Granger Causality and Dynamic Structural Systems

In a celebrated paper, Granger (1969) introduced a notion now known as Granger noncausality or, for brevity,“G noncausality.” Since itsintroduction, G noncausality andits counterpart, G-causality,have been the focus of intense attention and interest, both theoretically and inapplications. Hendry and Mizon (1999) discuss the sweeping extent to whichG-causalityhas permeated economics and econometrics. A recent citation count by Google Scholar shows almost4000 citations to Granger’s seminal article.

As Granger (1969) emphasizes, G-causality is based purely on the predictability of particular time series; it does not necessarily provide insight into whatever the underlying “true” causal relations may be. Some studies have been scrupulous in adhering to Granger’s original predictive notions. Examples are Sims’s (1972) seminal investigation of the relations between money and income and the study of health and socioeconomic status by Adams et al. (2003) . Other studies, too numerous to cite, have either explicitly or implicitly drawn structural or policy conclusions from theirs or others’ G-causality findings. Unfortunately, such conclusions are entirely unfounded, as G-causality, by itself, has no structural meaning.

It is not constructive to point out difficulties without also proposing remedies. Accordingly, our goal here is to provide direct links, previously missing, between G-causality and aspects of structural causality that emerge naturally from an explicit system of dynamic structural equations compatible with a wide range of economic data-generating processes.

Our analysis provides explicit guidance as to how to properly apply G-causality to gain structural insight. Specifically, when defining and testing G-causality, certain variables in addition to the dependent variables (Y , say) and “potential G-causes” (Q, say) play a crucial role. For convenience, denote these supplemental variables S and call them “covariates.” Our results establish that if Q G-causes Y , then Q structurally causes¹Y (and vice versa) when, among other things, the covariates S are variables that

• are not themselves structurally caused by either Y or Q

• structurally cause Q or Y or are proxies for unobserved structural causes of Q or Y

• may include leads of proxies for unobserved structural causes of Q or Y

As we explain, covariate leads play a purely predictive role in the back-casting sense; their presence does not violate the causal direction of time. In fact, as we discuss, they create the opportunity for more reliable structural inference.

An interesting consequence of our analysis is the emergence of new structurally informative extensions of the classical notion of G-causality. Depending on the context, these notions correspond to properties of direct or total (i.e., both direct and indirect) structural causality. In each case, we give a structural characterization of G-causality: we provide conditions ensuring specific forms of structural causality if and only if the corresponding versions of G-causality hold. We pay particular attention to finite-order G-causality, the version typically tested in practice, and we show that it characterizes a specific form of direct structural causality.

We obtain our results by making use of a system of general dynamic structural equations analyzed by White and Kennedy (2009) (WK). These systems permit data-generating processes (DGPs) that may be nonlinear as well as nonseparable and nonmonotonic between observables and unobservables. They may generate stationary or nonstationary (e.g., cointegrated) processes. Moreover, these systems support straightforward notions of structural causality both for structural vector autoregressions (VARs) and for recursive structures representing time-series natural experiments. Identification of structural effects is closely tied to a certain conditional form of exogeneity, as discussed, for example, in White (2006a) and WK. Here, this conditional exogeneity plays the key role in specifying the properties of the covariates S that ensure our characterizations. In fact, we prove that conditional exogeneity is necessary for valid structural inference. We also show that, in the absence of structural causality, any given version of G noncausality is equivalent to a matching version of conditional exogeneity.

All of these results have implications for drawing structural inferences from tests for G-causality or conditional exogeneity, and we propose new methods designed for this purpose. A major concern here is practicality. Our proposed tests can be implemented by essentially any econometric software that can generate random variables and perform linear weighted least squares regression inside a loop. Intricate matrix manipulations are not necessary. On the other hand, our tests are designed to have power against a relatively broad range of alternatives.

The plan of the paper is as follows. In Section 2, we review classical notions of G-causality. In Section 3, we introduce a structural dynamic DGP and distinguish structural VARs from time-series natural experiments. We provide notions of direct and total structural causality pertaining to both, and we characterize the relations between direct and total structural causality. Section 4 analyzes the links between G-causality and structural causality, using specific conditional exogeneity restrictions. For concreteness, the focus there is on the natural experiment case. A main result is a structural characterization of classical G-causality that proves its equivalence to a specific form of direct structural causality. Section 4 also introduces structurally informative extensions of classical G-causality and provides their structural characterizations. Of critical importance for applications is the structural characterization of finite-order G-causality, the version almost always tested in practice. Section 5 provides parallel results for structural VARs.

In Section 6, we describe our proposed G-causality tests. Section 7 analyzes the role of conditional exogeneity, focusing on the structural VAR case. Parallel results hold for the natural experiment case. We show that conditional exogeneity is necessary for valid structural inference, and we characterize its relation to G-causality. Section 7 also provides results supporting tests of conditional exogeneity and details their implementation. When neither conditional exogeneity nor G noncausality is a maintained assumption, then it is not possible to directly test for structural causality. To handle such cases, Section 8 introduces an indirect test for structural causality that combines G-causality and conditional exogeneity tests. Section 9 illustrates our methods, with applications to gasoline and oil prices (as in WK), monetary policy and industrial production (as in Romer and Romer, 1989 and Angrist and Kuersteiner, 2004), and stock returns and macroeconomic announcements (as in Chen, Roll, and Ross, 1986 and Flannery and Protopapadakis, 2002). Section 10 contains a summary and concluding remarks. Formal proofs of results are collected into the Mathematical Appendix.

