Hypothesis testing for $\sigma^2$ in linear regression

Question

I'm trying to figure out a way to test hypotheses about variance in the linear regression model. The hypotheses I want to test are: \begin{align*} &H_0:\sigma^2=1\\ &H_1:\sigma^2<1 \end{align*}

The linear regression model is given by $$Y_i=aX_i+b+\varepsilon_i,\quad i=\overline{1,n}$$ where $\varepsilon_i\in\mathcal{N}(0,\sigma^2)$ are independent random variables. If $\hat{a},\hat{b}$ are least-square estimates of $a,b$, then the least-square estimate of $\sigma^2$ is: $$\hat{\sigma^2}=\frac{1}{n-2}\sum_{j=1}^n(Y_j-\hat{a}X_j-\hat{b})^2$$ Based on the conversation in this thread, $$Z(X)=\frac{(n-2)\hat{\sigma^2}}{\sigma^2}\in\chi^2({n-2})$$ but it depends on $\sigma^2$ as a function and is not a statistic. However, its distribution is $\chi^2_{n-2}$ no matter the value of $\sigma^2$. I try to find a confidence interval for $\sigma^2$ and a given statistical significance $\alpha$.

I need to find $\sigma_a^2,\sigma_b^2:$ $$\mathbb{P}_0(\sigma_a^2\leq\sigma^2\leq\sigma_b^2)=\mathbb{P}_0(\chi^2_{\alpha}({n-2})\leq Z\leq\chi^2_{(1-\alpha)}(n-2))=1-\alpha$$ Using the expression for $Z$ to transform the inequalities I get: $$\mathbb{P}_0(\sigma_a^2\leq\sigma^2\leq\sigma_b^2)=\mathbb{P}_0(\frac{(n-2)\hat{\sigma^2}}{\chi^2_{(1-\alpha)}}\leq \sigma^2\leq\frac{(n-2)\hat{\sigma^2}}{\chi^2_{\alpha}})=1-\alpha$$ which explicitly gives the borders of an interval.

My question is, can I use this for interval for hypothesis testing? If not, how do I test hypotheses for $\sigma^2$? Is there a better way to do it?

Answer 1

As pointed out in the comments, the result you give for the pivotal quantity in this case is already enough to perform the hypothesis test. (Remember that when you do a classical hypothesis test you look at the distribution of the test statistic assuming that the null hypothesis is true.) If you assume normality of the error term, the standard way to test this is using the pivotal quantity you have identified:$^\dagger$

$$(n-2) \cdot \frac{\hat{\sigma}^2}{\sigma^2} \sim \text{ChiSq}(n-2) = \text{Ga}(\tfrac{n-2}{2}, \tfrac{1}{2}).$$

Consequently, taking $\sigma=1$ and scaling gives the null distribution for your test, which is:

$$\hat{\sigma}^2 \sim \text{Ga}(\tfrac{n-2}{2}, \tfrac{n-2}{2}).$$

Usually you would implement the test by computing the two-sided p-value:

$$p(\hat{\sigma}^2) = \int \limits_0^\infty \text{Ga}(r | \tfrac{n-2}{2}, \tfrac{n-2}{2}) \cdot \mathbb{I} \Big( \text{Ga}(r | \tfrac{n-2}{2}, \tfrac{n-2}{2}) \leqslant \text{Ga}(\hat{\sigma}^2 | \tfrac{n-2}{2}, \tfrac{n-2}{2}) \Big) \ dr.$$

Of course, you can proceed using the alternative method of computing a confidence interval for $\sigma^2$ at a particular confidence level $1-\alpha$ and then use this as a means of making inferences about the error variance. (Indeed, most statisticians prefer confidence intervals to hypothesis tests, due to a number of well-known limitations on the latter.) If you form a confidence interval then I would recommend you do so using the HDR criterion (i.e., setting the endpoints so that the density of your pivotal quantity is the same at each endpoint). For more information on this, see e.g., O'Neill (2022).

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A warning: Although the above is the standard method for the Gaussian linear regression model, you should bear in mind that this test/interval is highly sensitive to the assumption of a mesokurtic error distribution. This is true both for the hypothesis test and the corresponding confidence interval. In either case, the kurtosis of the error term strongly affects the variability of the error variance estimator $\hat{\sigma}^2$, so it is usually best to use more robust tests that account for the sample kurtosis in the residual distribution (as an estimate of the true kurtosis of the error distribution). If you would like to learn more about how to adjust your test/interval to account for the kurtosis of the error distribution, see e.g., O'Neill (2014).

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$^\dagger$ To avoid ambiguity, we note that in the gamma distribution we use the parameterisation of shape then rate.