QMLE for Quadratic ARCH Model with Long Memory

We discuss parametric quasi‐maximum likelihood estimation for quadratic autoregressive conditional heteroskedasticity (ARCH) process with long memory introduced in Doukhan emphet al. (2016) and Grublyt? and ?karnulis (2016) with conditional variance involving the square of inhomogeneous linear combination of observable sequence with square summable weights. The aforementioned model extends the quadratic ARCH model of Sentana ( 1995 ) and the linear ARCH model of Robinson ( 1991 ) to the case of strictly positive conditional variance. We prove consistency and asymptotic normality of the corresponding quasi‐maximum likelihood estimates, including the estimate of long memory parameter 0 < d < 1/2. A simulation study of empirical mean‐squared error is included.     

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.     

On geometric ergodicity of CHARME models

In this article we consider a CHARME model, a class of generalized mixture of nonlinear nonparametric AR‐ARCH time series. To provide sets of conditions under which such processes are geometrically ergodic and, therefore, satisfy some mixing conditions, we apply the theory of Markov chains to derive asymptotic stability of this model. These results form the basis for deriving an asymptotic theory for nonparametric estimation. As an illustration, neural network sieve estimates for the autoregressive and volatility functions are considered, and consistency of the parameter estimates is obtained.     

Bootstrapping a weighted linear estimator of the ARCH parameters

Abstract.  A standard assumption while deriving the asymptotic distribution of the quasi maximum likelihood estimator in ARCH models is that all ARCH parameters must be strictly positive. This assumption is also crucial in deriving the limit distribution of appropriate linear estimators (LE). We propose a weighted linear estimator (WLE) of the ARCH parameters in the classical ARCH model and show that its limit distribution is multivariate normal even when some of the ARCH coefficients are zero. The asymptotic dispersion matrix involves unknown quantities. We consider appropriate bootstrapped version of this WLE and prove that it is asymptotically valid in the sense that the bootstrapped distribution (given the data) is a consistent estimate (in probability) of the distribution of the WLE. Although we do not show theoretically that the bootstrap outperforms the normal approximation, our simulations demonstrate that it yields better approximations than the limiting normal.     

Testing the Fit of a Vector Autoregressive Moving Average Model

Abstract. A new procedure for testing the fit of multivariate time series model is proposed. The method evaluates in a certain way the closeness of the sample spectral density matrix of the observed process to the spectral density matrix of the parametric model postulated under the null and uses for this purpose nonparametric estimation techniques. The asymptotic distribution of the test statistic is established and an alternative, bootstrap‐based method is developed in order to estimate more accurately this distribution under the null hypothesis. Goodness‐of‐fit diagnostics useful in understanding the test results and identifying sources of model inadequacy are introduced. The applicability of the testing procedure and its capability to detect lacks of fit is demonstrated by means of some real data examples.     

Negative Binomial Quasi‐Likelihood Inference for General Integer‐Valued Time Series Models

Two negative binomial quasi‐maximum likelihood estimates (NB‐QMLEs) for a general class of count time series models are proposed. The first one is the profile NB‐QMLE calculated while arbitrarily fixing the dispersion parameter of the negative binomial likelihood. The second one, termed two‐stage NB‐QMLE, consists of four stages estimating both conditional mean and dispersion parameters. It is shown that the two estimates are consistent and asymptotically Gaussian under mild conditions. Moreover, the two‐stage NB‐QMLE enjoys a certain asymptotic efficiency property provided that a negative binomial link function relating the conditional mean and conditional variance is specified. The proposed NB‐QMLEs are compared with the Poisson QMLE asymptotically and in finite samples for various well‐known particular classes of count time series models such as the Poisson and negative binomial integer‐valued GARCH model and the INAR(1) model. Application to a real dataset is given.     

Testing Parameter Change in General Integer‐Valued Time Series

We consider the structural change in a class of discrete valued time series, which the conditional distribution belongs to the one‐parameter exponential family. We propose a change point test based on the maximum likelihood estimator of the model's parameter. Under the null hypothesis (of no change), the test statistic converges to a well‐known distribution, allowing the calculation of the critical value of the test. The test statistic diverges to infinity under the alternative, meaning that the test has asymptotic power one. Some simulation results and real data applications are reported to show the effectiveness of the proposed procedure.     

