).

These properties play an important role in my inventing a theory-based interpretation of Commodity-exchange. Roughly speaking, the bridge principles are valid in the given empirical context if and only if all the data admissible MPDs are congruent. When the bridge principles are valid, my interpretation of Commodity-exchange is empirically relevant only if the set of parameter vectors of data admissible MPDs has a non-empty intersection with the 95% confidence band around the estimated values of the corresponding TPD parameters.

I have added axioms of a theory universe, axioms of a data universe, bridge principles, and statistical arguments to Frisch’s analysis. When the bridge principles are empirically relevant, they provide a bridge between the theory universe and the data universe. In addition, the two universes, my bridge principles, and my statistical arguments, when fitted together, constitute a bridge between the model world and the real world. In the paper, I shall walk this bridge with the searched-for theory and give economic meanings to the variables and parameters of Commodity-exchange.

The bridge I build in this paper has several analogues. Two of them are relevant here: Trygve Haavelmo’s imaginary bridge (Haavelmo, 1944), and Aris Spanos’ completion of Haavelmo’s Bridge Pier (Spanos, 2015, p. 177). Haavelmo’s imaginary bridge linked theoretical variables in Frisch’s model world with the corresponding variables in his observational world. In his mind Haavelmo traversed the bridge by identifying the values of theoretical variables with the true values of the corresponding data variables. In (Stigum, 2025) I have with applied formal-econometric arguments built a bridge that has all the properties of Haavelmo’s imaginary bridge. Haavelmo can walk across that bridge with a model-world theory and give economic meaning to the variables and parameters of a given observational-world econometric model.

In two very interesting papers on the intrinsic properties of simultaneous equation systems (Spanos, 1989, 2015) Spanos builds a bridge between an observational world of data generating processes and a world of economic theories whose variables roam in Spanos’ observational world. Spanos seems to accept the Cowles Commission’s idea that a properly formulated economic theory is a simultaneous equation system with two parts, a structural form and a reduced form, but he interprets the reduced form primarily as a statistical model. His bridge links the variables in the reduced form with the variables in the structural form. I do not believe that a Cowles Commission researcher with an economic theory about a few observational-world variables can cross a Spanos-kind of bridge with his theory and use it to give economic meaning to the variables and parameters of an econometric model whose variables are the same as his own theoretical variables.

There are good reasons to believe that Frisch’s idea of an unbridgeable gap between the model world and the observational world and the Cowles Commission’s gallant skip of Haavelmo’s imaginary bridge have caused mainstream econometricians to base their empirical analyses on economic theories about observational-world variables. Both the economic-theoretical and the statistical analysis become easier when the basic theory resides in the real world instead of in a model world. I hope that now when there exists a prototype of a bridge between a model world and an observational world, econometricians will mend their ways.

The empirical relevance of the searched-for theory and the theory-based interpretation is proved at the end of Section 2. The empirical validity of the bridge principles that I use in my theory-based interpretation of Commodity-exchange is established in Section 3. I generated the data with the help of STATA’s Release 16 Number Generators.

2. A Theory-Based Interpretation of an Econometric Model

According to Haavelmo (1944, p. 4), it is “never possible—strictly speaking—to avoid ambiguities in classifications and measurements of real phenomena. Not only is our technique of physical measurements imprecise, but in most cases we are not even able to give an unambiguous description of the method of measurement to be used, nor are we able to give precise rules for the choice of things to be measured in connection with a certain theory.” Thus, in the words of Sismondo (1993, p. 516) the references of the data that economists have belong in a socially constructed World of Ideas—a world that has little in common with the real world.

The import of Haavelmo’s comments in this context is as follows: In an empirical analysis, the econometrician in charge knows what the data variables supposedly measure, but has only vague ideas about what they actually measure. The econometrician gives the data variables names, and uses the means of applied econometrics to ascertain whether the ideas have empirical relevance. The present simple example shows how an econometrician goes about that.

2.1. Commodity-Exchange: An Empirical Model for Three Variables

I have 400 observations of three variables, $y_1$, $y_2$, and $y_3$. To me they are observations of a perfectly competitive market’s equilibrium configurations of the demand and supply of a single commodity, y. I take $y_{1t}$ to be an observation of the actual sales of y in period t, $t=1,\dots ,400$. The corresponding price of y is $y_{2t}$. The third variable, $y_3$, I take to be an auxiliary variable that has a significant influence on the supply of y. Whether this interpretation is empirically relevant remains to be seen.

I will use ordinary least-squares analysis to determine what is the most reasonable description of the data generating process. I begin by regressing $y_1$ on $y_2$ and $y_3$, and by regressing $y_2$ on $y_1$ and $y_3$. These regressions suggest that $y_3$ has a statistically significant positive effect on $y_1$, and a statistically significant negative effect on $y_2$. Hence, it is possible that the components of $y_1$ measure the supply of y, and that the components of $y_2$ measure the price of y during the respective periods.

