Testing the equality of several linear regression models

Abstract

The linear regression models are widely used in different research fields, and often there is the need to analyze if there are similarities between two or more different linear models or to verify if a given relation between two variables remains the same in different intervals of time, in particular in cases where small differences might make a big difference. Motivated by these problems the authors consider a test of equality of k linear regression models which is a simultaneous test of equality of slopes, intercepts and variances. In order to overcome the extreme difficulties that exist in the use of the exact distribution of the likelihood ratio test (LRT) statistic and to make this test reliable and easy to use, we propose the use of near-exact distributions to approximate the distribution of the LRT statistic, under $H_0$, in the balanced case, and of new asymptotic approximations for the unbalanced case. The near-exact approximations are built by approximating one factor of an adequate factorization of the characteristic function of the logarithm of the LRT statistic and may be easily implemented. The asymptotic approximations are developed using an expansion for the ratio of gamma functions. The quality of these approximations is analyzed and confirmed. Power studies are conducted in order to better assess the performance of the test. Finally to illustrate the applicability of the test we consider a real data set of gross domestic product at market prices and final consumption expenditure in European countries and one tests the existence of similarities between countries.

Appendix: Empirical power values

1.1 Balanced case

See Tables 14, 15, 16, 17, 18 and 19.

Table 14 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\alpha _1=\alpha _2=0$ and significance level 0.1

Table 15 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\beta _1=\beta _2=2$ and significance level 0.1

Table 16 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\sigma _1^2=\sigma _2^2=1$ and significance level 0.1

Table 17 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\alpha _1=\alpha _2=0$ and significance level 0.01

Table 18 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\beta _1=\beta _2=2$ and significance level 0.01

Table 19 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\sigma _1^2=\sigma _2^2=1$ and significance level 0.01

1.2 Unbalanced case

See Tables 20, 21, 22, 23, 24 and 25.

Table 20 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\alpha _1=\alpha _2=0$ and significance level 0.1

Table 21 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\beta _1=\beta _2=2$ and significance level 0.1

Table 22 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\sigma _1^2=\sigma _2^2=1$ and significance level 0.1

Table 23 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\alpha _1=\alpha _2=0$ and significance level 0.01

Table 24 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\beta _1=\beta _2=2$ and significance level 0.01

Table 25 Empirical power values of the test, assuming $\alpha _1=0$, $\beta _1=2$ and $\sigma _1^2=1$ and for $\sigma _1^2=\sigma _2^2=1$ and significance level 0.01