Inference for the Fourth‐Order Innovation Cumulant in Linear Time Series

The rescaled fourth‐order cumulant of the unobserved innovations of linear time series is an important parameter in statistical inference. This article deals with the problem of estimating this parameter. An existing nonparametric estimator is first discussed, and its asymptotic properties are derived. It is shown how the autocorrelation structure of the underlying process affects the behaviour of the estimator. Based on our findings and on an important invariance property of the parameter of interest with respect to linear filtering, a pre‐whitening‐based nonparametric estimator of the same parameter is proposed. The estimator is obtained using the filtered time series only; that is, an inversion of the pre‐whitening procedure is not required. The asymptotic properties of the new estimator are investigated, and its superiority is established for large classes of stochastic processes. It is shown that for the particular estimation problem considered, pre‐whitening can reduce the variance and the bias of the estimator. The finite sample performance of both estimators is investigated by means of simulations. The new estimator allows for a simple modification of the multiplicative frequency domain bootstrap, which extends its considerable range of validity. Furthermore, the problem of testing hypotheses about the rescaled fourth‐order cumulant of the unobserved innovations is also considered. In this context, a simple test for Gaussianity is proposed. Some real‐life data applications are presented.     

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.     

Poisson QMLE of Count Time Series Models

Regularity conditions are given for the consistency of the Poisson quasi‐maximum likelihood estimator of the conditional mean parameter of a count time series model. The asymptotic distribution of the estimator is studied when the parameter belongs to the interior of the parameter space and when it lies at the boundary. Tests for the significance of the parameters and for constant conditional mean are deduced. Applications to specific integer‐valued autoregressive (INAR) and integer‐valued generalized autoregressive conditional heteroscedasticity (INGARCH) models are considered. Numerical illustrations, Monte Carlo simulations and real data series are provided.     

Estimation and testing for the parameters of ARCH(q) under ordered restriction

Abstract.  In this paper, we study a stationary ARCH( q ) model with parameters α ₀, α ₁, α ₂,…, α _q . It is known that the model requires all parameters α _i to be non-negative, but sometimes the usual algorithm based on Newton–Raphson's method leads us to obtain some negative solutions. So this study proposes a method of computing the maximum likelihood estimator (MLE) of parameters under the non-negative restriction. A similar method is also proposed for the case where the parameters are restricted by a simple order: α ₁≥ α ₂≥⋯≥ α _p . The strong consistency of the above two estimators is discussed. Furthermore, we consider the problem of testing homogeneity of parameters against the simple order restriction. We give the likelihood ratio (LR) test statistic for the testing problem and derive its asymptotic null distribution.     

Bootstrap Approximation to Prediction MSE for State–Space Models with Estimated Parameters

Abstract. We propose simple parametric and nonparametric bootstrap methods for estimating the prediction mean square error (PMSE) of state vector predictors that use estimated model parameters. As is well known, substituting the model parameters by their estimates in the theoretical PMSE expression that assumes known parameter values results in underestimation of the true PMSE. The parametric method consists of generating parametrically a large number of bootstrap series from the model fitted to the original series, re‐estimating the model parameters for each series using the same method as used for the original series and then estimating the separate components of the PMSE. The nonparametric method generates the series by bootstrapping the standardized innovations estimated for the original series. The bootstrap methods are compared with other methods considered in the literature in a simulation study that also examines the robustness of the various methods to non‐normality of the model error terms. Application of the bootstrap method to a model fitted to employment ratios in the USA that contains 18 unknown parameters, estimated by a three‐step procedure yields unbiased PMSE estimators.     

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.     

Statistical analysis of masked data in a hybrid system based on copula theory under progressive hybrid censoring

This article considers the reliability analysis of a hybrid system with dependent components, which are linked by a copula function. Based on Type I progressive hybrid censored and masked system lifetime data, we drive some probability results for the hybrid system and then the maximum likelihood estimates as well as the asymptotic confidence intervals and bootstrap confidence intervals of the unknown parameters are obtained. The effects of different dependence structures on the estimates of the parameter and the reliability function are investigated. Finally, Monte Carlo simulations are implemented to compare the performances of the estimates when the components are dependent with those when the components are independent.     

