Anders Bredahl Kock

Oracle Inequalities for High Dimensional Vector Autoregressions

This paper establishes non-asymptotic oracle inequalities for the prediction error and estimation accuracy of the LASSO in stationary vector autoregressive models. These inequalities are used to establish consistency of the LASSO even when the number of parameters is of a much larger order of magnitude than the sample size. Furthermore, it is shown that under suitable conditions the number of variables selected is of the right order of magnitude and that no relevant variables are excluded. Next, non-asymptotic probabilities are given for the Adaptive LASSO to select the correct sign pattern (and hence the correct sparsity pattern). Finally conditions under which the Adaptive LASSO reveals the correct sign pattern with probability tending to one are given. Again, the number of parameters may be much larger than the sample size. Some maximal inequalities for vector autoregressions which might be of independent interest are contained in the appendix.

Forecasting with Nonlinear Time Series Models

In this paper, nonlinear models are restricted to mean nonlinear parametric models. Several such models popular in time series econometrics are presented and some of their properties discussed. This includes two models based on universal approximators: the Kolmogorov-Gabor polynomial model and two versions of a simple artificial neural network model. Techniques for generating multi-period forecasts from nonlinear models recursively are considered, and the direct (non-recursive) method for this purpose is mentioned as well. Forecasting with complex dynamic systems, albeit less frequently applied to economic forecasting problems, is briefly highlighted. A number of large published studies comparing macroeconomic forecasts obtained using different time series models are discussed, and the paper also contains a small simulation study comparing recursive and direct forecasts in a particular case where the data-generating process is a simple artificial neural network model. Suggestions for further reading conclude the paper.

Oracle Efficient Variable Selection In Random And Fixed Effects Panel Data Models

This paper generalizes the results for the Bridge estimator of Huang, Horowitz, and Ma ( 2008 ) to linear random and fixed effects panel data models which are allowed to grow in both dimensions. In particular, we show that the Bridge estimator isoracle efficient. It can correctly distinguish between relevant and irrelevant variables and the asymptotic distribution of the estimators of the coefficients of the relevant variables is the same as if only these had been included in the model, i.e. as if an oracle had revealed the true model prior to estimation.

CONSISTENT AND CONSERVATIVE MODEL SELECTION WITH THE ADAPTIVE LASSO IN STATIONARY AND NONSTATIONARY AUTOREGRESSIONS

We show that the adaptive Lasso is oracle efficient in stationary and nonstationary autoregressions. This means that it estimates parameters consistently, selects the correct sparsity pattern, and estimates the coefficients belonging to the relevant variables at the same asymptotic efficiency as if only these had been included in the model from the outset. In particular, this implies that it is able to discriminate between stationary and nonstationary autoregressions and it thereby constitutes an addition to the set of unit root tests. Next, and important in practice, we show that choosing the tuning parameter by Bayesian Information Criterion (BIC) results in consistent model selection. However, it is also shown that the adaptive Lasso has no power against shrinking alternatives of the form c/T if it is tuned to perform consistent model selection. We show that if the adaptive Lasso is tuned to perform conservative model selection it has power even against shrinking alternatives of this form and compare it to the plain Lasso.

Estimation and Forecasting of Large Realized Covariance Matrices and Portfolio Choice

In this paper we consider modeling and forecasting of large realized covariance matrices by penalized vector autoregressive models. We propose using Lasso-type estimators to reduce the dimensionality to a manageable one and provide strong theoretical performance guarantees on the forecast capability of our procedure. To be precise, we show that we can forecast future realized covariance matrices almost as precisely as if we had known the true driving dynamics of these in advance. We next investigate the sources of these driving dynamics for the realized covariance matrices of the 30 Dow Jones stocks and find that these dynamics are not stable as the data is aggregated from the daily to the weekly and monthly frequency. The theoretical performance guarantees on our forecasts are illustrated on the Dow Jones index. In particular, we can beat our benchmark by a wide margin at the longer forecast horizons. Finally, we investigate the economic value of our forecasts in a portfolio selection exercise and find that in certain cases an investor is willing to pay a considerable amount in order get access to our forecasts.

Forecasting Macroeconomic Variables Using Neural Network Models and Three Automated Model Selection Techniques

When forecasting with neural network models one faces several problems, all of which influence the accuracy of the forecasts. First, neural networks are often hard to estimate due to their highly nonlinear structure. To alleviate the problem, White (2006) presented a solution (QuickNet) that converts the specification and nonlinear estimation problem into a linear model selection and estimation problem. We shall compare its performance to that of two other procedures building on the linearization idea: the Marginal Bridge Estimator and Autometrics. Second, one must decide whether forecasting should be carried out recursively or directly. This choice is investigated in this work. The economic time series used in this study are the consumer price indices for the G7 and the Scandinavian countries. In addition, a number of simulations are carried out and results reported in the article.

Sharp Threshold Detection Based on Sup-norm Error rates in High-dimensional Models

We propose a new estimator, the thresholded scaled Lasso, in high-dimensional threshold regressions. First, we establish an upper bound on the ℓ∞ estimation error of the scaled Lasso estimator of Lee, Seo, and Shin. This is a nontrivial task as the literature on high-dimensional models has focused almost exclusively on ℓ1 and ℓ2 estimation errors. We show that this sup-norm bound can be used to distinguish between zero and nonzero coefficients at a much finer scale than would have been possible using classical oracle inequalities. Thus, our sup-norm bound is tailored to consistent variable selection via thresholding. Our simulations show that thresholding the scaled Lasso yields substantial improvements in terms of variable selection. Finally, we use our estimator to shed further empirical light on the long-running debate on the relationship between the level of debt (public and private) and GDP growth. Supplementary materials for this article are available online.