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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?
Richard Hardy
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  • I am also looking for answers to the following related questions: ["Is Ljung-Box test applicable on residuals from MA(q) models?"](https://stats.stackexchange.com/questions/500812) and ["Testing whether h-step-ahead forecast errors are at most MA(h−1)"](https://stats.stackexchange.com/questions/500808). – Richard Hardy Jan 07 '21 at 11:58
  • Related: ["Testing for autocorrelation: Ljung-Box versus Breusch-Godfrey"](https://stats.stackexchange.com/questions/148004), ["How many lags to use in the Ljung-Box test of a time series?"](https://stats.stackexchange.com/questions/6455). – Richard Hardy Jan 07 '21 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. – Alecos Papadopoulos Jan 09 '21 at 16:49
  • @AlecosPapadopoulos, I hope I will! Thank you very much for the reference! – Richard Hardy Jan 09 '21 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. – Richard Hardy Jan 13 '21 at 08:34
  • p. 1297, middle, above eq. (18). But from the start it mentions that it will examine also MA errors, e.g. p.1294, eq. 3 – Alecos Papadopoulos Jan 13 '21 at 17:41
  • @AlecosPapadopoulos, MA(q) model plays a different role in my question than in Godfrey's paper. He uses it as an alternative for testing *residuals from an ARX model*. I do not care very much about a specific alternative, but my *residuals are generated by an MA(q) model*. I then want to test whether they are serially correlated. E.g. if I were to follow Godfrey's paper as well as I can, I would want to test whether residuals from my MA(q) model have zero autocorrelation vs. an alternative that they come from an MA(p) or AR(p) model. Do I make sense? – Richard Hardy Jan 13 '21 at 18:01
  • Not much. It would be helpful to include in your post the actual equation(s) that represents your model. – Alecos Papadopoulos Jan 13 '21 at 18:24
  • @AlecosPapadopoulos, I have updated my post. – Richard Hardy Jan 13 '21 at 19:25
  • Your model is the usual regression model but where the regressor matrix $X$ contains only a constant term. So what Godfrey writes in the penultimate paragraph of Section 3 page 1298 is directly applicable to your model. Godfrey notes that the test cannot distinguish between AR and MA errors (read the beginning of p. 1298). And caution: read very slowly and carefully how the various quantities in the test statistic are constructed, it is easy to get misled. Not also the Godfrey's $n$ is your $q$. – Alecos Papadopoulos Jan 13 '21 at 20:04
  • @AlecosPapadopoulos, if my model were a usual regression model where $X$ only contained a constant term, I would be doing the Ljung-Box test. However, my $X$ contains several lags of $u_t$. I do not think I can use the LB test as the model is an MA(q) model and thus the LB test does not seem to work (I have linked a question about that above). I guess I will have to read Godfrey several more times, but currently I just do not see him discussing my case. He has an ARX, not MA model at the base and considers residuals from that model. How these residuals are modelled further is another matter. – Richard Hardy Jan 13 '21 at 20:12
  • Richard, the lags of $u$ are not part of the regressor matrix in an MA model. These lags are part of the error term and their actual existence is what you want to test for. Godfery has a standard regression model and wants to test whether the error term is either autoregressive or moving average. An (possible) MA model with a constant term is equivalent to a regression model with only the constant regressor plus an error term that we want to test whether it is AR, MA or neither. – Alecos Papadopoulos Jan 13 '21 at 22:40
  • @AlecosPapadopoulos, I have an MA(q) model and I want to assess its adequacy for my data $y_t$. I will do this by checking whether the model's assumptions are satisfied. One assumption is that the errors $u_t$ are i.i.d. One way this could be violated is if $u_t$ were autocorrelated. I want to test $H_0\colon u_t\ \text{is not autocorrelated}$ against an alternative that they are. I thought I could use the BG test for that, as the LB test is likely inappropriate. I do not see Godfrey addressing my problem in his paper. I also do not see that following Godfrey would solve my original problem. – Richard Hardy Jan 14 '21 at 07:28
  • Regarding *the lags of $u$ are not part of the regressor matrix in an MA model*: the MA model is not estimated via OLS and there is no $X$ matrix, so in that sense I agree. Regarding *These lags are part of the error term and their actual existence is what you want to test for*: I doubt that. My MA model is fixed, I am not considering changing it just at this stage. What I seek is to evaluate its adequacy. Suppose it was another person who built the MA model and just gave me its residuals for diagnostics. I am about to do the diagnostics and am looking for help doing that. – Richard Hardy Jan 14 '21 at 07:33
  • @RichsardHardy To me "evaluate its adequacy" is not really different to "test whether the model _is_ MA(q)" _given that you are discussing specification testing_. I am clarifying this because, in your last example-sentence you are alluding that maybe "adequacy" should be about prediction/forecasting accuracy and things like that, which is a different matter. – Alecos Papadopoulos Jan 14 '21 at 11:20
  • @AlecosPapadopoulos, by *adequacy* I mean statistical adequacy of the model as Aris Spanos would put it (if I got him right). I.e. the model's assumptions should be satisfied. I think in my case this *is* specification testing. My $H_0$ is "the data generating process is MA(q)" against the alternative "it is not". I wonder if Godfrey's paper can help with that; he has MA(q) as the alternative, not the null. And no, I am not interested in forecast accuracy. – Richard Hardy Jan 14 '21 at 12:14
  • @It certainly is applicable Richard. When "our heart's desire" is for the model to be $MA(q)$ we set as the null hypothesis the opposite - that it really is not. This is not a scientific result, but a historically inherited socio-philosophical position that permeates statistical testing that dictates that we should be reluctant to accept that something exists (your MA model in our case). So we set the null hypothesis to be "White noise", we guard it with a low Type I error, and if we reject it, than we say "we have strong statistical evidence that our MA model is true/statistically adequate". – Alecos Papadopoulos Jan 14 '21 at 12:46
  • @AlecosPapadopoulos, interesting. The philosophical idea is logical, but I do not remember it being part of any major econometric philosophy. Also, I wonder if rejecting the null provides unambiguous evidence for the particular alternative. When rejecting $H_0$ based on the first $q$ autocorrelations, I can see both MA(q-1) and MA(q+1) (as well as more distant relatives of MA(q)) as plausible alternatives. All of them can lead to a rejection of $H_0$. Hence, I am quite far from proving MA(q) is **the** one. Might this be why this line of though is rarely used in model specification testing? – Richard Hardy Jan 24 '21 at 05:41
  • @AlecosPapadopoulos, also, despite your embargo, could you please give at least a Yes/No answer (comment) to ["Is Ljung-Box test applicable on residuals from MA(q) models?"](https://stats.stackexchange.com/questions/500812)? I presume it is easy for you since you have already answered a more general question about the case of ARMA instead of MA. Thank you! – Richard Hardy Jan 24 '21 at 05:50
  • Hi Richard and Alecos, I read the referenced paper by Alecos. I am very new to time-series and using different models. I was looking for a similar formulation of B-G test for ARIMA models (vs Ljung-Box test, used frequently). Based on the discussion in comments, what I am understanding is that, the general equation of *auxiliary regression* would remain the same for ARIMA models (like for AR model), given that $x_t$ may include lags from dependent variable $y_t$ as well. Please correct me if I am wrong. – Chintan Rajvir Mar 08 '21 at 12:36

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