In this paper we extend the Baillie and Baltagi (1999) paper (Prediction from the regression model with one-way error components. In Analysis of Panels and Limited Dependent Variables Models, Hsiao C, Lahiri K, Lee LF, Pesaran H (eds). Cambridge University Press, Cambridge, UK). In particular, we derive six predictors for the two-way error components model, as well as their associated asymptotic mean squared error (AMSE) of multi-step prediction. In addition, we also provide both theoretical and simulation evidence as to the relative effi ciency of our six alternative predictors. The adequacy of the prediction AMSE formula is also investigated by the use of Monte Carlo methods which indicate that the ordinary optimal predictors perform well for various accuracy criteria.

Causal inference methods have been developed for longitudinal observational study designs where confounding is thought to occur over time. In particular, one may estimate and contrast the population mean counterfactual outcome under specific exposure patterns. In such contexts, confounders of the longitudinal treatment‐outcome association are generally identified using domain‐specific knowledge. However, this may leave an analyst with a large set of potential confounders that may hinder estimation. Previous approaches to data‐adaptive model selection for this type of causal parameter were limited to the single time‐point setting. We develop a longitudinal extension of a collaborative targeted minimum loss‐based estimation (C‐TMLE) algorithm that can be applied to perform variable selection in the models for the probability of treatment with the goal of improving the estimation of the population mean counterfactual outcome under a fixed exposure pattern. We investigate the properties of this method through a simulation study, comparing it to G‐Computation and inverse probability of treatment weighting. We then apply the method in a real‐data example to evaluate the safety of trimester‐specific exposure to inhaled corticosteroids during pregnancy in women with mild asthma. The data for this study were obtained from the linkage of electronic health databases in the province of Quebec, Canada. The C‐TMLE covariate selection approach allowed for a reduction of the set of potential confounders, which included baseline and longitudinal variables.

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In this paper we extend the works of Baillie and Baltagi (1999, in Analysis of Panels and Limited Dependent Variables Models, Hsiao C et al. (eds). Cambridge University Press: Cambridge, UK; 255-267) and generalize certain results from the Baltagi and Li (1992, Journal of Forecasting 11: 561-567) paper accounting for AR(1) errors in the disturbance term. In particular, we derive six predictors for the one-way error components model, as well as their associated asymptotic mean squared error of multi-step prediction in the presence of AR(1) errors in the disturbance term. In addition, we also provide both theoretical and simulation evidence as to the relative efficiency of our alternative predictors. The adequacy of the prediction AMSE formula is also investigated by the use of Monte Carlo methods and indicates that the ordinary optimal predictor performs well for various accuracy criteria.