A COMPANION TO Theoretical Econometrics

Second-Generation RCMs

As previously noted, second-generation RCMs are concerned with relaxing the usual restrictions concerning the direct effects of explanatory variables on the explained variables, functional forms, measurement errors, and use of an addi­tive error term to proxy excluded variables. If these restrictions are violated - for example, if there are measurement errors when no measurement errors are as­sumed or a specified functional form is incorrect - the resulting estimates are subject to specification errors. In two path-breaking papers, Pratt and Schlaifer (1984, 1988) have demonstrated that, in order to assess the plausibility of these restrictions, we need "real-world" interpretations of the coefficients in equa­tion (19.1). In essence, two questions need to be addressed: (i) what are these real-world interpretations? (ii) are the above restrictions consistent with these interpretations? In what follows we show that any equation relating observable variables cannot coincide with the corresponding true economic relationship if each of its coefficients is not treated as the sum of three parts - one correspond­ing to a direct effect of the true value of a regressor on the true value of the dependent variable, a second part capturing omitted-variable biases, and a third part capturing the effect of mismeasuring the regressor. We also show that the true functional form of the equation is a member of a class of functional forms.

To explain, consider the following model of money demand:

mt = Yo + yr + y2yt + ut, (19.8)

where rt is the logarithm of an interest rate (i. e. opportunity cost) variable, yt is the logarithm of real income, and u is an error term. It is assumed that the Ys are constant and ut has mean zero and is mean independent of rt and yt. As before, mt is the logarithm of real money balances. Note that the variables in equation (19.8) are observed values. Because of measurement errors, they are unlikely to rep­resent true values. Also, equation (19.8) is unlikely to coincide with the "true" money demand equation, as we now show.

Consider the implications arising from a typical errors-in-the-variables model. Thus, suppose that mt = m* + v0t, rt = r* + v1t, and y t = y* + v2t, where mt, rt, and yt

are the observed values of the variables, the variables with an asterisk represent

the true values, and the vs represent the errors made in the measurement of the variables. For example, y* could be permanent income and r* could be the opportunity cost of holding money that is implied by the definition of permanent income. One consequence of using the observed rather than the true values is that a set of random variables - the vs - is incorporated into the equation determining mt. Additionally, we show below that the existence of measurement errors con­tradicts the assumption that the coefficients of equation (19.8) can be constants.

For the time being, suppose we are fortunate enough to know the true values of the variables. In that case, the true functional form of the money demand equation is a member of the class:

nt

m* = aot + aur* + a2ty* + ^ ajtx* (t = 1,..., T), (19.9)

і=3

where the xj* are all the determinants of m* other than r* and y* and where the as and nt are coefficients and the number of explanatory variables, respectively; they are time-varying, as indicated by their time subscripts. The variables r* and y*t are called the included variables and the variables xj*t are called the excluded variables. Temporal changes in the set of excluded variables change n over time. Equation (19.9) is not necessarily linear, since the time-varying quality of the coefficients permits the equation to pass through every data point even when the number of observations on its variables exceeds nt + 1. Thus, with time-varying coefficients the equation can be nonlinear. Equation (19.9) will have different functional forms for different sequences of its coefficients (or for different paths of variation in its coefficients) and the class of functional forms it represents is unrestricted as long as its coefficients are unrestricted. This lends support to our speculation that a member of this class is true. Equation (19.9) with un­restricted coefficients is more general than even the true money demand function, m* = f(r*, y*, x*t,..., x*t), whose functional form is unknown. However, for a certain (unknown) pattern of variation in its coefficients, (19.9) coincides with the true equation. The essential feature of equation (19.9) is that, since it encom­passes time-varying coefficients, the true values of variables, the correct func­tional form, and any omitted variables, it covers the true function determining money demand as a special case. The a coefficients that follow the true pattern of variation would represent the true elasticities of the determinants of money demand. We call a1t and a2t (representing the true pattern of variation) the direct effects on m* of r* and y*, respectively, since they are not contaminated by any of the four specification errors discussed above.

Two fundamental problems are involved in any attempt to estimate equa­tion (19.9). First, we may not know much (if anything) about the x*. Whatever we know may not be enough to prove that they are uncorrelated with the other explanatory variables in equation (19.9) (Pratt and Schlaifer, 1984, pp. 11-12). Second, the observed (as opposed to the true) values of the variables, are likely to contain measurement errors. A way to resolve the former problem is to assume that the xj* are correlated with the other explanatory variables as follows:

The coefficient has a straightforward interpretation. It is the portion of x* (i. e. an excluded variable) remaining after the effects of the variables r* and y* have been removed. The remaining у coefficients represent the partial correlations between the excluded and the included variables. As with equation (19.9), the coefficients of equation (19.10) will not be constants unless this equation is known with certainty to be linear.

