Sieve 2SLS Instrumental Variable Estimation =========================================== The sieve based instrumental variable estimator :class:.SieveTSLS is based on a two-stage least squares estimation procedure. The user must specify the sieve basis for :math:T, :math:X and :math: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: .. math:: 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 where :math:\{\psi_d\} is the sieve basis for :math:Y with degree :math:d^Y, :math:\{\rho_k\} is the sieve basis for :math:X, with degree :math:d^X, :math:\{\phi_d\} is the sieve basis for :math:T with degree :math:d^T, :math:Z are the instruments, :math:(X,W) is the horizontal concatenation of :math:X and :math:W, and :math:u and :math:\varepsilon may be correlated. Each of the :math:\psi_d is a function from :math:\dim(T) into :math:\mathbb{R}, each of the :math:\rho_k is a function from :math:\dim(X) into :math:\mathbb{R} and each of the :math:\phi_d is a function from :math:\dim(Z) into :math:\mathbb{R}. Our goal is to estimate .. math:: \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 :math:\E[\psi_d(T)|X,Z,W] by linear projection of :math:\psi_d(t_i) onto the features :math:\{\phi_d(z_i) \rho_k(x_i) \} and :math:(x_i,w_i). We will then project :math:y_i onto these estimated functions and :math:(x_i,w_i) again to arrive at an estimate :math:\hat{\beta}^Y whose individual coefficients :math:\beta^Y_{d,k} can be used to return our estimate of :math:\tau.