The Effect of a Treatment History

For an individual country, the causal effect of a particular history of welfare spending, x _{1: t}, relative to some other history of spending, x′_{1: t}, is the difference Y_it(x _{1: t}) − Y_it(x′_{1: t}). That is, it is the difference in the potential or counterfactual level of terrorism when the country follows history x _{1: t} minus the counterfactual outcome when it follows history x′_{1: t}. Given the number of possible treatment histories, there can be numerous causal effects to investigate, even with a simple binary treatment. As the length of time under study grows, so does the number of possible comparisons. In fact, with a binary treatment, there are 2^t different potential outcomes for the outcome in period t. This large number of potential outcomes allows for a very large number of comparisons and a host of causal questions: Does the stability of spending over time matter for the impact on the incidence of terrorism? Is there a cumulative impact of welfare spending or is it only the current level that matters?

These individual-level causal effects are difficult to identify without strong assumptions, so we often focus on estimating the average causal effect of a treatment history (Robins, Greenland, and Hu Reference Robins, Greenland and Hu1999; Hernán, Brumback, and Robins Reference Hernán, Brumback and Robins2001):

(1) $$\begin{equation} \tau (x_{1:t}, x^{\prime }_{1:t}) = E[Y_{it}(x_{1:t}) - Y_{it}(x^{\prime }_{1:t})]. \end{equation}$$

Here, the expectations are over the units so that this quantity is the average difference in outcomes between the world where all units had history x _{1: t} and the world where all units had history x′_{1: t}. For example, we might be interested in the effect of a country having always high welfare spending versus a country always having low spending levels. Thus, this quantity considers the effect of treatment at time t, but also the effect of all lagged values of the treatment as well. A graphical depiction of the pathways contained in τ(x _{1: t}, x′_{1: t}) is presented in Figure 1, where the dotted arrows correspond to components of the effect. These arrows represent all of the effects of X _it, X _{i, t − 1}, X _{i, t − 2}, and so on, that end up at Y _it. Note that many of these effects flow through the time-varying covariates, Z _it. This point complicates the estimation of causal effects in this setting and we return to it below.

FIGURE 1.

Directed acyclic graph (DAG) of typical TSCS data. Dotted lines are the causal pathways that constitute the average causal effect of a treatment history at time t.

Marginal Effects of Recent Treatments

As mentioned above, there are numerous possible treatment histories to compare when estimating causal effects. This can be daunting for applied researchers who may only be interested in the effects of the first few lags of welfare spending. Furthermore, any particular treatment history may not be well-represented in the data if the number of time periods is moderate. To avoid these problems, we introduce causal quantities that focus on recent values of treatment and average over more distant lags. We define the potential outcomes just intervening on treatment the last j periods as Y _it(x _{t − j:t}) = Y _it(X _{i, 1:t − j − 1}, x _{t − j:t}). This “marginal” potential outcome represents the potential or counterfactual level of terrorism in country i if we let welfare spending run its natural course up to t − j − 1 and just set the last j lags of spending to x _{t − j:t}.Footnote ³

With this definition in hand, we can define one important quantity of interest, the contemporaneous effect of treatment (CET) of X _it on Y _it:

$$\begin{eqnarray*} \tau _{c}(t) &=& E[Y_{it}(X_{i,1:\,t-1}, 1) - Y_{it}(X_{i,1:\,t-1},0)] , \nonumber \\ &=& E[Y_{it}(1) - Y_{it}(0)], \end{eqnarray*}$$

Here we have switched from potential outcomes that depend on the entire history to potential outcomes that only depend on treatment in time t. The CET reflects the effect of treatment in period t on the outcome in period t, averaging across all of the treatment histories up to period t. Thus, it would be the expected effect of switching a random country from low levels of welfare spending to high levels in period t. A graphical depiction of a CET is presented in Figure 2, where the dotted arrow corresponds to component of the effect. It is common in pooled TSCS analyses to assume that this effect is constant over time so that τ_c(t) = τ_c.

FIGURE 2.

DAG of a TSCS setting where the dotted line represents the contemporaneous effect of treatment at time t.