In order to take account of the correlations among the r* and y* and the x*, substitute equation (19.10) into equation (19.9):

(19.11)

Equation (19.11) expresses the time-varying relationship between the true values of the variables, where the time-varying effects of the excluded variables (i. e. the xjt) are included in each coefficient on the right-hand side. For example, the remaining portion of x* after the effects of the variables r* and y* have been removed is captured in the first coefficient (via y0jt), while the effects of r* on the x* are captured in the second term (via y1jt).

Equation (19.11) involves a relationship between the true variables, which are unobservable. Accordingly, to bring us closer to estimation substitute the observ­able counterparts of these variables into equation (19.11) to obtain:

mt = Ynt + Tub + Y^yt, (19.12)

where Ynt = (ant + з UjWjt + vw), Y1t = (a 1t + з а^Ш - it), and y2t = (a2t +

з a0j) (1 - f-).

The coefficients of equation (19.12) have straightforward real-world inter­pretations corresponding to the direct effect of each variable and the effects of omitted variables and measurement errors. Consider, for example, the coefficient Y1t on rt. It consists of three parts: a direct effect, a1t, of the true interest rate (r*) on the true value of real money balances (m*) given by equation (19.9); a term (XjL 3a jtv1 jt) capturing an indirect effect or omitted-variable bias (recall, ajt with j > 3 is the effect of an omitted variable on m*, and y1jt is the effect of r* on that omitted variable); and a term capturing the effect of measurement error, - (a 1t + Xnj‘= 3ajtViji)(v1t/rt) (recall that v1t is the measurement error associated with the interest rate). The coefficient y2t can be interpreted analogously. The direct effects provide economic explanations. The term Ynt also consists of three parts and these include the intercepts of equations (19.9) and (19.1n), the effects of omitted variables on m*, and the measurement error in mt. It is the connection between Ynt and the intercepts of equations (19.9) and (19.Ю) that demonstrates the real-world origin of Ynt. In other words, all the coefficients in equation (19.12) have been derived on the basis of a set of realistic assumptions which directly confront the problems that arise because of omitted explanatory variables, their correlations with the included explanatory variables, measurement errors, and unknown functional
forms. When these problems are present, as they usually are, a necessary condition for equation (19.12) to coincide with the true money demand function is that its coefficients are the sums of three parts stated below equation (19.12). At least two of these parts (omitted-variable biases and measurement error effects) cannot be constant and hence it may not be reasonable to assume that the constant coeffi­cients of equation (19.8) are the sums of these three parts. Thus, equation (19.8)'s premises are inconsistent with the real-world interpretations of the coefficients of equation (19.12), and equation (19.8) cannot coincide with the true money demand function. These results are false if (i) v1t and v2t are equal to zero for all t (i. e. there are no measurement errors in rt and yt), (ii) y1jt and y2jt are equal to zero for all j > 3 and t (i. e. the included variables are independent of excluded variables), (iii) a 1t and a2t are constant (i. e. the direct effects of r* and y* on m* are constant), and (iv) y0t = y0 + ut (i. e. the intercept of equation (19.12) is equal to the intercept plus the error of equation (19.8)). Though under these conditions, equations (19.8) and (19.12) coincide with equation (19.9) and no inconsistencies arise, the difficulty is that these conditions are shown to be false by Pratt and Schlaifer (1984, pp. 11-12).

Equation (19.12) may be correct in theory, but we need to implement it em­pirically. Ideally, we would like to have empirical estimates of the direct effects, but as shown above, the direct effects are commingled with mismeasurement effects and omitted-variable biases. It should also be observed that equation (19.12) is more complicated than a structural equation without exogenous variables since Y0t, Y1t, and y2t are correlated both with each other and with the variables rt and yt. These correlations arise because y1t and y2t are functions of rt and yt, respectively, and y0t, y1t, and y2t have a common source of variation in ajt, j =

3,. .., nt. Instrumental variable estimation (IVE) - intended to deal with the problem of correlations between y0t and rt and yt when y1t and y2t are constant

- of equation (19.12) does not "purge" its coefficients of mismeasurement effects and omitted-variable biases and, hence, cannot be used. IVE is designed neither to decompose the у s into direct, indirect, and mismeasurement effects nor to deal with the correlations between the included explanatory variables and their coefficients.

