Invariant Causal Mechanisms

https://doi.org/10.5281/zenodo.823264

David Waldner

University of Virginia

[**Original document page numbers preserved in brackets for citation purposes.**]

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David Waldner is Associate Professor of Politics at the University of Virginia. He can be reached at daw4h@virginia.edu. An earlier version of this paper was presented the annual conference of the American Political Science Association, Philadephia, PA, September 2016, where I had the benefit of comments from David Collier and Sid Tarrow and the pleasure of learning from my co-panelists, Andy Bennett, Derek Beach and Sherry Zaks. Post-conference conversations with Gerardo Munck, Sayres Rudy, and Alan Jacobs improved the essay considerably.

Why Causal Mechanisms are Not Variables

Qualitative methodologists generally treat process tracing methods and a mechanistic view of causation as natural allies. Two conjoined propositions form the basis for this alliance. The first proposition is that the identification of causal mechanisms is the sine qua non of distinguishing causal relations from mere correlations. The second proposition is that process-tracing methods are uniquely qualified to identify these critical causal mechanisms. In one admirably pithy formulation, Gary Goertz and James Mahoney state categorically: “No strong causal inference without process tracing.”There appears to be a tacit consensus that process tracing is both necessary and sufficient for causal inference.

One can applaud the development of process-tracing methods without making such strong claims on its behalf. It would behoove us, first of all, to distinguish causal inference from causal explanation. We make inferences about the existence of a causal relationship by claiming that an observed association is not merely correlational. We can do this qualitatively, by making claims about necessary and sufficient conditions, or we can do this quantitatively, by making claims about unbiased estimates of causal effects. Philosophers who are in general sympathetic to mechanistic conceptions of causation have been quick to point out that inferences can and frequently are made without explicit reference to causal mechanisms.2 Even as we affirm that the analysis of causal mechanisms can be a powerful tool in causal inference, there is no reason to reject experiments and design-based statistical studies as equally powerful instruments of causal inference.3 We make causal explanations, on the other hand, by invoking causes. In previous work, I have argued that a complete causal explanation requires the identification of the relevant mechanisms, but even on this point, philosophers sympathetic to a mechanistic perspective on mechanisms dissent; Harold Kincaid, to give one example, has argued cogently that no universal claim for mechanism-based explanation is reasonable.4

Moreover, reading various literatures on the discovery of causal mechanisms in disparate scientific disciplines makes it hard to credit the claim that “process-tracing methods are arguably the only method that allows us to study causal mechanisms.”5 It may be the case that process-tracing methods are particularly well-suited to studying mechanisms in social science fields like Political Science, but this claim requires an auxiliary argument about why the social sciences differ from natural sciences and why that difference makes process tracing particularly well suited for studying social science mechanisms.

Finally, it is worth revisiting the vexing question of whether our current conceptual understanding of causal mechanisms is adequate to the tremendous inferential and explanatory bur- dens we process tracers place upon them. In a 2001 review essay, James Mahoney identified approximately two-dozen definitions of causal mechanisms. Somewhat alarmingly, Mahoney noted that many of these definitions do not clearly distinguish causation from correlation.6 In the intervening years, political scientists have made some important progress defining causal mechanisms, yet skeptics remain, even among political scientists who are sympathetic to process tracing.7 Much more work remains to be done: a recent commentary on process-tracing methods by a sympathetic philosopher concludes that “the social science literature has been preoccupied more with how mechanisms are found or theorized, and with the role they play in explanation and theory, and less with developing a definition that captures the essential elements of mechanisms or that applies to all scientific contexts.”8

Yet it is not clear that we can make much progress by way of definition. Many of us, myself included, endorse a “generative” definition of causal mechanisms, as entities that generate an outcome of interest.9 There are various definitions in the philosophical literature that expand on this core definition, but all of them ultimately contain some version of the claim that causal mechanisms produce an outcome. Thus, we end up defining causation in terms of causal mechanisms and causal mechanisms in terms of generating or producing outcomes—that is to say, in terms of causation.

