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Conventional higher-order spatial autoregressive models usually assume that the relationship between the response variable and its associated explanatory variables is linear, which is rather restrictive and unrealistic in some empirical applications. In this paper, we introduce a new class of higher-order spatial autoregressive models by allowing the response variable to depend on some of the explanatory variables in a nonparametric way while linearly relate to its spatial lags and the remaining explanatory variables. We propose an estimation method for the proposed model and investigate asymptotic properties of resulting estimators. Furthermore, we propose a testing method to check whether the parameters in the parametric component of the proposed model satisfy certain linear constraint conditions and derive asymptotic distributions of the resulting test statistic. The finite sample performance of the proposed estimation and testing methods is evaluated through simulation studies and illustrated with a real data example. [ABSTRACT FROM AUTHOR]