I have seasonal time series (with frequency of 30). I am fitting ARIMA models using R library
My first ARIMA model would be (1,0,1)(1,1,0) with fitted parameters:
Coefficients: ar1 ma1 sar1 drift 0.6957 0.2992 -0.4496 0.8204 s.e. 0.0266 0.0398 0.0252 0.0597 sigma^2 estimated as 512.7: log likelihood=-5943.27 AIC=11896.53 AICc=11896.58 BIC=11922.42
Then I try second model, which is same model only now I perform seasonal differencing manually before ARIMA. The second model which ran on seasonally differenced data (1,0,1)(1,0,0) is fitted:
Coefficients: ar1 ma1 sar1 intercept 0.6959 0.2990 -0.4497 25.3671 s.e. 0.0266 0.0398 0.0252 1.8514 sigma^2 estimated as 500.8: log likelihood=-5943.27 AIC=11896.54 AICc=11896.58 BIC=11922.54
Then I calculate some future forecasts from both models (the forecasts from second model are transformed back via "inverse differencing") and compare the forecasts.
The prediction interval for the first model is very narrow while the prediction interval for the second model is "wide open". Why does the prediction intervals differ so much when it is "same" model?
The primary goal of task I am working on is to deliver prediction interval. But I am confused which prediction interval is now correct ?
The R code to fit both models for time series Y1 and to forecast 10 seasons ahead:
m1=Arima(y = Y1,order = c(1,0,1),seasonal=c(1,1,0),include.drift = T) f1=forecast(m1,level=c(.99),h=frequency(Y1)*10) sY1=diff(Y1,frequency(Y1)) m2=Arima(y = sY1,order = c(1,0,1),seasonal=c(1,0,0),include.mean = T) f2=forecast(m2,level=c(.99),h=frequency(Y1)*10) plot(f1$lower[,1],t="l",ylim=c(1e3,5e3)) lines(f1$upper[,1]) lines(diffinv(f2$lower[,1],lag=frequency(Y1),xi=tail(Y1,frequency(Y1)))[-(1:frequency(Y1))],col="red") lines(diffinv(f2$upper[,1],lag=frequency(Y1),xi=tail(Y1,frequency(Y1)))[-(1:frequency(Y1))],col="red")
To rephrase my question more generally:
Having two alternative models:
- Model M1 is an ARIMA(p,1,q) on time series Y
- Model M2 is an ARIMA(p,0,q) on time series Z=differenced(Y)
The AR and MA coefficients are equal for both models.
Then forecasting for future horizon H having two forecasted time series:
The mean forecasts is equal F1=F2 for the whole H horizon. But while the prediction interval of F1 are very narrow, the prediction interval of F2 is much wider.
Which prediction interval is correct ?