The restricted likelihood ratio test for autoregressive processes

The restricted likelihood is known to produce estimates with significantly less bias in AR(p) models with intercept and/or trend. In AR(1) models, the corresponding restricted likelihood ratio test (RLRT), unlike the t‐statistic or the usual LRT, has been recently shown to be well approximated by the chi‐square distribution even close to the unit root, thus yielding confidence intervals with good coverage properties. In this article, we extend this result to AR(p) processes of arbitrary order p by obtaining the expansion of the RLRT distribution around that of the limiting chi‐squared and showing that the error is bounded even as the unit root is approached. Next, we investigate the correspondence between the AR coefficients and the partial autocorrelations, which is well known in the stationary region, and extend to the more general situation of potentially multiple unit roots. In the case of one positive unit root, which is of most practical interest, the resulting parameter space is shown to be the bounded p‐dimensional hypercube (?1, 1] × (?1, 1)^p?1. This simple parameter space, in addition with a stable algorithm that we provide for computing the restricted likelihood, allows its easy computation and optimization as well as construction of confidence intervals for the sum of the AR coefficients. In simulations, the RLRT intervals are shown to have not only near exact coverage in keeping with our theoretical results, but also shorter lengths and significantly higher power against stationary alternatives than other competing interval procedures. An application to the well‐known Nelson–Plosser data yields RLRT based intervals that can be markedly different from those in the literature.     

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.     

RESULTS ON ESTIMATION AND TESTING FOR A UNIT ROOT IN THE NONSTATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODEL

Abstract. We review the limiting distribution theory for Gaussian estimation of the univariate autoregressive moving-average (ARMA) model in the presence of a unit root in the autoregressive (AR) operator, and present the asymptotic distribution of the associated likelihood ratio (LR) test statistic for testing for a unit root in the ARMA model. The finite sample properties of the LR statistic as well as other unit root test procedures for the ARMA model are examined through a limited simulation study. We conclude that, for practical empirical work that relies on standard computations, the LR test procedure generally performs better than other standard procedures in the presence of a substantial moving-average component in the ARMA model.     

Boundary Limit Theory for Functional Local to Unity Regression

This article studies functional local unit root models (FLURs) in which the autoregressive coefficient may vary with time in the vicinity of unity. We extend conventional local to unity (LUR) models by allowing the localizing coefficient to be a function which characterizes departures from unity that may occur within the sample in both stationary and explosive directions. Such models enhance the flexibility of the LUR framework by including break point, trending, and multidirectional departures from unit autoregressive coefficients. We study the behavior of this model as the localizing function diverges, thereby determining the impact on the time series and on inference from the time series as the limits of the domain of definition of the autoregressive coefficient are approached. This boundary limit theory enables us to characterize the asymptotic form of power functions for associated unit root tests against functional alternatives. Both sequential and simultaneous limits (as the sample size and localizing coefficient diverge) are developed. We find that asymptotics for the process, the autoregressive estimate, and its t‐statistic have boundary limit behavior that differs from standard limit theory in both explosive and stationary cases. Some novel features of the boundary limit theory are the presence of a segmented limit process for the time series in the stationary direction and a degenerate process in the explosive direction. These features have material implications for autoregressive estimation and inference which are examined in the article.     

Estimation in nonstationary random coefficient autoregressive models

Abstract.  We investigate the estimation of parameters in the random coefficient autoregressive (RCA) model X _k  = ( φ  +  b _k ) X _{k −1} +  e _k , where ( φ ,  ω ²,  σ ²) is the parameter of the process,     ,     . We consider a nonstationary RCA process satisfying E  log | φ  +  b ₀| ≥ 0 and show that σ ² cannot be estimated by the quasi-maximum likelihood method. The asymptotic normality of the quasi-maximum likelihood estimator for ( φ ,  ω ²) is proven so that the unit root problem does not exist in the RCA model.     

TESTING FOR A UNIT ROOT IN AN AR(1) TIME SERIES USING IRREGULARLY OBSERVED DATA

Abstract. For an AR(1) model having a unit root with nonconsecutively observed or missing data we consider the ordinary least squares estimator, the one-step Newton-Raphson estimator and an ordinary least squares type estimator which is a simple approximation of the Newton-Raphson estimator. It is shown that the limiting distributions of these estimators of the unit root are the same as those of the regression estimators as tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Statist. Assoc. 74 (1979), 427–31) for the complete data situation. Simulation results show that our proposed unit root tests perform very well for small samples.     