1 Granger Causality

Granger (1969) defined G-causality in terms of conditional expectations. Granger and Newbold (1986) gave a definition using conditional distributions. We work with the latter, as it is this that turns out to relate generally to structural causality. In what follows, we adapt Granger and Newbold’s notation but otherwise preserve the conceptual content.

If Equation (1) does not hold, Granger and Newbold say that Q is a “prima facie” G-cause for Y . Granger and Newbold (1986) drop the phrase “prima facie” when σ(Q^t−1, S^t−1, Y^t−1) is the “universal” information set (that generated by the past of all variables), but since, as they note, using this information set is not practical, we will take “prima facie” as implicit.

Granger and Newbold (1986, p. 221) caution that

Not everyone would agree that causation is the correct term to use for this situation, but we shall continue to do so as it is both simple and clearly defined.

As Granger and Newbold (1986, p. 222) further note,

It has been suggested, for example, that causation can only be accepted if the empirical evidence is associated with a clear and convincing theory of how the cause produces the effect. If this viewpoint is accepted, then “smoking causes cancer” would not be accepted.

Granger and Newbold do not require a clear and convincing theory of how causes produce effects. Instead, as Granger (1969, p. 430) notes, “The definition of causality used above is based entirely on the predictability of some series [emphasis added].” Thus, G-causality can relate any variables whatsoever, regardless of underlying structural relations. Knowledge about underlying structural relations may be helpful in investigating G-causality but is not necessary.

Of main interest here is the reciprocal fact, established in what follows, that knowledge about G-causality can provide structural or policy insight. This idea is often implicit in empirical applications of G-causality, without any firm foundation. Our results below make the relations between G-causality and structural causality explicit and precise.

As Florens and Mouchart (1982) and Florens and Fougère (1996) note, G noncausality is a form of conditional independence. Following Dawid (1979), we write 𝒳 ⊥𝒴 | ℱ when 𝒳 and 𝒴 are independent given ℱ. We work with the following definition of Granger causality:

G noncausality is clearly a pure forecasting concept: Q^t provides no information useful in predicting Y_t beyond the information contained in the histories S^t and Y^t−1.

Granger and Newbold (1986) reference Y_t+k, k ≥ 0, instead of Y_t in their definition; the k=0 case treated here is implicit in Granger (1969) and explicit in Granger (1980, 1988). We focus entirely on the k=0 case, as k ≥ 1 cases have not been important in applications, nor are they important in relating G-causality and structural causality.

In Equation (1), Q and S appear with a lag relative to Y ; a literal translation would require Q^t−1 and S^t−1 to appear in Definition 1.1 instead of Q^t and S^t. The lag in Equation (1) enforces Granger’s (1969; 1988) and Granger and Newbold’s (1986, p. 221) firm views on instantaneous causation:

Whether all IC (instantaneous causality) can be explained in terms of data inadequacies is unclear…. However, it is clear that it is not possible, in general, to differentiate between instantaneous causation in either direction and instantaneous feedback. Thus, the idea of instantaneous causality is of little or no practical value.

We strongly agree that instantaneous causality is not relevant, as economic responses invariably take some time, no matter how little. But responses need not take the entire period of observation; they may instead take place within the period. For example, with monthly observations, a change in Q_t may occur in the third week of January and the causal response embodied in Y_t may occur in the fourth. Both observations are properly labeled with an index t referencing January, but the response is not instantaneous. Instead, we call it contemporaneous. By this, we mean that the response necessarily follows the change in the underlying cause, but both the cause and the response have the same time index.

With this understanding, we specify Q^t in Definition 1.1 instead of Q^t−1. If, however, the response does require a full time period, we can just view Q_t as being observed at t − 1.

An interesting aspect of G-causality that appears to have been overlooked is that the role of S^t is not inherently structural; conditioning on S^t ensures that its role is primarily predictive. Thus, different considerations apply to specifying its time index relative to Y_t and Q_t. We further explore this below. For now, however, we just refer to S^t, parallel to our reference to Q^t.

2 A Dynamic DGP and Structural Causality

We now specify the DGP as a recursive dynamic structure in which “predecessors” structurally determine “successors,” but not vice versa. More precisely, successors cannot determine predecessors, whereas predecessors may or may not determine successors. In particular, future variables cannot precede present or past variables. This enforces the causal direction of time. We write 𝒴⇐𝒳 to denote that 𝒴 succeeds 𝒳 (𝒳 precedes 𝒴 ).

We specify a version of the structure analyzed by WK. For this, we write ℕ ≡{0, 1, …}.

Such structures are well suited to representing the interactions of intertemporally optimizing agents or the evolution of markets or other segments of an economy. (See, e.g., Chow, 1997.)

Empirically feasible procedures must be expressible as functions of observables. Throughout, we suppose that realizations of D_t, W_t, Y_t, and Z_t are observed, whereas realizations of U_t are not. Because D_t, U_t, W_t, or Z_t may have dimension zero, their presence is optional. Usually, however, some or all will be present. Indeed, there may be a countable infinity of unobservables.

In some cases, the relevant W_t’s or Z_t’s may not be directly observable. Instead, they may be “asymptotically observable,” in that they can be consistently estimated from observables, as in Imbens and Newey (2009). Here, we treat these as observable, enhancing generality. Treating their estimation is a statistical issue beyond our scope here, however.

Consistent with our discussion of contemporaneous response above, we understand that although D_t, Z_t, and U_t appear as arguments of q_t, their values are realized prior to those of Y_t.

This structure is general: the structural relations may be nonlinear and nonmonotonic in their arguments and nonseparable between observables and unobservables. This system may generate stationary processes, nonstationary processes, or both.

