Despite the ongoing philosophical debate regarding whether any relationship can be deemed causal, a significant share of quantitative research in sociology attempts to establish causal effects. Regression coefficients, while often not explicitly termed causal effects, are generally interpreted as indicating how much the dependent variable would increase or decrease under an intervention in which the value of a particular independent variable is changed by one unit, while the values of the other independent variables are held constant (Blalock 1961:17). Whether or not a regression model has been properly specified does not, however, justify the interpretation that a coefficient is a causal effect rather than a partial association without explicit attention to the conditions under which estimates should or should not be interpreted as causal effects. Freedman (1987), for example, offers this sharp criticism of the regression approach commonly practiced in sociology.

All statements about causality can be understood as counterfactual statements (Lewis 1973). The potential outcome approach to causal inference extends the conceptual apparatus of randomized experiments to the analysis of nonexperimental data, with the goal of explicitly estimating causal effects of particular “treatments” of interest. This approach has early roots in experimental designs (Neyman 1935) and economic theory (Roy 1951), but it has been extended and formalized for observational studies in statistics (Holland 1986; Rosenbaum and Rubin 1984, 1983; Rubin 1974) and in economics (Heckman 2005; Heckman, Ichimura, and Todd 1997, 1998; Manski 1995). The potential outcome approach has recently gained attention in sociological research (Brand and Halaby 2006; Harding 2003; Winship and Morgan 1999; Winship and Sobel 2004).

According to the potential outcome causal model, a “treatment” is defined as an intervention that can, at least in principle, be given to or withheld from a unit under study. Each unit has a response or outcome that would have been observed had the unit received the treatment, *y _{i}^{t}*, and a response that would have been observed had the unit received the control,

*y*, given

_{i}^{c}*n*observations (

*i*= 1, …,

*n*). The effect caused by the treatment in place of the control is a comparison of

*y*and

_{i}^{t}*y*. If both

_{i}^{c}*y*and

_{i}^{t}*y*could be observed for each unit, the causal effect could be directly calculated. However, each unit receives only one treatment and so only

_{i}^{c}*y*or

_{i}^{t}*y*

^{c}

_{i}is observed for each unit. The estimation of a causal effect therefore requires an inference about the response that would have been observed for a unit under a treatment condition it did not actually receive. Moreover, the existing literature on causal inferences assumes the stable unit treatment value assumption (SUTVA) (Rubin 1978), which means that the potential outcomes for one unit are unaffected by assignment mechanisms and assignment conditions of other units. It is as if potential outcomes were fixed attributes of the unit, with the observed assignment condition merely revealing one of them to the researcher.

As per the classic potential outcome approach, units of analysis are exposed to one of two possible values of the causal variable, treatment or control, at a given point in time, and values for an outcome are assessed some time subsequent to exposure.^{1} There is no time variation implicated in this setup, beyond the fact that the outcome is measured after exposure to the treatment. Robins and his associates (e.g., Robins, Hernan, and Brumback 2000) have extended the potential outcome approach to the time–varying case. Their emphasis is on recovering biases in epidemiological research that arise from endogenous time–varying covariates.

In this paper, we utilize the conceptual apparatus of the potential outcome framework, with its explicit attention to the comparisons needed in order to make causal claims. However, we examine a more general framework for longitudinal studies and consider the analysis of causal effects in which both exposure to treatment and the effects of treatment are time-varying. In this generalized setup, treatment of a unit can potentially take place at any point in time and the effect of treatment on an outcome can vary over time subsequent to treatment. We limit our paper only to the situation where treatment is dichotomous (*yes* or *no*), nonrepeatable, and nonreversible.^{2} That is, a unit can receive a treatment only once, and the treatment status stays “on” once a unit receives a treatment. Another way to visualize this is to imagine that each unit carries an indicator of being treated or not over time. The indicator can be turned “on” but not “off” once it is turned on. We are interested in the causal effects of whether and when the indicator is turned on.

Our limitation to nonrepeatable and nonreversible treatments in this paper makes our case qualitatively different from situations in which fixed-effects models are applied to longitudinal data. Fixed-effects models are powerful statistical tools for causal inference because they control for unobserved but time-invariant characteristics that may be confounders that affect both the causal variable and the outcome variable in observational studies (Allison 1994; Allison and Christakis 2006; Angrist and Krueger 2000; Winship and Morgan 1999). However, fixed-effect models capitalize on the condition that a treatment condition can be reversed. For a dichotomized treatment, a fixed-effect model utilizes information effectively only from units that change treatment status over time—that is, those that change the treatment indicator from “on” to “off” versus those that change from “off” to “on.” As shown by Chamberlain (1984), the comparison of the two-way transitions affords the researcher a particular leverage with which to net out unobserved but fixed attributes (see also Powers and Xie 2000, chap. 5) on longitudinal data. Since our setup does not permit units to transition from the “on” state to the “off” our conceptual framework is incongruent with the fixed-effects model.^{3}

Even for this restricted case, we need to consider a matrix of potential outcomes. Consequently, the causal framework for this setting, requires a complicated conceptualization of many potential counterfactuals. As we show below, consideration of time-varying treatments and time-varying outcomes gives rise to a large number of possible contrasts for potential outcome comparisons. Indeed, the number of such contrasts can become unmanageably large with even a moderate number of time points. Motivated by substantive considerations in sociological research, we propose a simplifying solution for the analysis of causal effects with time-varying treatments and time-varying outcomes.

The rest of the paper is organized as follows: (1) We provide notation for individual-level treatment effects in four scenarios: (a) classic potential outcome setup with two periods, (b) single-time treatment and time-varying outcomes, (c) time-varying treatments and single-time outcome, and (d) time-varying treatments and time-varying outcomes. (2) We define population-level mean treatment effects, including estimation under ignorability and comparison units utilized in the aforementioned settings. (3) We develop a composite causal effect, in which we decompose the expected value of the outcome for the comparison units with a “forward-looking sequential” approach. This approach involves a weighted combination of comparison units where the weights correspond to when the units are treated or not treated in the observation period. (4) We illustrate our approach with an empirical example demonstrating the causal effect of disability on unemployment using panel data from the Wisconsin Longitudinal Study (WLS). (5) We discuss a few possibilities of parametric modeling and nonparametric smoothing strategies. (6) We end the paper with concluding remarks.