MAXIMUM LIKELIHOOD ESTIMATION OF AUTOCOVARIANCE MATRICES FROM REPLICATED SHORT TIME SERIES

Abstract. The maximum likelihood estimation of an autocovariance matrix based on replicated observations of stationary times series is considered. A sufficient condition for the existence of the estimate, when the sample covariance matrix is singular, is given. An iterative method for its computation is proposed: it is based on some spectral decompositions of Toeplitz matrices. Simulation results show the superiority of the estimate over the usual empirical sample autocovariance matrix.     

The Effect of the Estimation on Goodness‐of‐Fit Tests in Time Series Models

Abstract. We analyze, by simulation, the finite‐sample properties of goodness‐of‐fit tests based on residual autocorrelation coefficients (simple and partial) obtained using different estimators frequently used in the analysis of autoregressive moving‐average time‐series models. The estimators considered are unconditional least squares, maximum likelihood and conditional least squares. The results suggest that although the tests based on these estimators are asymptotically equivalent for particular models and parameter values, their sampling properties for samples of the size commonly found in economic applications can differ substantially, because of differences in both finite‐sample estimation efficiencies and residual regeneration methods.     

Unit Root Tests Based on Unconditional Maximum Likelihood Estimation for the Autoregressive Moving Average

Unconditional maximum likelihood estimation is considered for an autoregressive moving average that may contain an autoregressive unit root. The limiting distribution of the normalized maximum likelihood estimator of the unit root is shown to be the same as that of the estimator for the first-order autoregressive process. A likelihood ratio test based on unconditional maximum likelihood estimation is proposed. In a Monte Carlo study for the autoregressive moving-average model of order (1, 1), the new test is shown to have better size and power than those of several other tests.     

ASYMPTOTIC PROPERTIES OF THE SAMPLE AUTOCORRELATIONS AND PARTIAL AUTOCORRELATIONS OF A MULTIPLICATIVE ARIMA PROCESS

Abstract. We shall investigate the asymptotic behaviour of the sample autocorrelations and partial autocorrelations of a multiplicative ARIMA process and derive their limiting distributions. Some simulations are presented to illustrate the results obtained.     

Approximate Conditional Unit Root Inference

Based on Cox and Reid (1987) adjustments of likelihood ratio (LR) tests for unit roots in higher-order autoregressive models are proposed. While unit root inference does not fit directly into the framework of Cox and Reid, the ideas are applied in models with multi-dimensional parameters of interest and only asymptotic orthogonality of parameters. The adjustments are very simple to apply in that they are of the degrees of freedom type. Detailed Monte Carlo experiments reveal that, for a wide range of admissible parameter values, adjusted LR statistics approximate the asymptotic percentiles of the unit root distributions at a much faster rate than unadjusted ones. In addition, the proposed adjustments are compared with simulated Bartlett type corrected LR tests. They behave equally well in a reasonable parameter region, while both fail on the boundary of the parameter region where an additional unit root is introduced.     

Testing for serial dependence in time series models of counts

Abstract. In analysing time series of counts, the need to test for the presence of a dependence structure routinely arises. Suitable tests for this purpose are considered in this paper. Their size and power properties are evaluated under various alternatives taken from the class of INARMA processes. We find that all the tests considered except one are robust against extra binomial variation in the data and that tests based on the sample autocorrelations and the sample partial autocorrelations can help to distinguish between integer-valued first-order and second-order autoregressive as well as first-order moving average processes.     

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.     

TESTING CHANGE-POINTS IN THE EXPLOSIVE GAUSSIAN AUTOREGRESSIVE PROCESSES

Abstract. A functional limit theorem with a particular function class and topology is derived for non-ergodic type time series. This limit theorem allows us to study the asymptotic law of the associated likelihood ratio test (LRT) statistic for testing the presence of a change in the covariance parameter in the explosive Gaussian autoregressive model. We show that the level of the LRT cannot be approximated without introducing appropriate normalization. The limit law of a particular weighted likelihood ratio test is examined through a simulation study and is compared with the well-known Kolmogorov distribution obtained in the stationary case; we conclude that for practical applications when the root is really close to unity one can use the same thresholds as in the stationary case. This procedure is applied to the study of three real time series known to be non-stationary.     

Asymptotic Relative Efficiency of Goodness‐Of‐Fit Tests Based on Inverse and Ordinary Autocorrelations

Abstract. We compare the performance of the inverse and ordinary (partial) autocorrelations for time series model identification. It is found that, both in terms of Bahadur's slope and Pitman's asymptotic relative efficiency, the inverse partial autocorrelations are more efficient than the ordinary autocorrelations for identification of moving‐average models. By duality, the partial autocorrelations turn out to be more powerful than the inverse autocorrelations to identify autoregressive models. Numerical experiments on both simulated and real data sets are presented to highlight the theoretical results.     

Generalized Sequential Probability Ratio Test for Separate Families of Hypotheses

Abstract

In this article, we consider the problem of testing two separate families of hypotheses via a generalization of the sequential probability ratio test. In particular, the generalized likelihood ratio statistic is considered and the stopping rule is the first boundary crossing of the generalized likelihood ratio statistic. We show that this sequential test is asymptotically optimal in the sense that it achieves asymptotically the shortest expected sample size as the maximal type I and type II error probabilities tend to zero.     

ON THE UNIMODALITY OF THE EXACT LIKELIHOOD FUNCTION FOR NORMAL AR(2) SERIES

Abstract. For stationary second-order autoregressive normal processes, the conjecture of uniqueness of the solution of the exact likelihood equations is examined. A sufficient condition for uniqueness is given; this condition is satisfied with very high probability if the number of observations is not extremely small. Moreover, it is shown that not more than two maxima may exist. Examples of data which actually produce a likelihood function with two local maxima are given.     

ON THE STRUCTURE OF THE LIKELIHOOD FUNCTION OF AUTOREGRESSIVE AND MOVING AVERAGE MODELS

Abstract. In finite order normal moving average models the maximum likelihood estimates always exist. For finite order normal autoregressive models sufficient conditions for the existence of maximum likelihood estimates is given. Some cases not satisfying the conditions are studied.     

STATE SPACE MODELS WITH DIFFUSE INITIAL CONDITIONS

Abstract. A state space model with diffuse initial conditions is considered. A simple and direct proof of the algorithm for computing the likelihood function and minimum mean square estimators of the state is given