The restricted likelihood ratio test at the boundary in autoregressive series

Abstract.  The restricted likelihood ratio test, RLRT, for the autoregressive coefficient in autoregressive models has recently been shown to be second-order pivotal when the autoregressive coefficient is in the interior of the parameter space and so is very well approximated by the     distribution. In this article, the non-standard asymptotic distribution of the RLRT for the unit root boundary value is obtained and is found to be almost identical to that of the     in the right tail. Together, these two results imply that the     distribution approximates the RLRT distribution very well even for near unit root series and transitions smoothly to the unit root distribution.     

NON‐STATIONARITY AND QUASI‐MAXIMUM LIKELIHOOD ESTIMATION ON A DOUBLE AUTOREGRESSIVE MODEL

This article first studies the non‐stationarity of the first‐order double AR model, which is defined by the random recurrence equation , where γ₀ > 0, α₀ ≥ 0, and {η_t}is a sequence of i.i.d. symmetric random variables. It is shown that the double AR(1) model is explosive under the condition . Based on this, it is shown that the quasi‐maximum likelihood estimator of (φ₀,α₀) is consistent and asymptotically normal so that the unit root problem does not exist in the double AR(1) model. Simulation studies are carried out to assess the performance of the quasi‐maximum likelihood estimator in finite samples.     

Improvement of the quasi‐likelihood ratio test in ARMA models: some results for bootstrap methods

Abstract. Quasi‐likelihood ratio tests for autoregressive moving‐average (ARMA) models are examined. The ARMA models are stationary and invertible with white‐noise terms that are not restricted to be normally distributed. The white‐noise terms are instead subject to the weaker assumption that they are independently and identically distributed with an unspecified distribution. Bootstrap methods are used to improve control of the finite sample significance levels. The bootstrap is used in two ways: first, to approximate a Bartlett‐type correction; and second, to estimate the p‐value of the observed test statistic. Some simulation evidence is provided. The bootstrap p‐value test emerges as the best performer in terms of controlling significance levels.     

Finite Sample Theory of QMLE in ARCH Models with Dynamics in the Mean Equation

Abstract. We provide simulation and theoretical results concerning the finite‐sample theory of quasi‐maximum‐likelihood estimators in autoregressive conditional heteroskedastic (ARCH) models when we include dynamics in the mean equation. In the setting of the AR(q)–ARCH(p), we find that in some cases bias correction is necessary even for sample sizes of 100, especially when the ARCH order increases. We warn about the existence of important biases and potentially low power of the t‐tests in these cases. We also propose ways to deal with them. We also find simulation evidence that when conditional heteroskedasticity increases, the mean‐squared error of the maximum‐likelihood estimator of the AR(1) parameter in the mean equation of an AR(1)‐ARCH(1) model is reduced. Finally, we generalize the Lumsdaine [J. Bus. Econ. Stat. 13 (1995) pp. 1–10] invariance properties for the biases in these situations.     

Asymptotic Theory and Unified Confidence Region for an Autoregressive Model

Although some unified inferences for the coefficient in an AR(1) model have been proposed in the literature, it remains open as to how to construct a unified confidence region for the intercept and the coefficient jointly without a prior on whether the sequence is stationary or unit root or near unit root or moderate deviations from a unit root or explosive and whether the sequence has a zero or nonzero constant intercept. After deriving the joint limit of the least squares estimator for all of these cases, this article proposes a unified empirical likelihood confidence region by first splitting the data into two parts and then constructing some weighted score equations. The good finite sample performance of the proposed method is demonstrated via a simulation study. Real data applications are provided as well.     

Testing for a Unit Root in Noncausal Autoregressive Models

This work develops maximum likelihood‐based unit root tests in the noncausal autoregressive (NCAR) model with a non‐Gaussian error term formulated by Lanne and Saikkonen (2011, Journal of Time Series Econometrics 3, Issue 3, Article 2). Finite‐sample properties of the tests are examined via Monte Carlo simulations. The results show that the size properties of the tests are satisfactory and that clear power gains against stationary NCAR alternatives can be achieved in comparison with available alternative tests. In an empirical application to a Finnish interest rate series, evidence in favour of an NCAR model with leptokurtic errors is found.     

Computer Algebra Derivation of the Bias of Linear Estimators of Autoregressive Models

Abstract. A symbolic method which can be used to obtain the asymptotic bias and variance coefficients to order O(1/n) for estimators in stationary time series is discussed. Using this method, the large‐sample bias of the Burg estimator in the AR(p) for p = 1, 2, 3 is shown to be equal to that of the least squares estimators in both the known and unknown mean cases. Previous researchers have only been able to obtain simulation results for the Burg estimator's bias because this problem is too intractable without using computer algebra. The asymptotic bias coefficient to O(1/n) of Yule–Walker as well as least squares estimates is also derived in AR(3) models. Our asymptotic results show that for the AR(3), just as in the AR(2), the Yule–Walker estimates have a large bias when the parameters are near the nonstationary boundary. The least squares and Burg estimates are much better in this situation. Simulation results confirm our findings.     

