Forecasting Levels of log Variables in Vector Autoregressions

Sometimes forecasts of the original variable are of interest although 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 under normality assumptions is derived. Despite its theoretical advantages the optimal forecast is shown to be inferior to the naive forecast if specification and estimation uncertainty are taken into account. Hence, in practice using the exponential of the log forecast is preferable to using the optimal forecast.