Measurement of causal effects

Abstract

Three types of stochastic coefficient models can be identified: models with “individually-varying”, “time-varying” and “individually- and time-varying” coefficients. A parameterized stochastic coefficient environment is well suited to measure the causal effects of the explanatory variables on the dependent variable of a law. This environment captures the unknown true functional form of the law as a special case. In the presence of omitted variables and measurement errors, each coefficient of the law stated in terms of only included variables is the sum of the bias-free term and omitted variables bias with or without measurement-error bias. A method of separating the bias-free term from these biases is developed. This method is shown to be different from the instrumental variables method by showing that the instrumental variables do not exist. Laws are also shown to be different from regression models. Models having lagged dependent variables and the current and lagged values of explanatory variables as its regressors are used for policy analysis and testing for Granger causality in the econometric literature. These models are shown to be non-parsimonious relative to certain types of time-varying coefficient models without lagged dependent variables and are shown to lead to identification problems.

Appendix

Proof of Theorem 1

All econometric models contain at least one error term. For simplicity, set K = 2 and L _t  = 3 so that there is only one included explanatory variable and one excluded variable. In this simple case, Eq. 2 of Sect. 2.1 with the correct functional form becomes

$$ y_{t}^{*} = \alpha_{0t}^{*} + \alpha_{1t}^{*} x_{1t}^{*} + u_{t} $$

(24)

where following the usual econometric practice the error term u _t is equated to $\alpha_{2t}^{*} x_{2t}^{*}$ (Greene 2008, p. 9). This equation is linear if its coefficients are constant and nonlinear otherwise. Equation 24 is adequate to prove Theorem 1. If there were no excluded variable, then there would be no need to enter an error term into (24).

It may appear that $x_{2t}^{*}$ is ‘the’ excluded variable. Now we will prove that the error term u _t and the coefficient of $x_{1t}^{*}$ are not unique and the definite article when referring to excluded variable is inappropriate. The operations of adding and subtracting $\alpha_{2t}^{*} x_{1t}^{*}$ on the right-hand side of the equality sign in (24) give

$$ y_{t}^{*} = \alpha_{0t}^{*} + (\alpha_{1t}^{*} + \alpha_{2t}^{*} )x_{1t}^{*} + \alpha_{2t}^{*} (x_{2t}^{*} - x_{1t}^{*} ) $$

(25)

This equation is the same as (24). Viewing (24) and (25) as regressions of $y_{t}^{*}$ on $x_{1t}^{*}$ shows that excluded variable in (25) is $\left( {x_{2t}^{*} - x_{1t}^{*} } \right)$, which is different from the excluded variable $x_{2t}^{*}$ in (24), and the coefficient of $x_{1t}^{*}$ in (25) is $\left( {\alpha_{1t}^{*} + \alpha_{2t}^{*} } \right)$, which is different from the coefficient $\alpha_{1t}^{*}$ of $x_{1t}^{*}$ in (24). Therefore, in these regressions of $y_{t}^{*}$ on $x_{1t}^{*}$, the coefficient of $x_{1t}^{*}$ and excluded variable are not unique. It is incorrect to say ‘the’ excluded variable when referring to nonuinque excluded variable. It is also incorrect to say that a regression with nonunique coefficients and error term is reality or data generating process, since the real-world relations “remain invariant against mere changes in the language we use to describe them” (Basmann 1988, p. 73).

Even though Eqs. 24 and 25 are the same, they can have different implications for the correlation between u _t and $x_{1t}^{*}$, as we now show. If u _t is equal to $\alpha_{2t}^{*} x_{2t}^{*}$ and the coefficient of $x_{1t}^{*}$ is equal to $\alpha_{1t}^{*}$ as in (24), then u _t may be independent of $x_{1t}^{*}$. On the other hand, if u _t is equal to $\alpha_{2t}^{*} (x_{2t}^{*} - x_{1t}^{*} )$ and the coefficient of $x_{1t}^{*}$ is equal to $\left( {\alpha_{1t}^{*} + \alpha_{2t}^{*} } \right)$ as in (25), then u _t is not independent of $x_{1t}^{*}$. One cannot know a priori which of the excluded variables in (24) and (25) u _t represents, since the coefficient of $x_{1t}^{*}$ is unknown. Without knowing whether u _t is equal to $\alpha_{2t}^{*} x_{2t}^{*}$ or equal to $\alpha_{2t}^{*} \left( {x_{2t}^{*} - x_{1t}^{*} } \right)$, it is “meaningless” to assume that u _t is independent of $x_{1t}^{*}$, as Pratt and Schlaifer (1984) pointed out. With meaningless assumptions about u _t , (24) does not give the correct causal inferences.

