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)]
In addition to what has been said, you might want to consider structural time series models. They account explicitly for one or more seasonalities and trend, and are very tolerant of missing data. A good starting point might be the R function StructTS(). More complex models can be fit with packages such as dlm, KFAS, and several others.
Best Answer
No need to subtract any seasonality.
I would recommend using a regression model with NO ARIMA component. Bring in 6 day of the week dummy variables, 11 month of the year variables, holiday variables while searching for outliers, level shifts, trend. Removing variables ("stepdown") that are not necessary and bringing ("stepup") in dummies for the outliers I listed.
We recommend using 3 years of data in order for holidays to exist on weekends and weekdays so that you can get an overall read on the lead and lag relationships. You can also search for day of the month dummy variables, but this is typically only necessary for CASH demand type problems. You can find the need to specify a week of the month dummy for special cases as well.
You can build a pivot table in Excel to compare the coefficients in the model to the % of the total for the day of the week and month of the year to get a "poor man's model" to confirm that the coefficients make sense.