With these comments in mind, I decided to adopt an ad-hoc econometric model for the three data variables that I call Commodity-exchange. The model describes for $t=1,\dots ,400$, how the variables relate to one another in the data generating process:

$$\begin{array}{cc}\hfill y_{1t}& =a_1+b_1{y}_{3t}+{\delta }_{1t},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill y_{2t}& =a_2+b_2{y}_{3t}+{\delta }_{2t}.\hfill \end{array}$$

(2)

In addition, I decide to keep the original interpretation of the variables; i.e., in the respective periods, the components of $y_1$ measure the sales of y, the components of $y_2$ record the price of y, and $y_3$ details the quantity of an auxiliary variable.

In the Introduction I described the idea of a true model of the data generating process, the TPD. For the present analysis I assume that my sample of observations is purely random, and that the data variables have finite means and finite positive variances in the TPD. If that is correct, one may obtain good estimates of TPD parameters. Below, Table 1 records estimates of the means and the correlation matrix of $y_1$, $y_2$, and $y_3$, and Table 2 records the least-squares estimates of Equations (1) and (2). I obtained the estimates in the two tables with Stata Release 16.

Table 1. Means and Correlation Matrix of y.

	Mean	Std. Err.	[95% Conf. Interval]	${\mathit{y}}_{\mathbf{1}}$	${\mathit{y}}_{\mathbf{2}}$	${\mathit{y}}_{\mathbf{3}}$
$y_1$	10.7225	0.0822	10.5608   10.8841	1.0000
$y_2$	5.2233	0.054	5.1172   5.3295	$-0.1658$	1.0000
$y_3$	2.9794	0.0615	2.8586   3.1003	0.3775	$-0.4342$	1.0000

Table 2. OLS Parameter Estimates of Equations (1) and (2).

Equation	Obs.	Parms.	RMSE	R-sq	F	P
(1)	400	2	1.5247	0.1425	66.1400	0.000
(2)	400	2	0.9737	0.1885	92.4416	0.000
Estimates of Equation (1)
	Coefficient	Std. Err.	t	P $>\left|t\right|$	[95% Conf. Interval]
$y_3$	0.5049	0.0621	8.13	0.000	0.3828	0.6270
Constant	9.2182	0.2001	46.08	0.000	8.8248	9.6115
Estimates of Equation (2)
	Coefficient	Std. Err.	t	P $>\left|t\right|$	[95% Conf. Interval]
$y_3$	−0.3812	0.0397	−9.61	0.000	−0.4591	−0.3033
Constant	6.3591	0.1278	49.77	0.000	6.1079	6.6103

In Table 2, RMSE is short for root mean squared error of the residual, R-sq is short for $R^2$, F designates the F-statistic, and P is short for Prob. $>F$. The true values of the estimates in Table 1 and Table 2 are TPD parameters. They lie with 95% certainty in their confidence intervals.

The estimated versions of Equations (1) and (2) are displayed in Equations (3) and (4) for $t=1,\dots ,400$.

$$\begin{array}{cc}\hfill y_{1t}& =9.2182+0.5049y_{3t}+f{e}_{1t},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill y_{2t}& =6.3591-0.3812y_{3t}+f{e}_{2t},\hfill \end{array}$$

(4)

The error terms in (3) and (4) have mean zero, finite positive variances, and are not normally distributed.

The non-normality of the distributions of the error terms makes my claim that the confidence intervals in Table 1 and Table 2 with 95% certainty contain the true values of the respective TPD parameters suspect Davidson and MacKinnon (1993, pp. 71–77). However, here as in the remainder of the paper, the numerical accuracy of my claims about the confidence intervals does not matter in the context of this paper. It is how I use the intervals in my arguments that is important.

2.2. A Search for a Theory-Based Interpretation of Commodity-Exchange

To provide a theory-based interpretation of Commodity-exchange, I invent an economic theory whose variables roam in Frisch’s model world, build a bridge between the model world and the world in which my data reside, walk the bridge with my theory, and use the theory to give economic meaning to the variables and parameters of my econometric model.

2.2.1. The Searched-For Economic Theory

Let x denote a commodity, and let p denote the price of x. I assume that the equations in (5) depict the equilibrium configurations of the market for x in period t.

$$x_{1t}=A+B{p}_t,\hspace{1em}\hspace{1em}{x}_{2t}=C{p}_t+D{x}_{3t},\hspace{1em}\hspace{1em}{x}_{1t}=x_{2t},t=1,\dots ,400,$$

(5)

where $x_t\in {\mathbb{R}}_{+}^3$, $p_t\in {\mathbb{R}}_{++}$, $A>0$, $B<0$, $C>0$, and $D>0$.

Here, the first equation describes the demand for x, $x_1$, as a linear function of its price. The second equation depicts the supply of x, $x_2$, as a linear function of its price and an auxiliary variable, $x_3$. The third equation insists that the values of p and $x_3$ ensure that in each period the demand for x equals the supply of x.

The equations in (5) constitute a simple simultaneous-equation system. They can be solved for $x_{1t}$ and $p_t$. Theorem 1 bears witness to that.

Theorem 1.