My recommendation is that we put an end to worrying about a unified and singular definition of causal mechanisms and that we instead focus on what property a mechanism must possess in order to perform its inferential and explanatory functions. By distinguishing properties from functions, my intention

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is to advance the debate without trafficking in tautology. The property I wish to emphasize is invariance. Definitionally, invariance is a property of a system that remains unchanged under some transformation. For example, the area of a surface remains unchanged if the surface is rotated in space, a property known as rotational invariance. In my usage, invariance is a property of an entity in a causal system that remains unchanged under intervention. We intervene on variables, setting them to new values; that intervention produces changes in other variables that are causal effects of the variable on which we intervened. Stated colloquially, wiggle the cause and the effect wiggles too; flip a light switch and the light turns on and off. To refer to mechanisms as possessing the property of invariance, then, is to claim that mechanisms are those elements of a causal system that we cannot wiggle.

To explore invariance further, let’s highlight what it is not. First, invariance is not determinism. Determinism is a property of a causal relationship between singular events (token causation) or variables (type causation).10 To claim that mechanisms are invariant neither implies determinism nor precludes it; to insist that invariance is equivalent to determinism is to make a category error.

Furthermore, mechanisms embodying the property of invariance are neither intervening variables nor systems of variables connected by arrows. Andrew Bennett, in this symposium, attributes to me the idea that mechanisms are systems of variables connected by arrows,11 but as the following discussion should make clear, I do not subscribe to either view of causal mechanisms. To see why not, it will be helpful to invoke the concept of a causal graph. A causal graph is nothing but a set of vertices or nodes, representing random variables, and a set of edges, representing postulated relations of statistical dependence. When edges have arrowheads, they denote direction and asymmetry; arrows represent postulated causal relations. Finally, when a path from a node does not return to itself, the graph is acyclic. These causal graphs are thus directed acyclic graphs.

It is common among both philosophers and social scientists to see references to causal mechanisms as mediators in a causal graph, as in X -> M -> Y, where X and Y are the independent and dependent variables, respectively, and M is the mechanism.12 There is great value in thinking about causal relations in terms of mediating variables. Doing so produces more extensive causal chains that have the methodological virtue of increasing the opportunity for falsification and the explanatory virtue of providing richer detail and depth. But it seems futile to identify these mediators with causal mechanisms, for they simply reproduce the relationship of correlation between variables that we are seeking to transcend. Indeed, there is nothing ontologically distinct about mediators as a type of random variable; whether a variable is a mediator or not depends on nothing but the construction of the causal model and the placement of the nodes. A variable that is a mediator in one causal graph may just as easily be a non-mediator in another.

Alternatively, we can think of the entire causal graph as a causal mechanism in terms of a system of interacting parts. This view is widely held among philosophers, especially in the literature on biological mechanisms. Rosa Runhardt, in this symposium. adopts this view, citing Machamer, Darden, and Craver 2000 to refer to the causal structure A -> B -> C as the schematic representation of a mechanism, with each of A, B, and C conceptualized as independent stages of the mechanism.13 Stuart Glennan defines a mechanism for a behavior as “a complex system that produces that behavior by the interaction of a number of parts, where the interactions between parts can be characterized by direct, invariant, change-relating generalizations.’’14 Social scientists have followed suit. Carly Knight and Christopher Winship define a mechanism as “a causal relationship involving one or more intervening variables between a treatment and an outcome.”15 Proponents of this approach, both philosophers and social scientists, offer the toy model of a car’s engine as this type complex mechanism or system. As Alexander Gebharter explains, “The question of why a car speeds up when the gas pedal is pressed can be answered by pointing at/describing the underlying mechanism (i.e., the motor and how it is connected to the gas pedal, the wheels, the gas tank, etc).”16 Derek Beach and Rasmus Pedersen offer the identical analogy, with X as the motor, Y the car’s movement, and “the driveshaft and wheels can be thought of as the causal mechanism that transmits forces from X (motor) to produce Y (movement).”17

