# Modeling seasonality in sales time series

Suppose I have 5 years of weekly sales data for a particular product. I also have other variables such as weekly unemployment rate, weekly coupon rates, and percentages of marketing amount spent on internet, TV, and mail advertising. Suppose I want to determine whether season has an effect on sales. Would it be best to treat season as a dummy variable where season takes values: Winter, Fall, Spring Summer, or would it be better to perform spectral analysis on the sales time series and determine seasonality that way? The current setup I have is:

$sales_{t} = week_{t}+ unemploy_{t} + coupon_{t} + internetprop_{t} + TVprop_{t} + mailprop_{t} + season_{t} + w_{t}$ where $w_{t}$ is white noise.

I run a linear regression first and depending on the ACF and PACF charts, I choose a model for the error term. Is the correct approach? I am unsure of how to model seasonality.

• If you have weekly data, there is no reason to assume that seasonality is best tackled by assuming quarterly shifts. At the same time, you don't have very much data to model seasonality in much detail. The best method has to reflect a trade-off between your precise goals, the dataset size, and which models work adequately. – Nick Cox Jul 20 '13 at 22:36
• So in order to detect seasonality, would spectral analysis on only the sales data reveal the seasonal lags given I had enough data. – phil12 Jul 20 '13 at 22:57
• My strong impression is that spectral analysis is data hungry. You have only 5 replications! My guess: graphing the raw data will tell you as much about seasonality as spectral analysis. stata-journal.com/sjpdf.html?articlenum=gr0025 and stata-journal.com/sjpdf.html?articlenum=gr0037 are reviews of relevant graphs. They are focused on Stata applications but the graphs are trivial in any decent software. – Nick Cox Jul 20 '13 at 23:05
• But you can still use the periodogram to identify smaller frequencies. You do not have enough information to identify yearly patterns, but you could identify monthly or weekly patterns. – phil12 Jul 20 '13 at 23:47
• Seasonality is in the first instance a matter of variation with time of year. If you have consistent variations within months, they might show up on a spectrum too but the irregularity of months and weeks doesn't help there. Clearly you have the data and we don't. Have you tried spectral analysis to see what it tells you? – Nick Cox Jul 20 '13 at 23:56