Bartlett Correction of Empirical Likelihood for Non‐Gaussian Short‐Memory Time Series

Bartlett correction, which improves the coverage accuracies of confidence regions, is one of the desirable features of empirical likelihood. For empirical likelihood with dependent data, previous studies on the Bartlett correction are mainly concerned with Gaussian processes. By establishing the validity of Edgeworth expansion for the signed root empirical log‐likelihood ratio statistics, we show that the Bartlett correction is applicable to empirical likelihood for short‐memory time series with possibly non‐Gaussian innovations. The Bartlett correction is established under the assumptions that the variance of the innovation is known and the mean of the underlying process is zero for a single parameter model. In particular, the order of the coverage errors of Bartlett‐corrected confidence regions can be reduced from O(n^?1) to O(n^?2).     

EXACT MAXIMUM LIKELIHOOD ESTIMATION IN AUTOREGRESSIVE PROCESSES

Abstract. The purpose of this paper is to complement the theory of exact maximum likelihood estimation in pure autoregressive processes by differentiating the exact Gaussian likelihood function with respect to the model parameters and obtaining a set of likelihood equations very similar in form to the Yule—Walker equations. The main contribution of this paper is a very simple expression for the derivatives and the resulting likelihood equations in terms of the components of a (p+ 1) x (p+ 1) function of the data, the model parameters (s?², φ) and the autocovariances at lags 0 through p. We propose an iterative algorithm for solving the likelihood equations by alternately solving two linear systems, first for (s?², φ) given current estimates of the autocovariances, then for updated estimates of the autocovariances given current estimates of (s?², φ). The number of operations per iteration is independent of the series length since the algorithm uses the data only through the value of the (p+ 1) x (p+ 1) sufficient statistic.     

A Gaussian Mixture Autoregressive Model for Univariate Time Series

The Gaussian mixture autoregressive model studied in this article belongs to the family of mixture autoregressive models, but it differs from its previous alternatives in several advantageous ways. A major theoretical advantage is that, by the definition of the model, conditions for stationarity and ergodicity are always met and these properties are much more straightforward to establish than is common in nonlinear autoregressive models. Another major advantage is that, for a pth‐order model, explicit expressions of the stationary distributions of dimension p + 1 or smaller are known and given by mixtures of Gaussian distributions with constant mixing weights. In contrast, the conditional distribution given the past observations is a Gaussian mixture with time‐varying mixing weights that depend on p lagged values of the series in a natural and parsimonious way. Because of the known stationary distribution, exact maximum likelihood estimation is feasible and one can assess the applicability of the model in advance by using a non‐parametric estimate of the stationary density. An empirical example with interest rate series illustrates the practical usefulness and flexibility of the model, particularly in allowing for level shifts and temporary changes in variance. Copyright © 2014 Wiley Publishing Ltd     

Local Likelihood for non‐parametric ARCH(1) models

Abstract. We propose a non‐parametric local likelihood estimator for the log‐transformed autoregressive conditional heteroscedastic (ARCH) (1) model. Our non‐parametric estimator is constructed within the likelihood framework for non‐Gaussian observations: it is different from standard kernel regression smoothing, where the innovations are assumed to be normally distributed. We derive consistency and asymptotic normality for our estimators and show, by a simulation experiment and some real‐data examples, that the local likelihood estimator has better predictive potential than classical local regression. A possible extension of the estimation procedure to more general multiplicative ARCH(p) models with p > 1 predictor variables is also described.     

Effects of outliers on the identification and estimation of GARCH models

Abstract. This paper analyses how outliers affect the identification of conditional heteroscedasticity and the estimation of generalized autoregressive conditionally heteroscedastic (GARCH) models. First, we derive the asymptotic biases of the sample autocorrelations of squared observations generated by stationary processes and show that the properties of some conditional homoscedasticity tests can be distorted. Second, we obtain the asymptotic and finite sample biases of the ordinary least squares (OLS) estimator of ARCH(p) models. The finite sample results are extended to generalized least squares (GLS), maximum likelihood (ML) and quasi‐maximum likelihood (QML) estimators of ARCH(p) and GARCH(1,1) models. Finally, we show that the estimated asymptotic standard deviations are biased estimates of the sample standard deviations.     

Frequency domain generalized empirical likelihood method

This paper is concerned with a version of empirical likelihood method for spectral restrictions, which handles stationary time series data via the frequency domain approach. The asymptotic properties of frequency domain generalized empirical likelihood are studied for either strictly stationary processes with vanishing cumulant spectral density function of order 4 or linear processes generated by iid innovations with possibly non‐zero fourth order cumulant. Several statistics for testing parametric restrictions, over‐identified spectral restrictions, and additional spectral restrictions are shown to have the limiting chi‐squared distributions. Some numerical results are presented to investigate the finite sample performance of the proposed procedures. Copyright © 2013 John Wiley & Sons, Ltd.     

