Bernd Droge

Estimators of the Mean Squared Error of Prediction in Linear Regression

If a linear regression model is used for prediction, the mean squared error of prediction (MSEP) measures the performance of the model. The MSEP is a function of unknown parameters and good estimates of it are of interest. This article derives a best unbiased estimator and a minimum MSE estimator under the assumption of a normal distribution. It compares the bias and the MSE of these estimators and some others. Similar results are presented for the case in which the model is used to estimate values of the response function.

Model Selection, Transformations and Variance Estimation in Nonlinear Regression

The results of analyzing experimental data using a parametric model may heavily depend on the chosen model for regression and variance functions, moreover also on a possibly underlying preliminary transformation of the variables. In this paper we propose and discuss a complex procedure which consists in a simultaneous selection of parametric regression and variance models from a relatively rich model class and of Box-Cox variable transformations by minimization of a cross-validation criterion. For this it is essential to introduce modifications of the standard cross-validation criterion adapted to each of the following objectives: 1. estimation of the unknown regression function, 2. prediction of future values of the response variable, 3. calibration or 4. estimation of some parameter with a certain meaning in the corresponding field of application. Our idea of a criterion oriented combination of procedures (which usually if applied, then in an independent or sequential way) is expected to lead to more ac...

Some Comments on Cross-Validation

A new variant of cross-validation, called full cross-validation, is proposed in order to overcome some disadvantages of the traditional cross-validation approach in general regression situations. Both criteria may be regarded as estimates of the mean squared error of prediction. Under some assumptions including normally distributed observations, the cross-validation criterion is shown to be outperformed by the full cross-validation criterion. Analogous modifications may be applied to the generalized cross-validation method, providing a similar improvement. This leads to the recommendation of replacing the traditional cross-validation techniques by the new ones for estimating the prediction quality of models or of regression function estimators.

On finite-sample properties of adaptive least squares regression estimates

This paper is mainly concerned with deriving finite-sample properties of least squares estimators for the regression function in a nonparametric regression situation under some simplifying assumptions such as normally distributed errors with a common known variance. The selection of basis functions to be used for the construction of an estimator may be regarded as a smoothing problem, and will usually be done in a data-dependent way, A straightforward application of a result by P. J. Kernpthorne yields that, under a squared error loss, all selection procedures are admissible. Furthermore, the minimax approach provides an interpolating estimator, which is often impractical, Therefore, within a certain class of selection procedures an optimal one is determined using the minimax regret principle. It can be seen to behave similarly to the procedure minimizing either an unbiased risk estimator or, equivalently, the Cp-criterion.

Panel cointegration testing in the presence of a time trend

A new likelihood-based panel cointegration test which allows a linear time trend in the data generating process is proposed. The test is an extension of the likelihood ratio type test with trend adjustment prior to testing to the panel data framework. Under the null hypothesis, the standardized statistic has a limiting normal distribution as the number of time periods and the number of cross-sections tend to infinity sequentially. Additionally, an approximation involving the moments based on a vector autoregressive process of order one is introduced. A Monte Carlo study demonstrates that the test has reasonable size and high power in finite samples.

Asymptotic optimality of full cross-validation for selecting linear regression models

For the problem of model selection, full cross-validation has been proposed as an alternative criterion to the traditional cross-validation, particularly in cases where the latter is not well defined. To justify the use of the new proposal we show that under some conditions, both criteria share the same asymptotic optimality property when selecting among linear regression models.

A note on estimating the msep in nonlinear regression

We consider the problem of estimating the mean squared error of prediction in the nonlinear regression case using the bootstrap and cross–validation approach.Emphasis is put on numerical economy in calculating the different estimates. For comparison,alterative proposals known from the literature are briefly discussed.Our in vestigation lead to the recommendation of some variants of bootstrap estimates.

Some Simulation Results on Cross-Validation and Competitors for Model Choice

The behaviour of model selection procedures based on different criteria such as cross-validation is investigated in a simulation study. Emphasis is on the relationship to the problem of estimating the prediction quality of a model.

Asymptotic properties of model selection procedures in linear regression

In the regression analysis, there is typically a large collection of competing models available from which we want to select an appropriate one. This article is concerned with asymptotic properties of procedures for selecting linear models, which are based on certain data-dependent criteria such as Mallows’ C p , cross-validation and the generalized information criterion. We avoid the assumption of an adequate (‘correct’) model and allow the maximal model dimension to increase with the sample size. General asymptotic concepts are introduced, covering the usual ones of consistency and asymptotic optimality. The focus is on conditions for penalizing the model complexity, which are necessary to obtain the different optimality properties. For example, the consistency of a procedure is decided by the interplay between these penalties, the complexity of the class of model candidates, and some quantity describing the ability to identify ‘wrong’ (pseudo-inadequate) models. Many results known from the literature appear as special cases or are slightly modified.

The relative importance of group-level effects on the performance of German companies

We examine the impact of performance groups on the estimation of the relative importance of firm, industry and other effects on corporate performance. Performance groups comprise firms from the same industry with a similar performance over a longer period of time. We present a statistical method which improves the procedure of variance decomposition by allowing firm effects and the interacting effects of firms and time to be unified into the group effects. Applied to a German data set of 219 companies observed over a period of eleven years (1987-1997) it appears that the majority of the firms can be ascribed to performance groups. The variance proportion of the group effects is about one half of the non-grouped firm effects. They explain about 17.9 percent of the total variance of the returns.