# Capturing seasonality in multiple regression for daily data

I have a daily sales data for a product which is highly seasonal. I want to capture the seasonality in the regression model. I have read that if you have quarterly or monthly data, in that case you can create 3 and 11 dummy variables respectively — but can I deal deal with daily data?

I have three years of daily data. The independent variables are price point, promotional flag (yes/no), and temperature. The dependent variable is sales of that product. I am not looking for a time series model as I am using a multiple regression model.

@Irishstat covered pretty much what I was about to say, but I would respond with my own personal experience in modeling these data with time series regression and OLS regression.

If it is a daily data then I would do the following:

Create a dummy variable for different seasonality:

• To capture day of the week seasonality, create 6 dummy variables.
• To capture day of the month seasonality, create 30 dummy variables
• To capture month of the year, create 11 dummy variables.

Create dummy variable for trend variables:

• If the time series exhibits linear trend, then add a time trend
variable.

• If the time series exhibits nonlinear trend, add a nonlinear time

• This is a time series data, so care should be taken about lead and
lag effects of independent varibales. For instance in your example,
you mention price point promotional flag, they might not have
immediate effect on your response, i.e., there may be lagging and
a decaying/permanent
effect. So for instance, if run a promotion
today, you might have a increase in sales today but the effect of
promotion decays after few days. There is no easy way to model this
using multiple regression, you would want to use transfer function
modeling which is parsimonoius and can handle any type of lead and
lag effects. See this example I posted earlier, where there is
an intervention(in your case price point) and there is an abrupt
increase, followed by a decaying effect. Having said that if you have a priori knowledge about the lead and lag effect, create additional variables in your case dummy variables before and after price point and (yes/no) promotion change.

• You would also need to add moving Holidays indicator variables, for example as Irishstat pointed out you would want to add Easter/Thanksgiving (in US) which are moving Holidays. Holidays that are fixed dates will be automatically taken care of if you are using dummy coding scheme for capturing seasonality.

• In addition, you would need to identify outliers such as additive/pulse (one time event) or level shift (permanent shift) and add them as regressors. Identifying outliers in multiple regression for time series data is nearly impossible; you would need time series outlier detection methods such as Tsay’s procedure or Chen and Liu’s procedure which has been incorporated in software such as AUTOBOX, SPSS, SAS or the tsoutlier package in R.

Potential Problems:

Following are the problems you would encounter if you model time series data using OLS multiple regression.

• Errors might be autocorrelated. See this nice website and this website explaining this issue. One way to avoid this is to use Generalized least
squares (GLS)
or ARIMAX approach vs. OLS multiple regression,
where you can correct for auto correlation.
• OLS model will not be parsimonoius. You have $6+30+11= 47$ dummy variables for seasonality.
• By using dummy variables, you are assuming that your seasonality is
deterministic i.e. it doesn’t change over time. Since you have only 3
years of data I would not worry about it, but still it is worthwhile
to plot the series and see if the seasonality doesn’t change.

And there are many more disadvantages of using multiple regression. If prediction is more important to you then I would hold out at least 6 months of data and test the predictive ability of your multiple regression. If your main goal is to explain the correlation between independent variables, then I would be cautious using multiple regression, and instead I would use a time series approach such as ARIMAX/GLS.

If you are interested, you could refer to the excellent text by Pankratz, for transfer function and dynamic regression modeling. For general time series forecasting please refer to Makridakis et al. Also, a good reference text would be by Diebold for regression and time series based forecasting.