# 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.

• How much data have you got? How many years worth? Do you have data on anything like temperature? What is the rest of your model like? What are your DV and IVs? – Peter Flom Jul 22 '14 at 16:07
• In addition to what Peter Flom asked, are you modeling your data as univariate time series or multivariate time series ? If it is multivariate, do you have other variables ? Do those variable exhibit seasonal behavior ? if so adding dummy variable would be unnecessary. Can you provide these additional information? – forecaster Jul 22 '14 at 16:13
• I have edited my question. Could you please provide any the solution. Thanks – Arushi Jul 22 '14 at 16:22
• – Rob Hyndman Jul 23 '14 at 1:38
• I totally agree with @IrishStat, we will not suppose to ignore time series models, a pretty good model out there which capturing multiple seasonality.I would suggest you can check out exponential smoothing state space model which is having a capability of handling multiple seasonality, trend,simultaneously.its exclusively in R.you can make use if forecast()package. – Karthi V Dec 30 '16 at 11:15

@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 trend variable such as quadratic/cubic/log

• 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.

• Very nice summary BUT I would like to add that you ignored particular week-of-the-month and particular day-of-the-month effects in addition to possible weekend effects all of which I have found to be very important. Furthermore pre-event and post-event effects are not to be ignored. Consider the activity around Easter and around other major holidays/events.There is often an individual response pattern which requires the incorporation of a LEAD specification. You should also note that parameters can and often change over time and one needs to validate the assumption of constancy of parameters. – IrishStat Jul 23 '14 at 11:28
• Thanks @Irishstat. You are right. I forgot bout moving Holidays and their lead and lag effects. – forecaster Jul 23 '14 at 13:36

What you need is a model that will incorporate daily effects, weekly effects, monthly effects,week of the month effects, day-of-the-month effects,lead and lag effects of the holidays, unspecified but empirically identifiable level/step shifts , local time trends, changes in seasonal pulses and pulses while incorporating ARIMA structure and possibly dealing with changes in parameters and error variance over time. This is called a Transfer Function and can be easily restated (BUT NOT PARSIMONIOUSLY) as a Multiple Linear Regression.

In specific a daily indicator would take 6 predictors. In general one has to carefully orchestrate(identify) what kind of predictors are needed. If you have a lot of time on your hands you can experiment with some of the structures I have mentioned. Alternatively you might need some advanced software/consultancy to get you to solve your problem in your lifetime.