A nonparametric test of conditional autoregressive heteroscedasticity for threshold autoregressive models

Threshold autoregressive models are widely used in time‐series applications. When building or using such a model, it is important to know whether conditional heteroscedasticity exists. The authors propose a nonparametric test of this hypothesis. They develop the large‐sample theory of a test of nonlinear conditional heteroscedasticity adapted to nonlinear autoregressive models and study its finite‐sample properties through simulations. They also provide percentage points for carrying out this test, which is found to have very good power overall.     

Nonlinear Autoregression with Positive Innovations

Use of nonlinear models in analyzing time series data is becoming increasingly popular. This paper considers a broad class of nonlinear autoregressive models where the autoregressive part is additive and the terms are nonlinear functions of the past data. Also, the innovation distribution is supported on the non-negative reals and satisfies a tail regularity condition. The linear parameters of the autoregression are estimated using a linear programming recipe which yields much more accurate estimates than traditional methods such as conditional least squares. Limiting distribution of the linear programming estimators is obtained. Simulation studies validate the asymptotic results and reveal excellent small sample properties of the LPE estimator.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

A Markov Model of Switching-Regime ARCH

In this article I present a new approach to model more realistically the variability of financial time series. I develop a Markov-ARCH model that incorporates the features of both Hamilton's switching-regime model and Engle's autoregressive conditional heteroscedasticity (ARCH) model to examine the issue of volatility persistence in the monthly excess returns of the three-month treasury bill. The issue can be resolved by taking into account occasional shifts in the asymptotic variance of the Markov-ARCH process that cause the spurious persistence of the volatility process. I identify two periods during which there is a regime shift, the 1974:2–1974:8 period associated with the oil shock and the 1979:9–1982:8 period associated with the Federal Reserve's policy change. The variance approached asymptotically in these two episodes is more than 10 times as high as the asymptotic variance for the remainder of the sample. I conclude that regime shifts have a greater impact on the properties of the data, and I cannot reject the null hypothesis of no ARCH effects within the regimes. As a consequence of the striking findings in this article, previous empirical results that adopt an ARCH approach in modeling monthly or lower frequency interest-rate dynamics are rendered questionable.     

On a mixture autoregressive model

We generalize the Gaussian mixture transition distribution (GMTD) model introduced by Le and co-workers to the mixture autoregressive (MAR) model for the modelling of non-linear time series. The models consist of a mixture of K stationary or non-stationary AR components. The advantages of the MAR model over the GMTD model include a more full range of shape changing predictive distributions and the ability to handle cycles and conditional heteroscedasticity in the time series. The stationarity conditions and autocorrelation function are derived. The estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The shape changing feature of the conditional distributions makes these models capable of modelling time series with multimodal conditional distributions and with heteroscedasticity. The models are applied to two real data sets and compared with other competing models. The MAR models appear to capture features of the data better than other competing models do.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.     

Out-of-Sample Performance of Discrete-Time Spot Interest Rate Models

We provide a comprehensive analysis of the out-of-sample performance of a wide variety of spot rate models in forecasting the probability density of future interest rates. Although the most parsimonious models perform best in forecasting the conditional mean of many financial time series, we find that the spot rate models that incorporate conditional heteroscedasticity and excess kurtosis or heavy tails have better density forecasts. Generalized autoregressive conditional heteroscedasticity significantly improves the modeling of the conditional variance and kurtosis, whereas regime switching and jumps improve the modeling of the marginal density of interest rates. Our analysis shows that the sophisticated spot rate models in the existing literature are important for applications involving density forecasts of interest rates.     

The Gaussian Mixture Dynamic Conditional Correlation Model: Parameter Estimation,Value at Risk Calculation,and Portfolio Selection

A multivariate generalized autoregressive conditional heteroscedasticity model with dynamic conditional correlations is proposed, in which the individual conditional volatilities follow exponential generalized autoregressive conditional heteroscedasticity models and the standardized innovations follow a mixture of Gaussian distributions. Inference on the model parameters and prediction of future volatilities are addressed by both maximum likelihood and Bayesian estimation methods. Estimation of the Value at Risk of a given portfolio and selection of optimal portfolios under the proposed specification are addressed. The good performance of the proposed methodology is illustrated via Monte Carlo experiments and the analysis of the daily closing prices of the Dow Jones and NASDAQ indexes.     

Time-varying coefficient models with ARMA–GARCH structures for longitudinal data analysis

Time-varying coefficient models with autoregressive and moving-average–generalized autoregressive conditional heteroscedasticity structure are proposed for examining the time-varying effects of risk factors in longitudinal studies. Compared with existing models in the literature, the proposed models give explicit patterns for the time-varying coefficients. Maximum likelihood and marginal likelihood (based on a Laplace approximation) are used to estimate the parameters in the proposed models. Simulation studies are conducted to evaluate the performance of these two estimation methods, which is measured in terms of the Kullback–Leibler divergence and the root mean square error. The marginal likelihood approach leads to the more accurate parameter estimates, although it is more computationally intensive. The proposed models are applied to the Framingham Heart Study to investigate the time-varying effects of covariates on coronary heart disease incidence. The Bayesian information criterion is used for specifying the time series structures of the coefficients of the risk factors.     

