Functional gradient descent boosting for additive non‐linear spatial autoregressive model (gaussian and probit)

In this working paper, I aim to establish a connection between the traditional mod- els of spatial econometrics and machine learning algorithms. The objective is to determine, within the context of big data, which variables should be incorporated into autoregressive nonlinear models and in what forms: linear, nonlinear, spatially varying, or with interactions with other variables. To address these questions, I propose an extension of boosting algorithms (Friedman, 2001; Buhlmann et al., 2007) to semi-parametric autoregressive models (SAR, SDM, SEM, and SARAR), formulated as additive models with smoothing splines functions. This adaptation primarily relies on estimating the spatial parameter using the Quasi-Maximum Like- lihood (QML) method, following the examples set by Basile and Gress (2004) and Su and Jin (2010). To simplify the calculation of the spatial multiplier, I propose two extensions. The first is based on the direct application of the Closed Form Estimator (CFE), recently proposed by Smirnov (2020). Additionally, I suggest a Flexible Instrumental Variable Approach/control function approach (Marra and Radice, 2010; Basile et al., 2014) for SAR models, which dynamically constructs the instruments based on the functioning of the functional gradient descent boosting algorithm. The proposed estimators can be easily extended to incorporate decision trees instead of smoothing splines, allowing for the identification of more complex variable interactions. For discrete choice models with spatial dependence, I extend the SAR probit model approximation method proposed by Martinetti and Geniaux (2018) to the nonlinear case using the boosting algorithm and smoothing splines. Using synthetic data, I study the finite sample properties of the proposed estimators for both Gaussian and probit cases. Finally, inspired by the work of Debarsy and LeSage (2018, 2022), I extend the Gaussian case of the nonlinear SAR model to a more complex spatial autoregressive multiplier, involving multiple spatial weight matrices. This extension helps determine the most geographically relevant spatial weight matrix. To illustrate the efficacy of functional gradient descent boosting for additive nonlinear spatial autoregressive models, I employ real data from a large dataset on house prices in France, assessing the out-sample accuracy.