Researchers are also often interested in how more distant changes to treatment affect the outcome. Thus, we define the lagged effect of treatment, which is the marginal effect of treatment in time t − 1 on the outcome in time t, holding treatment at time t fixed: E[Y _it(1, 0) − Y _it(0, 0)]. More generally, the j-step lagged effect is defined as follows:

(2) $$\begin{eqnarray} \tau _{l}(t, j) &=& E[Y_{it}(X_{i,1:\,t-j-1}, 1, 0_{j}) - Y_{it}(X_{i,1:\,t-j-1}, 0,0_{j})] , \nonumber \\ &=& E[Y_{it}(1,0_{j}) - Y_{it}(0_{j+1})], \end{eqnarray}$$

where 0_s is a vector of s zero values. For example, the two-step lagged effect would be E[Y _it(1, 0, 0) − Y _it(0, 0, 0)] and represents the effect of welfare spending two years ago on terrorism today holding the intervening welfare spending fixed at low levels. A graphical depiction of the one-step lagged effect is presented in Figure 3, where again the dotted arrows correspond to components of the effect. These effects are similar to a common quantity of interest in both time-series and TSCS applications called the impulse response (Box, Jenkins, and Reinsel Reference Box, Jenkins and Reinsel2013).

FIGURE 3.

DAG of a panel setting where the dotted lines represent the paths that constitute the lagged effect of treatment at time t − 1 on the outcome at time t.

Another common quantity of interest in the TSCS literature is the step response, which is the cumulative effect of a permanent shift in treatment status on some future outcome (Box, Jenkins, and Reinsel Reference Box, Jenkins and Reinsel2013; Beck and Katz Reference Beck and Katz2011). The step response function, or SRF, describes how this effect varies by time period and distance between the shift and the outcome:

(3) $$\begin{equation} \tau _s(t,j) = E[Y_{it}(1_{j}) - Y_{it}(0_{j})], \end{equation}$$

where 1_s has a similar definition to 0_s. Thus, τ_s(t, j) is the effect of j periods of treatment starting at time t − j on the outcome at time t. Without further assumptions, there are separate lagged effects and step responses for each pair of periods. As we discuss next, traditional modeling of TSCS data imposes restrictions on the data-generating processes, in part, to summarize this large number of effects with a few parameters.

Relationship to Traditional TSCS Models

The potential outcomes and causal effects defined above are completely nonparametric in the sense that they impose no restrictions on the distribution of Y _it. To situate these quantities in the TSCS literature, it is helpful to see how they are parameterized in a particular TSCS model. One general model that encompasses many different possible specifications is an ADL model:Footnote ⁴

(4) $$\begin{equation} Y_{it} = \beta _0 + \alpha Y_{i,t-1} + \beta _1 X_{it} + \beta _2X_{i,t-1} + \varepsilon _{it}, \end{equation}$$

where ε_it are independent and identically distributed errors, independent of X _is for all t and s. The key features of such a model are the presence of lagged independent and dependent variables and the exogeneity of the independent variables. This model for the outcome would imply the following form for the potential outcomes:

(5) $$\begin{equation} Y_{it}(x_{1:\,t}) = \beta _0 + \alpha Y_{i,t-1}(x_{1:\,t-1}) + \beta _1x_{t} + \beta _2x_{t-1} + \varepsilon _{it}. \end{equation}$$

In this form, it is clear to see what TSCS scholars have long pointed out: causal effects are complicated with lagged dependent variables (LDVs) since a change in x _{t − 1} can have both a direct effect on Y _it and an indirect effect through Y _{i, t − 1}. This is why even seemingly simple TSCS models such as the ADL imply quite complicated expressions for long-run effects.

The ADL model also has implications for the various causal quantities, both short term and long term. The coefficient on the contemporaneous treatment, β₁, is constant over time and does not depend on past values of the treatment, so it is equal to the CET, τ_c(t) = β₁. One can derive the lagged effects from different combinations of α, β₁, and β₂:

(6) $$\begin{equation} \tau _l(t,0) & = & \beta _1, \end{equation}$$\\

(7) $$\begin{equation} \tau _l(t,1) & = & \alpha \beta _1 + \beta _2, \end{equation}$$\\

(8) $$\begin{equation} \tau _l(t,2) & = & \alpha ^2\beta _1 + \alpha \beta _2. \end{equation}$$

(9) $$\begin{equation} \tau _s(t,0) & = & \beta _1, \end{equation}$$ \\

(10) $$\begin{equation} \tau _s(t,1) & = & \beta _1 + \alpha \beta _1 + \beta _2, \end{equation}$$\\

