Forecasting weekly demand: based on ACF and PACF, is ARIMA appropriate? I am new to the ADF test in statistics. I need to verify if ARIMA forecasting is suitable for this time series data. I have demand data in weeks that runs from 2012 to 2015. I need to find out if it is appropriate to use ARIMA to forecast. I have run the ACF and PACF but the problem is interpretation of the two graphs.

 A: The answer to your question is "YES" if you also include week-of-the-year dummy indicators, a few indicators for some anomalies (pulses) and a variable to reflect a change in one of the week-of-the-year variables.
Before the dawn of time analysts would take weekly data and analyze it with 51 weekly dummies (deterministic structure) and a possible time trend and call it quits. Your data is best modelled with this general approach while also incorporating a useful ARIMA component. Box and Jenkins introduced/popularized the concept of using memory i.e. previous values and construct a possible SARIMA model. Being careful they premised/stated that efficient SARIMA model identification/application required data that was free of deterministic structure i.e trends using the counting numbers , level shifts and seasonal dummies otherwise standard ACF/PACF identification would be flawed/difficult/incorrect/useless. It appears that this caveat has been widely ignored in the rush to construct SARIMA models.  But not everywhere !
AUTOBOX a piece of software that I have helped to develop actually approaches the problem using both approaches and then merges statistically significant structure. I ran the software in a totally automatic mode and obtained what I think is "useful model" .
Your model is a hybrid of 51 dummies , a few pulses reflecting anomalies and an ARIMA model of the form (1,0,0)(1,0,0)52. The problem is that the untreated deterministic structure of 51 dummies obfuscated/blocked the identification of the ARIMA structure. 
Here is the plot of the original data  and the final model's Actual/Fit and Forecast .   . The residuals from this model are presented here  . The ACF of the residuals is presented here   suggesting model sufficiency. The plot of the forecasts is here  . The Actual and Cleansed plot is here 
The model statistics are summarized here 
and the model is presented here in two images ..
 and   .
Hope this helps ... Note that the AR(1) coefficient is nearly 1.0 which suggest that an equivalent ARIMA model could easily be (0,1,0)(1,0,0)52 .
