On the Optimality of the LR Test for Mediation

Testing for mediation, or indirect effects, is empirically very important in many disciplines. It has two obvious symmetries that the testing procedure should be invariant to. The ordered absolute t-statistics from two ordinary regressions are maximal invariant under the associated groups of transformations. Sobel’s (1982) Wald-type and the LR test statistic are both functions of this maximal invariant and satisfy two logical coherence requirements: (1) size coherence: rejection at level α implies rejection at all higher significance levels; and (2) information coherence: more (less) evidence against the null implies continued (non) rejection of the null. The LR test statistic is simply the smallest of the two absolute t-statistics, and we show that the LR test is the Uniformly Most Powerful (information and size) Coherent Invariant (UMPCI) test. In short: the LR test for mediation is simple and best

Optimal Prediction in Loglinear Models

This paper introduces a Laplace inversion technique for deriving unbiased predictors in exponential families. This general technique is applied to derive the exact optimal unbiased predictor in loglinear models with Gaussian disturbances under quadratic loss. An exact unbiased estimator for its variance is also derived. The result generalizes earlier work and unifies expressions in terms of a simple hypergeometric function which has a number of advantages. Nonlinear models rarely admit exact solutions and we therefore compare the exact predictor with other predictors commonly used in nonlinear models. The naive predictor which is biased and inconsistent, can be best in terms of mean squared error, even for sample sizes of up to 40

Forecasting Levels in Loglinear Unit Root Models

This paper 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

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