A computationally convenient unit root test with covariates,conditional heteroskedasticity and efficient detrending

When testing for a unit root in a time series, in spite of the well‐known power problem of univariate tests, it is quite common to use only the information regarding the autoregressive behaviour contained in that series. In a series of influential papers, Elliott et al. (Efficient tests for an autoregressive unit root, Econometrica 64, 813–836, 1996), Hansen (Rethinking the univariate approach to unit root testing: using covariates to increase power, Econometric Theory 11, 1148–1171, 1995a) and Seo (Distribution theory for unit root tests with conditional heteroskedasticity, Journal of Econometrics 91, 113–144, 1999) showed that this practice can be rather costly and that the inclusion of the extraneous information contained in the near‐integratedness of many economic variables, their heteroskedasticity and their correlation with other covariates can lead to substantial power gains. In this article, we show how these information sets can be combined into a single unit root test.     

RELATIVE POWER OF t TYPE TESTS FOR STATIONARY AND UNIT ROOT PROCESSES

Abstract. This paper shows numerically that the lack of power and size distortions of the Dickey-Fuller type tests for unit roots (very well documented in the unit root literature) are similar to and in many situations even smaller than the lack of power and size distortions of the standard Student t tests for stationary roots of an autoregressive model.     

A Sieve Bootstrap For The Test Of A Unit Root

Abstract. In this paper, we consider a sieve bootstrap for the test of a unit root in models driven by general linear processes. The given model is first approximated by a finite autoregressive integrated process of order increasing with the sample size, and then the method of bootstrap is applied for the approximated autoregression to obtain the critical values for the usual unit root tests. The resulting tests, which may simply be viewed as the bootstrapped versions of Augmented Dickey–Fuller (ADF) unit root tests by Said and Dickey (1984 ), are shown to be consistent under very general conditions. The asymptotic validity of the bootstrap ADF unit root tests is thus established. Our conditions are significantly weaker than those used by Said and Dickey. Simulations show that bootstrap provides substantial improvements on finite sample sizes of the tests.     

Range Unit‐Root (RUR) Tests: Robust against Nonlinearities,Error Distributions,Structural Breaks and Outliers

Abstract. Since the seminal paper by Dickey and Fuller in 1979, unit‐root tests have conditioned the standard approaches to analysing time series with strong serial dependence in mean behaviour, the focus being placed on the detection of eventual unit roots in an autoregressive model fitted to the series. In this paper, we propose a completely different method to test for the type of long‐wave patterns observed not only in unit‐root time series but also in series following more complex data‐generating mechanisms. To this end, our testing device analyses the unit‐root persistence exhibited by the data while imposing very few constraints on the generating mechanism. We call our device the range unit‐root (RUR) test since it is constructed from the running ranges of the series from which we derive its limit distribution. These nonparametric statistics endow the test with a number of desirable properties, the invariance to monotonic transformations of the series and the robustness to the presence of important parameter shifts. Moreover, the RUR test outperforms the power of standard unit‐root tests on near‐unit‐root stationary time series; it is invariant with respect to the innovations distribution and asymptotically immune to noise. An extension of the RUR test, called the forward–backward range unit‐root (FB‐RUR) improves the check in the presence of additive outliers. Finally, we illustrate the performances of both range tests and their discrepancies with the Dickey–Fuller unit‐root test on exchange rate series.     

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.     

Least Squares Bias in Time Series with Moderate Deviations from a Unit Root

This articles derives the approximate bias of the least squares estimator of the autoregressive coefficient in discrete autoregressive time series where the autoregressive coefficient is given by α_T=1+c/k_T, with k_T being a deterministic sequence increasing to infinity at a rate slower than T, such that k_T=o(T) as T→∞. The cases in which c<0, c=0 and c>0 are considered correspond to (moderately) stationary, non‐stationary and (moderately) explosive series respectively. The result is used to derive the limiting distribution of the indirect inference method for such processes with moderate deviations from a unit root and for explosive series with a fixed coefficient, which does not depend on the sample size. Second, the result demonstrates why the jackknife estimator cannot be constructed for explosive time series for values of the autoregressive parameter close to unity in view of the discontinuity of the bias function, which the article derives. Finally, the expression is used to construct a bias‐corrected estimator, and simulations are carried out to assess the three estimators' bias reduction capabilities.     

Testing for a Unit Root in Autoregressive Moving-average Models with Missing Data

Testing for a single autoregressive unit root in an autoregressive moving-average (ARMA) model is considered in the case when data contain missing values. The proposed test statistics are based on an ordinary least squares type estimator of the unit root parameter which is a simple approximation of the one-step Newton–Raphson estimator. The limiting distributions of the test statistics are the same as those of the regression statistics in AR(1) models tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Stat. Assoc . 74 (1979), 427–31) for the complete data situation. The tests accommodate models with a fitted intercept and a fitted time trend.     