Consider the commodity market that the equations in (5) describe. In an equilibrium configuration of the market

$$x_{1t}={\alpha }_1+{\beta }_1{x}_{3t},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{p}_t={\alpha }_2+{\beta }_2{x}_{3t},t=1,\dots ,400,$$

(6)

where ${\alpha }_1=AC/(C-B)$, ${\beta }_1=-\left(BD\right)/(C-B)$, ${\alpha }_2=A/(C-B)$, and ${\beta }_2=-(D/(C-B))$.

Econometricians refer to the equations in (5) as the structural model of the theory. They refer to the equations above as the reduced form of the structural model.

Next, I will build a bridge between the model world and the world in which my data reside. I begin with the axioms for a theory universe.

2.2.2. The Axioms of a Theory Universe

The first axiom claims that the theory universe has eight variables, $x\in {\mathbb{R}}_{+}^3$, $p\in {\mathbb{R}}_{++}$, $z\in \mathbb{R}$, and $\eta \in {\mathbb{R}}^3$, that run around in the theory universe. The x and p have the same denotation as the x and p in the preceding theory, and the components of $\eta$ are error terms. The meaning of z is explicated in the third axiom.

The second axiom assumes that the $(x_{1t},x_{2t},p_t,x_{3t})$ in the theory universe satisfy the equations in (5).

The third axiom insists there exist two triples of constants, $(e_1,e_2,e_3)$ and $(c_1,c_2,1)$, and a sequence of purely random values of z, $z_t$, with mean zero and finite positive variance such that

$${(x_{1t},p_t,x_{3t})}^{\prime }={(e_1,e_2,e_3)}^{\prime }+{(c_1,c_2,1)}^{\prime }{z}_t,t=1,\dots ,400,$$

(7)

In formal econometrics, the probability distribution of $(x_{1t},p_t,x_{3t})$ which the equations in (6) and the $z_t$ in (7) induce, is the RPD distribution of $(x_{1t},p_t,x_{3t})$, where R is short for researcher, P for probability, and D for distribution.

The fourth axiom assumes that the RPD distributed vectors $(x_{1t},p_t,x_{3t},z_t,{\eta }_{1t},{\eta }_{2t},{\eta }_{3t})$, $t=1,\dots ,400$, constitute a purely random sample, and their components have finite means and finite positive variances. The error terms are distributed independently of each other and of z, and they have mean zero.

So much for the theory universe. Next, the three axioms of the data universe. The axioms are in accord with my description of Commodity-exchange.

2.2.3. The Axioms of a Data Universe

The first axiom insists that the data universe contains four hundred observations of each one of three variables, which I denote by $y_{1t}$, $y_{2t}$, and $y_{3t}$, $t=1,\dots ,400$.

The second axiom assumes that the vectors $(y_{1t},y_{2t},y_{3t})$ are TPD distributed and constitute a purely random sample. In the TPD the variables have finite means and finite positive variances.

The third axiom insists that there exist constants $a_1,b_1,a_2,b_2$, and four hundred TPD distributed purely random pairs $({\xi }_{1t},{\xi }_{2t})$, such that in the data generating process,

$$y_{1t}=a_1+b_1{y}_{3t}+{\xi }_{1t},$$

(8)

$$y_{2t}=a_2+b_2{y}_{3t}+{\xi }_{2t}.$$

(9)

The ${\xi }_{1t}$ and ${\xi }_{2t}$ are distributed independently of the $y_{3t}$ and of each other.

It remains to record the one and only axiom for the bridge between the two universes. Here it is:

2.2.4. The Axiom of a Bridge Between the Two Universes

The variables in the theory and data universes are linked as follows:

$$\begin{array}{c}\hfill x_{1t}=y_{1t}-{\eta }_{1t},p_t=y_{2t}-{\eta }_{2t},x_{3t}=y_{3t}-{\eta }_{3t},t=1,\dots ,400\end{array}$$

(10)

The RPD distribution of the two x’s and p and the three $\eta$’s in (10) induce a probability distribution of the components of y, the MPD, that I introduced in the Introduction. The MPD may differ from the TPD. However, the components of y have finite means and finite positive variances in both distributions. Moreover, later in Theorem 2 I will show that there are models of a data admissible MPD that satisfy the third data axiom. Consequently, a data admissible MPD has a model that encompasses the TPD.

The equations in (6), (7) and (10), the assumptions about the probability distribution of the sequence of vectors, $(z_t,{\eta }_{1t},{\eta }_{2t},{\eta }_{3t})$, and the assumption $(e_1,e_2,e_3)=(m{y}_1,m{y}_2,m{y}_3)$, provide a family of theory-based interpretations of the y in (1) and (2). A member of this family is empirically valid only if it satisfies several interesting conditions on the models of the market in (5) and on the models of the MPD distribution of $y_t$ in (10). Theorem 2 exhibits all of them, and the end of this section shows that there is at least one empirically valid member.

Theorem 2.

