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. 
 A: Have you seen this thread? What method can be used to detect seasonality in data? 
Keep in mind that, when using dummy variables, you should only include 3 seasons (e.g. Summer, Spring and Winter) and not all four, so as to avoid perfect multicollinearity (the so-called dummy variable trap). 
A: *

*"percentages of marketing amount spent on internet, TV, and mail advertising".  That doesn't sound like a good operationalization if they add up to 100% each week. Sales will respond to the level of these marketing activities, not percentage.

*Quarterly seasonality isn't nearly good enough if you have weekly data. You will need weekly estimates. If your product is holiday sensitive, you may need to do some hand adjustments for Christmas (big difference is it's early or late in the week). 

*In addition, you will have collinearity problems because marketing activity will, obviously, be higher during high seasons. One approach is (a) to depromote the weekly data for a SET of products individually, because the promotional activity will vary across products, then (b) use this depromoted data to estimate the seasonal factors for the TOTAL SET of products (if they can be assumed to have approximately the same seasonality). Then (c) use these seasonal factors in a model for the original data to better estimate the promotional effects. (the depromoted data in step a is only used in that step, and not later)