Indeed, an automobile’s mechanical components comport well with the idea of a complex system of interacting parts whose interactions can be characterized by direct, invariant, change-relating generalizations: under most circumstances, step on the accelerator and the car moves faster. But some- thing critical is missing from this conception of mechanisms and the toy model that instantiates it. Most automobiles are still powered by gasoline and gas-powered engines are internal combustion engines. To think of the mechanical parts of an engine as the causal mechanism is to exclude combustion from the causal explanation of how automobiles work. We can say something further: the mechanical parts of an automobile are variables, and as variables, we can intervene to turn each variable to a new value. Combustion, on the other hand, cannot be turned off; it is an exothermic chemical reaction that occurs under proper conditions. Each of those conditions—heat, oxygen, and a fuel—can be turned on or off, just as any random

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variable in a causal graph can be turned on or off. But if heat, oxygen, and a fuel are all present, combustion occurs. One cannot throw a lit match into a pool of gasoline in the presence of oxygen and the absence of any other flame retardant and then somehow intervene to turn off combustion. That makes combustion different from a random variable, for in a causal graph, to repeat, one can, in principle, intervene to set any of the variables to a new value.

In what way does combustion differ from the interconnected parts of an engine? As I describe below, combustion has the property of invariance. We invoke mechanisms to explain relations between variables; to perform this function, mechanisms cannot themselves be variables. Relations between variables, after all, are described by correlations, and we typically believe that causal inference and explanation are based on something beyond correlation. Therefore, we cannot be satisfied by defining mechanisms as intervening variables or as systems of variables.

What does it mean to define mechanisms as entities featuring the property of invariance? To understand invariance, we must recognize the critical distinction between causation and constitution. Causation refers to a particular type of relation of dependence between events (which can be coded as variables, at the aggregate level). Causal relations are temporal—they occur over time—and asymmetric, in that the manipulation of a cause produces an effect but the manipulation of the effect does not produce the cause. Causation implies etiology, in that we explain outcomes by detailing the causal path that led to them. Constitution, on the other hand, is about the fundamental nature of a phenomenon. As such, constitution is atemporal and provides no opportunity for manipulation. Consider the prosaic example of a batted baseball that strikes and shatters a glass window. The causal story has two events, the striking of the glass by the ball and the subsequent shattering of the glass. The two events are separated in time, albeit only briefly separated. There is asymmetry of manipulation: by striking the glass with the ball, the glass shatters; but shattering the glass does not cause it to be struck by the ball. The causal mechanism, on the other hand, is the inherent fragility of the glass, which is constituted by its molecular structure. “To be fragile is to have a particular molecular structure; the fragility is not a consequence of the molecular structure.”18 Similarly, combustion takes place when a fuel-oxygen mixture is exposed to heat due to the structure of dioxide molecules (their angular spin and momentum) such that oxygen molecules assault hydrocarbon molecules, converting them into molecules of water and carbon dioxide and releasing tremendous amounts of heat in the process.

Invariant Causal Mechanisms and Qualitative Causal Inference

Where does this discussion of invariant causal mechanisms get us? Elsewhere I have argued that process tracing should satisfy a “completeness standard,” consisting of four elements: a relatively complete causal graph representing the process being traced, an event-history map representing the events of a particular case that correspond to a particular realization of the causal graph, a set of descriptive inferences from the event- history map to the causal graph (it is here that Bayesian updating can be extremely useful), and, finally, a full set of causal mechanisms.19 What constitutes a full set of mechanisms? The arrows in the causal graph that connect random variables de- note the underlying causal mechanism. We can intervene on random variables but we cannot intervene on the causal mechanisms that generate causal connection. This is precisely the meaning of invariance. Thus, for a causal graph with N directed edges (connecting N + 1 random variables, in many cases), there must be N causal mechanisms.

