Time Series Forecasting with Daily Data: ARIMA with regressor

I'm using a daily time series of sales data that contains about 2 years of daily data points. Based on some of the online-tutorials / examples I tried to identify the seasonality in the data. It seems that there is a weekly, monthly and probably a yearly periodicity / seasonality.

For example, there are paydays, particularly on 1st payday of the month effect that lasts for few days during the week. There are also some specific Holiday effects, clearly identifiable by noting the observations.

Equipped with some of these observations, I tried the following:

1. ARIMA (with Arima and auto.arima from R-forecast package), using regressor (and other default values needed in the function). The regressor I created is basically a matrix of 0/1 values:

   • 11 month (n-1) variables
   • 12 holiday variables
   • Could not figure out the payday part...since it's little more complicated effect than I thought. The payday effect works differently, depending on the weekday of the 1st of month.

   I used 7 (i.e., weekly frequency) to model the time series. I tried the test - forecasting 7 days at a time. The results are reasonable: average accuracy for a forecast of 11 weeks comes to weekly avg RMSE to 5%.

2. TBATS model (from R-forecast package) - using multiple seasonality (7, 30.4375, 365.25) and obviously no regressor. The accuracy is surprisingly better than the ARIMA model at weekly avg RMSE 3.5% .

   In this case, the model without ARMA errors perform slightly better. Now If I apply the coefficients for just the Holiday Effects from the ARIMA model described in #1, to the results of the TBATS model the weekly avg RMSE improves to 2.95%

Now without having much background or knowledge on the underlying theories of these models, I'm in a dilemma whether this TBATS approach is even a valid one. Even though it's improving the RMSE significantly in the 11 weeks test, I'm wondering whether it can sustain this accuracy in the future. Or even if applying Holiday effects from ARIMA to the TBATS result is justifiable. Any thoughts from any / all the contributors will be highly appreciated.

Link for Test Data

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