The vector Y_t represents responses of interest. We will be interested in either (i) the effects on Y_t of D^t or (ii) with Y_t = (Y_{1, t}^′, Y_{2, t}^′)^′, the effects on Y_{1, t} of Y₂^t−1. The latter is the structural VAR case. In the former, D_t represents “exogenous” drivers or policy variables, analogous to treatments in the treatment effect literature (see White, 2006a), as recursivity ensures that lags of Y_t do not impact D_t. We call this the “time-series natural experiment” case. Depending on the case, we call either D_t or Y_{2, t−1} “causes of interest.” G-causality relates to both.

The histories U^t and Z^t contain drivers of Y_t whose effects are not of primary interest; we call U_t and Z_t “ancillary causes.” The history W^t may contain potential causes of D^t, as well as responses to U^t. Observe that W^t does not appear in the argument list for q_t, so W^t does not directly determine Y_t. Note also that W_t is not determined by either Y_t or D_t or by their lags. A useful convention is that W_t ⇐ (W^t−1, U^t, Z^t), so that W_t does not drive unobservables. If a structure does not have this property, then suitable substitutions can usually yield a derived structure satisfying this convention. Nevertheless, we do not require this, so W_t may also contain drivers of unobservable causes of Y_t and/or D_t.

A.1 supports a natural definition of direct structural causality, as White and Chalak (2009) discuss. For this, let d_s^t be the subvector of d^t with elements indexed by the nonempty set s ⊆{1, …, k_d}×{0, …, t}, and let d_(s)^t be the subvector of d^t with the elements of s excluded. Similarly, y_s^t−1 is the subvector of y^t−1 with elements indexed by the nonempty set s ⊆{1, …, k_y}×{0, …, t − 1}, and y_(s)^t−1 is the subvector of y^t−1 with elements of s excluded.

Because of its structural basis, we refer to this as “structural causality” to distinguish it from other notions. This is fully consistent with the causal notions of the Cowles Commission, and in particular with concepts explicitly articulated by Heckman (e.g., Heckman, 2008). See White and Chalak (2009) for further details. This is a “direct” notion of causality, as it does not account for indirect effects that can be propagated recursively. For brevity, we may refer to just “direct” or “structural” causality when the meaning is clear from the context.

We refer explicitly to Y_s^t−1 and D_s^t here because these are the main cases of interest. We can similarly define direct causality or noncausality of Z_s^t or U_s^t for Y_{j, t}, but we leave this implicit. We write, for example, D_s^tY_t when D_s^tY_{j, t} for some j ∈{1, …, k_y}.

For clarity, we just give a definition using D^t and Y_t, parallel to classical G-causality. This specifies a form of total causality, as it embodies both direct effects on Y_t of D^t and indirect effects, which propagate through Y^t−1 in Y_t = q_t(Y^t−1, D^t, Z^t, U^t). For brevity, we leave the “total” qualifier implicit. When it may not be clear from the context, we make it explicit.

Similar definitions for D_s^t ⇒_𝒮Y_t are implicit, as total causality is simply direct causality with respect to r_t. A similar notion for Y₀ is also implicit; this permits structural definition of impulse responses, but these are not a main focus here.

The relation between direct causality and (total) structural causality is fairly straightforward:

Part (ii) is a form of converse to (i). A literal converse to (i) does not hold because indirect effects can cancel direct effects, resulting in the absence of total effects. Such cases are special, so the converse of (i) can be thought heuristically to hold in most cases. Given some form of direct causality, the requirement of (ii) is plausible and rules out complete cancellation.

3 Granger Causality and Time-Series Natural Experiments

We now examine the relations between G-causality and structural causality. Several useful extensions of classical G-causality emerge naturally from this; we provide structural characterizations in each case. In this section, we analyze the natural experiment case. This yields concepts that then facilitate our treatment of structural VARs, the subject of the next section.

3.1 G-Causality, Conditional Exogeneity, and Direct Causality

White (2006a) shows how certain exogeneity restrictions permit identification of expected causal effects in dynamic structures. Although White (2006a) considers only binary-valued D_t, White and Chalak (2008) and WK show how analogous restrictions identify expected causal effects generally. Our first main result below shows that a specific form of exogeneity enables us to forge the missing link between direct causality and G-causality.

Let X_t ≡ (W_t, Z_t), t = 0, 1, … . We say D^t is “conditionally” exogenous when we have:

Strict exogeneity holds when conditioning is absent, motivating the term “conditional.”

Specifically, as explained in White (2006a), to help ensure A.2(a), covariates X_t should include variables whose lags (or leads, as we shall shortly see below) are useful for predicting either D_t or U_t, as the better a predictor X_t is, the less useful D_t will be as a predictor for U_t and vice versa. Specifically, X_t should include: (i) observable causes Z_t of Y_t; (ii) observable causes W_t and Z_t of D_t; and (iii) observable responses W_t to unobservable causes U_t of Y_t and/or V_t of D_t. The economics of the particular application typically suggest numerous such proxies.

We now give our first result linking structural causality and G-causality:

Thus, given conditional exogeneity of D^t, G-causality implies direct structural causality. With properly chosen S, any test that rejects G noncausality necessarily rejects direct structural noncausality (SN). A proper choice for S_t is X_t, where X_t contains

• variables that are not themselves structurally caused by either Y_t or D_t or their lags (A.1)

• variables that structurally cause D_t or Y_t or are observable proxies for unobserved structural causes of either D_t or Y_t (A.2(a))

Does G noncausality imply direct noncausality? Strictly speaking, the answer is no. This is essentially a sampling problem: even with direct causality, the realizations of Y^t−1, D^t, Z^t, and U^t that would otherwise reveal this may occur with zero probability. G noncausality then holds, despite the (undetectable) presence of direct causality. A simple example illustrates:

By construction, DY . But Y ⊥ D is the condition that D does not G-cause Y .