(11) $$\begin{equation} \tau _s(t,2) & = & \beta _1 + \alpha \beta _1 + \beta _2 + \alpha ^2\beta _1 + \alpha \beta _2. \end{equation}$$

Baseline Randomized Treatments

A powerful, if rare, research design for TSCS data is one that randomly assigns the entire history of treatment, X _{1: T}, at time t = 0. Under this assumption, treatment at time t cannot be affected by, say, previous values of the outcome or time-varying covariates. In terms of potential outcomes, the baseline randomized treatment history assumption is

(12) $$\begin{equation} \lbrace Y_{it}(x_{1:\,t}): t=1,\ldots ,T\rbrace \perp \!\!\!\perp X_{i, 1:\,t} | Z_{i0}, \end{equation}$$

where A⊥⊥B|C is defined as “A is independent of B conditional on C.” This assumes that the entire history of welfare spending is independent of all potential levels of terrorism, possibly conditional on baseline (that is, time-invariant) covariates. Hernán, Brumback, and Robins (Reference Hernán, Brumback and Robins2001) called X _{i, 1: t} causally exogenous under this assumption. The lack of time-varying covariates or past values of Y _it on the right-hand side of the conditioning bar in Equation (12) implies that these variables do not confound the relationship between the treatment and the outcome. For example, this assumes there are no time-varying covariates that affect both welfare spending and the number of terrorist incidents. Thus, baseline randomization relies on strong assumptions that are rarely satisfied outside of randomized experiments and is unsuitable for most observational TSCS studies.Footnote ⁵

Baseline randomization is closely related to exogeneity assumptions in linear TSCS models. For example, suppose we had the following distributed lag model with no autoregressive component:

(13) $$\begin{equation} Y_{it} = \beta _0 + \beta _1X_{it} + \beta _2X_{i,t-1} + \eta _{it}. \end{equation}$$

Here, baseline randomization of the treatment history, combined with the assumptions implicit in linear TSCS models, implies the usual identifying assumption in these models, strict exogeneity of the errors:

(14) $$\begin{equation} E[\eta _{it}|X_{i,1:\,T}]= E[\eta _{it}] = 0. \end{equation}$$

This is a mean independence assumption about the relationship between the errors, η_it, and the treatment history, X _{i, 1: T}.

Sequentially Randomized Treatments

Beginning with Robins (Reference Robins1986), scholars in epidemiology have expanded the potential outcomes framework to handle weaker identifying assumptions than baseline randomization. These innovations centered on sequentially randomized experiments, where at each period, X _it was randomized conditional on the past values of the treatment and time-varying covariates (including past values of the outcome). Under this sequential ignorability assumption, the treatment is randomly assigned not at the beginning of the process, but at each point in time and can be affected by the past values of the covariates and the outcome.

At its core, sequential ignorability assumes there is some function or subset of the observed history up to time t, V _it = g(X _{i, 1:t − 1}, Y _{i, 1:t − 1}, Z _{i, 1: t}), that is sufficient to satisfy no unmeasured confounders for the effect of X _it on future outcomes. Formally, the assumption states that, conditional on this set of variables, V _it, the treatment at time t is independent of the potential outcomes at time t:

Assumption 1 (Sequential Ignorability). For every treatment history x_{1: T} and period t,

(15) $$\begin{equation} \lbrace Y_{is}(x_{1:s}): s = t,, \ldots , T\rbrace \perp \!\!\!\perp X_{it} | V_{it}. \end{equation}$$

For example, a researcher might assume that sequential ignorability for current welfare spending holds conditional on lagged levels of terrorism, lagged welfare spending, and some contemporaneous covariates, so that V _it = {Y _{i, t − 1}, X _{i, t − 1}, Z _it}. Unlike baseline randomization and strict exogeneity, it allows for observed time-varying covariates like conflict status and lagged values of terrorism to confound the relationship between welfare spending and current terrorism levels, so long as we have measures of these confounders. Furthermore, these time-varying covariates can be affected by past values of welfare spending.

In the context of traditional linear TSCS models such as Equation (4), with their implicit assumptions, sequential ignorability implies the sequential exogeneity assumption:

(16) $$\begin{equation} E[\varepsilon _{it}|X_{i,1:\,t}, Z_{i,1:\,t}, Y_{i,1:\,t-1}] = E[\varepsilon _{it}|X_{it}, V_{it}] = 0. \end{equation}$$

According to the model in Equation (4), the time-varying covariates here would include the LDV. This assumption states that the errors of the TSCS model are mean independent of welfare spending at time t given the conditioning set that depends on the history of the data up to t. Thus, this allows the errors for levels of terrorism to be related to future values of welfare spending.