Let me give one quick example that should help us distinguish causal graphs from causal mechanisms. Figure 1 is the causal graph that I have previously reconstructed from Elisabeth Wood’s analysis of insurgency and democratic transitions in El Salvador and South Africa.20 The graph has seven nodes, each representing a random variable, and six directed edges, each connecting two of the random variables. By my account, this causal graph has to be supplemented by event- history maps supported by evidence (individual chapters of the book perform this task) and by six invariant causal mechanisms, one mechanism for each arrow. Let’s consider one such mechanism linking the third and fourth nodes in the graph: by what mechanism does insurgent collective action reshape economic structures to make them less reliant on the repression of labor? In her African case study, Wood articulates this causal mechanism as a decision-theoretic model of how investor confidence in the profitability of apartheid institutions was eroded by worker collective action leading to the suspension of investment in South Africa by both domestic and foreign capitalists. Wood derives two states of the world from the comparative statics of the model, one in which the political control of labor keeps wages below what they would be in a liberal market economy and one in which worker mobilization raises wages to above their level set by a liberal market economy. Worker mobilization thus changes elite preferences over labor market conditions and hence over political regimes. Wood then compiles considerable evidence that her model best depicts actual events, and is thus superior to several plausible alternative mechanisms.21

This brief summary does not do justice to either Wood’s theoretical model of the causal mechanism or her empirical confirmation of the mechanism. But it is sufficient to support two claims. First, causal mechanisms are distinct from either mediating nodes in a causal graph or the entire set of nodes in the causal graph. Second, causal mechanisms are invariant insofar as their fundamental constitutive features constrain their actions. Wood, like other advocates of rational-actor models, conceives of human beings as fundamentally endowed with intentionality and rationality and hence acting in ways

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[Insert Figure 1 Here]

that maximize some objective function. This is not the only possible interpretation of how the constitution of agency constrains action and so I concur with Rosa Runhardt’s observation, in this symposium, that “units in social science mechanisms are not as clearly defined as biological entities.”22

What is the payoff of conceptualizing process tracing in terms of causal graphs, event-history maps, and invariant causal mechanisms? Doing so gives process tracers a principled response to the fundamental problem of causal inference. The fundamental problem of causal inference is derived from the Rubin Causal Model, also known as the potential-outcomes framework, which defines a causal effect as the difference in a response variable between a unit under treatment and the same unit under control. The problem of inference at the unit level is a problem of missing data: one cannot observe a unit under treatment and under control simultaneously; one or the other state of the world cannot be observed and so one cannot observe the two quantities needed to infer the causal effect. This claim, if left uncontested, would seem to undermine the possibility of within-case causal analysis. Many qualitative scholars simply ignore this fundamental problem of causal inference, while others explicitly reject it as part of a quantitative perspective that is simply not relevant to the concerns or the philosophical commitments of qualitative scholars.23

My approach to process tracing—the combination of causal graphs, event-history maps, and invariant causal mechanisms, aka the “completeness standard”—is based on the pre- supposition that process tracing would only be strengthened by directly engaging the fundamental problem of causal inference. The outlines of a principled response to the fundamental problem of causal inference would rely on what Paul Holland calls the “scientific solution,” a third option to experiments with random assignment to treatment and design-based statistical inference. The scientific solution relies on exploiting “various homogeneity or invariance assumptions.”24 My claim is that a fully specified causal model that is fully identified by the

relevant invariant causal mechanisms can be used to satisfy this invariance assumption. Insofar as one has the full set of invariant causal mechanisms, one can use the fully identified causal model as a substitute for the missing observations about the counterfactual states of the world. To use the informal example favored by Donald Rubin, we can make unit-level causal inferences about the effects of aspirin on relief from the pain of a headache because we can produce causal graphs representing pathways from damaged cells to the production of a chain on enzymes to signals sent to the pain center of the brain, and also because we understand the nature of neurotransmitters and their receptors and how their activities are constrained by their fundamental structures. We can, in other words, tell the causal story in terms of etiology and in terms of constitution, and, in doing so, we can resolve the fundamental problem of causal inference.25

Response to Andrew Bennett’s Comments in This Symposium

In his contribution to this symposium, Andrew Bennett doubts my claim that the completeness standard constitutes a potential solution to the fundamental problem of causal inference.26 His critical interrogation has two components, a discussion of mechanisms and a discussion of evidence. On mechanisms, Bennett equates my approach to mechanisms to Judea Pearl’s explication of causal graphs and to James Woodward’s manipulation account of causation.27 I emphatically dissent from this characterization of my position. I consider causal mechanisms to be ontologically distinct from causal graphs; indeed, that distinction must be made since the completeness standard calls for both causal graphs and causal mechanisms, with one mechanism corresponding to each arrow in the causal graph.

