# Variance inhomogeneity in time series when forecasting

I am using a time series for monthly temperatures to predict future temperatures.

To this I am using the seasonal ARIMA model and Holt Winters forecast, and my results seems fine.

However, my data set shows that the variance depends on the month. In the winter the temperature changes a lot more over the years than in the summer.

Can I do anything to even out the variance and is it necessary to use SARIMA and HW i R?

• The variability depends on the location (e.g., latitude, longitude, continental vs coast line) as well as the season. In order to decrease the variability of prediction, multiple other factors would need to be incorporated into the model, so the short answer is no for the model being used.
– Carl
Dec 26 '18 at 18:58
• Campbell & Diebold "Weather Forecasting for Weather Derivatives" (2005) deal with this quite successfully with an ARMA-GARCH model. Dec 26 '18 at 19:12
• Carl: I am aware that a model that only consists of temperature date will not be a great model to predict temperatures in the future. At the moment I am just learning how to use theese tools and methods. Dec 27 '18 at 8:02

## 2 Answers

Make sure that the larger variance in particular months is not a resultant of untreated anomalies ala http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html. If after treating latent deterministic structure such as seasonal pulses , pulses , level/step shifts you still have unequal error variance by month then I suggest the following.

classify your model's residuals by month to create monte-carlo (bootstrapping) distributions for each forecast period . You then should apply the inflation factor using the psi-weights to correctly reflect the autoprojective model structure.

We have seen this phenomenon on daily data where not only does the forecast depend on the day pf the week BUT the forecast variance does also.

It sounds like this variance is real: that for your location, the temperature in February is less predictable than the temperature in July. To capture this, you need to include error bars in your predictions.

I suggest you plot the residuals (errors in prediction) by month. Are these normal? If so, you can usefully state a variance for each prediction. If they are not normal, you might just have to use the empirical error distributions.