Abstract
Reichenbach’s ‘principle of the common cause’ is a foundational assumption of some important recent contributions to quantitative social science methodology but no similar principle appears in econometrics. Reiss (Philos Sci 72:964–976, 2005) has argued that the principle is necessary for instrumental variables methods in econometrics, and Pearl (In Causality: Models, reasoning and inference, Cambridge: Cambridge University Press, 2000/2009) builds a framework using it that he proposes as a means of resolving an important methodological dispute among econometricians. Through analysis of instrumental variables methods and the issue of multicollinearity, we aim to show that the relationship of the principle to econometric methods is more nuanced than implied by previous work but nevertheless may make a valuable contribution to the coherence and validity of existing methods.
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16 May 2022
A Correction to this paper has been published: https://doi.org/10.1007/s11229-022-03729-4
Notes
Where \(P(A \wedge B)\) is the probability of A and B both occuring. We will use P(A|C) to represent the probability of A given that C is known to have occurred.
The Spirtes et al. (2000) approach has also been employed in the analysis of time-series data.
As Pearl (2009, pp. 194–195) notes, there are some nuanced distinctions to be made between confounders and confounding.
While macroeconometrics is mostly concerned with datasets containing observations over time (‘large T, small N’ in econometricians’ parlance), microeconometrics focuses on single, or repeated, cross-sections (‘large N, small T’).
See Woodward (1988) for what one might call a ‘traditional’ overview of the formalities of regression methods directed at philosophers. Our discussion relies more on the presentations by Manski (1991) and Wooldridge (2002a). A valuable additional reference is Morgan & Winship’s (2007) book, which provides an overview of regression and graph-based methods within a detailed discussion of causal inference based on counterfactuals.
Linearity in the parameters allows E(y|x) to be a non-linear function of the explanatory variables. E.g. We could have \(E(y|{\mathbf {x}}) = \beta _{0} + \beta _{1} x_{1} + \beta _2 x_{2} + \beta _3 x^{2}_{2}\). The linearity assumption is convenient though not essential.
The implicit assumption of causality, whether in the presentation of methods or their application, is usually observed in the interpretation or use of the estimates from the subsequent empirical analysis, as noted by Woodward (1988).
Specifically, “\(\varvec{\beta }\) can be written in terms of population moments in observable variables”(Wooldridge 2002a, (53). Using the vector of explanatory variables \({\mathbf {x}}\), we can write:
$$\begin{aligned} {\varvec{\beta }} = E[{\mathbf {x}}^{T}{\mathbf {x}}]^{-1}E({\mathbf {x}}^{T}y) \end{aligned}$$This is obtained by rewriting (3) as \(y = \mathbf {x} {\varvec{\beta }}+u\), premultiplying both sides by the transposed vector \({\mathbf {x}}^{T}\), applying the expectations operator to both sides and solving for \(\varvec{\beta }\) (noting that \(E[{\mathbf {x}}^{T} {\mathbf {u}}] = 0\)).
See Qin (2018) for a recent critique of that approach.
There is some disagreement about how this severance of relationships should be represented; see Pearl (2009, pp. 376–377).
This is identical in structure to a graph by Pearl (Figure 7.8(a), 2009, p. 248).
See Angrist & Pischke (2009, pp. 117–120) for a summary of this example in the context of an explanation of the instrumental variables approach.
The relevant expression is:
$$\begin{aligned} {\varvec{\beta }} = E[{\mathbf {z}}^{T}{\mathbf {x}}]^{-1}E({\mathbf {z}}^{T}y) \end{aligned}$$Where \({\mathbf {z}}\) is the \({\mathbf {x}}\) vector including z and excluding \(x_{j}\).
Indeed IV1 is a serious concern in the empirical literature because of theoretical results showing the negative consequences of ‘weak instruments’ (small value for \(corr(z, x_{j}|\mathbf {x_{\lnot j}})\)).
As another example, in a widely-cited survey article, Angrist & Krueger (2001, p. 73) state: “In our view, good instruments often come from detailed knowledge of the economic mechanism and institutions determining the regressor of interest”.
