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In the Arima() method, in the forecast package in R, I can provide a vector of parameters to the fixed argument, and the model is estimated while ensuring the provided parameters are fixed to the supplied values.

However, when I do this, the model returns no standard errors for these coefficients. Why is this the case? Is it not possible to estimate standard errors of coefficients that are manually provided? Would love an explanation as to why this might be the case.

Moreover, the forecast method still calculates confidence intervals when forecasting from a model that has fixed parameters. Are these intervals still statistically valid? I would have thought such would rely on the standard errors of the estimated coefficients, which it seems we may not know in the case of manually-entered parameters?

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    $\begingroup$ I'd say you have it backwards. If we enter coefficients, we believe that we know them. On what basis would we calculate standard errors, which are a measure of the parameter estimates' variability? For your 3rd paragraph, are you asking about confidence intervals (for parameters) or prediction intervals (for future realizations)? $\endgroup$ – Stephan Kolassa Jul 22 '16 at 11:29
  • $\begingroup$ Hi Stephan. But say these fixed coefficients are not known, but rather a good guess. Purely theoretically is it possible to calculate standard errors in this case? And I'm asking about prediction intervals for future realisations in my last paragraph. $\endgroup$ – nlml Jul 22 '16 at 11:33
  • $\begingroup$ Consider this example: I am trying to predict Sales as a function of GDP and Employment with a linear regression model. I search the entire parameter space and find there is one global minimum, where GDP has a negative coefficient and Employment has a large positive coefficient: due to my data sample and collinearity, one coefficient is offsetting the other. But say there is another minimum, where both have positive coefficients. I think this must be the 'correct' minimum, due to economic intuition, so I supply these coefficients as fixed parameters. Are confidence intervals still feasible? $\endgroup$ – nlml Jul 22 '16 at 11:44
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Fixed parameters entered into Arima() will be interpreted as known parameters. (Whether this makes sense is a different question.) As such, it would rather be questionable if standard errors were calculated for them - after all, SEs are measures of the parameter estimates' variability, and in this interpretation, there is none.

The prediction intervals should still be valid - of course, under the assumption of known parameters. They should therefore only include the variability from the noise or innovations term. As above, whether this makes sense or not is a different question.

(Note that there is a difference between a confidence interval, which refers to an unobservable parameter, and a prediction interval, which refers to a realization that is in principle observable later on.)

If the parameters you provide are not known, but rather "a good guess", then the best way of dealing with this is quantifying your uncertainty and running a Bayesian ARIMA model. This would allow the model to update your guesses, and resulting prediction intervals should reflect both your posterior uncertainty and the remaining noise terms. Unfortunately, I don't know of an implementation of Bayesian ARIMA, but I assume there is one.

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  • $\begingroup$ Thanks Stephan for a very clear answer. Yes, a Bayesian approach could indeed be the answer. If I think about it, what I'm really trying to do here is embed my prior economic intuition into the estimation process of an ARIMAX model. If you (or anyone else) has any thoughts on this I'd be greatly appreciative! $\endgroup$ – nlml Jul 22 '16 at 12:09
  • $\begingroup$ This may be helpful. $\endgroup$ – Stephan Kolassa Jul 22 '16 at 12:28

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