A TEST FOR CONDITIONAL HETEROSKEDASTICITY IN TIME SERIES MODELS

Abstract. When testing for conditional heteroskedasticity and nonlinearity, the power of the test in general depends on the functional forms of conditional heteroskedasticity and nonlinearity that are allowed under the alternative hypothesis. We suggest a test for conditional heteroskedasticity and nonlinearity with the nonlinear autoregressive conditional heteroskedasticity model of Higgins and Bera as the alternative. Standard testing procedures are not applicable since our nonlinear autoregressive conditional heteroskedasticity (ARCH) parameter is not identified under the null hypothesis. To resolve this problem, we apply the procedure recently proposed by Davies. Power and size of the suggested test are investigated through simulation, and an empirical application of testing for ARCH in exchange rates is also discussed.     

Consistent Estimation of the Value at Risk When the Error Distribution of the Volatility Model is Misspecified

A two‐step approach for conditional value at risk estimation is considered. First, a generalized quasi‐maximum likelihood estimator is employed to estimate the volatility parameter, then the empirical quantile of the residuals serves to estimate the theoretical quantile of the innovations. When the instrumental density h of the generalized quasi‐maximum likelihood estimator is not the Gaussian density, both the estimations of the volatility and of the quantile are generally asymptotically biased. However, the two errors counterbalance and lead to a consistent estimator of the value at risk. We obtain the asymptotic behavior of this estimator and show how to choose optimally h.     

Testing for co‐integration and nonlinear adjustment in a smooth transition error correction model

This article introduces a testing procedure for cointegration and nonlinear adjustment in a smooth transition vector error correction model. To overcome the unidentified parameters problem under the null of no‐cointegration, the Wald statistic is optimized over the unidentified parameter space. The asymptotic distribution of the test statistic is shown to be non‐standard but nuisance parameter‐free and hence critical values are obtained by simulations, Simulations show that the proposed test outperforms the alternatives in small sample sizes both in terms of size and power. Application to the exchange rate‐monetary fundamentals relationship show that the proposed test works considerably well. This article also finds that nonlinear adjustment dynamics are symmetric for some currencies and therefore the speed of adjustment depends on the size of the deviations and is asymmetric for others, hence, the adjustment dynamics depend not only on the size but also on the sign of the deviations.     

Testing the Null of Co‐integration in the Presence of Variance Breaks

Abstract. We show that changes in the innovation covariance matrix of a vector of series can generate spurious rejections of the null hypothesis of co‐integration when applying standard residual‐based co‐integration tests. A bootstrap solution to the inference problem is suggested which is shown to perform well in practice, redressing the size problems associated with the standard test but not losing power relative to the standard test under the alternative.     

Oracle M‐Estimation for Time Series Models

We propose a thresholding M‐estimator for multivariate time series. Our proposed estimator has the oracle property that its large‐sample properties are the same as of the classical M‐estimator obtained under the a priori information that the zero parameters were known. We study the consistency of the standard block bootstrap, the centred block bootstrap and the empirical likelihood block bootstrap distributions of the proposed M‐estimator. We develop automatic selection procedures for the thresholding parameter and for the block length of the bootstrap methods. We present the results of a simulation study of the proposed methods for a sparse vector autoregressive VAR(2) time series model. The analysis of two real‐world data sets illustrate applications of the methods in practice.     

Powerful Unit Root Tests Free of Nuisance Parameters

We propose a variance ratio‐type unit root test where the nuisance parameter cancels asymptotically under both the null of a unit root and a local‐to‐unity alternative. Critical values and asymptotic power curves can be computed using standard numerical techniques. Our test exhibits higher power compared with tests that share the virtue of being free of tuning parameters. In fact, the local asymptotic power curves of our procedure get close to the power functions of the point optimal test, where the latter suffers from the drawback of having to correct for a nuisance parameter consistently.