Empirical Likelihood Ratio Test for Seemingly Unrelated Regression Models

Due to the correlation of the model errors in regression equations, the SUR model allows one to estimate the regression coefficients more efficiently than each of the regression equations is estimated separately with the correlation is ignored. It is by now clear that for the traditional linear SUR model, the Generalized Least Squares (GLS) estimator is more efficient than its Ordinary Least Squares (OLS) counterpart. They are equivalence if the error covariance of the SUR model is diagonal. Therefore, the problem of testing independence of equations of a SUR model is important. Many testing approaches have been proposed for this problem. Breusch and Pagan (1980) proposed a Lagrange multiplier test statistic. Dufour and Khalaf (2002) extended the exact independence test method of Harvey and Phillips (1982) to the multi-equation framework. Tsay (2004) constructed a multivariate independent test statistic for SUR model with serially correlated errors.


Introduction
The Seemingly Unrelated Regression (SUR) of Zellner (1962) is an important tool to analyze multiple equations with correlated disturbances. SUR models have been studied extensively by statistician and econometrician and applied in many areas, more details can be found in Srivastava and Giles (1987) and Fiebig (2001).
Due to the correlation of the model errors in regression equations, the SUR model allows one to estimate the regression coefficients more efficiently than each of the regression equations is estimated separately with the correlation is ignored. It is by now clear that for the traditional linear SUR model, the Generalized Least Squares (GLS) estimator is more efficient than its Ordinary Least Squares (OLS) counterpart. They are equivalence if the error covariance of the SUR model is diagonal. Therefore, the problem of testing independence of equations of a SUR model is important. Many testing approaches have been proposed for this problem. Breusch and Pagan (1980) proposed a Lagrange multiplier test statistic. Dufour and Khalaf (2002) extended the exact independence test method of Harvey and Phillips (1982) to the multi-equation framework. Tsay (2004) constructed a multivariate independent test statistic for SUR model with serially correlated errors. Different to the above methods, we propose a empirical likelihood test statistic. The empirical likelihood of Owen (1988Owen ( ,1990 is an effective nonparametric inference method. More references can be found in Owen (2001).
The paper is organized as follows. The empirical log-likelihood ratio test statistic is given in Section 2. Section 3 conducts some simulation studies to illustrate the performance of the proposed method. An empirical example is also provided to demonstrate the usefulness of this test. Finally, conclusion is given in Section 4. The Appendix provides the proofs of the main results.
By the Lagrange multiplier technique, the empirical log-likelihood ratio can be represented as where λ = (λ 1 , λ 2 , · · · , λ n ) T is the solution of the equation International Journal of Statistics and Probability Vol. 10, No. 3;2021 The following theorem indicates that l n is asymptotically distributed as a χ 2 -distribution.
Theorem 2.1. Suppose the assumptions 1-2 given in Appendix hold, under the null hypothesis, as n→∞, we have Remark 2.1 For the testing problem (2.4), Breusch and Pagan (1980) proposed a Lagrange multiplier test statistic. This is based upon the sample correlation coefficients of the OLS residuals: whereρ i j is the sample estimate of the pair-wise correlation of the residuals. Specifically, Under the null hypothesis, LM has an asymptotic χ 2 N distribution, too.
In order to examine the empirical size of the proposed empirical likelihood (EL) test and the Lagrange multiplier (LM) test statistic, we set ρ = (ρ 12 , ρ 13 , ρ 23 ) = (0, 0, 0), and n = 30, 50, 100, 150, 200, 300, 400, 1000 replications were run and the rejection rate under a given significance level α(0.01, 0.05, 0.10) was computed as the empirical size of the test, and the results are reported in Table 3.1. From the results, we can see that the empirical size of the proposed EL test is quite large for small samples. The size distortion of the LM test is smaller than that of the EL test for small samples. The sizes of both the EL test and the LM test converge to the correct nominal levels when n grows, as would be expected. The fact that the size distortion of the EL test is relatively large indicates that the approximation of the finite sample distribution in small samples using the asymptotic χ 2 is relatively poor. The phenomenon was also reported by Dong and Giles (2007), Liu et al. (2008) and Liu et al. (2011) in other testing problems. According to Owen (2001), this may be improved by using Fisher's F-distribution, or Bartlett correction, or bootstrap sample.
To assess the power of the EL and the LM tests, we took the values of ρ to be each of the following values, (0.1,0,0),  Table 3.2. we can see that the power of the EL is bigger than that of the LM test for significance levels of 10%, 5%, and 1%.

A Real Example
Baltagi and Griffin (1983) considered the following gasoline demand equation where Gas/Car is motor gasoline consumption per auto, Y/N is real per capita income, PMG/PGDP is real motor gasoline price and Car/N denotes the stock of cars per capita. This panel consists of annual observations across 18 OECD countries, covering the period 1960-78. The data for this example can be found in package plm of the open source software R. Baltagi (2008) (P 244) considered the problem of testing the independence of the first two countries: Austria and Belgium.
The observed values of Breusch-Pagan (1980) Lagrange multiplier test statistic and the Likelihood Ratio test statistic for this problem are 0.947 and 1.778, respectively. The observed value of the proposed empirical likelihood test statistic is 2.343. All the three test statistics are distributed as χ 2 1 under the null hypothesis, and do not reject the null hypothesis.

Conclusion
This paper proposes a novel approach for the independence test for the disturbances of the SUR models based on the empirical-likelihood method. The proposed test statistic under the null hypothesis is shown to has an asymptotic chisquare distribution. Compared to the Lagrange multiplier test statistic, the simulation experiment demonstrates that the proposed method performs satisfactorily. Furthermore, our approach can be applied to the case that the model errors of one equation of SUR model are correlated.
International Journal of Statistics and Probability Vol. 10, No. 3;2021 Appendix: Proof of the main results We begin with the following assumptions required to derive the main results. These assumptions are quite mild and can be easily satisfied.
In order to prove that main results, we first introduce several lemmas.
By Lemma 3 in Owen (1990), we can prove that max 1≤i≤p,1≤k≤n x ik = o p (n 1/2 ), max Finally, combining Lemmas 1 and 2, we have l n D −→ χ 2 N as n → ∞. The theorem is then proved. Conflicts of Interest: The author declares no conflict of interest.