Sequential ignorability weakens baseline randomization to allow for feedback between the treatment status and the time-varying covariates, including lagged outcomes. For instance, sequential ignorability allows for the welfare spending of a country to impact future levels of terrorism and for this terrorism to affect future welfare spending. Thus, in this dynamic case, treatments can affect the covariates and so the covariates also have potential responses: Z _it(x _{1:t − 1}). This dynamic feedback implies that the lagged treatment may have both a direct effect on the outcome and an indirect effect through these covariates. For example, welfare spending might directly affect terrorism by reducing resentment among potential terrorists, but it might also have an indirect effect if it helps to increase levels of state capacity which could, in turn, help combat future terrorism.

In TSCS models, the LDV, is often included in the above time-varying conditioning set, V _it, to assess the dynamics of the time-series process or to capture the effects of longer lags of treatment in a simple manner.Footnote ⁶ In either case, sequential ignorability would allow the LDV to have an effect on the treatment history as well, but baseline randomization would not. For instance, welfare spending may have a strong effect on terrorism levels which, in turn, affect future welfare spending. Under this type of feedback, an LDV must be in the conditioning set V _it and strict exogeneity will be violated.

Unmeasured Confounding and Fixed Effects Assumptions

Sequential ignorability is a selection-on-observables assumption—the researcher must be able to choose a (time-varying) conditioning set to eliminate any confounding. An oft-cited benefit of having repeated observations is that it allows scholars to estimate causal effects in spite of time-constant unmeasured confounders. Linear fixed effects models have the benefit of adjusting for all time-constant covariates, measured or unmeasured. This would be very helpful if, for instance, each country had its own baseline level of welfare spending that was determined by factors correlated with terrorist attacks, but the year-to-year variation in spending within a country was exogenous. At first glance, this ability to avoid time-constant omitted variable bias appears to be a huge benefit.

Unfortunately, these fixed effects estimation strategies require within-unit baseline randomization to identify any quantity other than the CET (Sobel Reference Sobel2012; Imai and Kim Reference Imai and Kim2017). Specifically, standard fixed effects models assume that previous values of covariates like GDP growth or lagged terrorist attacks (that is, the LDV) have no impact on the current value of welfare spending. Thus, to estimate any effects of lagged treatment, fixed effects models would allow for time-constant unmeasured confounding but would also rule out a large number of TSCS applications where there is feedback between the covariates and the treatment. Furthermore, the assumptions of fixed-effects-style models in nonlinear settings can impose strong restrictions on over-time variation in the treatment and outcome (Chernozhukov et al. Reference Chernozhukov, Fernndez-Val, Hahn and Newey2013). For these reasons, and because there is a large TSCS literature in political science that relies on selection-on-observables assumptions, we focus on situations where sequential ignorability holds. We return to the avenues for future research on fixed effects models in this setting in the conclusion.

Structural Nested Mean Models

Our first class of models, SNMMs, can be seen as an extension of the ADL approach that allows for estimation of lagged effects in a relatively straightforward manner (Robins Reference Robins1986, Reference Robins and Berkane1997). At their most general, these models focus on parameterizing a conditional version of the lagged effects (that is, the impulse response function):Footnote ⁸

(18) $$\begin{eqnarray} b_t(x_{1:\,t}, j) &=& E[Y_{it}(x_{1:\,t-j},0_{j}) \nonumber\\ &&-\, Y_{it}(x_{1:\,t-j-1}, 0_{j+1})| X_{1:\,t-j} = x_{1:\,t-j}]. \nonumber\\ \end{eqnarray}$$

Robins (Reference Robins and Berkane1997) refers to these impulse responses as “blip-down functions.” This function gives the effect of a change from 0 to x _{t − j} in terms of welfare spending on levels of terrorism at time t, conditional on the treatment history up to time t − j. Inference in SNMMs focuses on estimating the causal parameters of this function. The conditional mean of the outcome given the covariates needs to be estimated as part of this approach, but this is seen as a nuisance function rather than the object of direct interest.