Suppose that the sequence of vectors, $(x_{1t},p_t,x_{3t},z_t)$, satisfies the equations in (6) and (7). Assume, also, that ${(x_{1t},p_t,x_{3t})}^{\prime }$, ${(y_{1t},y_{2t},y_{3t})}^{\prime }$, and ${({\eta }_{1t},{\eta }_{2t},{\eta }_{3t})}^{\prime }$ satisfy the equations in (10), and that $(e_1,e_2,e_3)=(m{y}_1,m{y}_2,m{y}_3)$. Then, there is a model of the equations in (6) such that, for $t=1,\dots ,400,$

$$\begin{array}{c}{(y_{1t}-m{y}_1,y_{2t}-m{y}_2,y_{3t}-m{y}_3)}^{\prime }={(c_1,c_2,1)}^{\prime }{z}_t+{({\eta }_{1t},{\eta }_{2t},{\eta }_{3t})}^{\prime }\\ x_{1t}=(m{y}_1-c_{1}m{y}_3)+c_1{x}_{3t}\end{array}$$

(11)

$$\begin{array}{c}{p}_t=(m{y}_2-c_{2}m{y}_3)+c_2{x}_{3t},and{x}_{3t}=z_t+m{y}_3.\end{array}$$

(12)

$$\begin{array}{c}{y}_{1t}=(m{y}_1-c_{1}m{y}_3)+c_1{y}_{3t}+({\eta }_{1t}-c_1{\eta }_{3t}),and\\ y_{2t}=(m{y}_2-c_{2}m{y}_3)+c_2{y}_{3t}+({\eta }_{2t}-c_2{\eta }_{3t}).\end{array}$$

(13)

Finally, the $({\alpha }_1,{\beta }_2,{\alpha }_2,{\beta }_2)$ in (6), and the $(a_1,b_1,a_2,b_2)$ in (1) and (2) with the MPD distribution of y, are such that,

$$\begin{array}{c}\hfill ({\alpha }_1,{\beta }_2,{\alpha }_2,{\beta }_2)=(a_1,b_1,a_2,b_2)=(c_3,c_1,c_4,c_2)\\ \hfill where{c}_3=(m{y}_1-c_{1}m{y}_3)and{c}_4=(m{y}_2-c_{2}m{y}_3)\end{array}$$

(14)

In an empirically valid model of Theorem 2’s assumptions, it follows from the equations in (13) that the vector of coefficients in (6) and (11) must satisfy the conditions,

$$\begin{array}{c}\hfill {\alpha }_1=(m{y}_1-c_{1}m{y}_3), {\alpha }_2=(m{y}_2-c_{2}m{y}_3), {\beta }_1=c_1\mathrm{and}{\beta }_2=c_2\end{array}$$

(15)

In addition, the inequalities in (5) insist that

$$\begin{array}{c}\hfill (m{y}_1/m{y}_3)\ge c_1,c_1>0, (m{y}_2/m{y}_3)\ge c_2,\mathrm{and}{c}_2<0.\end{array}$$

(16)

Finally, since $(m{y}_1,m{y}_2,m{y}_3)$ is an estimate of $(e_1,e_2,e_3)$, it must also be the case that the true values of $e_1$, $e_2$, and $e_3$ with high probability lie in the confidence intervals in Table 1, and satisfy the conditions,

$$\begin{array}{c}\hfill e_1>10.55084, e_2>5.11722,\mathrm{and}{e}_3<3.1003.\end{array}$$

(17)

To show that the given theory-based interpretation of Commodity-exchange has an empirically relevant model, I begin by estimating the coefficients in Equation (13).

2.2.5. The Statistical Arguments

I assume that the assumptions I made about $(x_{1t},p_t,x_{3t}),(y_{1t},y_{2t},y_{3t})$, and $({\eta }_{1t},{\eta }_{2t},{\eta }_{3t})$ in Theorem 2 are valid. In addition, I assume that the assumptions I have made about the probability distribution of the $(z_t,{\eta }_{1t},{\eta }_{2t},{\eta }_{3t})$ are valid. Finally, I assume that the covariance matrix of $({\eta }_{1t},{\eta }_{2t},{\eta }_{3t})$ is diagonal, and that $z_t$, and $({\eta }_{1t},{\eta }_{2t},{\eta }_{3t})$ are Gaussian. With these assumptions, the model in (13) becomes a restricted factor analysis model (see Lawley and Maxwell (1971, pp. 86–104) and Anderson and Rubin (1956)), and I can use factor analysis to estimate $c_1$ and $c_2$. Moreover, if l insist that the third component of the factor is 1 and not $-1$, $c_1$ and $c_2$ and the four variances of the model are identifiable.

