I took your 412 daily historical values an introduced them to AUTOBOX. After some interrogation they disclosed that not only was there memory in the data : ARIMA (1,0,0)(0,0,0)7 but that latent deterministic structure was found/detected .
The deterministic structure found was 2 significant months were higher (August and September) while the first day-of-the-week was significantly lower (curiously a Tuesday ! ).
In addition there were three level shifts (periods 42, 273 and 317 now visually obvious !) and some unusual one-time pulses.
The reasons that you din't discover this model was 1) you and your software of choice limited your analysis to memory only i.e. ARIMA and more importantly your tool of choice expressly ignored deterministic effects which are often predominant in daily data due to important human patterns or habits or some unknown possibly unspecified persistant effect.
Here is the model and the Actual/fit and forecast graph nicely projecting into the future 103 days. Here are the forecasts for your 112 days that you withheld from the analysis
The plot of the residuals suggests randomness which is supported by the acf of the residuals
Finally the Actual and Cleansed graph visually presents the Identified Interventions
In summary 1) no variance change(s) was detected 2) no parameter change(s) were detected . Note well that if you don't deal with level shifts/changes you can falsely accept the hypothesis that are error variance changes.
Finally there are 3 types of time series models
1) autoprojective using only the history
2) deterministic using fixed dummies ( pulses,level shifts, seasonal pulses,local time trends
3) models that integrate both 1) and 2) ...like yours !
Hope this helps you , your fellow students , and your instructor , and of course other readers of SE ...
For @Stats no limits on forecasts and no negative forecasts.....
@Stats asked for a graph showing constrained forecast and forecast limits .
The OP asked for direction as to how they could implement the solution I presented. One way is to acquire acquire AUTOBOX ..there is an R version with major discounts to Universities. AUTOBOX is the only program I am aware of that simultaneously optimizes/identifies both the minimally sufficient ARIMA structure and the statistically significant Intervention Detection structure for both a Univariate case and a Multivariate case while dealing with non-constant model parameters and non-constant error variance..
If you want to build such model or close to that( a model with memory and latent deterministic changes) , I can suggest the following articles;
2) Is it possible to automate time series forecasting?
3) How to determine order of sarima?