Volume 4, Issue 1

We derive the relation between the biases of correlograms and of estimates of auto-regressive AR(k) representations of stationary series, and we illustrate it with a simple AR example. The new relation allows for k to vary with the sample size, which is a representation that can be used for most stationary processes. As a result, the biases of the estimators of such processes can now be quantified explicitly and in a unified way.

In a fractionally cointegrated model, we analyze, both theoretically and by means of a Monte Carlo experiment, the performance of the most popular first stage estimation methods, including ordinary and narrow band least squares (Robinson, 1994), difference taper narrow band least squares (Chen and Hurvich, 2003a), instrumental variables (Robinson and Gerolimetto, 2006), and compare it with the behavior of a new proposal, the integrated ordinary least squares. An appropriate version of this latter estimator (and also of the instrumental variables one) achieves in all circumstances the fastest convergence rate (among other first stage methods) and performs well in finite samples. The use of improved first stage methods is most important in cases of low collective memory of regressor and cointegrating error. This is particularly relevant in multivariate settings, where the key parameters which rule the convergence properties of the estimators are the memories of adjacent cointegrating subspaces.

This article develops a new Markov breaks (MB) model for forecasting and making inference in linear regression models with breaks that are stochastic in both timing and magnitude. The MB model permits an arbitrarily large number of abrupt breaks in the regression coefficients and error variance, but it maintains a low-dimensional state space, and therefore it is computationally straightforward. In particular, the likelihood function can be computed analytically using a single iterative pass through the data and thereby avoids Monte Carlo integration. The model generates forecasts and conditional coefficient predictions using a probability weighted average over regressions that include progressively more historical data. I employ the MB model to study the predictive ability of the yield curve for quarterly GDP growth. I show evidence of breaks in the predictive relationship, and the MB model outperforms competing breaks models in an out-of-sample forecasting experiment.

We present some theoretical results on semi-parametric regression models in the presence of autocorrelated errors using design-adapted Haar wavelets. We prove that the risks for the linear and nonlinear estimators are asymptotically almost minimax when the errors have absolutely summable autocovariances. For the nonlinear estimator, we also need a strong mixing property with a specific coefficient and a condition on the errors' higher-order moments. Some simulations ilustrate the theoretical achievements.