Nevertheless, it is reasonable to expect that G noncausality can reveal the absence of detectable structural causality, properly defined. The next definition permits us to formalize this.

This notion of direct noncausality a.s. (D^tY_t) formalizes the idea that even though D^tY_t, so that d^t → q_t(y^t−1, d^t, z^t, u^t) is not a constant function for some values of (y^t−1, z^t, u^t), the joint distribution of Y^t−1, D^t, U^t, X^t can be such that this nonconstancy is never reflected in the sample data, as in Example 3.2(a). A continuation of this example gives further insight:

As we show next, Y ⊥̸D implies²DY . Thus, the distribution of the drivers of Y_t can hide or reveal structural causality. Clearly, though, D^tY_t implies D^tY_t.

We now give a main result structurally characterizing G-causality.

3.2 Retrospective Weak G-Causality and Total Structural Causality

In considering the identification and estimation of the total effects of D^t, WK introduce the notion of retrospective expected effects, in which expected counterfactual responses are conditioned on all information available at the present, time T. Identification of such retrospective effects is ensured by the retrospective weak conditional exogeneity of D^t:

First, note that A.2(b) conditions on Y₀ instead of Y^t−1, appropriate to consideration of total rather than direct effects. We call this “weak” conditional exogeneity, as conditioning on Y₀ is less informative (weaker). Second, while A.2(a) conditions on X^t, A.2(b) also conditions on future covariates, relative to t. As WK discuss, this does not violate the causal order of time enforced by A.1. The “retrospective” conditioning on X^T has no particular causal content. Instead, conditioning variables are inherently predictive; future covariates are predictive in the back-casting sense. Including leads can be especially useful, as the additional information in X^T may enable A.2(b) to hold when conditioning on X^t is insufficient.

When D_t = b_t ( D^t−1, X^t, V^t) , it suffices for A.2(b) that U^t ⊥ (V^t, D₀)∣Y₀, X^T.

We now introduce some structurally informative extensions of classical G-causality:

The third case corresponds to A.2(b). We present the others for completeness, as weak G noncausality is equivalent to total SN a.s. given D^t ⊥ U^t | Y₀, X^t (weak conditional exogeneity), and retrospective G noncausality is equivalent to direct noncausality a.s. given D^t ⊥ U^t |  Y^t−1, X^T (retrospective conditional exogeneity). There are no necessary relations among any of these extensions nor between any of these and classical G-causality.

An analog of Definition 3.3 appropriate here is (total) SN a.s.:

Analogous definitions hold for the weak and retrospective cases.

The structural characterization of retrospective weak G-causality is:

Thus, given retrospective weak conditional exogeneity of D^t, retrospective weak G noncausality implies (total) SN a.s. and vice versa. Again, the covariates must be properly chosen. But now the covariates can explicitly include leads of X_t. Because including leads can enable A.2(b) to hold when D^t ⊥ U^t | Y₀, X^t fails, more reliable inference becomes possible.

Specifically, suppose that in fact D^t does not structurally cause Y_t and suppose further that A.2(b) holds with suitably chosen leads of X_t, whereas, unknown to the researcher, conditional exogeneity fails when the necessary leads are omitted. Because conditional exogeneity is a maintained assumption in Theorem 3.7, tests using only X_t and its lags will tend to reject G noncausality, due not to structural causality, but to the failure of conditional exogeneity. Because this failure is unsuspected, the researcher will infer structural causality from the rejection of G noncausality, precisely the wrong inference. On the other hand, when A.2(b) holds with suitable leads of X_t, the test for retrospective G-causality will deliver the correct inference. In the current example, tests will tend not to reject G noncausality, consistent with both the absence of structural causality and the presence of retrospective conditional exogeneity.

A substantive implication of both Theorems 3.4 and 3.7 is that past inferences about structural causality based either implicitly or explicitly on tests rejecting G noncausality will require careful reexamination: it may be the failure of conditional exogeneity rather than structural causality that has been detected. Put another way, these results make clear why drawing structural or policy conclusions on the basis of tests for G-causality is potentially tenuous: proper consideration must be given to the economic structure and to the choice of covariates, sufficient to ensure the plausibility of the relevant conditional exogeneity assumption.

3.3 Finite-order G-Causality and Markov Structures

The classical notion of G-causality involves the histories Q^t, Y^t−1, and S^t. The weak and retrospective versions involve Q^t and S^t or S^T. Testing such hypotheses is challenging in time-series practice because there is only one data history {Y_t, Q_t, S_t}_t=0^T. Although de Jong (1996) and Escanciano and Velasco (2006) have proposed methods for testing certain mean independence hypotheses involving infinite histories of data, similar methods for testing the conditional independence represented by G noncausality are, to the best of our knowledge, as yet unavailable.

Given this challenge, researchers typically test G-causality by regressing Y_t on a finite number of its lags and on Q_t and S_t and a finite number of their lags. Of course, this does not give a test for classical G-causality but for a related property, finite-order G-causality. By itself, this is neither necessary nor sufficient for classical G-causality or any of its variations.

To define this, we define the finite histories Y_t−1 ≡ (Y_t−ℓ, …, Y_t−1) and Q_t ≡ (Q_t−k, …, Q_t) .

We call max(k, ℓ − 1) the “order” of the finite-order G noncausality.

Here, we understand that S_t may represent a finite history of covariates. These may include leads or lags with respect to time t. Leads correspond to the retrospective case.