Let me say a few more words to explain why I consider causal graphs and causal mechanisms to be irreducibly distinct elements of a causal explanation. In Pearl’s account, causal graphs represent relations of probabilistic dependence; these

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graphs are causal only insofar as we invoke the Causal Markov Condition, which states, informally, that in the causal graph X -> M -> Y, the value of Y depends only on its direct causal relation to M such that after conditioning on M, Y is statistically independent of X. Let me state emphatically: At no point in my account is causal inference predicated upon the Causal Markov Condition. I consider probabilistic dependence to be only a consequence of a causal relationship and not constitutive of that relationship.

    Woodward, on the other hand, presents a manipulationist account of causation. Recognizing that many relations of counterfactual dependence are not causal in nature, Wood- ward restricts causal relations to a subset of relations of counterfactual dependence such that X is a cause of Y only in the case that an ideal intervention on X—setting it to a new value—also changes the value of Y without changing the value of any other variable that is not on the pathway between X and

    Y. Woodward occasionally refers to his account as mechanistic. Yet the key distinction that Woodward preserves is between difference-making accounts of causation and production accounts of causation.28 Woodward’s account is unambiguously a difference-making account, which does not assign any role to mechanisms, while my own is a production account that requires causal mechanisms. One possible source of Bennett’s mistaken conflation of my approach with Wood- ward’s is that I use the term invariance, a term that is central to Woodward’s account as well. Yet we use that term in very distinct ways. For Woodward, invariance is a predicate of a generalization; empirical generalizations linking X and Y are causal if they are invariant under a range of interventions. In my account, on the other hand, invariance is a predicate of mechanisms; it is for this reason that I invoke the fundamental distinction between causation and constitution, as I explain above.

    While I reject the characterization of my approach to process tracing as reducible to the accounts offered by Pearl and Woodward, I do see my project of qualitative causal inference as affiliated with work by philosophers that seeks to integrate difference-making and production accounts. I would continue to insist, however, that production accounts are more fundamental; we can logically derive relations of counterfactual dependence from causal mechanisms but I do not believe that the converse is necessarily true. 29

    But to make ontological claims about mechanisms is not to deny the epistemological relevance of counterfactuals. There is a big difference between the claim that causation consists of a particular type of counterfactual dependence, a claim that I reject, and the claim that causation consists of causal mechanisms whose productive capacities logically imply counterfactual dependence, a claim I endorse. Therefore, when process tracers invoke mechanisms, it should not give them license to ignore the fundamental problem of causal inference, which is a fundamental problem precisely because observations about counterfactuals are missing data. Process tracers do themselves no favor when they reject the fundamental problem of causal inference as relevant only to quantitative analysis. The problem is a logical consequence of any causal relationship, regardless of the method used to investigate that relationship.

    In his contribution to this symposium, Bennett also ex- presses concern about the adequacy of the completeness standard, concern that stems from his skepticism that the standard sufficiently addresses the practice of process tracing. Bennett has done more than anybody to develop the practice of process tracing over the past dozen years and so his comments deserve very close attention. I believe his most fundamental point is that I underestimate the extent to which “explanations of cases and populations will always be potentially fallible.” Using slightly different language, however, I made the same claim in the chapter I contributed to his recent edited volume, where I wrote that the standard is intended to justify the claim that a causal explanation “has weathered sufficient scrutiny relative to its rivals and to the current state of theory and data gathering that belief in its approximate truth is more reasonable than disbelief but is also subject to revision in the face of future data gathering or theorizing.”30 I believe we agree on this point.

