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To study possibly nonlinear relationship between housing price index (HPI) and consumer price index (CPI) for individual states in the USA, accounting for the temporal lag interactions of the housing price in a given state and spatio‐temporal lag interactions between states could improve the accuracy of estimation and forecasting. There lacks, however, methodology to objectively identify and estimate such spatio‐temporal lag interactions. In this article, we propose a semiparametric data‐driven nonlinear time series regression method that accounts for lag interactions across space and over time. A penalized procedure utilizing adaptive Lasso is developed for the identification and estimation of important spatio‐temporal lag interactions. Theoretical properties for our proposed methodology are established under a general near epoch dependence structure and thus the results can be applied to a variety of linear and nonlinear time series processes. For illustration, we analyze the US housing price data and demonstrate substantial improvement in forecasting via the identification of nonlinear relationship between HPI and CPI as well as spatio‐temporal lag interactions. [ABSTRACT FROM AUTHOR]