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I estimated an ARIMA (2,1,3) model and found that AR(1), AR(2) had both significant coefficients however my MA tests were unexpected - both the MA(2) and MA(3) were significant yet the MA(1) was insignificant. Can someone please explain what that actually means? Should I remove one of the MA lags in my estimation?

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  • $\begingroup$ I've changed "AIRMA" to "ARIMA" - autoregressive integrated moving average. $\endgroup$ – Scortchi - Reinstate Monica Oct 13 '15 at 8:31
  • $\begingroup$ Your model is probably way over-parameterized. Basic descriptive statistics kike the AIC/BIC assume no pulses/level shifts/seasonal pulses and/or local time trends thus "get confused" about the correct/sufficient ARIMA structure.This condition that you have may reflect the presence of deterministic break/change points in the error variance or a coupling of the error variance with the expected value suggesting a power transform or perhaps changing coefficients over time.Post your data I will try and suggest a useful model that will be parsimonious but be sufficient to separate signal and noise. $\endgroup$ – IrishStat Oct 13 '15 at 11:49
  • $\begingroup$ I wouldn't remove the MA(1) term. There's more harm in removing it on the basis of a significance test than leaving it in. $\endgroup$ – Glen_b Oct 13 '15 at 23:24
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The below answer is restricted to the MA(2) coefficient which I am calling $\phi_2$. All of this may be generalized to the other coefficients which aren’t significant. Here $\phi_2$ represents the “true” value of the MA(2) coefficient which is unknown. Your estimate of $\phi_2$ is $\hat \phi_2$ which you do know because it came with the output of your ARIMA model. Just to be perfectly clear the hypothesis test being conducted in the background is $$H_0:\;\; \phi_2=0\;\;\;\;\;H_a:\;\; \phi_2 \neq 0$$ This is what is used by standard statistical packages when reporting p-values in output. If you are referring to some other test you will have to correct me.

Below is what failure to reject the above hypothesis and "insignificant" mean and do not mean in this context.

Things it means

  1. You cannot reject the null hypothesis that $\phi_2=0$ (holding all other coefficients constant of course). For example suppose $\alpha$=5%, and you estimated $\hat \phi_2= .1$. If you asked yourself: “if I assume a priori the the true value of the MA(2) coefficient was zero ($\phi_2=0$), what is the probably of getting $|\hat \phi_2|\geq .1$?” the answer would have to be greater than 5% (the p.value is the answer to this question). If the answer was 5% or lower, it would make you suspect of the null hypothesis and you would reject that $\phi_2=0$, but because it wasn’t, you are unable to reject the null that $\phi_2=0$ , thus the MA(2) coefficient is “insignificant”.

Things it does not mean:

I am including this only because most people end up making these incorrect assumptions at some point in time. The below may or may not be true given failure to reject $\phi_2=0$;

  1. $\phi_2$ is probably equal to zero, or approximately zero

  2. You would be better off leaving $\phi_2$ out of the model, a model without $\phi_2$ would offer better forecasts.

What this type of hypothesis test is useful for?

These types of tests are useful when rejected because they provide strong evidence that a non-zero coefficient exists where we had previously not thought one to exist before. In a time series setting an example may be showing that a shock to the unemployment rate persists for a certain number of lags, which may be important to policy makers at the federal reserve or something.

When a lot of coefficients in your model are insignificant, (this may be your case), it could be a potential sign of overfitting. This is because when you include an excessive number of parameters or partially redundant parameters, the standard error of all the parameters in the model have a tendency to increase (Note: this does mean that the inclusion of any additional parameter will cause standard errors to increase).

What this type of hypothesis test is not useful for ?

Though lots of insignificant coefficients hint towards an overfit model, they should not, in and of themselves, be used for model selection and/or removing coefficients, rather AIC, BIC, MSFE, and methods of that nature are built for that purpose. In my own experience I have discovered instances where models with a lot of insignificant coefficients outperform models with all significant coefficients.

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  • $\begingroup$ +1 for looking at $AIC$ (and $AIC_c$) together with p-values rather than blindly dropping variables based on significant/insignificant test! $\endgroup$ – Sergey Bushmanov Oct 13 '15 at 12:43
  • $\begingroup$ Since the standard error of a coefficient is partially based on the variance of the errors which gets smaller with increased parameters , it seems just as logical to assert that coefficients will appear to be more significant rather than less significant. $\endgroup$ – IrishStat Oct 13 '15 at 12:43
  • $\begingroup$ @IrishStat Yes your right it could go either way, but the point I was trying to make is that when you add way to many parameters the standard errors get big which causes a lot of variables to be insignificant. Perhaps I should reword that part. $\endgroup$ – Zachary Blumenfeld Oct 13 '15 at 12:46
  • $\begingroup$ When you add too many parameters you can approach a singularity/redundancy thus the system collapses and the standard errors become huge which argues for simplicity ala Occam's Razor. $\endgroup$ – IrishStat Oct 13 '15 at 12:54

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