Breusch-Godfrey test on residuals from an MA(q) model

Question

Consider testing for presence of autocorrelation of lag order up to $h$ in the residuals from a regression model $$ y_t = \mathbf x_t^\top \beta + u_t $$ where $\mathbf x_t$ may or may not include lags of $y_t$. The Breusch-Godfrey test would employ an auxiliary regression $$ \hat u_t = \mathbf x_t^\top \gamma + \varphi_1\hat u_{t-1} + \dots + \varphi_h\hat u_{t-h} + \varepsilon_t $$ and derive its test statistic from there.

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Instead of the usual regression model, consider an MA(q) model $$ y_t = \theta_0+\theta_1 u_{t-1} + \dots + \theta_q u_{t-q} + u_t. $$ How do I carry out the Breusch-Godfrey test on residuals from this model? (The null hypothesis being that the residuals are not autocorrelated against an alternative that autocorrelation of lag order up to $h$ is present in them.) Concretely:

1. How do I construct the auxiliary regression?
   Will it be $\hat u_t=\varphi_1\hat u_{t-1}+\dots+\varphi_s\hat u_{t-s}+\varepsilon_t$ where $s$ somehow depends on $h$ and $q$?
2. How do I construct the test statistic?
3. What is its asymptotic distribution under the null hypothesis of zero autocorrelation?

Answer 1

I had the exact same question and this was my train of thought: First I thought that the innovations 'u' are not the same as the regression residuals. While theoretically true, the actual innovations are not observed so you would replace them with the residuals when running the auxiliary reg. Then I realised that by fitting an MA or ARMA you are already controlling for residual correlation. So in the test you will then only check whether you left out correlations in lags that you did not include in your structural model. So basically if you originally fit an MA(2) model, you should then run MA(q) of higher order and compute the test statistics for the coefficients starting from q=3. No guarantee that this is correct.