The more important issue here, however, is whether finite-order G-causality has useful implications for structural causality. As we now show, by mildly restricting the DGP and by correspondingly modifying the conditional exogeneity requirement, we can provide a structural characterization of finite-order G-causality. This then leads to practical tests.

Specifically, we now restrict attention to certain Markov (i.e., finite-order) structures. To specify these, we write D_t ≡ (D_t−k, …, D_t) and Z_t ≡ (Z_t−m, …, Z_t).

Because of their relative feasibility for estimation, structures of this sort are those almost always used in applications. Note that we write U_t as the argument of q_t in B.1 for simplicity; the potentially countably infinite dimension of U_t ensures that the absence of lags of U_t in q_t does not necessarily result in any loss of generality.

Corresponding to this structure is the following finite-order conditional exogeneity requirement. For this, we write X_t ≡ (X_t−τ1, …, X_t+τ2).

When τ₂> 0, covariate leads are present, corresponding to the retrospective case.

Finite-order conditional exogeneity identifies the expected direct effects of D_t on Y_t. A notion of direct noncausality a.s. relevant here (now abbreviated for conciseness) is:

Note that direct SN implies D_tY_t.

The structural characterization of finite-order G-causality is:

This justifies practical tests of SN, based on tests for finite-order G-causality: Given conditional exogeneity of D_t, finite-order G noncausality implies direct noncausality a.s. and vice versa. As always, covariates must be properly chosen. Here, τ₁ should be sufficiently large that X_t includes all observable direct ancillary causes of Y_t.

Because of the relations between direct and total causality, it follows that if one rejects direct SN a.s., one must also necessarily reject direct SN and (total) SN (exact cancellations excepted).

4 Granger Causality and Structural VARs

Many studies have applied G-causality to attempt to gain such structural insights. Without proper care, such exercises are questionable at best. Nevertheless, just as before, we can ensure that suitable versions of G-causality support legitimate structural inferences.

As before, testing whether the full history Y₂^t−1G-causes Y_{1, t} using a single time-series sample is a major challenge. Nevertheless, tests for finite-order G-causality are generally feasible. Further, as we now show, finite-order G-causality can be given a structural characterization under suitable conditions, parallel to those for the previous cases.

For the structural VAR case, we consider Markov structures analogous to those of B.1. As above, we let Y_t−1 ≡ (Y_t−ℓ, …, Y_t−1) and Z_t ≡ (Z_t−m, …, Z_t). Alternatively, interpreting Y_t−1 as Y^t−1 and Z_t as Z^t will give results for classical or retrospective G-causality.

Matching this structure is a finite-order conditional exogeneity requirement. For this, we write Y_{1, t−1} ≡ (Y_{1, t−ℓ}, …, Y_{1, t−1}), Y_{2, t−1} ≡ (Y_{2, t−ℓ}, …, Y_{2, t−1}); as before, ⁠.

We state sufficient conditions for C.2 formally, as their verification is a bit more involved than for conditions ensuring A.2 or B.2.

Imposing U^t−1 ⊥ U_{1, t}∣Y₀, Z^t−1, X_t is the analog of requiring that serial correlation is absent when lagged dependent variables are present. Imposing Y_{2, t−1} ⊥ Y₀, Z^{t−τ₁−1}∣Y_{1, t−1}, X_t ensures that ignoring Y₀ and omitting distant lags of Z_t from X_t doesn’t matter.

Assumption C.2 ensures that expected direct effects of Y_{2, t−1} on Y_{1, t} are identified. The relevant notion of direct noncausality a.s. here is:

The structural characterization of finite-order G-causality for structural VARs is:

5 Testing Finite-Order G-Causality

To test finite-order G-causality, we require tests for conditional independence. Nonparametric tests for conditional independence consistent against arbitrary alternatives are readily available (e.g., Linton and Gozalo, 1997; Fernandes and Flores, 2001; Delgado and Gonzalez-Manteiga, 2001; Su and White, 2007a, 2007b, 2008; Huang and White, 2009). In principle, one can apply any of these to consistently test for finite-order G-causality.

Despite their appealing theoretical power properties, nonparametric tests are often not practical, due to the typically modest number of time-series observations available relative to the number of relevant observable variables. On the other hand, parametric methods for testing conditional independence are straightforward; their main potential drawback is that they may not have power against certain alternatives. Here, we propose methods designed to balance these competing concerns. For practicality, we propose parametric tests that resemble as closely as possible standard methods for testing finite-order G noncausality (e.g., Stock and Watson, 2007, p. 547). To ensure power against a broader range of alternatives, our new tests exploit the “QuickNet” procedures introduced by White (2006b).

For simplicity and concreteness, we take Y_t to be a scalar and focus on tests of zero-order G noncausality, that is, Y_t ⊥ Q_t | Y_t−1, S_t. Then Q_t = D_t and S_t = X_t when D_t is the cause of interest. For the VAR case, Y_t = Y_{1, t}, Q_t = Y_{2, t−1}, and S_t = X_t. The analogous procedures for the general finite-order case and with Y_t a vector will be obvious.

5.1 Testing Conditional Mean Independence with Linear Regression

Observe that with conditional exogeneity of Q_t (D_t or Y_{2, t−1} ), when CI Test Regression 1 is correctly specified, β₀ represents the direct structural effect of Q_t on Y_t. The remaining coefficients have no necessary structural interpretation.

