Two estimators of the long-run variance: Beyond short memory

This paper deals with the estimation of the long-run variance of a stationary sequence. We extend the usual Bartlett-kernel heteroskedasticity and autocorrelation consistent (HAC) estimator to deal with long memory and antipersistence. We then derive asymptotic expansions for this estimator and the memory and autocorrelation consistent (MAC) estimator introduced by Robinson [Robinson, P. M., 2005. Robust covariance matrix estimation: HAC estimates with long memory/antipersistence correction. Econometric Theory 21, 171–180]. We offer a theoretical explanation for the sensitivity of HAC to the bandwidth choice, a feature which has been observed in the special case of short memory. Using these analytical results, we determine the MSE-optimal bandwidth rates for each estimator. We analyze by simulations the finite-sample performance of HAC and MAC estimators, and the coverage probabilities for the studentized sample mean, giving practical recommendations for the choice of bandwidths.     

Adaptive forecasting in the presence of recent and ongoing structural change

We consider time series forecasting in the presence of ongoing structural change where both the time series dependence and the nature of the structural change are unknown. Methods that downweight older data, such as rolling regressions, forecast averaging over different windows and exponentially weighted moving averages, known to be robust to historical structural change, are found also to be useful in the presence of ongoing structural change in the forecast period. A crucial issue is how to select the degree of downweighting, usually defined by an arbitrary tuning parameter. We make this choice data-dependent by minimising the forecast mean square error, and provide a detailed theoretical analysis of our proposal. Monte Carlo results illustrate the methods. We examine their performance on 97 US macro series. Forecasts using data-based tuning of the data discount rate are shown to perform well.     

LARCH, Leverage, and Long Memory

We consider the long-memory and leverage properties of a modelfor the conditional variance of an observable stationary sequence X_t, where is the square of an inhomogeneous linear combination of X_s, s < t, withsquare summable weights b_j. This model, which we call linearautoregressive conditionally heteroskedastic (LARCH), specializes,when depends only on X_t–1, to theasymmetric ARCH model of Engle (1990, Review of Financial Studies3, 103–106), and, when depends only on finitely many X_s, to a version of the quadratic ARCH modelof Sentana (1995, Review of Economic Studies 62, 639–661),these authors having discussed leverage potential in such models.The model that we consider was suggested by Robinson (1991,Journal of Econometrics 47, 67–84), for use as a possiblylong-memory conditionally heteroskedastic alternative to i.i.d.behavior, and further studied by Giraitis, Robinson and Surgailis(2000, Annals of Applied Probability 10, 1002–1004), whoshowed that integer powers , =" BORDER="0">2 can have long-memory autocorrelations. We establish conditionsunder which the cross-autocovariance function between volatilityand levels, , decays in the manner of moving average weights of long-memory processes on suitable choiceof the b_j. We also establish the leverage property that h_t <0 for 0 < t k, where the value of k (which may be infinite)again depends on the b_j. Conditions for finiteness of thirdand higher moments of X_t are also established.     

Inference on stochastic time-varying coefficient models

Recently, there has been considerable work on stochastic time-varying coefficient models as vehicles for modelling structural change in the macroeconomy with a focus on the estimation of the unobserved paths of random coefficient processes. The dominant estimation methods, in this context, are based on various filters, such as the Kalman filter, that are applicable when the models are cast in state space representations. This paper introduces a new class of autoregressive bounded processes that decompose a time series into a persistent random attractor, a time varying autoregressive component, and martingale difference errors. The paper examines, rigorously, alternative kernel based, nonparametric estimation approaches for such models and derives their basic properties. These estimators have long been studied in the context of deterministic structural change, but their use in the presence of stochastic time variation is novel. The proposed inference methods have desirable properties such as consistency and asymptotic normality and allow a tractable studentization. In extensive Monte Carlo and empirical studies, we find that the methods exhibit very good small sample properties and can shed light on important empirical issues such as the evolution of inflation persistence and the purchasing power parity (PPP) hypothesis.