Hi I am working on a data representing daily sales of a product. I have five years of data and I need to forecast daily sales for next 30 days. As I have daily data I created my ts series as noted bellow;
x<-ts(project.shortt$x,frequency = 365.25,start=2014+1/365.25)
then I skipped to
auto.arima modelling and determine my model as ARIMA(0,1,5)+ exogenous holiday, day of week and fourier terms.
fit <- auto.arima(x.ts, xreg=cbind(fourier(x, K=7),dummies.training_week), seasonal=TRUE,biasadj = TRUE,lambda="auto")
by using fourier, I am able to cover seasonalities, but also for remaining seasonal effects I left argument,
However, unless I had a stationary residual series coming from ARIMAX, and they are also nearly normally distributed, I observe a very problematic ACF plot, which indicates statistically significant auto correlations for all lags.
I am not sure what's the problem but if I am not wrong an I(0) stationary is not expected to contain any auto correlations for any lags. Because finally it follows a white noise pattern. I am not sure what causes the problem, is there weird kind of seasonality in my series or not?
Moreover, after searching through questions on daily forecasts I come upon one of robjhyndman's answers.
After changing my ts's frequency as Professor Hyndman suggested my ACF figures differs significantly
y <- ts(x, frequency=7)
But my Ljung-Box statistics still refers that I have significant auto correlations in my data.
Ljung-Box test data: Residuals from Regression with ARIMA(3,0,2)(2,0,0) errors Q* = 118.89, df = 3, p-value < 2.2e-16 Model df: 28. Total lags used: 31
I am not sure what is the difference that caused by professor's suggestion. It's not a aggregation of my data to weekly frequency, because I am still working with 1903 daily observations. On the other hand if this transformation means seasonality should be in weekly frequency I am not sure how can I follow yearly cycles in my data.