With this exercise, you can learn more about time series analysis with ARMA. You can learn how to determine trends and seasonal effects and how to pick suitable parameters for an ARMA model based on the (partial) autocorrelation.

Libraries and Data

Your task in this exercise is to perform time series analysis, including detrending, seasonal adjustment, ARMA model fitting, and forecasting. You can find everything you need in pandas and statsmodels.

For this exercise set, we provide revenue data about online and mail retail business in the US. The data we provide is for the years 2002-2014. You can download a CSV file with the data here.

Detrending and seasonal adjustment with regression and mean values

Detrend the data using a linear regression and then do a seasonal adjustment by removing the monthly mean values from the data. Visualize the effects of these adjustments.

Detrending and seasonal adjustment with differencing

Use first order differencing for detrending the data and do a monthly seasonal adjustment for the detrended data. Visualize the effects of these adjustments.

Forecasting with ARMA

Fit an ARMA model with appropriate parameters $p$ and $q$ to the detrended and seasonal adjusted data (both versions) without the year 2013 (i.e., drop year 2013 for the model fitting). You can use plots of the autocorrelation and the partial autocorrelation for determining good values. Use the fitted models to forecast the values for the year 2013 and visualize the differences between the forecasted values and the actual values. You should do this with the actual values, i.e., including trend and seasonal effect.