SUMMARY
Thirty-six years ago, introducing a distinction between factors and concomitants in regressions, John W. Pratt and Robert Schlaifer determined that the error term in a regression represents the net effect of omitted relevant regressors. As this paper demonstrates, this assumption poses a problem whenever the purpose of a model is to explain an economic phenomenon, because the estimated coefficients as well as the error will be wrong in the sense that they are not unique. But a model that is not unique cannot be a causal description of unique events in the real world. For a remedy, this paper presents a methodology based on conditions under which the error term and the coefficients on regressors included in a model do become unique, where the latter represent the sums of direct and indirect effects on the dependent variable, with omitted but relevant regressors having been chosen to define both these effects. The two effects corresponding to any particular omitted relevant regressor can be learned only by converting that regressor into an included regressor. For those cases where omitted relevant regressors are not identified, thereby preventing a meaningful distinction between direct and indirect effects, we introduce so-called coefficient drivers and a feasible method of generalized least squares, permitting a “total-effect” causal interpretation of the coefficient on each regressor included in a model.
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Notes
Here, “mis-specifications-free models” describe “real-world relations,” and mis-specifications-free models with unique coefficients and error terms describe unique real-world relations. Since causality designates a property of the real-world, causal relations being unique in the real-world, causality is used here to designate a property of mis-specifications-free models with unique coefficients and error terms.
Throughout this paper, following Lehmann and Casella (1998, pp. 180-181), we maintain the distinction between each random variable and its value by using an upper-case symbol to denote the former and a lower-case symbol to denote the latter.
Sufficient conditions for its existence are given in Rao, C.R. (1973, p. 97).
This is a conclusion also reached by Freedman (2005).
Goldberger (1964, pp. 380-388) showed that only incomplete theories can have exogenous variables. Here, we have established that even incomplete theories represented by single-equation models with errors made up of omitted relevant regressors cannot have exogenous regressors.
Skyrms (1988, p. 57) pointed out that the relations between different conceptions of probability are of central importance to questions of probabilistic causation. For this reason, we need to emphasize here that we use frequentist probability.
A referee has thoughtfully suggested that this definition of causality may need some further discussion, as there are other definitions around that do not require uniqueness for a causal description. But are these other definitions correct or relevant? Basmann (1988, p. 99) answered this question with the observation that “None of the generally accepted meanings of ‘causality' fails to involve the notion that causation is a real-world, invariant relation between events rather than a mere property of a linguistic representation. To use ‘causality' in the latter sense may court eventual public … [devaluation] and dismissal of econometric research and econometricians.” Any relationship with non-unique coefficients and error term is by definition mis-specified. But how can a mis-specified relation reflect any kind of acceptable causation?
This echoes Freedman's (2005) assumption (i).
In this measure \( \sum \limits_{\ell =i}^L{\lambda}_{\mathit{\ell jt}}^{\ast }{\omega}_{\mathit{\ell t}}^{\ast } \) of the indirect effect, \( {\lambda}_{\mathit{\ell jt}}^{\ast } \) and \( {\omega}_{\mathit{\ell t}}^{\ast } \) are set equal to zero if the dependent variable of the ℓ-th equation in (8) is a relevant pre-existing condition. This restriction is needed because the vector wt is assumed to contain all relevant pre-existing conditions, and the effect of \( {x}_{jt}^{\ast } \) on each relevant pre-existing condition is not part of an indirect effect of \( {x}_{jt}^{\ast } \) on \( {y}_t^{\ast } \).
After clarifying the complications that arise in the Bayesian analyses of laws, PS (1988, p. 49) concluded that a Bayesian will do much better to search like a non-Bayesian for variables that absorb “proxy effects” for omitted regressors. These effects can be equal to the indirect-effect \( \sum \limits_{\ell =i}^L{\lambda}_{\mathit{\ell jt}}^{\ast }\ {\omega}_{\mathit{\ell t}}^{\ast } \) component of γjt, j = 1, …, K – 1. Taking a hint from this conclusion, we choose the coefficient drivers that absorb the indirect effects of included regressors in (10). This will be made clear in equation (15)(ii) below.
This stabilization does not take place if Φ is non-diagonal.
Time-series settings that involve time trends, polynomial time series, and trending variables give cases where this assumption is not satisfied. In these cases, we use Grenander's conditions presented in Greene (2012, p. 65).
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Paravastu, S., von zur Muehlen, P., Mehta, J.S. et al. THE STATE OF ECONOMETRICS AFTER JOHN W. PRATT, ROBERT SCHLAIFER, BRIAN SKYRMS, AND ROBERT L. BASMANN. Sankhya B 84, 627–654 (2022). https://doi.org/10.1007/s13571-021-00273-y
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DOI: https://doi.org/10.1007/s13571-021-00273-y
Keywords
- Unique time-varying coefficient and unique error term; direct effect
- Indirect effect; and total effect of a regressor
- Omitted relevant regressor
- Coefficient driver
- Measurement-error bias