forecastinginterpolationtime series
I would like to disaggregate monthly forecasts of sales into daily data.
I have historical data about daily sales over the past two years (which mainly depends on deterministic effects like day-of-the-week, holiday, end-of-month effect,..).
Which technique can I use to predict future daily sales starting from available monthly forecasts according to the historical distribution of sales by day and ensuring that the daily forecasts sum up to the monthly forecasts?
You should be evaluating models and forecasts from different origins across different horizons and not one one number in order to gauge an approach.
I assume that your data is from the US. I prefer 3+ years of daily data as you can have two holidays landing on a weekend and get no weekday read. It looks like your Thanksgiving impact is a day off in the 2012 or there was a recording error of some kind and caused the model to miss the Thanksgiving day effect.
Januarys are typically low in the dataset if you look as a % of the year. Weekends are high. The dummies reflect this behavior....MONTH_EFF01, FIXED_EFF_N10507,FIXED_EFF_N10607
I have found that using an AR component with daily data assumes that the last two weeks day of the week pattern is how the pattern is in general which is a big assumption. We started with 11 monthly dummies and 6 daily dummies. Some dropped out of the model. B**1 means that there is a lag impact the day after a holiday. There were 6 special days of the month (days 2,3,5,21,29,30----21 might be spurious?) and 3 time trends, 2 seasonal pulses (where a day of the week started deviating from the typical, a 0 before this data and a 1 every 7th day after) and 2 outliers (note the thanksgiving!) This took just under 7 minutes to run. Download all results here www.autobox.com/se/dd/daily.zip
It includes a quick and dirty XLS sheet to check to see if the model makes sense. Of course, the XLS % are in fact bad as they are crude benchmarks.
Try estimating this model:
Y(T) = .53169E+06
+[X1(T)][(+ .13482E+06B** 1)] M_HALLOWEEN
+[X2(T)][(+ .17378E+06B**-3)] M_JULY4TH
+[X3(T)][(- .11556E+06)] M_MEMORIALDAY
+[X4(T)][(- .16706E+06B**-4+ .13960E+06B**-3- .15636E+06B**-2
- .19886E+06B**-1)] M_NEWYEARS
+[X5(T)][(+ .17023E+06B**-2- .26854E+06B**-1- .14257E+06B** 1)] M_THANKSGIVI
+[X6(T)][(- 71726. )] MONTH_EFF01
+[X7(T)][(+ 55617. )] MONTH_EFF02
+[X8(T)][(+ 27827. )] MONTH_EFF03
+[X9(T)][(- 37945. )] MONTH_EFF09
+[X10(T)[(- 23652. )] MONTH_EFF10
+[X11(T)[(- 33488. )] MONTH_EFF11
+[X12(T)[(+ 39389. )] FIXED_EFF_N10107
+[X13(T)[(+ 63399. )] FIXED_EFF_N10207
+[X14(T)[(+ .13727E+06)] FIXED_EFF_N10307
+[X15(T)[(+ .25144E+06)] FIXED_EFF_N10407
+[X16(T)[(+ .32004E+06)] FIXED_EFF_N10507
+[X17(T)[(+ .29156E+06)] FIXED_EFF_N10607
+[X18(T)[(+ 74960. )] FIXED_DAY02
+[X19(T)[(+ 39299. )] FIXED_DAY03
+[X20(T)[(+ 27660. )] FIXED_DAY05
+[X21(T)[(- 33451. )] FIXED_DAY21
+[X22(T)[(+ 43602. )] FIXED_DAY29
+[X23(T)[(+ 68016. )] FIXED_DAY30
+[X24(T)[(+ 226.98 )] :TIME TREND 1 1/ 1 1/ 3/2011 I~T00001__010311stack
+[X25(T)[(- 133.25 )] :TIME TREND 423 61/ 3 2/29/2012 I~T00423__010311stack
+[X26(T)[(+ 164.56 )] :TIME TREND 631 91/ 1 9/24/2012 I~T00631__010311stack
+[X27(T)[(- .42528E+06)] :SEASONAL PULSE 733 105/ 5 1/ 4/2013 I~S00733__010311stack
+[X28(T)[(- .33108E+06)] :SEASONAL PULSE 370 53/ 6 1/ 7/2012 I~S00370__010311stack
+[X29(T)[(- .82083E+06)] :PULSE 326 47/ 4 11/24/2011 I~P00326__010311stack
+[X30(T)[(+ .17502E+06)] :PULSE 394 57/ 2 1/31/2012 I~P00394__010311stack
+ + [A(T)]
This is not the full model. There are a total of 74 variables. It looks like you have already have been drinking at the trough of daily data. What is there to be learned differently from your question here that wasn't answered here? Decomposition of daily time series (several years) with multiple seasonal patterns
Best Answer
Ideally, you want to forecast at the level you need forecasts for. One alternative is using Temporal Hierarchical forecasting via: http://robjhyndman.com/papers/temporalhierarchies.pdf - (implented in the R thief package). This uses multiple aggregation levels (daily, weekly, biweekly, etc.), forecasts out each and reconciles the forecasts. Alternatively, you could try multiple seasonal TBATS or double seasonal Holt-Winters.
Lastly, you could also try moving averages of the Day of Week pattern, index day of month, or seasonal index day of month (i.e. Jan. 2016 index 1 received 2% of total monthly volume, index day 2 received 1.4%, etc.). With all of these methods, it would be wise to use test sets to see which have the best predictive accuracy so you can select the one that is best for your data.