In an attempt to estimate a1t and a2t, we need to introduce some additional terminology.7 To derive estimates of a1t and a2t, we will attempt to estimate the ys using concomitants. A formal definition of concomitants is provided in foot­note 7. Intuitively, these may be viewed as variables that are not included in the equation used to estimate money demand, but help deal with the correlations between the ys and the explanatory variables (in this example, interest rates and real income). This notion can be stated more precisely in the form of the follow­ing two assumptions:

Assumption 1. The coefficients of equation (19.12) satisfy the stochastic equations

p

Ykt = Ям + X KkjZjt + єи (k = 0, 1, 2),

j=1

where the concomitants zjt explain the variation in the уи, E(ekt |zt) = E(ekt) = 0 for all t and each k, and the ekt satisfy the stochastic equation

ekt = 9kkek, t-1 + akt, (19.14)

where for k, k' = 0, 1, 2, -1 < фкк < 1, and akt are serially uncorrelated with E(akt) = 0 and E aktakt = akk - for all t.

Assumption 2. The explanatory variables of equation (19.12) are indepen­dent of the ekt, given any values of the concomitants zjt, and condition (iii) of footnote 7 holds.

Equation (19.14) is not needed if t indexes individuals and is needed if t indexes time and if the ekt in equation (19.13) are partly predictable. It can also be as­sumed that Ekt and ek, t-1 with k Ф k' are correlated. The explanatory variables of equation (19.12) can be independent of their coefficients conditional on a given value of the zs even though they are not unconditionally independent of their coefficients. This property provides a useful procedure for consistently estima­ting the direct effects contained in the coefficients of equation (19.12). The criti­cism of Assumption 1 contained in the last paragraph of Section 2 (or the errors in the specification of equation (19.13)) can be avoided by following the criteria laid out in Section 4.

To illustrate the procedure, suppose a money demand specification includes two explanatory variables - real income and a short-term interest rate. Also, suppose two concomitants (so that p = 2) are used to estimate the у s - a long-term interest rate (denoted as z1t) and the inflation rate (denoted as z2t). A straightfor­ward interpretation of the use of these concomitants is the following. The direct effect (a1t) component of the coefficient (i. e. y1t) on the short-term interest-rate variable rt in equation (19.12) is represented by the linear function (n10 + n11z1t) of the long-term rate. The indirect and mismeasurement effects are captured by using a function (n12z2t + e1t) of the inflation rate and e1t. In this example, the measure of the direct effects (a2t) contained in y2t (the coefficient on real income) is represented in (n20 + n21z1t); the measure of indirect and mismeasurement effects contained in y2t is represented in (n22z2t + e2t). These definitions do not impose any zero restrictions, but may need to be extended (see Section 4).

Substituting equation (19.13) into equation (19.12) gives an equation in estim­able form:

p p

mt = П00 + X П0jzjt + П 10rt + ^ П 1jzjtrt + n2oVt

j=1 j=1

p

+ x n2jzjtVt + Єоt + e1trt + ZitVt (t = 1 2 . . . , T). (19.15)

j=1

A computer program developed by Chang, Swamy, Hallahan and Tavlas (1999) can be used to estimate this equation. Note that equation (19.15) has three error terms, two of which are the products of es and the included explanatory variables of equation (19.8). The sum of these three terms is both heteroskedastic and serially correlated. Under Assumptions 1 and 2, the right-hand side of equation (19.15) with the last three terms suppressed gives the conditional expectation of the left-hand side variable as a nonlinear function of the conditioning variables. This conditional expectation is different from the right-hand side of equation (19.8) with ut suppressed. This result demonstrates why the addition of a single error term to a mathematical formula and the exclusion of the interaction terms on the right-hand side of equation (19.15) introduce inconsistencies in the usual situations where measurement errors and omitted-variable biases are present and the true functional forms are unknown.

In these usual situations, equation (19.8) can be freed of its inconsistencies by changing it to equation (19.12) and making Assumptions 1 and 2. A similar approach does not work for probit and logit models which are also based on assumptions that are inconsistent with the real-world interpretations of their coefficients. As regards switching regressions, Swamy and Mehta (1975) show that these regressions do not approximate the underlying true economic relation­ships better than random coefficient models.

Note the validity of our above remarks regarding IVE. There cannot be any instrumental variables that are uncorrelated with the error term of equation (19.15) and highly correlated with the explanatory variables of equation (19.12) because these explanatory variables also appear in the error term.

Second-generation RCMs have been applied in recent years to a wide variety of circumstances and with much success in terms of forecasting performance relative to models of the type in equation (19.8) (Akhavein, Swamy, Taubman, and Singamsetti, 1997; Leusner, Akhavein, and Swamy, 1998; Phillips and Swamy, 1998; and Hondroyiannis, Swamy, and Tavlas, 1999).