Given the chosen lag length to study, a researcher must only specify the parameters of the impulse response up to that many lags. If we chose a lag length of 1, for example, then we might parameterize the impulse response function as

(19) $$\begin{equation} b_t(x_{1:\,t}, j; \gamma ) = \gamma _{j}x_{t-j}, \qquad j \in \lbrace 0,1\rbrace . \end{equation}$$

Here, γ_j is the impulse effect of a one-unit change of welfare spending at lag j on levels of terrorism, which does not depend on the past treatment history, x _{1:t − 1} or the time period t. Keeping the desired lag length, we could generalize this specification and have an impulse response that depended on past values of the treatment:

(20) $$\begin{eqnarray} b_t(x_{1:\,t}, j; \gamma ) = \gamma _{1j}x_{t-j} + \gamma _{2j}x_{t-j}x_{t-j-1}, \qquad j \in \lbrace 0,1\rbrace , \nonumber\\ \end{eqnarray}$$

where γ_2j captures the interaction between contemporaneous and lagged values of welfare spending. Note that, given the definition of the impulse response, if x _t-j = 0, then b _t = 0 since this would be comparing the average effect of a change from 0 to 0. Choosing this function is similar to modeling X _{i, t − j} in a regression—it requires the analyst to decide what nonlinearities or interactions are important to include for the effect of treatment. If Y _it is not continuous, it is possible to choose an alternative functional form (such as one that uses a log link) that restricts the effects to the proper scale (Vansteelandt and Joffe Reference Vansteelandt and Joffe2014).

Note that the noninteractive impulse response function in Equation (19) can be seen as an alternative parameterization of the ADL (1,1) in Equation (4). When j = 0 in Equation (19) and an ADL (1,1) model holds, then the contemporaneous effect of γ₀ corresponds to the β₁ parameter from the ADL model. When j = 1 in Equation (19) and an ADL (1,1) model holds, then the impulse response effect of γ₁ corresponds to the αβ₁ + β₂ combination of parameters from the ADL model. We derive this connection in more detail below, but one important difference can be seen in this example. The SNMM approach directly models the lagged effects while the ADL model recreates these effects from all constituent path effects.

The key to the SNMM identification approach is that problems of post-treatment bias can be avoided by using a transformation of the outcome that leads to easy estimation of each conditional impulse responses (γ_j). This transformation is

(21) $$\begin{equation} \widetilde{Y}^j_{it} = Y_{it} - \sum ^{j-1}_{s=0} b_{t}(X_{i,1:\,t}, s), \end{equation}$$

which, under the modeling assumptions of Equation (19), would be

(22) $$\begin{equation} \widetilde{Y}^j_{it} = Y_{it} - \sum ^{j-1}_{s=0} \gamma _{s}X_{i,t-s}. \end{equation}$$

These transformed outcomes are called the blipped-down or demediated outcomes. For example, the first blipped-down outcome, which we will use to estimate first lagged effect, subtracts the contemporaneous effect for each unit off of the outcome, $\widetilde{Y}^1_{it} = Y_{it} - \gamma _0X_{it}$. Intuitively, the blip-down transformation subtracts off the effect of j lags of treatment, creating an estimate of the counterfactual level of terrorism at time t if welfare spending had been set to zero for j periods before t. Robins (Reference Robins1994) and Robins (Reference Robins and Berkane1997) show that, under sequential ignorability, the transformed outcome, $\widetilde{Y}^j_{it}$, has the same expectation as this counterfactual, Y _it(x _{1:t − j}, 0_j), conditional on the past. Thus, we can use the relationship between $\widetilde{Y}^j_{it}$ and X _{i, t − j} as an estimate of the j-step lagged effect of treatment, which can be used to create $\widetilde{Y}^{j+1}_{it}$ and estimate the lagged effect for j + 1. This recursive structure of the modeling is what gives SNMM the “nested” moniker.

We focus on one approach to estimating the parameters called sequential g-estimation in the biostatistics literature (Vansteelandt Reference Vansteelandt2009).Footnote ⁹ This approach is similar to an extension of the standard ADL model in the sense that it requires modeling the conditional mean of the (transformed) outcome to estimate the effect of each lag under study. In particular, for lag j the researcher must specify a linear regression of $\widetilde{Y}^{j}_{it}$ on the variables in the assumed impulse response function, b _t(x _{1: t}, j; γ) and whatever covariates are needed to satisfy sequential ignorability.

