Sieve 2SLS Instrumental Variable EstimationΒΆ

The sieve based instrumental variable estimator SieveTSLS is based on a two-stage least squares estimation procedure. The user must specify the sieve basis for \(T\), \(X\) and \(Y\) (Hermite polynomial or a set of indicator functions), and the number of elements of the basis expansion to include. Formally, we now assume that we can write:

\[\begin{split}Y =~& \sum_{d=1}^{d^Y} \sum_{k=1}^{d^X} \beta^Y_{d,k} \psi_d(T) \rho_k(X) + \gamma (X,W) + \epsilon \\ T =~& \sum_{d=1}^{d^T} \sum_{k=1}^{d^X} \beta^T_{d,k} \phi_d(Z) \rho_k(X) + \delta (X,W) + u\end{split}\]

where \(\{\psi_d\}\) is the sieve basis for \(Y\) with degree \(d^Y\), \(\{\rho_k\}\) is the sieve basis for \(X\), with degree \(d^X\), \(\{\phi_d\}\) is the sieve basis for \(T\) with degree \(d^T\), \(Z\) are the instruments, \((X,W)\) is the horizontal concatenation of \(X\) and \(W\), and \(u\) and \(\varepsilon\) may be correlated. Each of the \(\psi_d\) is a function from \(\dim(T)\) into \(\mathbb{R}\), each of the \(\rho_k\) is a function from \(\dim(X)\) into \(\mathbb{R}\) and each of the \(\phi_d\) is a function from \(\dim(Z)\) into \(\mathbb{R}\).

Our goal is to estimate

\[\tau(\vec{t}_0, \vec{t}_1, \vec{x}) = \sum_{d=1}^{d^Y} \sum_{k=1}^{d^X} \beta^Y_{d,k} \rho_k(\vec{x}) \left(\psi_d(\vec{t_1}) - \psi_d(\vec{t_0})\right)\]

We do this by first estimating each of the functions \(\E[\psi_d(T)|X,Z,W]\) by linear projection of \(\psi_d(t_i)\) onto the features \(\{\phi_d(z_i) \rho_k(x_i) \}\) and \((x_i,w_i)\). We will then project \(y_i\) onto these estimated functions and \((x_i,w_i)\) again to arrive at an estimate \(\hat{\beta}^Y\) whose individual coefficients \(\beta^Y_{d,k}\) can be used to return our estimate of \(\tau\).