Forecasting Levels in Loglinear Unit Root Models
研究了当数据序列取自然对数后建模为带漂移的随机游走时，如何对原始水平值进行无偏预测，推导了增长率和水平值的无偏预测公式，并证明条件无偏预测不存在，最后应用于比特币价格和英国工业产出预测。
This article considers unbiased prediction of levels when data series are modeled as a random walk with drift and other exogenous factors after taking natural 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. 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 Mean Squared Forecast Error (MSFE) than usual forecasts. The results are applied to Bitcoin price levels and a disaggregated eight sector model of UK industrial production.