Implements the iterative backfitting algorithm with Cochrane-Orcutt updates as described in Malabanan et al. (2022). This function handles the core estimation loop, separating the non-parametric spline component from the parametric and temporal components.
.semipar_internal(y, x_high, z_low, w_mat, max_iter = 50, tol = 1e-04, ...)A list containing:
List of scalar estimates: beta (covariate effect), gamma (spatial effect), and rho (temporal autocorrelation).
List containing the spline object and f_hat (estimated non-parametric component).
Matrix (N x T) of pure structural residuals \((Y - \hat{Y})\).
Matrix (N x T) of fitted values.
Vector of MSPE values per iteration.
Number of iterations performed.
Numeric matrix (N x T). The response variable in wide format (rows=locations, cols=time).
Numeric array (N x T x K). The high-frequency covariate.
N: Number of spatial locations.
T: Number of low-frequency time points (matching y).
K: Frequency ratio (e.g., 3 for monthly data predicting quarterly response).
Numeric matrix (N x T). The parametric covariate (same frequency as y).
Numeric matrix (N x T). The spatial neighborhood covariate (e.g., spatial lag of Z).
Integer. Maximum number of backfitting iterations. Default is 50.
Numeric. Convergence tolerance based on percentage change in Mean Squared Prediction Error (MSPE). Default is 1e-4.
Additional arguments passed to smooth.spline (e.g., spar).
Malabanan, V. A., Lansangan, J. R. G., & Barrios, E. B. (2022). Semiparametric Spatiotemporal Model with Mixed Frequencies: With Application in Crop Forecasting. Science & Engineering Journal, 15(2), 90-107.