For example, suppose we focused on the contemporaneous effect and the first lagged effect of welfare spending and we adopted the simple impulse response b _t(x _{1: t}, j; γ) = γ_jx _{t − j} for both of these effects. As above, we assume that sequential ignorability held conditional on V _it = {X _{i, t − 1}, Y _{i, t − 1}, Z _it}. Sequential g-estimation involves the following steps:

1. 1. For j = 0, we would regress the untransformed outcome on {X _it, X _{i, t − 1}, Y _{i, t − 1}, Z _it}, just as we would for the ADL model. If the modeling is correctly specified (as we would assume with the ADL approach), the coefficient on X _it in this regression will provide an estimate of the blip-down parameter, γ₀ (the contemporaneous effect).

2. 2. We would use $\widehat{\gamma }_0$ to construct the one-lag blipped-down outcome, $\widetilde{Y}^1_{i,t} = Y_{it} - \widehat{\gamma }_0X_{it}$.

3. 3. This blipped-down outcome would be regressed on {X _{i, t − 1}, X _{i, t − 2}, Y _{i, t − 2}, Z _{i, t − 1}} to estimate the next blip-down parameter, γ₁ (the first lagged effect).

If more than two lags are desired, we could use $\widehat{\gamma }_1$ to construct the second set of blipped-down outcomes, $\widetilde{Y}^2_{i,t} = \widetilde{Y}^1_{i,t} - \widehat{\gamma }_1X_{i,t-1}$, which could then be regressed on {X _{i, t − 2}, X _{i, t − 3}, Y _{i, t − 3}, Z _{i, t − 2}} to estimate γ₂. This iteration can continue for as many lags as desired. This approach avoids including a post-treatment covariate when estimating a particular lagged effect. That is, when estimating the effect of welfare spending at lag j, only variables causally prior to welfare spending at that point are included in the regression. Standard errors for all of the estimated effects can be estimated using a consistent variance estimator presented in the Supplemental Material or via a block bootstrap.

This sequential g-estimation approach requires the correct specification of the relationship between the (transformed) outcome and the covariate and treatment histories. It thus requires a similar regression model to the ADL approach described above. More complicated SNMM estimators can incorporate a model for the treatment process, providing some robustness to the modeling choices for the outcome. These estimators are consistent for the parameters of the SNMM when either the model for the (transformed) outcome or the model for the treatment process is correctly specified. This property is called double robustness because there are “two shots” to achieve consistency. Vansteelandt and Joffe (Reference Vansteelandt and Joffe2014) provides a review of these methods for SNMMs.

Relationship to the ADL model

As we mentioned above, the ADL approach and the sequential g-estimation version of SNMM presented above are very similar when the time-varying covariates, Z _it, are not affected by treatment. One intuition for this result is that the ADL model and the SNMM with the linear model are equivalent when there are no covariates aside from the LDV. To see this, suppose that the ADL model in Equation (4) is correct and perform the first transformation from step 2 above, noting, as above, that the contemporaneous effect is the same for both models γ₀ = β₁:

(23) $$\begin{equation} Y_{it} - \gamma _0X_{it} = Y_{it} - \beta _1X_{it} \end{equation}$$

(24) $$\begin{equation} = \beta _0 + \alpha Y_{i,t-1} + \beta _2X_{i,t-1} + \varepsilon _{it} \end{equation}$$

(25) $$\begin{eqnarray} &=& \beta _0 + \alpha (\beta _0 + \alpha Y_{i,t-2} + \beta _1X_{i,t-1} + \beta _2X_{i,t-2} \nonumber\\ &&+\, \varepsilon _{i,t-1}) + \beta _2X_{i,t-1} + \varepsilon _{it} \end{eqnarray}$$

(26) $$\begin{eqnarray} &=& (\beta _0 + \alpha \beta _0) + \alpha ^2 Y_{i,t-2} + \underbrace{(\alpha \beta _1 + \beta _2)}_{\gamma _1}X_{i,t-1} \nonumber\\ &&+\, \alpha \beta _2X_{i,t-2} + (\alpha \varepsilon _{i,t-1} + \varepsilon _{it}). \end{eqnarray}$$

From this, we can see that the coefficient on X _{i, t − 1} for this transformed outcome is simply the impulse response at lag 1, which is exactly the quantity that the SNMM targets. Given the ADL and SNMM assumptions above, this quantity will be αβ₁ + β₂ for the ADL model and γ₁ for the SNMM. Of course, this correspondence will continue for all lagged effects and Table 1 shows how the two sets of quantities relate for various lags.

