When should a time series model be used

#1
Consider the following scenario:
  1. i have a set of responses y(t),y(t-1),.....y(1), where Y's are continuous and fall between [0,1] , and "t" is the time.
  2. A set of predictors x(t),x(t-1),....x(1). Some of these predictors include day of the week, month of the year, etc that "t" falls into.
  3. The response and predictors are collected on a daily basis (one per day)
  4. Objective: multiple-ahead forecasting of y(t)
  5. Question: (a) What criteria should the data possess to make ARIMA a suitable model ? (b) are weak (insignificant) auto-correlation values based on ACF/PACF plots a sign that ARIMA models are not appropriate ?

After fitting regression model Y = bX + e, i plotted ACF and PACF of the regression residuals. The residuals of the fitted model (e-hat) show weak auto-correlations. For example only lag one is barely significant in ACF/PACF.

I tried using various configurations of ARIMAX in R to combine both the y = bx + e model with ARIMA, where the X part was the regression model:

arima(Y, order = c(p1, p2, p3), seasonal = list(order = c(p4, p5, p6), period = 7), xreg = X)

also,

arima(Y, order = c(p1, p2, p3), seasonal = list(order = c(p4, p5, p6), period = 7), xreg = E[Y|X])

where E[X|Y] is the expected response using the regression model y = bx + e.

but they do not seem to improve predictions beyond the linear regression and the forecasts from the model above are very similar to the prediction from the regression model.

I am concluding the ARIMA is not adding any additional value, and simple linear regression seems an appropriate model, but i wanted to double check in case i have missed something in my analyses.


Thank you very much
 
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#2
I think you can use simple progressive linear regression. I faced the same situation like yours. I used progressive simple linear equation and it seemed to work. For example, take 5 days to predict the 6th day and take 6 days data to predict the 7th day etc. Also, try higher order linear regression models.