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2021-12-16

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We can construct the regression:$$beginbmatrix y_1 y_2 endbmatrix

beginbmatrix x_1 & 0 x_2 & x_2 endbmatrix

beginbmatrix beta_1 Delta endbmatrix

beginbmatrix e_1 e_2 endbmatrix$$ When estimating the parameters, we should keep in mind that $sigma^2_1$ need not equal $sigma^2_2$, which introduces a simple, well-structured, heteroskedasticity into the estimation, which can be addressed via weighted least squares.The null hypothesis is $H_0: Delta 0$, and the alternative is, evidently, $H_A: Delta neq 0$. In the case of Gaussian errors, the obvious test is an $F$-test, which would be exact in finite samples if it were not for the likely mild) disruption caused by having to estimate two variance terms instead of one. To see how much of a disruption the heteroskedasticity causes to the distribution of the $F$-statistic, we construct an example with $n_1 n_2 100$ and four regressors in both $x_1$ and $x_2$. We set $sigma^2_2 4sigma^2_1$, and estimate the regression using iteratively reweighted least squares. We then calculate the p-value of the $F$-test, which, under $H_0$, should be distributed according to a Uniform distribution. Repeating the entire process 10,000 times allows us to test the distribution of the 10,000 p-values against the Uniform distribution, in this case using a Kolmogorov-Smirnov test:which indicates, at least in this case, that the $F$-statistic's distribution is pretty close to the theoretical distribution, even for a not-terribly-large sample size.In the case of non-Gaussian errors, it seems not unlikely that the $F$-test will break down, due to its known sensitivity to exactly this situation. In that case, an alternative would be to construct a permutation or bootstrap test (https://en.wikipedia.org/wiki/Resampling_(statistics)) based on the $F$-statistic, but I should point out that if you were to do so, there would likely be no particular advantage to sticking with the $F$-statistic as the test statistic of choice. An asymptotically equivalent permutation test can be applied if $n_1$ and $n_2$ together are too large for an exact permutation test; the asymptotics are such that the proper significance level is obtained (asymptotically) as long as exchangeability assumptions are met (and of course the null hypothesis is true.)You can also rely on the asymptotic distribution of the $F$-statistic under non-Gaussianity, as outlined here: Does the F-test for multivariable regression work with non-normal residuals but large sample size?. However, IIRC the power of the $F$-test can be severely affected even if its significance level is approximately correct, hence my recommendation for the use of a permutation-based test.

This question is a follow-up to Testing equality of coefficients from two different regressions.

Consider the two data generating processes $$y_1x_1'beta_1e_1$$ and $$y_2x'_2beta_2e_2$$ where $x_1$ and $x_2$ are vectors of the same length. Assume that $x_j$ is independent of $e_j$ for $j1,2$, and that we have two independent iid samples of sizes $n_1$ and $n_2$ from the first and second data generating process, respectively. Assume $n_1>n_2$ (or $n_1neq n_2$). For us to be able to use asymptotic theory for least squares I also assume that $E(y^4)

I believe the test statistics proposed in Testing equality of coefficients from two different regressions do not have known asymptotic distributions.

I tried to use multivariate regression and SUR to create a test statistic, but could not derive relevant asymptotic results once $n_1neq n_2$.

Â·OTHER ANSWER:

This question is a follow-up to Testing equality of coefficients from two different regressions.

Consider the two data generating processes $$y_1x_1'beta_1e_1$$ and $$y_2x'_2beta_2e_2$$ where $x_1$ and $x_2$ are vectors of the same length. Assume that $x_j$ is independent of $e_j$ for $j1,2$, and that we have two independent iid samples of sizes $n_1$ and $n_2$ from the first and second data generating process, respectively. Assume $n_1>n_2$ (or $n_1neq n_2$). For us to be able to use asymptotic theory for least squares I also assume that $E(y^4)

I believe the test statistics proposed in Testing equality of coefficients from two different regressions do not have known asymptotic distributions.

I tried to use multivariate regression and SUR to create a test statistic, but could not derive relevant asymptotic results once $n_1neq n_2$.

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