Model Specifications

FastOLS

Model Specification

The model is given by: \[ \begin{equation} \mathbf{Y} = T \beta + \mathbf{Q}\mathbf{\Gamma} + \left(T \circ \mathbf{Q}\right)\mathbf{\Omega} + \mathbf{W}\mathbf{\Psi} + \mathbf{E} \tag{1} \end{equation} \]

where

• \(\mathbf{Y}_{n \times p}\) is the matrix of \(p\) outcomes
• \(T_{n \times 1}\) is the treatment variable
• \(\mathbf{Q}_{n \times (j+l)} = \bigl[\mathbf{X} \; \mathbf{G} \bigr]\) is the horizontal stack matrix of \(j\) covariates and \(l\) group variables
• \(\mathbf{W}_{n \times m}\) is the matrix of \(m\) control covariates
• \(\beta_{1 \times p}\) is the vector of coefficients on \(T\)
• \(\mathbf{\Gamma}_{(j+l) \times p}\) is the matrix of coefficients on \(\mathbf{Q}\)
• \(\mathbf{\Omega}_{(j+l) \times p}\) is the matrix of coefficients on the interaction terms between \(T\) and \(\mathbf{Q}\)
• \(\mathbf{\Psi}_{m \times p}\) is the matrix of coefficients on \(\mathbf{W}\)
• \(\mathbf{E}_{n \times p}\) is the error term matrix

\(\mathbf{Q}\) contains the covariates and group variables used to model treatment effect heterogeneity via interaction terms.

Treatment Effect Estimation & Inference

Our average treatment effects (ATE) \(\tau\) for a binary treatment variable \(T\) is defined as:

\[ \tau = \mathbb{E}_n[\mathbf{Y}_1 - \mathbf{Y}_0] \]

where \(\mathbf{Y}_1\) and \(\mathbf{Y}_0\) are the potential outcomes. Assuming exogeneity in \(T\), the ATEs are identified and can be estimated as follows:

\[ \tau = \mathbb{E}_n\left[\mathbb{E}\left[\mathbf{Y} \mid T = 1\right] - \mathbb{E}\left[\mathbf{Y} \mid T = 0\right]\right] \]

Within the context of (1), this can be estimated via:

\[ \mathbf{\tau} = \mathbf{\Theta'}\bar{d} \]

where \(\mathbf{\Theta'} = \left[\beta' \; \mathbf{\Gamma'} \; \mathbf{\Omega'} \; \mathbf{\Psi'}\right]\) is the horizontally concatenated matrix of transposed coefficient matrices, and \(\bar{d} = \mathbb{E}_n\left[D_{T=1} - D_{T=0}\right]\) is the the average difference in the design matrix \(D\) of (1) from toggling the treatment variable across all observations.

Furthermore, for each outcome \(k \in \{1,2,...,p\}\), we can estimate the standard error of the ATE as follows: \[ \text{SE}(\tau_k) = \sqrt{\bar{d}'\text{VCV}(\mathbf{\Theta}_k)\bar{d}} \]

where \(\text{VCV}(\mathbf{\Theta}_k)\) is the variance-covariance matrix of the estimated coefficients for the \(k\)-th outcome.

This logic extends naturally to the estimation of GATEs and CATEs (e.g., \(\bar{d} = \mathbb{E}_n\left[D_{T=1} - D_{T=0} | \mathbf{G}=g\right]\), \(\bar{d} = \mathbb{E}_n\left[D_{T=1} - D_{T=0} | \mathbf{G}=g, \mathbf{X}=x\right]\), \(\dots\), etc.) and to continuous treatments (e.g., \(\bar{d} = \mathbb{E}_n\left[D_{T=t+1} - D_{T=t}\right]\), \(\dots\), etc.).