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

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.

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?
• I am also looking for answers to the following related questions: "Is Ljung-Box test applicable on residuals from MA(q) models?" and "Testing whether h-step-ahead forecast errors are at most MA(h−1)". Jan 7, 2021 at 11:58
• Jan 7, 2021 at 11:58
• I have a self-imposed embargo on posting new material here (for a good cause). But I believe you will get your answer by the master himself, Godfrey, L. G. (1978). Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables. Econometrica, 46(6) 1293-1301. Jan 9, 2021 at 16:49
• @AlecosPapadopoulos, I hope I will! Thank you very much for the reference! Jan 9, 2021 at 17:05
• @AlecosPapadopoulos, I read the paper once so far and got the impression it does not address my question. The paper considers testing residuals from an ARX model, not an MA model. Meanwhile, I need to test residuals from an MA model. Jan 13, 2021 at 8:34