With the given assumption about the probability distribution of $(z_t,{\eta }_{1t},{\eta }_{2t},{\eta }_{3t})$, the sample covariance matrix, $S\left(n\right)$, multiplied by $(n-1)$, has the Wishart distribution with $(n-1)$ degrees of freedom and the log-likelihood function,

$$\begin{array}{c}\hfill \mathrm{Log}(S\left(n\right),\mathsf{\Sigma })=g(y\left(1\right),\dots ,y\left(n\right))-((n-1)/2)\left[\log |\mathsf{\Sigma }|+tr.\left(S\left(n\right){\mathsf{\Sigma }}^{-1}\right)\right]\end{array}$$

(18)

where $g(.)$ is a function of the observations, $\left|\right(.\left)\right|$ and $tr.(.)$ denote, respectively, the determinant and trace of $(.)$, $\mathsf{\Sigma }=ES\left(n\right)$ in the probability distribution of $(z_t,{\eta }_{1t},{\eta }_{2t},{\eta }_{3t})$, $my\left(n\right)=n^{-1}{\sum }_{1\le t\le n}y\left(t\right)$, and

$$\begin{array}{c}\hfill S\left(n\right)={(n-1)}^{-1}{\displaystyle \sum _{1\le t\le n}}{\left[y\left(t\right)-my\left(n\right)\left]\right[y\left(t\right)-my\left(n\right)\right]}^{\prime }\end{array}$$

One obtains factor analytic estimates of $c_1$, $c_2$, and the variances of the components of $(z_t,{\eta }_{1t},{\eta }_{2t},{\eta }_{3t})$ by minimizing the value of $\left[\mathrm{Log}\left(S\right(n),\mathsf{\Sigma })-\left|S\right(n\left)\right|-3\right]$ with respect to the six parameters subject to the conditions in Equation (19), where the entries of the three-dimensional diagonal matrix must be non-negative.

$$\begin{array}{c}\hfill \mathsf{\Sigma }=\lambda {\sigma }_z^2{\lambda }^{\prime }+\gamma ,\lambda ={(c_1,c_2,l)}^{\prime },\mathrm{and}\gamma =\left(\begin{array}{ccc}{\sigma }_{\eta 1}^2& 0& 0\\ 0& {\sigma }_{\eta 2}^2& 0\\ 0& 0& {\sigma }_{\eta 3}^2\end{array}\right)\end{array}$$

(19)

The Wishart distribution in (18) has many models. In a case where the ${\sigma }_{\eta 3}^2$ in (19) is zero, the equations in (19) will have a solution only if ${\mathsf{\Sigma }}_{12}={\mathsf{\Sigma }}_{13}{\mathsf{\Sigma }}_{32}/{\mathsf{\Sigma }}_{33}.$

Table 3 records the Stata Release 16 maximum-likelihood estimates of $c_1,c_2$ and the variances of z and $\eta$. The estimated chi-squared statstic is, Chi2(3,0) = 0.00, for an ‘LR test of model vs. saturated.’ Moreover, the estimated log likelihood equals −1940.9322.

Table 3. Factor loadings and variances of z and the error terms.

Variables	Coef.	Std. Err.	z	$\mathit{P}>|\mathit{z}|$	[95% Conf Interval]
$y_1-m{y}_1$	0.5190	0.1427	3.64	0.000	0.2395	0.7986
$y_2-m{y}_2$	−0.3919	0.1048	−3.74	0.000	−0.5972	−0.1865
$y_3-m{y}_3$
Var(z)	1.4668	0.3785			0.8846	2.4323
Var(${\eta }_1$)	2.3024	0.1899			1.9586	2.7064
Var(${\eta }_2$)	0.9373	0.0866			0.7820	1.1234
Var(${\eta }_3$)	0.0411	0.3632			1.23 × 10−9	1.3741

The table and the equations in (13) imply that, for all $t=1,\dots ,400$,

$$\begin{array}{cc}\hfill y_{1t}& =m{y}_1+0.5190z_t^{*}+{\eta }_{1t}^{*};\hfill \\ \hfill y_{2t}& =m{y}_2-0.3919z_t^{*}+{\eta }_{2t}^{*};\hfill \\ \hfill y_{3t}& =m{y}_3+z_t^{*}+{\eta }_{3t}^{*}.\hfill \end{array}$$

(20)

Moreover, Table 3 suggests that I cannot reject the possibility that the variance of ${\eta }_3^{*}$ is zero; i.e., that my observations of $x_3$ may be accurate.

2.2.6. A Theory-Based Interpretation of Commodity-Exchange

With the searched-for economic theory I have crossed the bridge I built between Frisch’s two worlds, and I am about to enter the real world to give economic meaning to the variables and parameters of Commodity-exchange.

Here I find that Table 3 suggests that I cannot reject the possibility that the variance of ${\eta }_3^{*}$ is zero; i.e., my observations of $x_3$ may be accurate. Since $y_3$ is an auxiliary variable, I decide to assume that my observations of $x_3$ are accurate.

