Testing for Spatial Lag Effects in Varying Coefficient Spatial Autoregressive Models

1 Introduction

In the last three decades, spatial autoregressive models as a popular spatial econometric method have received much attention in the literature. For a long time, however, its main theory has centered around parametric models in which the relationship between the response and the covariates is assumed to be linear. In recent years, many useful semiparametric spatial autoregressive models have been proposed to relax traditional parametric models. Su and Jin[1] proposed a partially linear spatial autoregressive model and studied the properties of the profile quasi-maximum likelihood estimators for the parameters in the model. Li and Mei[2] proposed a statistical test procedure to check a polynomial relationship of the non-parametric component in partially linear spatial autoregressive models. Su[3] studied a nonprametric spatial autoregressive model that the spatially lagged response variable enters the model linearly while the covariates enter the model nonparametrically.

Like parametric models, semiparametric models have various forms. An alternative approach to relax the conditions imposed on traditional parametric models and explore the hidden structure is varying coefficient models, which was introduced by Cleveland et al.[4] and popularized by Hastie and Tibshirani[5]. Varying coefficient model is a useful extension of linear regression model, by allowing all the regression coefficients to vary as unknown functions of other factors. Due to its flexibility, varying coefficient model has been studied in many different contexts and has been successfully applied to nonlinear time series analysis, longitudinal and functional data analysis, and time-varying models in finance, see Fan and Zhang[6] for a comprehensive survey. In this paper, we consider the following varying coefficient spatial autoregressive model by combining the spatial autoregressive model and the varying coefficient regression model,

where Yi,s are responses, X_i = (X_i1, X_i2, · · ·, X_ip)^T and U_i are associated covariates, α(·) = (α₁(·), α₂(·), · · ·, α_p(·))^T is a p-dimensional vector of unknown functions. W = (w_ij), 1 ≤ i, j ≤ n is a specified n × n spatial weight matrix, εi,s are independent and identically distributed random errors with zero mean and finite variance σ². Obviously, when α(·) = α, model (1.1) becomes the standard spatial autoregressive model, which was studied by Cliff and Ord[7], Anselin[8], Kelejian and Prucha[9] and Lee[10,11]. When ρ = 0, the model (1.1) reduces to the varying coefficient model.

For model (1), Li and Chen[12] proposed a profile likelihood approach based on the local linear method to estimate the unknown coefficient functions α(·) and the spatial lag parameter ρ. As we all know, an important question of spatial autoregressive model is to test the existence of the spatial effects. This leads to the following testing problem

For the standard linear spatial autoregressive models, Likelihood ratio test and Rao’s Score (Lagrange Multiplier ) test can be applied to solve this problem, details can be found in Anselin[8]. However, the test problem (2) is a semiparametric hypothesis versus another semiparametric hypothesis testing problem. Many traditional tests cannot be directly applied to the above hypothesis. For this kind of testing problem, Fan and Huang[13] proposed a profile generalized likelihood ratio (PGLR) test based on the partially linear varying coefficient model. Thus, we are motivated to extend the PGLR test procedure for the testing problem (2) of model (1).

The rest of this paper is organized as follows. In Section 2, the profile maximum likelihood method to fit the varying coefficient spatial autoregressive model is briefly described to facilitate our consequent discussions. In Section 3, the test statistic is proposed and the residual-based bootstrap procedure is suggested to derive the p-value of the test. Simulations are conducted in Section 4 to examine the finite sample performance of the proposed test procedure. Conclusion is presented in Section 5.

2 Profile Likelihood Estimation Procedure

For the need of constructing the test statistic, we first introduce the profile likelihood estimating approach of model (1) proposed by Li and Chen[12]. Let us work with the matrix notation. Denote

Then model (1) can be written as

Assume that ε ∼ N(0, σ²I_n), let θ = (ρ, σ²)^T, the log-likelihood function of model (3) is

where I_n is the identity matrix of order n.

If the parameter ρ is known, then model (1) can be written as

where (Y1∗,Y2∗,⋯,Yn∗)=YY−ρWWYY=(IIn−ρWW)YY. This transforms the varying coefficient spatial autoregressive model (1) into the standard varying coefficient model (5). Now, we apply a local linear regression technique to estimate the varying coefficient functions {α_j(·), j = 1, 2, · · ·, p} in the model (5). For u in a small neighborhood of u₀, one can approximate α_j(·) locally by a linear function

This leads to the following weighted local least-squares problems: find αj(u0),αj′(u0) to minimize

where K is a kernel function, h is a bandwidth and K_h(·) = K(·/h)/h.

Let

with 0_1×p is the 1 × p zero matrix, and

The solution of the problem (61) is given by

Then, we have

where 0_p is the p × p zero matrix.

We take u₀ to be each of U₁, U₂, · · ·, U_n, then we can obtain the estimators of α̂(U_j), j = 1, 2, · · ·, p. Then we can define the estimator for M as

Replacing M of (4) by M̄, we obtain the following profile log-likelihood function

Given ρ, by differentiating log L_n(Y | θ) with respect to σ², we obtain the following equation:

Then, the profile maximum likelihood estimator of σ² can be obtained as

Furthermore, the concentrated log-likelihood function of ρ is

Maximizing log L_n(ρ) leads to the estimator ρ̂ of ρ. Then, we can define the final estimator of σ² as σ̂² = σ̂²(ρ̂). Substituting ρ̂ into (8), we can obtain the estimator of M as

Finally, the residual vector is

3 Profile Generalized Likelihood Ratio Test

4 Simulation Studies

In this section, we shall conduct some simulations to examine the performance of the proposed estimate procedure.

Assume that the observations are collected from a uniform, two-dimensional grid consisting of m × m lattice points with unit distance between any two neighboring points along the horizontal and vertical axes. These m² points are arranged in an orthogonal coordinate system. We generate the spatial weight matrix W according to the principle of Rook contiguity.

The data are generated from the following varying coefficient spatial autoregressive model

where x_1i ∼ U(–2, 2), x_2i ∼ N(1, 1), u_i ∼ U(0, 1), and α1(ui)=sin⁡(2πui)+1,α2(ui)=2e−2(2ui−1)2+3ui. The spatial lag parameter ρ took each of the values in the set (–0.15, –0.1–0.05, 0, 0.05, 0.1, 0.15). To gain an idea of the effect of the distribution of the error on our results, we take the following two different types of the error distribution, 1) ε_i ∼ N(0, 1), 2) εi∼U(−3,3). In the simulation, the kernel function was taken to be the Gaussian kernel K(z)=12πexp⁡(−z22), and the smoothing parameter h was taken as sun−1/5 for simplicity, where s_u is the sample standard deviation of u₁, u₂, · · ·, u_n.

For each given value of ρ and each type of error distribution and bandwidth, 1000 replications with n = 10², 15² were run and the rejection rate at the significance level α = 0.05 was computed as the simulated power of our proposed test procedure. And for each replication, the p-value was computed based on m = 500 bootstrap samples. The results are shown in Table 1.

We summarize our findings as follows. When the null hypothesis is true (that is ρ = 0), the rejection frequencies (estimated sizes) of our proposed test are quite good and close to their nominal level 0.05 under different error distributions. Under the alternative hypothesis, the rejection rate seems very robust to the variation of the type of error distribution, and increases rapidly as the alternative hypothesis deviates from the null hypothesis.

5 Conclusion

In this paper, a test approach is proposed to check the existence of spatial effects in varying coefficient spatial autoregressive models, in which a residual-based bootstrap procedure is suggested to derive the p-value of the test. The simulation experiment demonstrates that the proposed test performs satisfactorily.