How to determine the critical values of ACF?

I have a sample of 1000 data points and I used it as the training sample to forecast with Timeseries. My lecture suggested me comparing the ACF with its critical values (upper and lower) numerically rather than looking at the graph.

Here are my ACF values:

Question: How do I come up with the upper and the lower critical values for the ACF? Is there any function in R to yield these values?

• You mean the range()? – user81847 Dec 7 '15 at 8:35
• @Pascal No. I mean, how to determine cut off or not by ACF value? – Shieryn Dec 7 '15 at 8:53
• So it is not upper and lower. Please edit. – user81847 Dec 7 '15 at 8:54
• @Pascal What is it called? – Shieryn Dec 7 '15 at 8:57
• Are you looking for critical values for ACF so that you could determine the statistical significance of each lag? Like in a graph, you would have ACF bars and a line representing the 95% critical value; the bars that stick out are statistically significant. – Richard Hardy Dec 7 '15 at 9:32

Based on this source, it looks like under the null the autocorrelation is asymptoticaly standard normal. The 5% critical values of the autocorrelation at any given lag $d$ ($d \neq 0$) are

$$\pm \frac{1.96}{\sqrt{T-d}}$$

where $T$ is the sample size.

In your case, $T=1000$, so the critical values for lag 1 are $\pm \frac{1.96}{\sqrt{1000-1}} \approx 0.06201$, for lag 2 are $\pm \frac{1.96}{\sqrt{1000-2}} \approx 0.06204$, and so on.

Mind also a note from another source:

Additionally, in small sample conditions ... this test may be overly conservative such that the null hypothesis is rejected (residuals indicated as non-white) less often than indicated by the chosen significance level (Lutkepohl, 2006).

However, it is not likely to be relevant for a sample as large as 1000.

• Is that different critical value between lag 1 , lag 2, lag 3 and further? i thought the critical value in one graph is the same. :D Because in my experience, the graph that i used to make, The line is constant. is there any syntax in R to help me get the critical value? – Shieryn Dec 8 '15 at 1:24
• Yeah, that is a bit puzzling since the line in ACF graphs is typically flat. The critical values are almost the same, at least when the sample size is large and you do not consider very distant lags (just the first few). – Richard Hardy Dec 8 '15 at 7:43
• How many sample is it that called large? in R, it's flat. which formula to calculate the critical value that flat? how about the PACF? do you have any references? – Shieryn Dec 11 '15 at 9:49

Since the standard deviation of the acf is approximately = 1/SQRT(NOB) it is so approximate that it is practically useless for large sample sizes . If your "reason" for obtaining critical values is to automatically identify the form of the ARIMA model , you can stop right now ! . Identification of a reasonable starting model for the ARIMA structure is better conducted via approaches like the Inverse Autocorrelation Function http://www.jstor.org/stable/2982488?seq=1#page_scan_tab_contents which is the basis of how AUTOBOX (a piece of software that I have helped develop) effectively solves the riddle.

• Regarding your first sentence, isn't the approximation working better when the sample size is large than when the sample size is small? I thought it works fine for large samples (e.g. 1000 observations). – Richard Hardy Dec 7 '15 at 21:37
• I don;t think so after having a long history analyzing simulated data and "obsering/cataloging" over-modelling suggested by this "rule of thumb" . . The larger the number of observations normally leads to "more correct" estimates but not in this case as there is a vast over-statement regarding the possible significance of sample acf's or pacf;s – IrishStat Dec 7 '15 at 21:46
• There is nonsense and there is nonsense but the most non-sensical thing of all is statistical nonsense ! – IrishStat Dec 7 '15 at 21:50

According to this source, there are two kinds of critical value for ACF. 2/SQRT(N), where N is the sample size, is a simple approximate confidence interval to judge whether the series is significantly random under the null hypothesis. But, if you want to determine cut-off or not, you can use large-lag standard error that calculate the standard error of ACF at the shorter lags. For more details, please visit the link I provide above.

• It would be nice for readers of this answer to get more details without having to check the linked reference, – Xi'an Feb 9 '16 at 10:21