# Variations on rolling forecasts

#### Hyndsight

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)
``````