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If the autocorrelation below confidence for time series, does it mean, that there is no sense to you past (lagged) data for the forecasting?

And probably the best result is a distribution mean or median (depends on metric mse or mape).

Example:

enter image description here

Richard Hardy
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Michael D
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3 Answers3

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The confidence bands give a range in which we would expect sample autocorrelation to vary based on random noise alone, if the data generating process does not include any "real" autocorrelation. As such, your ACF is consistent with no AR dynamics, and the overall mean or median is probably a strong contender in forecasting. This is a frequent finding.

Modern approaches to ARIMA order selection rely more on optimizing information criteria than on inspecting (P)ACF plots, so I'd encourage you to look at auto.arima() in the forecast package for R or similar.

Incidentally, the median is the optimal point forecast for the MAE, not for the MAPE. The optimal point forecast for the MAPE is the (-1)-median, per Gneiting (2011, JASA, p. 752 with $\beta=-1$). You may find What are the shortcomings of the Mean Absolute Percentage Error (MAPE)? useful.

Stephan Kolassa
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  • In such case does it have any sense to use past observations as features in any regression models such as Linear Regression, Decession Tree regression and etc.? – Michael D Oct 27 '20 at 14:36
  • You can try. It's usually easier to just pump your data through `auto.arima()`, and surprisingly hard to beat that. In the end, it comes down to how much time you have to tweak your models (and even if you have a lot of time, you might end up just overfitting to the test set). – Stephan Kolassa Oct 27 '20 at 14:44
  • Regarding auto.arima, does it have sense to use it, in case if there is no auto correlation (pacf)? My thoughts, regarding LR and DT, were to use on many time series, and probably it's possible to learn something from one time series to another.... – Michael D Oct 27 '20 at 14:50
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    (1) I'd much more trust `auto.arima()` than a (P)ACF plot. [Interpreting them is non-trivial (scroll down to the last section).](https://otexts.com/fpp2/non-seasonal-arima.html) (2) Cross-learning is a promising approach, but you will need to do a lot more work to make it run. Even if you do decide to do cross-learning, it makes sense to compare the results to some very simple benchmark, like each series' mean and an `auto.arima()` forecast. – Stephan Kolassa Oct 27 '20 at 14:55
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It means that with the default 5% significance level your series doesn't depend on it's past, so the true model is probably just the mean, yes. But sometimes for prediction, you don't want the true model, but the one that makes smaller prediction errors, so you might still be interested in modelling the series with it's past.

Ricardo Semião e Castro
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The lag $0$ autocorrelation ($cor(Y_t, Y_t)$) will always be $1$ ($Y_t$ is perfectly correlated with itself) and then lag $1$ autocorrelation $cor(Y_t, Y_{t-1})$ will in general be non-zero but potentially quite small. In your plot the lag $1$ autocorrelation appears to be non-significant which does indeed suggest that $Y_{t-1}$ is not useful for predicting $Y_{t}$.

Another way to verify this is to plot $Y_t$ against $Y_{t-1}$. Any clear relationship between the two suggests that $Y_{t-1}$ is useful in predicting $Y_{t}$.

It would also be useful to see the raw time series so other stackexchange users can see use their experiences to help out

jcken
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