Similar approaches are also utilised in macroeconometrics, however the nature of much macroeconomic data-in the form of time-series at the country level-is such that there are additional complications which are outside the scope of the present paper.
Causal transitivity means that, “For any three variables A, B, and C, if A causes B and B causes C, then A causes C” (Reiss 2005, p. 969). And Reiss defines functional correctness as: “A structural equation is functionally correct if and only if it represents the true functional (but not necessarily causal) relations among its variables”. In essence, this can be understood as meaning that an equation correctly represents the relationship between the magnitudes of variables in the equation, but those relations need not be causal.
The word ‘definitional’ is intended to indicate that these assumptions are not ‘implicit’ in the sense of being wholly unstated (as Reiss suggests later in his paper). Rather, they follow directly from initial definitions of the problem, such as specification of the structural equations.
The second assumption could be strengthened by requiring that the structural equation specify all causes, but this is not necessary for causal inference.
Hoover (2007) draws attention to an analogous problem in Reiss’s reasoning about ‘collider’ variables.
The fact that an instrument cannot belong in the true structural equation is an assumption made clear in a number of texts, e.g. Wooldridge (2002b, p. 517) and Pearl (Figure 7.8 (d), 2009, p. 248). Reiss actually proposes this later in that paper (see condition CIV-2, Reiss (2005, 973)) as one of a set of assumptions that would justify the econometrician’s approach, but this assumption clearly is made both in theory and in practice.
There are two additional problems with Reiss’s argument. First, his proposed stage 2 assumption is in fact the standard way of interpreting the error term in a hypothesised structural equation. Econometric textbooks often make this interpretation explicit–see for instance Greene (2003, p. 8)-and it is recognised by Woodward (1988, p. 261). Pearl endorses a conceptual understanding of such error terms as representing omitted factors since it is a useful guide “when building, evaluating and thinking about causal models” (Pearl 2009, p. 162-163). Second, Reiss fails to note that one can describe the IV logic without reference to causality per se. Contra to Reiss’s criticism that “textbooks contain ‘recipes’ for econometric inference that give the impression that econometrics can proceed without causal background assumptions” (Reiss 2005, p. 966), econometrics can proceed without such assumptions. But, as per Cartwright’s mantra: ‘no causes in, no causes out’, not if the interest is in causal inference.
We exclude Woodward’s reference to these as ‘causal’ assumptions since that remains a moot point.
It may be useful for some readers to note that within the discipline it is known that implicit in such presentations in the econometrics literature is that if \( y \) was a cause of any explanatory variables–i.e. there was ‘simultaneous causation’–then an explicit system of equations would be required.
No particular justification for this assertion is provided, though in principle it should be testable; standard statistical tests can be used to examine the likelihood of a given correlation being due to chance. Yet such commentary typically makes no mention of examining the presumption of spuriousness.
In matrix representations the assumption of no perfect collinearity is clearly stated and known as ‘the rank condition’.
The relevant derivations are available from the author, but the result should be unsurprising given that the presence of causal relationships does not alter the statistical results.
The case of a common cause is omitted since it will suffice for us to demonstrate that at least one system may exist that would require a reinterpretation of estimated coefficients.
Implicit in this graph is that, following Pearl (2009), all variables affecting more than one other variable in the system are shown.
A partial exception is reflected in one historical practice of conducting and reporting results from multiple regressions. The practice was to report regression results in such a way as to show how the incremental inclusion of every additional explanatory variable (‘control’) altered the estimated parameter on the explanatory variable of primary interest.
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Acknowledgements
I am grateful to two anonymous reviewers for detailed comments and suggestions that led to improvements in this manuscript, and to Martin Wittenberg for helpful comments on the very first draft of this work. Participants at a SALDRU seminar in the School of Economics at the University of Cape Town and other anonymous reviewers provided comments that informed the first working paper version (Muller 2012). The usual disclaimers apply.
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Muller, S.M. Econometric methods and Reichenbach’s principle. Synthese 200, 185 (2022). https://doi.org/10.1007/s11229-022-03684-0
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DOI: https://doi.org/10.1007/s11229-022-03684-0