If I accept that I have accurate observations of $x_3$ and note that, with $c_3^{*}=(m{y}_1-0.5190m{y}_3)=9.1762$, and $c_4^{*}=(m{y}_2+0.3919m{y}_3)=6.3910$, I find that, for $t=1,\dots ,400$,

$$\begin{array}{cc}\hfill y_{1t}& =9.1762+0.5190y_{3t}+{\eta }_{1t}^{*},\hfill \\ \hfill y_{2t}& =6.3910-0.3919y_{3t}+{\eta }_{2t}^{*},\hfill \end{array}$$

(21)

But, if that is so, I can—with the x and p in the searched-for theory—identify $x_{1t}$ with $y_{1t}-{\eta }_{1t}^{*}$, $p_t$ with $y_{2t}-{\eta }_{2t}^{*}$, $x_{3t}$ with $y_{3t}$, and deduce that

$$\begin{array}{cc}\hfill x_{1t}& =9.1762+0.5190x_{3t},\hfill \\ \hfill p_t& =6.3910-0.3919x_{3t},t=1,\dots ,400.\hfill \end{array}$$

(22)

When I identify $x_{1t}$ with $y_{1t}-{\eta }_{1t}^{*}$, $p_t$ with $y_{2t}-{\eta }_{2t}^{*}$, and $x_{3t}$ with $y_{3t}$, I take $y_{1t}-{\eta }_{1t}^{*}$, $y_{2t}-{\eta }_{2t}^{*}$, and $y_{3t}$ to be possible observed true values of the respective components of y in the MPD distributed data generating process. As such, the identification becomes the first step in the process of giving economic meaning to the components of y and the parameters of Commodity-exchange.

The next step consists in demonstrating that the equations in (22) constitute a reduced form of a structural form that satisfies the inequality constraints that I listed in (5). For that purpose I need Theorem 3.

Theorem 3.

The quadruple $({\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2)$ in the equations in (6) is the associated quadruple of a model of the commodity market in (5) if and only if the coefficients in (5) satisfy the equations

$$\begin{array}{c}C={\displaystyle \frac{{\alpha }_1}{{\alpha }_2}},B={\displaystyle \frac{{\beta }_1}{{\beta }_2}},C-B={\displaystyle \frac{{\alpha }_1{\beta }_2-{\beta }_1{\alpha }_2}{{\alpha }_2{\beta }_2}}\hfill \\ A={\displaystyle \frac{{\alpha }_1{\beta }_2-{\beta }_1{\alpha }_2}{{\beta }_2}},D=-{\displaystyle \frac{{\alpha }_1{\beta }_2-{\beta }_1{\alpha }_2}{{\alpha }_2}}\hfill \end{array}$$

From Theorem 3 and easy calculations it follows that the $(c_3^{*},c_1^{*},c_4^{*},c_2^{*})$ in (22) comprises the reduced-form parameters of a model of the equations in (5) in which

$$\begin{array}{c}\hfill A=17.6398,B=-1.3243,C=1.4358,D=1.0817,\end{array}$$

(23)

i.e., in which

$$\begin{array}{c}{x}_{1t}=17.6398-1.3243p_t,x_{2t}=1.4358p_t+1.0817x_{3t},\\ \mathrm{and}\\ x_{1t}=x_{2t},t=1,\dots ,400.\end{array}$$

(24)

The economic meanings which the first two steps in the interpretation process give to the variables and parameters of Commodity-exchange is economically meaningful. It remains to show that it, also, is statistically adequate. That takes two more steps—one for the empirical relevance of the searched-for economic theory, and one for the empirical relevance of the theory-based interpretation of Commodity-exchange.

In the third step I note that with $z_t^{*}=y_{3t}-m{y}_3$ and $(c_1^{*},c_2^{*})=(0.5190,-0.3919)$, and with the estimates of the three means in Table 1 and the four variances in Table 3, the components of the vector,

$$\begin{array}{c}\hfill (c_1^{*},c_2^{*},m{y}_1,m{y}_2,m{y}_3,{\sigma }_z^{*2},{\sigma }_{\eta 1}^{*2},{\sigma }_{\eta 2}^{*2},{\sigma }_{\eta 3}^{*2}),\end{array}$$

(25)

are the parameters of a data admissible MPD distribution of y. Consequently, being an economic theory with a data admissible model of its MPD, the searched-for theory is empirically relevant in the empirical context of Commodity-exchange.

In the fourth step I will demonstrate that my theory-based explication of Commodity-exchange is, also, empirically relevant. The preceding arguments demonstrate that an MPD with the parameter vector in (25) is statistically meaningful. Hence, to show that my theory-based explication is empirically relevant, it remains to show that the reduced-form parameters in (23), $(9.1762,0.5190,6.3910,-0.3919)$, belong in the confidence intervals for $(a_1,b_1,a_2,b_2)$ in Table 2. To wit: $9.17616\in (8.8248,9.6115)$; $0.5190\in (0.3829,0.6269)$; $6.3910\in (6.1080,6.6102)$; $-0.3919\in (-0.4592,-0.3033)$.

3. The Empirical Validity of the Bridge Principles in (10)

The interpretation of Commodity-exchange that I give in Section 2 looks good, but it is not well founded. The bridge principles seem arbitrary since the probability distribution of ${\eta }^{*}$ comes from nowhere. Besides, the probability distribution of y in an MPD may be very different from its probability distribution in the TPD. In this section I will take care of both problems by showing that all the statistically meaningful data admissible MPDs are congruent. The fact that all the statistically meaningful data admissible MPDs are congruent implies that the MPD in my interpretation of Commodity-exchange can account for all the relevant statistical results of the TPD. In addition, it implies (Stigum, 2016, p. 7) that the bridge principles that I use in my theory-based interpretation of the parameters of Commodity-exchange are empirically valid.