5.2 Testing Conditional Mean Independence with A More Flexible Regression

5.3 Testing Conditional Independence Using Nonlinear Transformations

6 Conditional Exogeneity

In each result relating structural causality to G-causality, a corresponding conditional exogeneity assumption is maintained. In this section, we address two related questions. First, can we dispense with conditional exogeneity? As it turns out, the answer is negative. Given this, our second question is whether we can test conditional exogeneity. We provide straightforward tests valid under assumptions common in practice.

As practical tests for G-causality are for the finite-order case, we focus solely on this case. To balance the attention given the natural experiment case in Section 4, our focus in this section is on the structural VAR case. Analogous results hold in both cases.

6.1 The Crucial Role of Conditional Exogeneity

As we now show, there is a precise sense in which conditional exogeneity plays a crucial role in the relation between structural and G-causality.

This shows that conditional exogeneity is crucial in that if it does not hold, then there are always structures that generate data exhibiting finite-order G-causality, even in the absence of direct structural causality. Because q_{1, t} is unknown, such worst case scenarios can never be discounted.

In fact, as the proof shows, the class of worst case structures includes precisely those usually assumed in applications, namely separable structures (e.g., Y_{1, t} = q_{1, t}(Y_{1, t−1}, Z_t) + U_{1, t} ), as well as more general classes of invertible structures analogous to those considered in the static case by, for example, Altonji and Matzkin (2005). Thus, in the cases typically assumed in the literature, the failure of conditional exogeneity guarantees G-causality in the absence of structural causality. We state this formally as a corollary.

Together with Theorem 4.3, this establishes that, in the absence of direct causality and for the class of invertible structures that dominate in applications, finite-order conditional exogeneity is necessary and sufficient for finite-order G noncausality. The same holds for the natural experiment case with B.1 and B.2 in place of C.1 and C.2.

6.2 Separability and Finite-order Conditional Exogeneity

Testing finite-order conditional exogeneity is hampered by the unobservability of U_{1, t}. Despite this, for the separable cases common in applications, we can readily construct feasible tests. We base these on the consequences of conditional exogeneity developed next.

Because μ_t is generally not identified without further conditions, recovering U_{1, t} is generally not possible. Nevertheless, recovering and estimating ϵ_t are typically straightforward. Specifically, one can estimate E(Y_{1, t}|Y_t−1, X_t), either parametrically or nonparametrically. Let denote the resulting estimator, and define the estimated residuals A test of Y_{2, t−1} ⊥ U_{1, t}∣Y_{1, t−1}, X_t can then be performed by testing the implication Y_{2, t−1} ⊥ ϵ_t∣Y_{1, t−1}, X_t, using observations on ⁠, and X_t. We provide details in the next section.

Tests of finite-order conditional exogeneity for the general separable case follow from:

Tests based on this result can detect the failure of C.2 or nonseparability. For now, we treat separability as a maintained assumption. We relax this below. Note, however, that under the null of direct noncausality, q_{1, t} is necessarily separable, as then ζ_t is the zero function.

Given an estimator of ζ_t (see, e.g., Linton and Nielsen (1995) for a marginal integration method or Newey, Powell, and Vella (1999) for a series estimation method), one can then form⁴ and test Y_t−1 ⊥ U_{1, t}∣X_t using observations on ⁠, Y_t−1, and X_t.

A result for the general finite-order case is the following:

Significantly, however, the recovery of U_{1, t} relies on C.2. Simple examples of the failure of C.2 (e.g., Example 6.A of the Appendix) show that, in the absence of further information, such failures can be undetectable, blocking recovery of U_{1, t}. Thus, even in the favorable separable case, there is no feasible way to construct tests of conditional exogeneity consistent against arbitrary alternatives without adding other identifying assumptions to the maintained hypotheses.

Further, although this result does recover U_{1, t}, observe that ϵ_t = U_{1, t} − μ_t(X_t) implies Y_t−1 ⊥ ϵ_t∣X_t if and only if Y_t−1 ⊥ U_{1, t}∣X_t. Thus, one can equally well construct tests of C.2 using ⁠, which may require less computation. It is not clear that working with offers any advantages, especially when one considers the additional assumptions involved.

6.3 Testing Conditional Exogeneity

The results of the previous subsection motivate tests of conditional exogeneity based on either or ⁠, If we could use ϵ_t or U_{1, t} instead of or ⁠, we could apply any of CI Test Regressions 1-3 with the same regressors, but with ϵ_t or U_{1, t} as the dependent variable. We would then test β₀ = 0 just as before, but now we would have a test of conditional exogeneity.

The imposed memory condition is often plausible, as it says that the more distant history (Y^t−ℓ−1, X^{t−τ₁−1}) is not predictive for U_{1, t}, given the more recent history 𝒳_t of (Y^t−1, X^t+τ₂). Note that separability is not needed here.

The details of C₀ can be involved, especially with choices like ⁠. But since this is a standard m-estimation setting, we can avoid explicit estimation of C₀ : the bootstrap delivers asymptotically valid critical values, even without the martingale difference property (see, e.g., Gonçalves and White, 2004; Kiefer and Vogelsang, 2002, 2005; Politis, 2009).

Of these two methods, the first may be preferable, as although it requires more operations to compute (O(nlnn) vs. O(n)), the second requires some additional regularity conditions for its validity, and some care may be required in constructing ⁠, as Gonçalves and White (2005) note in a related context⁷.

Because the given asymptotic distribution is joint for and ⁠, the same methods apply to testing G noncausality, that is, 𝕊₁θ₀ = 0, where 𝕊₁ selects β_{0, 1} from θ₀.

Of course, more sophisticated bootstrap procedures with potentially better finite sample properties are readily available. We limit our discussion here to these straightforward methods to concisely make clear the key ideas and to provide some easy-to-implement methods.

