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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?
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