Empirical Limits for Time Series Econometric Models

This paper characterizes empirically achievable limits for time series econometric modeling and forecasting. The approach involves the concept of minimal information loss in time series regression and the paper shows how to derive bounds that delimit the proximity of empirical measures to the true probability measure (the DGP) in models that are of econometric interest. The approach utilizes joint probability measures over the combined space of parameters and observables and the results apply for models with stationary, integrated, and cointegrated data. A theorem due to Rissanen is extended so that it applies directly to probabilities about the relative likelihood (rather than averages), a new way of proving results of the Rissanen type is demonstrated, and the Rissanen theory is extended to nonstationary time series with unit roots, near unit roots, and cointegration of unknown order. The corresponding bound for the minimal information loss in empirical work is shown not to be a constant, in general, but to be proportional to the logarithm of the determinant of the (possibility stochastic) Fisher–information matrix. In fact, the bound that determines proximity to the DGP is generally path dependent, and it depends specifically on the type as well as the number of regressors. For practical purposes, the proximity bound has the asymptotic form (K/2)log n, where K is a new dimensionality factor that depends on the nature of the data as well as the number of parameters in the model. When ‘good’ model selection principles are employed in modeling time series data, we are able to show that our proximity bound quantifies empirical limits even in situations where the models may be incorrectly specified. One of the main implications of the new result is that time trends are more costly than stochastic trends, which are more costly in turn than stationary regressors in achieving proximity to the true density. Thus, in a very real sense and quantifiable manner, the DGP is more elusive when there is nonstationarity in the data. The implications for prediction are explored and a second proximity theorem is given, which provides a bound that measures how close feasible predictors can come to the optimal predictor. Again, the bound has the asymptotic form (K/2)log n, showing that forecasting trends is fundamentally more difficult than forecasting stationary time series, even when the correct form of the model for the trends is known.     

On implementing Jeffreys' substitution likelihood for Bayesian inference concerning the medians of unknown distributions

When statisticians are uncertain as to which parametric statistical model to use to analyse experimental data, they will often resort to a non-parametric approach. The purpose of this paper is to provide insight into a simple approach to take when it is unclear as to the appropriate parametric model and plan to conduct a Bayesian analysis. I introduce an approximate, or substitution likelihood, first proposed by Harold Jeffreys in 1939 and show how to implement the approach combined with both a non-informative and an informative prior to provide a random sample from the posterior distribution of the median of the unknown distribution. The first example I use to demonstrate the approach is a within-patient bioequivalence design and then show how to extend the approach to a parallel group design.     

3-6岁儿童比例推理、空间类比与近似数量系统的发展及关系

为了探究3-6岁儿童比例推理、空间类比与近似数量系统的发展特点，以及三者之间存在的关系及变化,选取120名3-6岁儿童为被试用“草莓酱任务”“找鸡蛋任务”以及Panamath进行测查，结果发现，3岁儿童就已经具有初步的比例推理、空间类比能力；儿童在3-6岁期间，比例推理、空间类比、近似数量系统随年龄增长而提高；比例推理、空间类比与近似数量系统三者之间相关性显著，但这种相关性在3-6岁阶段随年龄增长而减弱。控制性别与年龄后，3-6岁儿童的比例推理与空间类比可以预测儿童的近似数量系统，比例推理能够调节空间类比对近似数量系统的影响。