Local Whittle estimation of multi‐variate fractionally integrated processes

This article derives a semi‐parametric estimator of multi‐variate fractionally integrated processes covering both stationary and non‐stationary values of d. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the multi‐variate local Whittle estimator of Shimotsu (2007) to cover non‐stationary values of d. Consistency and asymptotic normality is shown for d ∈ (?1/2,∞). A simulation study illustrates the performance of the proposed estimator for relevant sample sizes. Empirical justification of the proposed estimator is shown through an empirical analysis of log spot exchange rates. We find that the log spot exchange rates of Germany, United Kingdom, Japan, Canada, France, Italy and Switzerland against the US Dollar for the period January 1974 until December 2001 are well decribed as I(1) processes.     

Limiting distributions of unconditional maximum likelihood unit root test statistics in seasonal time–series models

Abstract.  The likelihood function of a seasonal model, Y _t  =  ρ Y _{t − d}  +  e _t as implemented in computer algorithms under the assumption of stationary initial conditions is a function of ρ which is zero at the point ρ  = 1. It is a smooth function for ρ in the above seasonal model with a well-defined maximum regardless of the data-generating mechanism. Gonzalez-Farias (PhD Thesis, North Carolina State University, 1992) proposed tests for unit roots based on maximizing the stationary likelihood function in nonseasonal time series. We extend it to seasonal time series. The limiting distribution of seasonal unit root test statistics based on the unconditional maximum likelihood estimators are shown. Models having a single mean, seasonal means, and a single-trend variable across the seasons are considered.     

Stability of nonlinear AR‐GARCH models

Abstract. This article studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p [AR(p)] with the conditional variance specified as a nonlinear first‐order generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and β‐mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance, and only require mild moment conditions.     

An approximate likelihood function for panel data with a mixed ARMA(p, q) remainder disturbance model

Abstract. An approximate likelihood function for panel data with an autoregressive moving‐average (ARMA)(p, q) model remainder disturbance is presented and Whittle's approximate maximum likelihood estimator (MLE) is used to yield an asymptotic estimator. Although an asymptotic approach, the power test is quite successful for estimating and testing. In this approach, we do not need to calculate the transformation matrix in exact form. Through the Riemann sum approach, we can construct a simple approximate concentrated likelihood function. In addition, the model is also extended to the restricted maximum likelihood (REML) function, in which the package of Gilmour, Thompson and Cullis [Biometrics (1995) Vol. 51, pp. 1440–1450] is applied without difficulty. In the case study, we implement the model on the characteristic line for the investment analysis of Taiwanese computer motherboard makers.     

UNIFIED INTERVAL ESTIMATION FOR RANDOM COEFFICIENT AUTOREGRESSIVE MODELS

The consistency of the quasi‐maximum likelihood estimator for random coefficient autoregressive models requires that the coefficient be a non‐degenerate random variable. In this article, we propose empirical likelihood methods based on weighted‐score equations to construct a confidence interval for the coefficient. We do not need to distinguish whether the coefficient is random or deterministic and whether the process is stationary or non‐stationary, and we present two classes of equations depending on whether a constant trend is included in the model. A simulation study confirms the good finite‐sample behaviour of our resulting empirical likelihood‐based confidence intervals. We also apply our methods to study US macroeconomic data.     

Discrete‐Time Approximation of a Cogarch(p,q) Model and its Estimation

In this article, we construct a sequence of discrete‐time stochastic processes that converges in the Skorokhod metric to a COGARCH(p,q) model. The result is useful for the estimation of the COGARCH(p,q) on irregularly spaced time series data. The proposed estimation procedure is based on the maximization of a pseudo log‐likelihood function and is implemented in the yuima package.     

Penalised Complexity Priors for Stationary Autoregressive Processes

The autoregressive (AR) process of order p(AR(p)) is a central model in time series analysis. A Bayesian approach requires the user to define a prior distribution for the coefficients of the AR(p) model. Although it is easy to write down some prior, it is not at all obvious how to understand and interpret the prior distribution, to ensure that it behaves according to the users' prior knowledge. In this article, we approach this problem using the recently developed ideas of penalised complexity (PC) priors. These prior have important properties like robustness and invariance to reparameterisations, as well as a clear interpretation. A PC prior is computed based on specific principles, where model component complexity is penalised in terms of deviation from simple base model formulations. In the AR(1) case, we discuss two natural base model choices, corresponding to either independence in time or no change in time. The latter case is illustrated in a survival model with possible time‐dependent frailty. For higher‐order processes, we propose a sequential approach, where the base model for AR(p) is the corresponding AR(p?1) model expressed using the partial autocorrelations. The properties of the new prior distribution are compared with the reference prior in a simulation study.