TABLE 1.

The lagged effects, or impulse responses, under the ADL (1,1) in Equation (4) and SNMM in Equation (19).

Furthermore, in the Supplemental Material, we show that the sequential g-estimation estimator with no covariates except an LDV is nearly mechanically equivalent to a traditional ADL estimator with one lag. The difference is that the traditional ADL model relies on an assumption that the contemporaneous effect is constant over time, whereas sequential g-estimation relaxes this assumption. This provides a useful interpretation of the ADL model in terms of counterfactual causal effects. It is important to note, however, that this equivalence also relies on the form of the ADL model, which uses only three parameters regardless of the number of lags, while the SNMM in this version uses a new parameter for every lag. Additionally, the near equivalence disappears once there is an additional time-varying covariate (Z _it) in the model.

Marginal Structural Models

One potential downside of the SNMM approach is that it requires the analyst to correctly model the relationship between the time-varying covariates and the outcome. This can be difficult when the outcome is a complicated process and there is little theoretical guidance for specifying the outcome-covariate relationships. An alternative that relies instead on modeling the treatment-covariate relationship is called a marginal structural model or MSM (Robins, Hernán, and Brumback Reference Robins, Hernán and Brumback2000).Footnote ¹⁰ To specify an MSM, we first choose a potential outcome lag length to study and write a model for the marginal mean of those potential outcomes in terms of the treatment history. At the most general, then, an MSM would be the following:

(27) $$\begin{equation} E[Y_{it}(x_{1:\,t})] = g(x_{1:\,t};\beta ), \end{equation}$$

where the function g operates similarly to a link function in a generalized linear model.Footnote ¹¹ These models are similar to the impulse response functions in the SNMM approach, b _t, because they provide structure for the treatment-outcome relationship. For instance, suppose that we were focused on the contemporaneous effect and the effect of the first two lags and so we had to model E[Y _it(x _{t − 2:t})] = g(x _{t − 2:t}; β), marginalizing over further lags and other covariates. If Y _it were approximately continuous, as in the case of the number of terrorist incidents, we might take g to be linear and focus on the additive effects of each period of treatment:Footnote ¹²

(28) $$\begin{equation} g(x_{t-2:t};\beta ) = \beta _0 + \beta _1x_{t} + \beta _2x_{t-1} + \beta _3x_{t-2}.\vspace*{-2pt} \end{equation}$$

If Y _it were binary, we might instead assume g to have a logistic form:

(29) $$\begin{eqnarray} g(x_{t-2:t};\beta ) = \frac{\exp (\beta _0 + \beta _1x_{t} + \beta _2x_{t-1} + \beta _3x_{t-2})}{1 + \exp (\beta _0 + \beta _1x_{t} + \beta _2x_{t-1} + \beta _3x_{t-2})}. \nonumber\\\vspace*{-2pt} \end{eqnarray}$$

In both of these cases, we have restricted our attention to the last three periods of treatment and so we cannot answer questions about longer-term effects with these models. On the other hand, as we increase the number of lags under study, the number of parameters needed to summarize the effects grows and the model can become unwieldy. Thus, we may consider focusing on the effect of the cumulative number of treated periods, ∑^t_{s = 1}x_is. This allows for the entire history of treatment to affect the outcome in a structured, low-dimensional way. Under any of these models, the average causal effect becomes

(30) $$\begin{equation} \tau (x_{1:\,t},x^{\prime }_{1:\,t}) = g(x_{1:\,t};\beta ) - g(x_{1:\,t}^{\prime };\beta ). \vspace*{-2pt} \end{equation}$$

Of course, the MSM specification will place restrictions on the average causal effects. An MSM that is a function of only the cumulative treatment, for instance, implies that τ(x _{1: t}, x′_{1: t}) = 0 if x _{1: t} and x′_{1: t} have the same number of treated periods, even if their sequence differs.

How can a researcher estimate an MSM? If one blindly follows model Equation (28) and regresses Y _it on {X _{i, t − 2}, X _{i, t − 1}, X _it} using ordinary least squares, there will be omitted variable bias in the estimated coefficients. But as we have seen above, simply including time-varying covariates in these models can lead to post-treatment bias. Fortunately, the causal parameters of these models are estimable using an IPTW approach where we adjust for time-varying covariates using the propensity score weights, not the outcome model itself, avoiding post-treatment bias (Robins, Hernán, and Brumback Reference Robins, Hernán and Brumback2000). The weighting balances the distribution of the time-varying covariates across values of the treatments, so that omitting these variables in the reweighted data produces no omitted variable bias.

