Diagnostic analysis for a vector autoregressive model under Student′s t‐distributions

In this paper, we use the local influence method to study a vector autoregressive model under Student^′s t‐distributions. We present the maximum likelihood estimators and the information matrix. We establish the normal curvature diagnostics for the vector autoregressive model under three usual perturbation schemes for identifying possible influential observations. The effectiveness of the proposed diagnostics is examined by a simulation study, followed by our data analysis using the model to fit the weekly log returns of Chevron stock and the Standard & Poor's 500 Index as an application.     

Minimax optimality of CUSUM for an autoregressive model

Different change point models for AR(1) processes are reviewed. For some models, the change is in the distribution conditional on earlier observations. For others, the change is in the unconditional distribution. Some models include an observation before the first possible change time – others not. Earlier and new CUSUM type methods are given, and minimax optimality is examined. For the conditional model with an observation before the possible change, there are sharp results of optimality in the literature. The unconditional model with possible change at (or before) the first observation is of interest for applications. We examined this case and derived new variants of four earlier suggestions. By numerical methods and Monte Carlo simulations, it was demonstrated that the new variants dominate the original ones. However, none of the methods is uniformly minimax optimal.     

The exact moments of the least squares estimator for the autoregressive model

Exact mean and variance of the least squares estimate of the stationary first-order autoregressive coefficient, i.e., β in y_t=α+βx_t+u_t are evaluated algebraically as well as numerically. It turns out that the least squares estimate is seriously biased for the sample of two-digits sizes typically dealt with in econometrics if the mean of the process is unknown, i.e., if the equation has a non-zero intercept (α≠0). Kendall's approximation to the mean and Barlett's approximation to the variance are shown to be fairly good. Also, our numerical results confirm Orcutt and Winokur's (Econometrica, Vol. 37) based on Monte Carlo experiments.     

Testing for linear vector autoregressive dynamics under multivariate generalized autoregressive heteroskedasticity

In this paper, we propose a fixed design wild bootstrap procedure to test parameter restrictions in vector autoregressive models, which is robust in cases of conditionally heteroskedastic error terms. The wild bootstrap does not require any parametric specification of the volatility process and takes contemporaneous error correlation implicitly into account. Via a Monte Carlo investigation, empirical size and power properties of the method are illustrated for the case of white noise under the null hypothesis. We compare the bootstrap approach with standard ordinary least squares (OLS)-based, weighted least squares (WLS) and quasi-maximum likelihood (QML) approaches. In terms of empirical size, the proposed method outperforms competing approaches and achieves size-adjusted power close to WLS or QML inference. A White correction of standard OLS inference is satisfactory only in large samples. We investigate the case of Granger causality in a bivariate system of inflation expectations in France and the United Kingdom. Our evidence suggests that the former are Granger causal for the latter while for the reverse relation Granger non-causality cannot be rejected.     

The conditional autoregressive Wishart model for multivariate stock market volatility

We propose a Conditional Autoregressive Wishart (CAW) model for the analysis of realized covariance matrices of asset returns. Our model assumes an autoregressive moving average structure for the scale matrix of the Wishart distribution. It accounts for positive definiteness of covariance matrices without imposing parametric restrictions, and can be estimated by Maximum Likelihood. We also propose extensions of the CAW model obtained by including a Mixed Data Sampling (MIDAS) component and Heterogeneous Autoregressive (HAR) dynamics for long-run fluctuations. The CAW models are applied to realized variances and covariances for five New York Stock Exchange stocks.     

Estimation and testing for a Poisson autoregressive model

This article considers statistical inference for a Poisson autoregressive model. A condition for ergodicity and a necessary and sufficient condition for the existence of moments are given. Asymptotics for maximum likelihood estimator and weighted least squares estimators with estimated weights or known weights of the parameters are established. Testing conditional heteroscedasticity and testing the parameters under a simple ordered restriction are noted. A simulation study is also given.     

Asymptotic inference for a one-dimensional simultaneous autoregressive model

A nonstationary simultaneous autoregressive model \({X^{(n)}_k=\alpha \Big(X^{(n)}_{k-1}+X^{(n)}_{k+1}\Big)+\varepsilon_k, k=1, 2, \ldots , n-1}\), is investigated, where \({X^{(n)}_0}\) and \({X^{(n)}_n}\) are given random variables. It is shown that in the unstable case α = 1/2 the least squares estimator of the autoregressive parameter converges to a functional of a standard Wiener process with a rate of convergence n ², while in the stable situation |α| < 1/2 the estimator is biased but asymptotically normal with a rate n ^1/2.     

The cointegrated vector autoregressive model with general deterministic terms

In the cointegrated vector autoregression (CVAR) literature, deterministic terms have until now been analyzed on a case-by-case, or as-needed basis. We give a comprehensive unified treatment of deterministic terms in the additive model $X_t=\gamma Z_t+Y_t$, where $Z_t$ belongs to a large class of deterministic regressors and $Y_t$ is a zero-mean CVAR. We suggest an extended model that can be estimated by reduced rank regression, and give a condition for when the additive and extended models are asymptotically equivalent, as well as an algorithm for deriving the additive model parameters from the extended model parameters. We derive asymptotic properties of the maximum likelihood estimators and discuss tests for rank and tests on the deterministic terms. In particular, we give conditions under which the estimators are asymptotically (mixed) Gaussian, such that associated tests are ${\chi }^2$-distributed.     

