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Rolling forecasts are commonly used to compare time series models. Here are a few of the ways they can be computed using R. I will use ARIMA models as a vehicle of illustration, but the code can easily be adapted to other univariate time series models.

### One-step forecasts without re-estimation

The simplest approach is to estimate the model on a single set of training data, and then compute one-step forecasts on the remaining test data. This can be handled by applying the fitted model to the whole data set, and then extracting the “fitted values” which are simply one-step forecasts.

 ```library(fpp) train <- window(hsales,end=1989.99) fit <- auto.arima(train) refit <- Arima(hsales, model=fit) fc <- window(fitted(refit), start=1990)```

### Multi-step forecasts without re-estimation

For multi-step forecasts, a loop is required. The following example computes 5-step forecasts:

 ```h <- 5 train <- window(hsales,end=1989.99) test <- window(hsales,start=1990) n <- length(test) - h + 1 fit <- auto.arima(train) fc <- ts(numeric(n), start=1990+(h-1)/12, freq=12) for(i in 1:n) { x <- window(hsales, end=1989.99 + (i-1)/12) refit <- Arima(x, model=fit) fc[i] <- forecast(refit, h=h)\$mean[h] }```

### Multi-step forecasts with re-estimation

An alternative approach is to extend the training data and re-estimate the model at each iteration, before each forecast is computed. This is what I call “time series cross-validation” because it is analogous to leave-one-out cross-validation for cross-sectional data. This time, I will store the forecasts from 1– to 5-steps ahead at each iteration.

 ```# Multi-step, re-estimation h <- 5 train <- window(hsales,end=1989.99) test <- window(hsales,start=1990) n <- length(test) - h + 1 fit <- auto.arima(train) order <- arimaorder(fit) fcmat <- matrix(0, nrow=n, ncol=h) for(i in 1:n) { x <- window(hsales, end=1989.99 + (i-1)/12) refit <- Arima(x, order=order[1:3], seasonal=order[4:6]) fcmat[i,] <- forecast(refit, h=h)\$mean }```

A variation on this also re-selects the model at each iteration. Then the second line in the loop is replaced with

 `refit <- auto.arima(x)`