Forecasting growth and levels in loglinear unit root models

This paper considers unbiased prediction of growth and levels when data series are modelled as a random walk with drift and other exogenous factors after taking logs. We derive the unique unbiased predictors for growth and its variance. Derivation of level forecasts is more involved because the last observation enters the conditional expectation and is highly correlated with the parameter estimates, even asymptotically. This leads to conceptual questions regarding conditioning on endogenous variables and we prove that no conditionally unbiased forecast exists. We derive forecasts that are unconditionally unbiased and take into account estimation uncertainty, non-linearity of the transformations, and the correlation between the last observation and estimate which is quantitatively more important than estimation uncertainty and future disturbances together. The exact unbiased forecasts are shown to have lower MSFE than usual forecasts. We derive exact unbiased estimators of the MSFE and show that they can succesfully be used in the construction of forecast intervals. The results are applied to a disaggregated eight sector model of UK industrial production.