To establish the empirical validity of my bridge principles, I must introduce new notation, define the concepts of encompassing and congruence properly, and record a useful mathematical detail. First, the mathematical detail: Formally, the data variables constitute a vector-valued random process, $Y=\left\{y\right(t,\omega );t\in N\}$, on a probability space, $({\mathsf{\Omega }}_p,{\aleph }_p,P_p(.))$, where $N=0,1,2,\dots$, ${\mathsf{\Omega }}_p$ is a subset of a vector space, ${\aleph }_p$ is a $\sigma$ field of subsets of ${\mathsf{\Omega }}_p$, and $P_p(.):{\aleph }_p\to [0,1]$ is a probability measure. The family of finite-dimensional probability distributions of the members of Y relative to $P_p(.)$ is the true probability distribution of the data variables, the TPD. Now, according to Kolmogorov’s consistency theorem (Kolmogorov, 1933), one can, also, with an MPD associate a probability measure on $({\mathsf{\Omega }}_p,{\aleph }_P)$, $P_M(.):{\aleph }_p\to [0,1]$, such that Y has the probability distribution MPD relative to $P_M(.)$, and such that Y becomes a vector-valued random process on the probability space, $({\mathsf{\Omega }}_p,{\aleph }_p,P_M(.))$.

Next the notation: Let $s_n$ denote a finite set of observations of $y_1$, $y_2$, $y_3$, and let $m_M^0(.)$ and $m_P^0(.)$, respectively, be the factor-analytic estimator of the MPD and the least-squares estimator of the TPD. Then, in the given empirical context $m_P^0\left(s_n\right)$ records the least-squares estimates of the parameters of Commodity-exchange in Equations (1) and (2); i.e., the values of

$$\begin{array}{c}\hfill (a_1,b_1,a_2,b_2,m{y}_1,m{y}_2,m{y}_3,{\sigma }_{y1}^2,{\sigma }_{y2}^2,{\sigma }_{y3}^2),\end{array}$$

(26)

which according to Table 1 and Table 2 equal

$$\begin{array}{c}\hfill (9.2182,0.5049,6.359,-0.3812,10.722,5.223,2.979,1.5079,0.0058,0.0024).\end{array}$$

(27)

Moreover, $m_M^0\left(s_n\right)$ records the MPD parameters of Commodity-exchange in Equations (13) and (15); i.e., the values of

$$\begin{array}{c}\hfill (c_3,c_1,c_4,c_2,m{y}_1,m{y}_2,m{y}_3,{\sigma }_z^2,{\sigma }_{\eta 1}^2,{\sigma }_{\eta 2}^2,({\sigma }_{\eta 3}^2=0)),\end{array}$$

(28)

which according to Table 1, Table 2 and Table 3–with ${{\sigma }_{y3}^{*}}^2={{\sigma }_z^{*}}^2$—equal

$$\begin{array}{c}\hfill (9.1762,0.519,6.391,-0.3919,10.722,5.223,2.979,1.4668,2.302,0.937,0).\end{array}$$

(29)

I assume that $m_M^0(.)$ and $m_P^0(.)$ are both consistent estimators. In addition, I let $(M1,m_M^0(.))$ and $(M2,m_P^0(.))$ be two econometric models in which M1 denotes a model of MPD and M2 denotes a model of TPD. I assume that the parameters of M1 and M2 are, respectively, as specified in (28) and (26), and that the M2 is the TPD. Finally, let ${\theta }_M$ and ${\theta }_P$ denote different values of the parameters in (28) and (26). I assume that there are two subsets of $R^{10}$, AM and AP, such that ${\theta }_M\in$ AM and ${\theta }_P\in$ AP, and I let ${\theta }_M^0$ and ${\theta }_p^0$ be, respectively, the $P_{\lim }$ of $m_M^0\left(s_n\right)$ and $m_P^0\left(s_n\right)$ in $P_P^0(.)$-measure.

Now, the definitions of encompassing and congruence: Let ${\theta }_M^{*}$ be the parameter of a statistically meaningful data admissible MPD, and let $P_M^{*}(.)$ be the probability measure on $({\mathsf{\Omega }}_p,{\aleph }_p)$ that corresponds to this MPD. Then, in the present empirical context $(M1,m_M^0(.))$ encompasses $(M_2,m_P^0(.))$ if and only if there exists a mapping, $\Gamma (.):AM\to AP$, such that

$$\begin{array}{c}\hfill \Gamma \left({\theta }_M^{*}\right)=P_{\lim }{m}_P^0\left(s_n\right)\mathrm{in}{P}_M^{*}(.)\text{-}\mathrm{measure},\mathrm{and}\Gamma \left({\theta }_M^0\right)={\theta }_P^0\end{array}$$

(30)

If M1 is coherent with the a-priori theory of Commodity-exchange and if $(M1,m_M^0(.))$ encompasses $(M2,m_P^0(.))$ and $M2$ is the TPD, then ($M1,m_M^0(.))$ is a congruent model of TPD.