To use IPTW, a researcher must develop a model for the probability of treatment in period t given the variables that satisfy sequential ignorability. For example, suppose that sequential ignorability holds conditional on some conditioning set V _it. If X _it is binary, then we must obtain a consistent estimate of $\Pr [X_{it} = 1 \;|\; V_{it} = v]$. This might be a pooled logit, a generalized additive model with a flexible functional form, a boosted regression (McCaffrey, Ridgeway, and Morral Reference McCaffrey, Ridgeway and Morral2004), or a covariate-balancing propensity score (CBPS) model (Imai and Ratkovic Reference Imai and Ratkovic2015). The IPTW approach requires this model to provide consistent estimates of the conditional predicted probability of treatment.Footnote ¹³ In spite of this requirement, some methods for propensity score estimation such as CBPS have good finite-sample properties in the face of model misspecification (Imai and Ratkovic Reference Imai and Ratkovic2015).

We use the predicted probabilities from this treatment model to construct weights for each country-year. For example, suppose that V _it included lagged levels of terrorism, Y _{i, t − 1}, lagged welfare spending, X _{i, t − 1}, and a set of time-varying covariates, Z _it. Then, for a binary treatment, we would construct the weights as

(31) $$\begin{equation} \widehat{SW}_{it} = \prod _{t = 1}^t \frac{\widehat{\Pr }[X_{it} \;|\; X_{i,t-1}; \widehat{\gamma }]}{\widehat{\Pr }[X_{it} \;|\; Z_{it}, Y_{i,t-1}, X_{i,t-1}; \widehat{\alpha }]}. \end{equation}$$

The denominator of each term in the product is the predicted probability of observing unit i’s observed treatment status in time t (X _it), conditional on the covariates that satisfy sequential ignorability.Footnote ¹⁴ When we multiply this over time, it is the probability of seeing this unit’s treatment history conditional on the past. The numerators here are the marginal probability of the observed treatment history and stabilize the weights to make sure they are not too variable which can lead to poor finite sample performance (Cole and Hernán Reference Cole and Hernán2008). For instance, to construct this numerator we might run a pooled logistic regression of welfare spending in year t on welfare spending in year t − 1, omitting any time-varying covariates or LDVs. While this choice of numerator is not required for consistency of the estimator (it can be replaced with 1, for instance), it can help to stabilize weights that are highly variable and thus increase efficiency.

Under these assumptions, the expectation of Y _it conditional on X _{i, 1: t} in the reweighted data converges to the true MSM:

(32) $$\begin{equation} E_{\widehat{SW}}[Y_{it}|X_{i,1:\,t} = x_{1:\,t}] \stackrel{p}{\rightarrow } E[Y_{it}(x_{1:\,t})]. \end{equation}$$

Here $E_{\widehat{SW}}[\cdot ]$ is the expectation in the reweighted data. For example, if we used the linear MSM in Equation (28), then we can estimate the causal parameters of MSM by running a weighted least squares (WLS) regression of the outcome, Y _it on {X _{i, t − 2}, X _{i, t − 1}, X _it} with $\widehat{SW}_{it}$ as the weights. If sequential ignorability holds, the coefficients on the components of X _{i, 1: t} from this regression will have a causal interpretation, though they may depend on the particular modeling choices of the MSM (Robins, Hernán, and Brumback Reference Robins, Hernán and Brumback2000). Standard errors can be estimated via a block bootstrap of units. Note that, unlike the ADL and SNMMs, this approach does not require a model for the relationship between the time-varying covariates and the outcome.

Finally, when the conditional probability of treatment is close to zero or one, the IPTW approach can have large and unstable weights, leading to high variance and sometimes small sample biases (Imai and Ratkovic Reference Imai and Ratkovic2015). SNMMs, on the other hand, tend to be more stable in this setting. And while MSMs and SNMMs can accommodate general types of covariates, SNMMs also tend to be more stable when the treatment is continuous since weighting by a continuous density (as would be required with IPTW) is sensitive to small perturbations in the data (Goetgeluk, Vansteelandt, and Goetghebeur Reference Goetgeluk, Vansteelandt and Goetghebeur2008).

Modeling Checklist

In this section, we review the key modeling choices required to implement these methods.