Laurent Callot

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.

Regularized estimation of structural instability in factor models: The US macroeconomy and the Great Moderation

This paper shows that the parsimoniously time-varying methodology of Callot and Kristensen (2015) can be applied to factor models. We apply this method to study macroeconomic instability in the US from 1959:1 to 2006:4 with a particular focus on the Great Moderation. Models with parsimoniously time-varying parameters are models with an unknown number of break points at unknown locations. The parameters are assumed to follow a random walk with a positive probability that an increment is exactly equal to zero so that the parameters do not vary at every point in time. The vector of increments, which is high dimensional by construction and sparse by assumption, is estimated using the Lasso. We apply this method to the estimation of static factor models and factor augmented autoregressions using a set of 190 quarterly observations of 144 US macroeconomic series from Stock and Watson (2009). We find that the parameters of both models exhibit a higher degree of instability in the period from 1970:1 to 1984:4 relative to the following 15 years. In our setting the Great Moderation appears as the gradual ending of a period of high structural instability that took place in the 1970s and early 1980s.

Vector Autoregressions with Parsimoniously Time Varying Parameters and an Application to Monetary Policy

This paper studies vector autoregressive models with parsimoniously time-varying parameters. The parameters are assumed to follow parsimonious random walks, where parsimony stems from the assumption that increments to the parameters have a non-zero probability of being exactly equal to zero. We estimate the sparse and high-dimensional vector of changes to the parameters with the Lasso and the adaptive Lasso. The parsimonious random walk allows the parameters to be modelled non parametrically, so that our model can accommodate constant parameters, an unknown number of structural breaks, or parameters varying randomly. We characterize the finite sample properties of the Lasso by deriving upper bounds on the estimation and prediction errors that are valid with high probability, and provide asymptotic conditions under which these bounds tend to zero with probability tending to one. We also provide conditions under which the adaptive Lasso is able to achieve perfectmodel selection. We investigate by simulations the properties of the Lasso and the adaptive Lasso in settings where the parameters are stable, experience structural breaks, or follow a parsimonious random walk. We use our model to investigate the monetary policy response to inflation and business cycle fluctuations in the US by estimating a parsimoniously time varying parameter Taylor rule. We document substantial changes in the policy response of the Fed in the 1970s and 1980s, and since 2007, but also document the stability of this response in the rest of the sample.

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.

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.

Natural Funnel Asymmetries: A Simulation Analysis of the Three Basic Tools of Meta Analysis

Meta-analysis studies a set of estimates of one parameter with three basic tools: The funnel diagram is the distribution of the estimates as a function of their precision; the funnel asymmetry test, FAT; and the meta average, where PET is an estimate. The FAT-PET MRA is a meta regression analysis, on the data of the funnel, which jointly estimates the FAT and the PET. Ideal funnels are lean and symmetric. Empirical funnels are wide, and most have asymmetries biasing the plain average. Many asymmetries are due to censoring made during the research-publication process. The PET is tooled to correct the average for censoring. We show that estimation faults and misspecification may cause natural asymmetries, which the PET does not correct. If the MRA includes controls for omitted variables, the PET does correct for omitted variables bias. Thus, it is important to know the reason for an asymmetry.