    Furthermore, I think Bennett’s claim betrays some misapprehension of the goal of the completeness standard. The goal of the standard is not to achieve fallibility and certainty. The goal of the standard is to justify unit-level causal inference in light of the fundamental problem of causal inference, which denies that such inferences are possible. The appropriate contrast, then, is not certain inference versus uncertain inference but rather valid inference versus invalid inference. Insofar as the standard accomplishes its goal, it permits unit-level causal inferences that, like all inferences, will be fallible; that is to say, we will always be uncertain about our inferences precisely because they are inferences. Indeed, the standard allows us to form a typology of the sources of uncertainty; we may have uncertainty about the causal graph itself, uncertainty about inferences from the event-history maps to the causal graphs, and uncertainty about the identification and validation of causal mechanisms. Uncertainty about any element of the standard— concerns about the sufficiency of the causal graph, about the degree of evidentiary support for the event-history maps, or uncertainty about mechanisms—will translate directly into uncertainty about the explanation. But, to use a Bayesian frame- work, we can still update our posterior beliefs about the validity of the explanation.31 All I wish to say at this point is to make the conditional claim that if the standard is met, then unit-level causal inference can be justified.

    Therefore, I agree with Bennett that no explanation is ever truly complete.32 We make fallible judgments about the sufficiency

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    of a causal graph, about the adequacy of evidence for descriptive inferences, about the credibility of alternative hypothesis, and about the significance of omitted variable bias. Bennett’s work is absolutely invaluable in thinking about these crucial topics. The completeness standard is an aspirational standard against which we judge specific process-tracing research and therefore judge it as more or less complete and therefore more or less supportive of a valid causal inference. It is true that all causal models simplify and thus leave some relevant causal knowledge unstated; causal graphs, after all, encode qualitative relationships. Bennett criticizes me for not stating explicitly that combustion requires gravity; but if we take Bennett at his word, all of our causal arguments are going to be burdened by enumerating relevant background conditions like gravity, oxygenation, and pretty much all physical laws. Causal models may also, of course, suffer from omitted variable bias; this is the existential fact of all science.33

    Bennett thus raises critical points that have not yet been fully addressed in the completeness standard. I believe his concerns can be accommodated in future iterations of the standard, in part by drawing on his work, and so I thank him deeply for raising them. I do not agree, however, that these concerns add up to a fundamental critique of the approach as a standard of causal inference.

    References

    Beach, Derek and Rasmus Brun Pedersen. 2013. Process-Tracing Methods: Foundations and Guidelines. Ann Arbor: University of Michigan Press.

    Beach, Derek and Rasmus Brun Pedersen. 2016. Causal Case Study Methods: Foundations and Guidelines for Comparing, Matching, and Tracing. Ann Arbor: University of Michigan Press.

    Bennett, Andrew and Jeffrey T. Checkel. 2015. “Process Tracing: From Philosophical Roots to Best Practices.” In Process Tracing: From Metaphor to Analytic Tool, edited by Andrew Bennett and Jeffrey T. Checkel. Cambridge: Cambridge University Press: 3–37.

    Bennett, Andrew. 2016. “Do New Accounts of Causal Mechanisms offer Practical Advice for Process Tracing?” Qualitative and Multi- Method Research: Newsletter of the American Political Science Association’s QMMR Section vol. 14, no. 1–2.

    Dunning, Thad. 2012. Natural Experiments in the Social Sciences: A Design-Based Approach. Cambridge: Cambridge University Press. Gebharter, Alexander. 2014. “A Formal Framework for Representing

    Mechanisms.” Philosophy of Science vol. 81, no. 1: 138–153. George, Alexander L. and Andrew Bennett. 2005. Case Studies and

    Theory Development in the Social Sciences. Cambridge: MIT Press. Gerring, John. 2008. “The Mechanismic Worldview: Thinking Inside the Box.” British Journal of Political Science vol. 38, no. 1: 161–79.