The C-component of Definition 1 on page 6 in Stigum (2016) gives a precise definition of the pertinent idea of encompassing. In the present proof, $\Gamma (.)$ is taken to be the identity matrix, $I(.)$. The corresponding definition of congruence is part of Definition 2 on page 6 in Stigum (2016). For a comprehensive discussion of encompassing and congruence see Mizon (1995), Bontemps and Mizon (2003, 2008), and pages 322–333 in Stigum (2015).

With these definitions in mind I can show that my bridge principles are valid in the present empirical context. In a given sample in which I have accurate observations of the $x_{3t}$, the functions which estimate the respective components of the vector in (26), are the same as the functions which estimate the respective components of the vector in (28). The estimates differ because the functions have the TPD distribution in (26) and the estimated MPD distribution in (28). To show that the functions, which estimate the respective components of the vector in (26), are the same as the functions, which estimate the respective components of the vector in (28), it suffices to show that the functions that estimate $c_1$, $c_2$ and ${\sigma }_z^2$ in (28) are the same as the functions that estimate $b_1$, $b_2$ and ${\sigma }_{y3}^2$ in (26).

In a model of the Wishart distribution in (18) in which the ${{\sigma }_{\eta 3}}^2$ in (19) is zero, the equations in (19) will have a solution only if ${\mathsf{\Sigma }}_{12}={\mathsf{\Sigma }}_{13}{\mathsf{\Sigma }}_{32}/{\mathsf{\Sigma }}_{33}$. In addition, it is a fact that, if the equations in (31) have a solution, my factor-analytic estimates of $\lambda$, ${\sigma }_z^2$, and $\gamma$ will satisfy the equations in (31).

$$\begin{array}{cc}\hfill S\left(n\right)& =\lambda {\sigma }_z^2{\lambda }^{\prime }+\gamma ; \lambda ={(c_1,c_2,l)}^{\prime }, \mathrm{and}\gamma =\left(\begin{array}{ccc}{\sigma }_{\eta 1}^2& 0& 0\\ 0& {\sigma }_{\eta 2}^2& 0\\ 0& 0& {\sigma }_{\eta 3}^2\end{array}\right)\hfill \\ \hfill 0& \le ({\sigma }_{\eta 1}^2,{\sigma }_{\eta 2}^2,{\sigma }_{\eta 3}^2)\mathrm{and}0={\sigma }_{\eta 3}^2.\hfill \end{array}$$

(31)

Consequently, when I have accurate observations on the $x_{3t}$, it follows from the equations in (31) that the factor-analytic estimates of $c_1$, $c_2$, and ${\sigma }_z^2$ satisfy the equations

$$\begin{array}{cc}\hfill c_1& ={\displaystyle \sum _{1\le t\le 400}}(y_{1t}-m{y}_1)(y_3-m{y}_3)/{\displaystyle \sum _{1\le t\le 400}}{(y_{3t}-m{y}_3)}^2;\hfill \\ \hfill c_2& ={\displaystyle \sum _{1\le t\le 400}}(y_{2t}-m{y}_2)(y_3-m{y}_3)/{\displaystyle \sum _{1\le t\le 400}}{(y_{3t}-m{y}_3)}^2; \mathrm{and}\hfill \\ \hfill {\sigma }_z^2& ={(n-1)}^{-1}{\displaystyle \sum _{1\le t\le 400}}{(y_{3t}-m{y}_3)}^2.\hfill \end{array}$$

These equations are—as was to be shown—the least-squares equations for the estimates of $b_1$, $b_2$, and ${\sigma }_{y3}^2$.

Next, let M2 be the TPD, let Ml be a statistically meaningful data-admissible MPD, let ${\theta }_M$ be the corresponding values assigned to the parameters in (31), and let $P_M(.)$ be the probability measure on $({\mathsf{\Omega }}_P,{\aleph }_P)$ determined by this ML. Then it is the case that in both $P_M(.)$-measure and $P_p(.)$-measure, $m_M^0\left(s_n\right)=m_P^0\left(s_n\right)$ almost everywhere. Hence, ${\theta }_M=P_{\lim }{m}_P^0\left(s_n\right)$ in $P_M(.)$-measure, and ${\theta }_M^0={\theta }_P^0$ in $P_P(.)$-measure. Consequently, $(M1,m_M^0(.))$ encompasses $(M2,m_P^0(.))$. Moreover, $M1$ is coherent with the a-priori theory of Commodity-exchange in the sense that there is an MPD model of Equations (1) and (2). Finally, since M2 is the TPD, $(Ml,m_M^0(.))$ is a congruent model of the TPD. It follows that $Ml$ is congruent. Since Ml was an arbitrarily chosen statistically meaningful data-admissible MPD, it follows that all statistically meaningful data-admissible MPDs are congruent, and that my bridge principles in (10) are valid in the present empirical context.