I am using R and am trying to figure out how to merge some forecasts I've done via the vars package, back to my original xts data, which is in levels.
In previous work, I would have used only level data and used rbind to merge the old data with the forecasts, e.g.:
series <- rbind(orig, fcst)
However, the new forecasts I have are in differences, and I'm kind of at a loss as to how to merge the differenced data with the original data in levels. Does anyone have any suggestions?
If you have the last observed value $y_t$ and a foreast for the differenced series $\widehat{\Delta y}_{t+1} = \hat y_{t+1} - y_t $, then clearly the forecast of $y_{t+1}$ is $\hat y_{t+1} := y_t + \widehat{\Delta y}_{t+1}$.
Similarly, $\hat y_{t+2} := y_t + \widehat{\Delta y}_{t+1} + \widehat{\Delta y}_{t+2}$ and so on until $\hat y_{t+h} := y_t + \sum_{i=1}^h \widehat{\Delta y}_{t+i}$.
Thus use cumulative sums of the forecasts for the first differences added to the initial value $y_t$ to obtain forecasts for levels of $y_t$.
The R implementation would be apply(rbind(orig,fcst),2,cumsum).
