1. Set the target: The theoretical estimand (\(\tau\))

This is the theoretical estimand, which requires a substantive argument. We care about the average difference in the potential outcome of foreign aid each aid-eligible country \(i\) would receive if that country increased its legal restrictions on NGOs in the previous year (or the past in general, depending on number of lags?) versus if it did not increase legal restrictions:

\[ \begin{aligned} \tau =& \underbrace{{\textstyle \frac{1}{n}}\ {\textstyle \sum_{i=1}^n}}_{\substack{\text{mean over} \\ \text{all countries} \\ \text{eligible for aid}}} \bigg[ \underbrace{Y_{it} (\text{New law}_{t-1})}_{\substack{\text{potential outcome} \\ \text{with change in} \\ \text{NGO law}}} - \underbrace{Y_{it} (\text{No new law}_{t-1})}_{\substack{\text{potential outcome} \\ \text{with no change in} \\ \text{NGO law}}} \bigg] & [\text{Theoretical estimand, binary treatment}] \end{aligned} \]

We can also look at a continuous treatment… somehow… and say that we care about the foreign aid that each aid-eligible country \(i\) would receive if the count of legal restrictions in the previous year took a particular value. This is similar to how @LundbergJohnsonStewart:2021 describe Chetty et al.’s unit-specific quantity in Figure 2, p. 4.

\[ \begin{aligned} \tau =& \underbrace{{\textstyle \frac{1}{n}}\ {\textstyle \sum_{i=1}^n}}_{\substack{\text{mean over} \\ \text{all countries} \\ \text{eligible for aid}}} \bigg[ \underbrace{Y_{it} (x_{t-1})}_{\substack{\text{potential outcome} \\ \text{with overall} \\ \text{NGO legal regime}}} - \underbrace{Y_{it} (x^\prime_{t-1})}_{\substack{\text{potential outcome} \\ \text{with alternative} \\ \text{NGO legal regime}}} \bigg] & [\text{Theoretical estimand, continuous treatment}] \end{aligned} \]

Key definitions:

With this TSCS kind of data, though, it’s also a little more complicated than these equations. Marginal potential outcome represents the counterfactual level of foreign aid in country \(i\) if NGO laws run their natural course from the beginning of the data up to \(t - 2\) and then the last lag of NGO laws remains the same, with no changes (see p. 3 in @BlackwellGlynn:2018). In other words, it would be the expected effect of a random country increasing its anti-NGO laws in period \(t-1\) on foreign aid in period \(t\).

In the biostats world, they write these potential outcome specifications with g-estimation-based models, like this:

\[ \begin{aligned} \tau =&\ g(x_{t-1}; \beta) - g(x^\prime_{t-1}; \beta) & [\text{g-estimation-based theoretical estimand}]\\ \end{aligned} \]

It is not possible to observe both potential outcomes for each country, so the theoretical estimand \(\tau\) is unmeasurable and not empirical.

3. Learn from the data (\(\hat{\theta}\))

We find \(\hat{\theta}\) by defining an estimation strategy and providing statistical evidence. We estimate \(\hat{\theta}\) using a marginal structural model (MSM) and inverse probability of treatment weights (IPTW). There’s a whole literature in epidemiology and biostats about how these work and how they provide unbiased estimates of causal parameters through g-estimation. In summary (see equation 32 in @BlackwellGlynn:2018), using stabilized inverse probability weights in a MSM makes it so that the expectation of \(Y_{it}\) conditional on \(X_{i, t-1}\) converges to the true theoretical estimand:

\[ \begin{aligned} \hat{\theta} =&\ \underbrace{\textbf{E}_\text{SW}[Y_{it} \mid X_{i, t-1} = x_{t-1}]}_{\substack{\text{observed average aid} \\ \text{given reweighted treatment}}} \xrightarrow{p} \underbrace{\textbf{E}[Y_{it} (x_{t-1})]}_{\substack{\text{this is kinda} \\ \text{like } \tau \text{ or } \theta?}} & [\text{Estimate of estimand, binary and continuous}] \end{aligned} \]

The biostats g-estimation way of writing this MSM looks like this:

\[ \begin{aligned} \hat{\theta} =&\ g(x_{t-1}; \beta) = \beta_0 + \underbrace{\beta_1}_{\substack{\text{average} \\ \text{causal} \\ \text{effect}}} x_{t-1} & [\text{Estimate of estimand, marginal structural model}] \end{aligned} \]

We can thus obtain an unbiased estimate of the causal parameter \(\beta_1\) of the MSM by fitting the model, by giving each country a time-specific weight, based on the confounders.