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    Machamer, Peter, Lindley Darden and Carl F. Craver. 2000. “Thinking about Mechanisms.” Philosophy of Science vol. 67, no. 1: 1–25. Mahoney, James. 2001. “Beyond Correlational Analysis: Recent In-

    novations in Theory and Method.” Sociological Forum vol. 16, no. 3: 575–593.

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    Steel, Daniel. 2004. “Social Mechanisms and Causal Inference.” Philosophy of the Social Sciences vol. 34, no. 1 : 55–78.

    Waldner, David. 2007. “Transforming Inferences into Explanations: Lessons from the Study of Mass Extinctions.” In Theory and Evidence in Comparative Politics and International Relations, edited by Richard New Lebow and Mark Lichbach. New York: Pal- grave Macmillan: 145–175.

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    Waldner, David. 2015a. “What Makes Process Tracing Good? Causal Mechanisms, Causal Inference, and the Completeness Standard in Comparative Politics.” In Process Tracing: From Metaphor to Analytic Tool, edited by Andrew Bennett and Jeffrey T. Checkel. Cambridge: Cambridge University Press: 126–152.

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    Footnotes

    1 Goertz and Mahoney 2012, 103. See also Bennett and Checkel 2015.

    2 Steel 2004.

    3 Dunning 2012.

    4 Kincaid 1996, 181; Waldner 2007.

    5 Beach and Pedersen 2013, 1-2.

    6 Mahoney 2001.

    7 Gerring 2010.

    8 Runhardt 2015, 453.

    9 Mahoney 2001, 580; George and Bennett 2005, 137; Waldner 2012.

    10 Many philosophers claim that all instances of type causation can be reduced to token causation. I agree, but see little harm in making claims about causation involving variables.

    11 Bennett 2016.

    12 Gerring 2008; Kincaid 2012.

    13 Runhardt 2016.

    14 Glennan 2002, S344.

    15 Knight and Winship 2013, 283.

    16 Gebharter 2014, 139.

    17 Beach and Pedersen 2013, 30. Without invoking the theory of causal graphs, Beach and Pedersen represent causal mechanisms pre- cisely by drawing boxes connected by arrows and referring to the entire system as a causal mechanism. See their Figures 4.2 and 4.3 for examples.

    18 Ylikoski 2012, 34; Ylikoski 2013.

    19 Waldner 2015a, 2015b, 2015c.

    20 Wood 2000. For my discussion of this work as an exemplar of process tracing, see Waldner 2015a, 137–141.

    21 Wood 2000, 143–168.

    22 Runhardt 2016.

    23 See Goertz and Mahoney (2012) for the claim that qualitative and quantitative approaches represent two distinct cultures. See Beach and Pedersen (2016, 3) for the claim that case-study researchers should eliminate the “vestiges” of variance-based thinking embedded in such terms as “independent and dependent variables.”

    24 Holland 1986, 948.

    25 This is obviously an abridged version of how invariant causal mechanisms can satisfy an invariance assumption and permit unit- level causal inference. For the full account, see Waldner 2015c.

    26 Bennett 2016.

    27 Pearl 2000; Woodward 2002 and 2003.

    28 See especially Woodward 2011.

    29 Waldner 2012.

    30 Waldner 2015a, 129–130, emphasis added.

    31 Our updated beliefs will be a distribution. I thank Alan Jacobsfor helping me think about this point.

    32 Ironically, I coined the term “completeness standard” after reading some of Bennett’s early and still wonderfully influential work with Alexander George on process tracing, in which he claimed that “[A]ll the intervening steps in a case must be predicted by a hypothesis or else that hypothesis must be amended...” See George and Bennett 2005, 207. This type of continuity criterion has distinguished pedigree in the philosophical literature and forms the basis for Runhardt’s (this symposium) criterion of “productive continuity.”

    33 Bennett is critical of my approach for omitting an error term from causal graphs. Given that my approach is qualitative, it is not clear what adding that term accomplishes, for we do not estimate standard errors and confidence intervals and we do not make assumptions about the distribution of errors. But it is easy enough to add an exogenous error term to remind readers of possible omitted variables.

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