Solved – R: forecast function accuracy for ARIMA models

accuracyarimaforecastingr

I have a problem with the forecast function for ARIMA models in R. It calls predict that calls KalmanForecast. Ok…here's the deal.

the mean one-step forecast of the Arima object produced by this call

forecast(Arima, h=1)$mean[[1]]

is often significantly different from the result of a manual forecast by conditional expected value (best linear predictor).

For example a non seasonal Arima(1,1,1) without drift has of course the structure

y[t] = y[t-1] + AR1*(y[t-1] - y[t-2]) + MA1*epsilon[t-1] + epsilon[t]

so the one-step prediction is very straightforward 

y[t] = y[t-1] + AR1*(y[t-1] - y[t-2]) + MA1*epsilon[t-1]

but this result is always different from the result of the forecast function call.
Is it due to approximation errors in the Kalman recursion?

Try yourself with this code, it only needs the forecast package

  x = arima.sim(n = 1000, list(ar = c(0.8897, -0.4858), ma = c(-0.2279, 0.2488)),sd=sqrt(0.1796))
  t = length(x) + 1
 Arimafit = Arima(x = x, order = c(1,1,1), seasonal = list(order = c(0,0,0), period =     1), include.mean = FALSE,include.drift = FALSE)
 manualforecast = x[t-1] + coef(Arimafit)[["ar1"]]*(x[t-1] - x[t-2]) + coef(Arimafit)    [["ma1"]]*Arimafit$residuals[t-1]
     autoforecast = forecast(Arimafit, h = 1)$mean[[1]]

autoforecast is always different from manualforecast, sometimes significantly.

Best Answer

In this example, the model is mis-specified, and so the fitted coefficients might be close to the boundary of the stationarity or invertibility regions. This will cause instability in the computations.

I tried running your code twice. The first time, the estimated MA coefficient was -0.999999824 which is on the edge of the invertibility region. In that case, the two forecasts differed:

> manualforecast
[1] -0.4033295
> autoforecast
[1] -0.4036089

The second time I ran the code, both coefficients were well inside the boundaries, and the forecasts matched:

> manualforecast
[1] -0.777368178314693
> autoforecast
[1] -0.777368178314693
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