Bootstrapping unit root tests for integrated processes

Abstract. In this paper, we consider two bootstrap algorithms for testing unit roots under the condition that the observed process is unit root integrated. The first method consists of generating the resampled data after fitting an autoregressive model to the first differences of the observations. The second method consists of applying the stationary bootstrap to the first differences. Both procedures are shown to give methods that approach the correct asymptotic distribution under the null hypothesis of a unit root. We also present a Monte-Carlo study comparing the two methods for some ARIMA models.     

Bayesian Inference on Periodicities and Component Spectral Structure in Time Series

We detail and illustrate time series analysis and spectral inference in autoregressive models with a focus on the underlying latent structure and time series decompositions. A novel class of priors on parameters of latent components leads to a new class of smoothness priors on autoregressive coefficients, provides for formal inference on model order, including very high order models, and leads to the incorporation of uncertainty about model order into summary inferences. The class of prior models also allows for subsets of unit roots, and hence leads to inference on sustained though stochastically time-varying periodicities in time series. Applications to analysis of the frequency composition of time series, in both time and spectral domains, is illustrated in a study of a time series from astronomy. This analysis demonstrates the impact and utility of the new class of priors in addressing model order uncertainty and in allowing for unit root structure. Time-domain decomposition of a time series into estimated latent components provides an important alternative view of the component spectral characteristics of a series. In addition, our data analysis illustrates the utility of the smoothness prior and allowance for unit root structure in inference about spectral densities. In particular, the framework overcomes supposed problems in spectral estimation with autoregressive models using more traditional model-fitting methods.     

THE ZERO-CROSSING RATE OF AUTOREGRESSIVE PROCESSES AND ITS LINK TO UNIT ROOTS

Abstract. The asymptotic zero-crossing rate (ZCR) of the general second-order autoregressive process is investigated. When the associated characteristic polynomial has a unit root e ^iθ (0 ≤θ≤π), the ZCR converges in mean square to θ/π and the rate of convergence is very fast regardless of the noise level.     

Weak identification in the ESTAR model and a new model

Determining good parameter estimates in (exponential smooth transition autoregressive) models is known to be difficult. We show that the phenomena of getting strongly biased estimators is a consequence of the so‐called identification problem, the problem of properly distinguishing the transition function in relation to extreme parameter combinations. This happens in particular for either very small or very large values of the error term variance. Furthermore, we introduce a new alternative model – the TSTAR model – which has similar properties as the ESTAR model but reduces the effects of the identification problem. We also derive a linearity and a unit root test for this model.     

ALTERNATIVE ESTIMATORS AND UNIT ROOT TESTS FOR THE AUTOREGRESSIVE PROCESS

Abstract. We compare several estimators for the second-order autoregressive process and compare the associated tests for a unit root. Monte Carlo results are reported for the ordinary least squares estimator, the simple symmetric least squares estimator and the weighted symmetric least squares estimator. The weighted symmetric least squares estimator of the autoregressive parameters generally has smaller mean square error than that of the ordinary least squares estimator, particularly when one root is close to one in absolute value. For the second-order model with known zero intercept, the one-sided ordinary least squares test for a unit root is more powerful than the symmetric tests. For the model with an estimated intercept, the one-sided weighted symmetric least squares test is the most powerful test.     

Parameter change test for random coefficient integer-valued autoregressive processes with application to polio data analysis

Abstract.  In this paper, we consider the problem of testing for a parameter change in a first-order random coefficient integer-valued autoregressive [RCINAR(1)] model. We employ the cumulative sum (CUSUM) test based on the conditional least-squares and modified quasi-likelihood estimators. It is shown that under regularity conditions, the CUSUM test has the same limiting distribution as the supremum of the squares of independent Brownian bridges. The CUSUM test is then applied to the analysis of the monthly polio counts data set.     

Dickey–Fuller, Lagrange Multiplier and Combined Tests for a Unit Root in Autoregressive Time Series

In this paper we investigate (augmented) Dickey–Fuller (DF) and Lagrange multiplier (LM) type unit root tests for autoregressive time series through comprehensive Monte Carlo simulations. We consider two sorts of null and alternative hypotheses: a unit root without drift versus level stationarity and a unit root with drift versus trend stationarity. The DF-type coef ficient tests are found to show the best overall performance in both cases, at least if the sample size is sufficiently large. How ever, it is also found that the DF and LM tests are roughly complementary with regard to their finite-sample power. We therefore consider combining these two types of unit root tests to obtain ( ad hoc 'but') 'robust' test procedures. Critical values for the proposed tests are provided