On Thu, 23 Mar 2017, Sven Schreiber wrote: Thanks for the full example. There are two things going on here. 1) The convention when calculating BIC from a model estimated by least squares is to set "k" to the number of regression coefficients (leaving aside the error variance), while the convention under MLE is to include the variance estimator in k. (Or at least I think that's a fair statement of the case.) This is one way a $bic discrepancy could arise (for the same value of the loglikelihood). This raises a question about our "arma" command: for a pure "I" model (p=q=0), in which case OLS = MLE under normality, should we report the information criteria compatibly with "ols", or compatibly with arma variants estimated via ML? 2) An ARIMA(1,1,0) model can be estimated via OLS on differenced data but that's not equivalent to exact ML (it's equivalent to conditional ML), meaning that you'll get a difference in the log-likelihood (and hence the BIC). I've checked that gretl's ARIMA(1,1,0) output matches that of X-13-ARIMA: they both differ from OLS in the same way. Now in your comparison of two ARIMA(p,1,0) models, you got (with * indicating "best" according to BIC): 1) Using OLS p = 0: BIC = 1923.8 p = 1: BIC = 1922.6* 2) Using the "arma" command p = 0: BIC = 1923.8* p = 1: BIC = 1929.8 In light of the comments above, what's happening here? Well, at present the BIC value shown by "arma" for p = 0 is as per OLS. Calculating it compatibly with ML means adding ln(T), in which case the comparison becomes: p = 0: BIC = 1928.4* p = 1: BIC = 1929.8 So BIC under exact ML still favors the p = 0 model, though only marginally. That seems OK: note that the ML estimate of phi is not significant at the 5 percent level, and the improvement in the log likelihood relative to p = 0 is very small. For the benefit of hansl code that uses AIC or BIC to select ARIMA specification, I guess we should switch to ML-compatible calculation of information criteria when we end up using least squares under "arma". Allin

On Thu, 23 Mar 2017, Sven Schreiber wrote: OK, that's now done in git. Maybe I'll add a little note on this to the chapter on Information Criteria in the User's Guide. Allin

On Fri, 24 Mar 2017, Sven Schreiber wrote: I don't know of any canonical source of the convention, and in fact it's not universal. Some writers argue for including the variance parameter in the "k" count for least squares, but it seems that most software doesn't do that (Stata, SAS, SPSS at least). R does include the extra term, however. William Greene doesn't include it, in his account of info criteria in Econometric Analysis, but he doesn't comment on the matter. I guess using k = (number of regressors) in the least squares case is motivated by the fact that k in that sense is the standard measure of loss of degrees of freedom in estimation. Allin