7 An Indirect Test for Structural Causality

Theorems 3.10 and 4.3 imply that if we test and reject finite-order G noncausality, then we must reject either direct SN or finite-order conditional exogeneity (or both). In what follows, we drop explicit references to the “finite-order” or “direct” qualifiers for brevity, but these will be understood. If conditional exogeneity is a maintained assumption, then we can directly test SN by testing G-causality; otherwise, a direct test is not available.

Similarly, maintaining the usual separability assumption, Corollary 6.2 and its natural experiment analog imply that if we test and reject conditional exogeneity, then we must reject either SN or G noncausality (or both). If G noncausality is maintained, then we can directly test SN by testing conditional exogeneity; otherwise, a direct test is not available.

When neither conditional exogeneity nor G noncausality is maintained, no direct test of SN is possible. Nevertheless, it is possible to test structural causality indirectly when the results of the G-causality and conditional exogeneity tests can be combined to isolate the source of any rejections. We propose the following indirect test for structural causality:

1. 1.

   Reject SN if either:

   1. (i)

      the conditional exogeneity test fails to reject and the G noncausality test rejects; or

   2. (ii)

      the conditional exogeneity test rejects and the G noncausality test fails to reject.

If these rejection conditions do not hold, however, we will not just decide to “accept” (i.e., fail to reject) SN. To see why, let CE denote conditional exogeneity and GN G noncausality. When the above rejection conditions fail, there are two possibilities. The first is that CE and GN both hold; in this case, the decision not to reject SN is appropriate. But when CE and GN both fail, difficulties arise. We could well have structural causality together with the failure of CE, both contributing to the failure of GN. Failing to reject SN here thus runs the risk of Type II error. On the other hand, rejecting SN runs the risk of Type I error because Corollary 6.2 ensures that, when SN holds, the failure of CE alone implies the failure of GN.

We resolve this dilemma by specifying the further rules:

1. 2.

   Fail to reject SN if the CE and GN tests both fail to reject;

2. 3.

   Make no decision as to SN if the CE and GN tests both reject.

In the latter case, we conclude only that CE and GN both fail, thereby obstructing structural inference. This sends a clear signal that the researcher needs to revisit the model specification, with particular attention to specifying covariates sufficient to ensure conditional exogeneity.

In the former case, the empirical evidence accords not with the absence of direct causality per se, but the absence of detectable direct causality, that is, direct noncausality a.s. Nevertheless, given separability, more can be said about the possible causal structures. In particular, one can verify that given separability as in Proposition 6.3, Y_{2, t−1}Y_{1, t}if and only if ⁠, say, almost surely.

For simplicity and clarity, we motivate these rules by taking the null hypotheses CE and GN at face value. In practice, the DGP can have properties other than those the null explicitly specifies that are impossible to detect with a given test (or even any test). Example 6.A is a case in point. Such DGPs are part of the “implicit” null of the test. Even tests that theoretically have power against every alternative primarily have power concentrated in certain directions (Escanciano, 2009); the other alternatives are nearly members of the implicit null. For DGPs in or near the implicit null, the test’s power equals or is near its level. We therefore caution that when interpreting the results of the indirect test proposed here (as for any test), the researcher should bear in mind the power consequences, should a given DGP belong not to the explicit null but to the vicinity of the implicit null of the implemented GN or CE test.

Our next result provides useful bounds on these probabilities.

Essentially, the probability of wrongly making a decision decreases with the powers of the CE and GN tests, whereas the probability of wrongly making no decision decreases with the levels. The restricted level α^* decreases with the levels of the CE and GN tests, whereas the restricted power π^* increases with the powers of the underlying tests. This accords well with intuition.

In applications, tests are usually conducted using asymptotic critical values. The exact level and power of a given test are then unknown but typically converge to known limits. Proposition 7.1 implies asymptotic properties (T →∞) for the sample-size T values of the probabilities defined above, p_T, q_T, α_T^*, and π_T^*. When the CE and GN tests are consistent, we get especially straightforward results.

When π_1T → 1 and π_2T → 1, one can also typically ensure α₁ = 0 and α₂ = 0 by suitable choice of an increasing sequence of critical values. In this case, q_T → 0, α_T^*→ 0, and π_T^*→ 1. Because GN and CE tests will not be consistent against every possible alternative, weaker asymptotic bounds on the level and power of the indirect test hold for these cases by Proposition 7.1. Thus, whenever possible, one should carefully design GN and CE tests to have power against particularly important or plausible alternatives.

For simplicity, we treated separability as a maintained assumption in our CE test. As implemented, our GN tests also maintain separability. But what if separability fails? Interestingly, our inference rules are robust to this, as a case-by-case analysis in the Appendix shows: following these inference rules without maintaining separability yields the same inferences about direct causality. As we show, our rules sequester the nonseparable case within the “no decision” action. Thus, when no decision is taken, the researcher should also carefully consider the possibility of nonseparability. This motivates development of tests of structural separability (see Lu, 2009), as well as tests of GN and CE that do not maintain separability. Given the significant challenges involved, these developments constitute topics properly left for research elsewhere.

8 Illustrative Applications

8.1 Crude Oil and Gasoline Prices

WK apply their methods for estimating retrospective causal effects to estimate the expected total effects of crude oil prices on gasoline prices. Here, we address a related but more fundamental question by testing whether crude oil prices directly cause gasoline prices. We let Y_t be the (natural) logarithm of the spot price for US Gulf Coast conventional gasoline per gallon; our cause of interest, D_t, is the logarithm of the Cushing OK WTI spot crude oil price per barrel. U_t represents all other gasoline price drivers, as in WK, so Z_t has dimension zero.