To avoid all these difficulties, we do not make the assumption that u _t is independent of $x_{1t}^{*}$. If $x_{2t}^{*}$ is the symbol used to denote a nonunique excluded variable, then we assume

$$ x_{2t}^{*} = \lambda_{0t}^{*} + \lambda_{1t}^{*} x_{1t}^{*} $$

(26)

This is a special case of Eq. 3 in Sect. 2.1. Replacing $x_{2t}^{*}$ in (24) with the right-hand side of Eq. 26 gives

$$ y_{t}^{*} = \alpha_{0t}^{*} + \alpha_{1t}^{*} x_{1t}^{*} + \alpha_{2t}^{*} \left( {\lambda_{0t}^{*} + \lambda_{1t}^{*} x_{1t}^{*} } \right) = \alpha_{0t}^{*} + \alpha_{2t}^{*} \lambda_{0t}^{*} + \left( {\alpha_{1t}^{*} + \alpha_{2t}^{*} \lambda_{1t}^{*} } \right)x_{1t}^{*} $$

(27)

This is a special case of Eq. 4 in Sect. 2.1.

If $\left( {x_{2t}^{*} - x_{1t}^{*} } \right)$ is the symbol used to denote a nonunique excluded variable, then an equivalent form of (26) is

$$ \left( {x_{2t}^{*} - x_{1t}^{*} } \right) = \lambda_{0t}^{*} - x_{1t}^{*} + \lambda_{1t}^{*} x_{1t}^{*} $$

(28)

Substituting the expression on the right-hand side of the equality sign in Eq. 28 for $\left( {x_{2t}^{*} - x_{1t}^{*} } \right)$ in (25) gives

$$ y_{t}^{*} = \alpha_{0t}^{*} + \left( {\alpha_{1t}^{*} + \alpha_{2t}^{*} } \right)x_{1t}^{*} + \alpha_{2t}^{*} \left( {\lambda_{0t}^{*} - x_{1t}^{*} + \lambda_{1t}^{*} x_{1t}^{*} } \right) = \alpha_{0t}^{*} + \alpha_{2t}^{*} \lambda_{0t}^{*} + \left( {\alpha_{1t}^{*} + \alpha_{2t}^{*} \lambda_{1t}^{*} } \right)x_{1t}^{*} $$

(29)

This equation is exactly the same as Eq. 27. Thus, Eq. 26 makes Eq. 27 invariant to rewriting Eq. 24 in different but equivalent forms.

Now the unique coefficients of Eq. 27 can have plausible interpretations. The term $\alpha_{0t}^{*}$ is the true intercept, $\alpha_{2t}^{*} \lambda_{0t}^{*}$ is the correct function of a sufficient portion of excluded variable, $\alpha_{1t}^{*}$ is the bias-free component of the coefficient of $x_{1t}^{*}$, and $\alpha_{2t}^{*} \lambda_{1t}^{*}$ is the omitted-variable bias in the coefficient of $x_{1t}^{*}$. These interpretations are the same as those of the coefficients of model (4) in Sect. 2.1. If the coefficients of any model do not seem to contain omitted-variable biases, then it may be because the meaningless assumption that the arbitrarily introduced error term of the model is independent of its explanatory variables is made.

Under Assumptions I–V, $\alpha_{1t}^{*}$ in (27) is unique because it is a fact about the real world that remains unaltered when (24) is rewritten in different but equivalent forms in terms of $x_{1t}^{*}$ and a function $x_{2t}^{*}$ and $x_{1t}^{*}$ (Pratt and Schlaifer 1984). This proof can be extended straightforwardly to any K and L _t . $\square$