Proposed separability restriction tests using nonparametric regression methods

Abstract

This article proposes some tests for separability restriction incorporating nonparametric regression methods, as well as offering their general statistic characteristics. An effective separability restriction test is essential for appropriate model specification or appropriate implementation of semi-parametric estimation. In this article, I describe two procedures to yield the estimated residuals, which is very sensitive to separability restriction, upon which one test statistics is proposed. In some benchmark models of sine/cosine functions, I simulate out the probability density function of test statistics in a small sample. These presented results and analysis show that the proposed estimator is robust and effective to variable functional form of regression curves and to variable scale factors, broader than the ‘optimal’ level, and can be put conveniently and widely into a practical use.

Notes

¹In Aoki (2005), I called them ‘Test I/II’ in stead of ‘Procedure I/II’, respectively.

² is a N × M matrix, the (n,m)′th element of which is a scale factor from x_ni -on- y_mi regression, , say.

³ and are the scale factor vectors on X_i and , respectively.

⁴Therefore x_i follows a uniform distribution on [0, d ], too.

⁵Here I assume the homoscedasticity for both ∊ _i and v_i .

⁶ T = 100 is a typical finite small sample case.

⁷For one example, the Nadaraya–Watson, leave-one-out estimator of z on x is calculated as , where K_h (·) is a Gaussian kernel with scale factor h.

⁸For example, Step 3 of Procedure I regresses the estimated residual of z_i on Y_i , further on X_i and , the latter of which contains, as well as the former, considerably enough information of X_i . Also, in Procedure II, the sample mean along one axis of estimated envelopes along the other axis makes its variance smaller.

Fig. 1. (a) Sample mean of test statistics W Case 1 (Model A and Procedure I). (b). Sample std of test statistics W Case 1 (Model A and Procedure I)

Fig. 2. (a) Sample mean of test statistics W Case 2 (Model B and Procedure I). (b) Sample std of test statistics of W Case 2 (Model B and Procedure I)

Fig. 3. (a) Sample mean of test statistics W Case 3 (Model A and Procedure II). (b) Sample std. of test statistics W Case 3 (Model A and Procedure II)

Fig. 4. (a) Sample mean of test statistics W Case 4 (Model B and Procedure II). (b) Sample std. of test statistics W Case 4 (Model B and Procedure II)

Fig. 5. Simulated probability density function (Case 1, σ = 1.0, ρ = 0, h  = 0.1, T  = 200 and L  = 200)

⁹ .

¹⁰Under these general conditions, E(f(X_i )Y_i ) becomes a deterministic function of .