Our sample covers from January 1987 through December 1997, 132 observations. Over this interval, both crude and gasoline prices were relatively stable. Specifically, augmented Dickey-Fuller tests for Y_t and D_t reject the unit root null hypothesis. Nevertheless, we cannot reject the unit root null for certain of the covariates (the 10 year Treasury note rate, the 3 month T-Bill rate, the logarithm of the electricity price index, and the index of the foreign exchange value of the dollar). We enter these in first differences.

Thus, apart from addressing a different question and testing rather than assuming CE, there are several other differences between our use of these data and that of WK: (i) the sample period is different; (ii) we consider a stationary process with contemporaneous causation, whereas WK’s empirical analysis involves a cointegrated relation with a lag; and (iii) we use the Federal Reserve’s Index of the Foreign Exchange Value of the Dollar instead of the Yen-US dollar and British pound-US dollar exchange rates to avoid multicollinearity.

To test SN, we apply the procedure of Section 8. Specifically, we test finite-order retrospective G noncausality by testing Y_t ⊥ D_t∣Y_t−1, X_t, and we test CE analogously. For GN, we perform CI Test Regressions 1-3 with various combinations of τ₁, τ₂, and r, taking ψ, ψ_y1, and ψ_q to be ridgelet functions and with ψ_y2 the identity. For CE, we perform CI Test Regression 3, and let ψ_y1 and ψ_q be the absolute value, square, and ridgelet functions.

We soundly reject GN with p-value = 0.000 for all τ₁ = τ₂ = 0, 1, 2, 3 and r = 0, …, 5. We do not provide a table, as the entries would all be zero. On the other hand, we fail to reject CE for almost all τ₁ = τ₂ = 0, 1, 2, 3 and r = 0, …, 5. The results are reported in Table 1.

For comparison, we also tested finite-order nonretrospective GN and CE(τ₂ = 0). Theresults are very similar to the retrospective case, so we do not tabulate them.

Applying the inference rules of Section 8, we therefore soundly reject direct SN of oil prices for gasoline prices. Although it would be surprising if we found otherwise, these results have further substantive implications that must not be overlooked: First, the fact that we test and do not reject CE suggests that we have proper covariates and can therefore draw valid structural inferences, not just predictive inferences. Second, as the discussion of Section 8 implies, our results are consistent with a separable relation between oil and gasoline prices, a new and interesting structural finding.

8.2 Monetary Policy and Industrial Production

Angrist and Kuersteiner (2004) study the causal relationship between the Federal Reserve’smonetary policy and industrial output using Romer and Romer’s (1989) data. Romer andRomer (1989, 1994) construct a monetary policy shock variable using a narrative approach.They examine the Federal Open Market Committee minutes to identify dates when theFed took a marked anti-inflationary stance. There are six such periods between 1948 and 1991 (Romer and Romer, 1994), the “Romer dates.” The total number of observations is528.

The null is that monetary policy (Y_{2, t−1})has no causal effect on output growth (Y_{1, t}).The key retrospective CE assumption is Y_{2, t−1} ⊥ U_{1, t}∣Y_{1, t−1}, X_t.This says that given Y_{1, t−1}and X_t,Y_{2, t−1} cannothelp predict U_{1, t},and vice versa. That is, given past output growth and past and future unemployment andinflation rates, the Fed’s policy is as good as randomly assigned. Fed decisions usually targetfuture inflation and/or unemployment rates. To the extent that Fed expectations are drivenby past and present values of unemployment and inflation, current and lagged values ofX_t should predictY_{2, t−1} well, leaving little rolefor U_{1, t} in predictingY_{2, t−1}. Moreover, if futurevalues of X_t are driven byU_{1, t}, then these may beuseful in back-casting U_{1, t},leaving little role for Y_{2, t−1} inpredicting U_{1, t}. Both of thesefeatures help to ensure Y_{2, t−1} ⊥ U_{1, t}∣Y_{1, t−1}, X_t.Of course, we will not take this on faith but perform a CE test.

An important criticism of Romer and Romer’s (1989) approach is that it neglects theforward-looking aspects of monetary policy (e.g., Shapiro, 1994; Leeper, 1997). By conditioning oncovariate leads as well as lags, we exploit rather than neglect the forward-looking aspects of monetarypolicy. This makes use of the retrospective approach especially appealing.

We apply the procedures of Section 8, as for the previous example. For GN, we testY_{2, t−1} ⊥ Y_{1, t}∣Y_{1, t−1}, X_t using CI Test Regressions 1-3with various combinations of τ₁, τ₂, andr, takingψ,ψ_y1, andψ_q to be ridgeletfunctions and with ψ_y2the identity. For CE, we perform CI Test Regression 3, and letψ_y1 andψ_q be theabsolute value, square, and ridgelet functions.

In Table 2-1, we see that GN is rejected for allτ₁ = τ₂ = τ = 0, …, 8 andr = 0, …, 5. ForCE (Table 2-2), the overall Bonferroni-Hochberg (BH) adjusted p-value is borderline significant(p ≈ 0. 05) forτ = 0; in thiscase, we would make no decision. Nevertheless, the overall BH p-values exceed 0.05 forτ = 1, …, 8,illustrating the importance of the lead and lag covariates.

For comparison, we also conduct nonretrospective GN and CE tests(τ₂ = 0);see Tables 2-3 and 2-4. Again, the GN tests reject. Now, however, we seep < 0. 10 forτ = 0,3, and 8and p < 0. 05 forτ = 5 and6,demonstrating the importance of covariate leads to achieving CE.