multiplicative treatment effects with standard errors

Question

I simplified this a fair bit after finding a draft version of the Imbens and Rubin chapter.

I am interested in estimating a constant multiplicative treatment effect from a randomized experiment. I believe the DGP for my observed data to be something like \begin{equation} y_i=y_i^C\cdot(1+\beta)^T\cdot\varepsilon_i, \end{equation} where $y_i^C$ is the potential outcome in the untreated state and $T\in \{0,1\}$.

Keele et al. (2012) cite Imbens and Rubin (2008) and suggest that you can interpret \begin{equation} \Delta_M=\left(\frac{1}{N_T}\sum_{T=1} \ln y_i^T\right)-\left(\frac{1}{N_C}\sum_{T=0} \ln y_i^C \right) \end{equation} as a constant multiplicative treatment effect. Imbens and Rubin actually make the assumption that $y_i=y_i^C\cdot\exp\{\beta\}^T,$ which is similar since $\exp\{\beta\}\approx1+\beta$.

In my case, I think the derivation looks like this:
\begin{equation} \Gamma_M=\frac{\left(\frac{1}{N_T}\sum \ln y_i^T\right)-\left(\frac{1}{N_C}\sum \ln y_i^C \right)}{\left(\frac{1}{N_C}\sum \ln y_i^C \right)}=\frac{\left(\frac{1}{N_T}\sum \ln y_i^C \cdot(1+\beta)\right)-\left(\frac{1}{N_C}\sum \ln y_i^C \right)}{\left(\frac{1}{N_C}\sum \ln y_i^C \right)}, \end{equation} which simplifies to $\beta$ if average untreated outcome for those in the treated group (potential and unobserved) is the same as the average outcome for the control group if untreated (observed). That should be the case with decent randomization. This is different than the Rubin/Imbens equation, but somewhat more complicated, so I will use their exponential model.

Here's my question. How do I get a confidence interval for $\Delta_M$? Can I just run a regression of $\ln y$ on a treatment dummy to get standard errors?

Answer 1

Basically, yes. Your DGP is a little weird. Normally, you would not both subscript the $Y_i^C$ and add and error. I'm going to take away the subscript on the $Y^C$. Observe:

\begin{align} Y_i &= Y^C \cdot exp(T_i\beta) \cdot \epsilon_i\\ ln(Y_i) &= ln(Y^C) + T_i \beta + \epsilon_i \end{align}

That's just a regression equation (and, it's true since it is derived from the assumed true DGP by a legal step of algebra). $T$ and $\epsilon$ are uncorrelated as long as the randomization was not broken. That suffices to get $\hat{\beta}_{OLS}$ to be unbiased, consistent, and asymptotically normal (subject to some weak regularity conditions). The only worry you have left is whether the usual OLS standard errors are right. If it's reasonable in your application to assume that $\epsilon$ is homoskedastic and not serially correlated then the usual OLS standard errors are right. That is, yes, just run a regression and use the default standard errors to make confidence intervals.

If you think $\epsilon$ might be heteroskedastic, you can either use Huber-White (aka heteroskedasticity robust) standard errors or you can use case bootstrapping to calculate standard errors. And you might as well just use BCa confidence intervals if you are going to bootstrap.