# significant difference between two ARIMA models on full- and cut-time series

I have a financial time-series, its length equal to $n=252$ points. I have built an autoregressive integrated moving average (ARIMA) model. I'm going to predict the future points of time-series. To determine the order of the ARIMA model the Bayesian Information Criterion (BIC) was applied. In my case the model is ARIMA(2, 1, 0).

I would like to reduce the length of time-series n as much as possible and build an ARIMA model again. In this case I have a) the full (original) time-series, length $n$ and b) the reduced time-series, length $n-k$, here $k$ is the number of ommited points from the start of the time series.

Question. What is a rule can I use to estimate the acceptable length, $n_{new}=n-k$?

More generally: How to measure if there is a significant difference between the ARIMA($p_1, d_1, q_1$) model and a ARIMA($p_2, d_2, q_2$) model?

Edit after Richard Hardy's comment.

1. Now I want to shorten the series as much as possible in academic interest (?may be for bootstrapping on time-series). With practical application in future.
2. I guess the difference between ARIMA models should be based on some stastical test while paraetrs $p_1=p_2, d_1=d_2, q_1=q_2$ are fixed.
• 1.Why do you want to shorten the series as much as possible? (The answer will depend on this.) 2. How do you define the difference between two ARIMA models? – Richard Hardy Sep 5 '16 at 8:09
• @RichardHardy, thanks for the comment, I have tried to answer on your questions. – Nick Sep 5 '16 at 9:26
• Regarding 1., let me rephrase: what do you want to do with the series? You can cut it to length 1 or 2, nobody can prohibit that, but depending on the goal it may not be smart to do. Regarding 2., you seem to keep the same model order ($p_1=p_2$ etc.) but allow different coefficients. Then the possible question to ask is whether the full-series coefficients are statistically different from the cut-series coefficients. Is that what you want? Meanwhile, I am afraid it does not make sense to compare BIC when the sample is not exactly the same (one shorter and one longer). – Richard Hardy Sep 5 '16 at 16:43
• @RichardHardy, thanks for comment. Regarding 2. Can I compare R-squared values of full- and cut-series to answer is whether the full-series coefficients are statistically different from the cut-series coefficients? – Nick Sep 6 '16 at 2:03
• Regarding 1, yes, but be aware that you need to preserve the time ordering of the observations while bootstrapping and so only special bootstrap methods developed for time series will work. Regarding 2, no, you cannot. Very different coefficients may lead to very similar $R^2$ values, so $R^2$ cannot tell whether the coefficients are statistically significantly different. – Richard Hardy Sep 6 '16 at 7:18