Since there is some guesswork in deciding which ARIMA(p,d,q) process to use (i.e. which value to choose for p, q and d), we may need to look at various processes. As for other regression processes, Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC), aka Schwarz Information Criterion (SIC) or Bayesian Information Criteria (BIC), can be used for this purpose. Generally, the process with the lower AIC or BIC value should be selected. Here
where n = the sample size and k = the number of parameters (including σ2) = p+q+c+1 where c = 1 if there is a constant term and c = 0 otherwise.
The above measures leave out the constant term n(1+LN(2π).
Example 1: Which produces a better model for the data in Example 1 of Real Statistics ARMA Tool, the ARIMA(2,0,1) model with constant or the ARIMA(2,1,1) model with zero constant.
Based on the Akaike Information Criterion, AIC = 16.682 for the ARIMA(2,0,1) model (see Figure 2 of Real Statistics ARMA Tool), while AIC = 26.768 for the ARIMA(2,1,1) model (see Figure 1 of Calculating ARIMA Model Coefficients). Thus, we choose the ARIMA(2,0,1) as the better model.
We get the same result using the Bayesian Information Criteria: BIC = 29.856 vs. BIC = 37.267.
Hi Charles,
I’ve noticed that the AIC and BIC values are given in OLS regressions. I’m holding a single variable x constant in ten models, but varying all the remaining variables j through t-5 to t+5. As such these are causal models where I test the effect of j on x and x on j. I am wondering whether the AIC and BIC values will be useful for comparison between the models, as an indicator of an ‘optimal lag’, where the smallest AIC/BIC values imply those lag periods of which the causal effect is strongest? (assuming the variables are found to be statistically significant in the regressions)
Thank you
I should add that all the variables in each of the models are identical, also in terms of their transformations. The only difference between each of the ten models is the time period of the j variables.
Hi Gareth,
AIC and BIC are used to compare models. They are especially useful when the models are not hierarchical, in which case a chi-square test can be used as described at https://real-statistics.com/multiple-regression/testing-significance-extra-variables-regression-model/
Charles
Thank you Charles. I think AIC and BIC are suitable for this purpose then.
ARE THERE EXCEL FORMULAS IN NOTES FOR “PARAMETER ESTIMATES OF STANDARD ERRORS” IN ARIMA MODEL?
Hello Mike,
I don’t what you are referring to when you write “PARAMETER ESTIMATES OF STANDARD ERRORS”
Charles
Charles,
I am looking for formulas on “standard errors” on ARIMA parameters?
Many Thanks
Mike
Michael,
I don’t have a formula. It depends on the Hession matrix in a manner similar to that described for multiple linear regression.
In particular, for multiple linear regression, the standard errors can be calculated as the square roots of the values on the diagonal of the matrix in Property 4 of the following webpage:
https://real-statistics.com/multiple-regression/multiple-regression-analysis/multiple-regression-using-matrices/
Charles