Forecasting Levels of log Variables in Vector Autoregressions

Sometimes forecasts of the original variable are of interest al- though a variable appears in logarithms (logs) in a system of time series. In that case converting the forecast for the log of the variable to a naive forecast of the original variable by simply applying the exponential transformation is not optimal theoretically. A simple expression for the optimal forecast un- der normality assumptions is derived. Despite its theoretical advantages the optimal forecast is shown to be inferior to the naive forecast if speci¯cation and estimation uncertainty are taken into account. Hence, in practice using the exponential of the log forecast is preferable to using the optimal forecast.