Efficient estimation of the semiparametric spatial autoregressive model

Efficient semiparametric and parametric estimates are developed for a spatial autoregressive model, containing non-stochastic explanatory variables and innovations suspected to be non-normal. The main stress is on the case of distribution of unknown, nonparametric, form, where series nonparametric estimates of the score function are employed in adaptive estimates of parameters of interest. These estimates are as efficient as the ones based on a correct form, in particular they are more efficient than pseudo-Gaussian maximum likelihood estimates at non-Gaussian distributions. Two different adaptive estimates are considered, relying on somewhat different regularity conditions. A Monte Carlo study of finite sample performance is included.     

The adequacy of asymptotic approximations in the near-integrated autoregressive model with dependent errors

We consider the normalized least squares estimator of the parameter in a nearly integrated first-order autoregressive model with dependent errors. In a first step we consider its asymptotic distribution as well as asymptotic expansion up to order O_p(T⁻¹). We derive a limiting moment generating function which enables us to calculate various distributional quantities by numerical integration. A simulation study is performed to assess the adequacy of the asymptotic distribution when the errors are correlated. We focus our attention on two leading cases: MA(1) errors and AR(1) errors. The asymptotic approximations are shown to be inadequate as the MA root gets close to −1 and as the AR root approaches either −1 or 1. Our theoretical analysis helps to explain and understand the simulation results of Schwert (1989) and DeJong, Nankervis, Savin, and Whiteman (1992) concerning the size and power of Phillips and Perron's (1988) unit root test. A companion paper, Nabeya and Perron (1994), presents alternative asymptotic frameworks in the cases where the usual asymptotic distribution fails to provide an adequate approximation to the finite-sample distribution.     

Testing for autoregressive against moving average errors in the linear regression model

This paper presents a new approach to hypotheses testing problems which are non-nested in the classical sense and which concern the covariance matrix of the disturbance vector of the linear regression model. In particular, the application of the approach to testing for AR(1) disturbances against MA(1) disturbances is explored in some detail. Practical difficulties are discussed and selected upper bounds for the test's five percent significance points are tabulated. The small sample power of four versions of the new test are compared empirically and a clear conclusion is made in regard to the best overall test.     

Multinomial distributions applied to random sampling of particulate materials

When sampling a batch consisting of particulate material, the distribution of a sample estimator can be characterized using knowledge about the sample drawing process. With Bernoulli sampling, the number of particles in the sample is binomially distributed. Because this is rarely realized in practice, we propose a sampling design in which the possible samples have a nearly equal mass. Expected values and variances of the sample estimator are calculated. It is shown that the sample estimator becomes identical to the Horvitz–Thompson estimator in the case of a large batch-to-sample mass ratio and a large sample mass. Simulations and experiments were performed to test the theory. Simulations confirm that the round-off error due to the discrete nature of particles is negligible for large sample sizes. Sampling experiments were carried out with a mixture of PolyPropylene (PP) and PolyTetraFluorEthylene (PTFE) spheres suspended in a viscous medium. The measured and theoretical variations are in good agreement.     

Several efficient two-step estimators for the dynamic simultaneous equations model with autoregressive disturbances

Several asymptotically efficient methods are suggested on both the full and the limited information approach to estimate the simultaneous equations model in which the lagged endogenous variables and the autoregressive disturbances coexist. They are two-step procedures and do not involve iterations. A method is suggested also for the case where any portion of the autoregressive parameter matrix is specified to be zero. Since the consistency and efficiency depend upon the asymptotic, local identifiability, the necessary and sufficient condition is derived for it. It does not depend on the exclusion of the lagged endogenous variables.     

Representation,estimation and forecasting of the multivariate index-augmented autoregressive model

We examine the conditions under which each individual series that is generated by a vector autoregressive model can be represented as an autoregressive model that is augmented with the lags of a few linear combinations of all the variables in the system. We call this multivariate index-augmented autoregression (MIAAR) modelling. We show that the parameters of the MIAAR can be estimated by a switching algorithm that increases the Gaussian likelihood at each iteration. Since maximum likelihood estimation may perform poorly when the number of parameters increases, we propose a regularized version of our algorithm for handling a medium–large number of time series. We illustrate the usefulness of the MIAAR modelling by both empirical applications and simulations.     

Predictors for the first-order autoregressive process

The error made in predicting a first-order autoregressive process with unknown parameters is investigated. It is shown that the least squares predictor is unbiased for symmetric error distributions. Alternative predictors for stationary and non-stationary processes are studied using the Monte Carlo method. The ordinary least squares statistics perform reasonably well for one period predictions with samples as small as ten for both stationary and non-stationary processes. It is demonstrated that there is a considerable loss in efficiency when outdated estimators are used to construct predictors.     

Analyzing reciprocal relationships by means of the continuous-time autoregressive latent trajectory model

Over the past decades, several analytic tools have become available for the analysis of reciprocal relations in a non-experimental context using structural equation modeling (SEM). The autoregressive latent trajectory (ALT) model is a recently proposed model [BOLLEN and CURRAN Sociological Methods and Research (2004) Vol. 32, pp. 336–383; CURRAN and BOLLEN New Methods for the Analysis of Change (2001) American Psychological Association, Washington, DC], which captures features of both the autoregressive (AR) cross-lagged model and the latent trajectory (LT) model. The present article discusses strengths and weaknesses and demonstrates how several of the problems can be solved by a continuous-time version: the continuous-time autoregressive latent trajectory (CALT) model. Using SEM to estimate the exact discrete model (EDM), the EDM/SEM continuous-time procedure is applied to a CALT model of reciprocal relations between antisocial behavior and depressive symptoms.