14 Dynamic Causal Effect
As in all fields of science, we are perpetually interested in understanding the causal effect of one thing on another. In economics, we want to understand how monetary policy affects output and inflation, how exchange rate affects import and export, and so on. However, causality is something much easier said than done. In reality, there are multiple forces at work simultaneously that leads to the consequences we observed. It is challenging both conceptually and statistically to isolate the causality of a variable of particular interest.
In cross-sectional analysis, causality is defined counterfactually. That is, the causal effect of a treatment is defined as the difference between the treated outcome and the untreated outcome assuming that they would be otherwise the same without the treatment. In practice, that involves working with a large number of \(iid\) observations that are similar on average only differentiated by the status of the treatment. This approach, however, does not work well with many macroeconomic studies. For example, suppose we want to figure out the causal effect of monetary policy on inflation rate. The cross-sectional approach would entail finding a large number of almost identical countries, each with independent monetary policy. And a random subset of them tighten their monetary policies while others do not. Then we work out the different economic outcomes between these two groups. This is clearly infeasible. The question we posed concerns only one country with inflation and interest rates observed through time. We would need a definition of causal effect that encompasses observations over time not across individuals.
Suppose \(\epsilon_t\) denote a random treatment happened at time \(t\). Then the causal effect on an outcome variable \(y_{t+h}\), \(h\) periods ahead, of a unit shock in \(\epsilon\) is defined as
\[ \mathbb{E}[y_{t+h}|\epsilon_t=1]-\mathbb{E}[y_{t+h}|\epsilon_t=0]. \tag{14.1}\]
We require the randomness of the treatment \(\epsilon_t\) in a sense that it is uncorrelated with any other variables that could possible have an impact on the outcome. Therefore, \(\epsilon_t\) happens or not does not affect other forces that shape the outcome. The difference in the outcomes is solely attributable to \(\epsilon_t\). It is this randomness that guarantees a causal interpretation.
Our example of monetary policy above clearly does not meet this requirement. The monetary authority does not set the interest rate randomly, but based on the economic conditions of the time, which makes it correlated with other economic variables that could also have an impact on inflation. A qualified random shock may be a change in weather conditions. Weather has huge impact on agricultural production, but it is determined independent of any human activity. If \(\epsilon_t\) denotes a rainy day at time \(t\), and \(y_{t+h}\) be the agricultural production, Equation 14.1 could be a plausible causal effect. However, most variables of interest in economics are endogenously determined. How to estimate the causal effect in such cases is an art in itself. We will come back to this point later.
The conceptual definition of Equation 14.1 can not be computed directly as the counterfactual is not observed. What we have is a sample of experiments over time, in which the treatment happens randomly at some points but not others, \(\{\epsilon_1=0, \epsilon_2=1, \epsilon_3=0,\dots\}\). We could envision that if we have long enough observations, by comparing the outcomes when the shock happens and when it does not, it gives us an reasonable estimation of the causal effect because all other factors that contributing to the outcome, despite they are changing over time, would be averaged out provided the randomness of the treatment.
Assuming linearity and stationarity, the causal effect of Equation 14.1 can be effectively captured by a regression framework,
\[ y_{t+h} = \theta_h\epsilon_t + u_{t+h}, \]
where \(u_{t+h}\) represents all other factors contributing to the outcome variable. Since \(\epsilon_t\) is random, it holds that \(\mathbb{E}(u_{t+h}|\epsilon_t) = 0\). Therefore,
\[ \theta_h = \mathbb{E}(y_{t+h}|\epsilon_t=1)-\mathbb{E}(y_{t+h}|\epsilon_t=0). \]
Thus, \(\theta_h\) captures the causal effect of one unit shock of \(\epsilon_t\) on \(y_{t+h}\). The path of the causal effects mapped out by \(\{\theta_0, \theta_1, \theta_2, \dots\}\) is called the dynamic causal effect, in a sense that it is the causal effects through time.