Efficient inference for parameters of unobservable periodic autoregressive time series

This paper considers estimating the model coefficients when the observed periodic autoregressive time series is contaminated by a trend. The proposed Yule–Walker estimators are obtained by a two-step procedure. In the first step, the trend is estimated by a weighted local polynomial, and the residuals are obtained by subtracting the trend estimates from the observations; in the second step, the model coefficients are estimated by the well-known Yule–Walker method via the residuals. It is shown that under certain conditions such Yule–Walker estimators are oracally efficient, i.e., they are asymptotically equivalent to those obtained from periodic autoregressive time series without a trend. An easy-to-use implementation procedure is provided. The performance of the estimators is illustrated by simulation studies and real data analysis. In particular, the simulation studies show that the proposed estimator outperforms that obtained from the residuals when the trend is estimated by kernel smoothing without taking the heteroscedasticity into consideration.     

The Message in Daily Exchange Rates: A Conditional-Variance Tale

Formal testing procedures confirm the presence of a unit root in the autoregressive polynomial of the univariate time series representation of daily exchange-rate data. The first differences of the logarithms of daily spot rates are approximately uncorrelated through time, and a generalized autoregressive conditional heteroscedasticity model with daily dummy variables and conditionally t-distributed errors is found to provide a good representation to the leptokurtosis and time-dependent conditional heteroscedasticity. The parameter estimates and characteristics of the models are found to be very similar for six different currencies. These apparent stylized facts carry over to weekly, fortnightly, and monthly data in which the degree of leptokurtosis and time-dependent heteroscedasticity is reduced as the length of the sampling interval increases.     

Inference for Box–Cox Transformed Threshold GARCH Models with Nuisance Parameters

Abstract. Generalized autoregressive conditional heteroscedastic (GARCH) models have been widely used for analyzing financial time series with time‐varying volatilities. To overcome the defect of the Gaussian quasi‐maximum likelihood estimator (QMLE) when the innovations follow either heavy‐tailed or skewed distributions, Berkes & Horváth (Ann. Statist., 32, 633, 2004) and Lee & Lee (Scand. J. Statist. 36, 157, 2009) considered likelihood methods that use two‐sided exponential, Cauchy and normal mixture distributions. In this paper, we extend their methods for Box–Cox transformed threshold GARCH model by allowing distributions used in the construction of likelihood functions to include parameters and employing the estimated quasi‐likelihood estimators (QELE) to handle those parameters. We also demonstrate that the proposed QMLE and QELE are consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Asymptotic Inefficiency of Mean-Correction on Parameter Estimation for a Periodic First-Order Autoregressive Model

A common practice in time series analysis is to fit a centered model to the mean-corrected data set. For stationary autoregressive moving-average (ARMA) processes, as far as the parameter estimation is concerned, fitting an ARMA model without intercepts to the mean-corrected series is asymptotically equivalent to fitting an ARMA model with intercepts to the observed series. We show that, related to the parameter least squares estimation of periodic ARMA models, the second approach can be arbitrarily more efficient than the mean-corrected counterpart. This property is illustrated by means of a periodic first-order autoregressive model. The asymptotic variance of the estimators for both approaches is derived. Moreover, empirical experiments based on simulations investigate the finite sample properties of the estimators.     

Some asymptotic properties in INAR(1) processes with Poisson marginals

The first-order integer-valued autoregressive (INAR(1)) process with Poisson marginal distributions is considered. It is shown that the sample autocovariance function of the model is asymptotically normally distributed. We derive asymptotic distribution of Yule-Walker type estimators of parameters. It turns out that our Yule-Walker type estimators are better than the conditional least squares estimators proposed by Klimko and Nelson (1978) and Al-Osh and Alzaid (1987). also, we study the relationship between the model andM/M/∞ queueing system.     

Flat-Top Realized Kernel Estimation of Quadratic Covariation With Nonsynchronous and Noisy Asset Prices

This article develops a general multivariate additive noise model for synchronized asset prices and provides a multivariate extension of the generalized flat-top realized kernel estimators, analyzed earlier by Varneskov (2014), to estimate its quadratic covariation. The additive noise model allows for α-mixing dependent exogenous noise, random sampling, and an endogenous noise component that encompasses synchronization errors, lead-lag relations, and diurnal heteroscedasticity. The various components may exhibit polynomially decaying autocovariances. In this setting, the class of estimators considered is consistent, asymptotically unbiased, and mixed Gaussian at the optimal rate of convergence, n^1/4. A simple finite sample correction based on projections of symmetric matrices ensures positive definiteness without altering the asymptotic properties of the estimators. It, thereby, guarantees the existence of nonlinear transformations of the estimated covariance matrix such as correlations and realized betas, which inherit the asymptotic properties from the flat-top realized kernel estimators. An empirically motivated simulation study assesses the choice of sampling scheme and projection rule, and it shows that flat-top realized kernels have a desirable combination of robustness and efficiency relative to competing estimators. Last, an empirical analysis of signal detection and out-of-sample predictions for a portfolio of six stocks of varying size and liquidity illustrates the use and properties of the new estimators.     

Semiparametric ARCH Models: An Estimating Function Approach

We introduce the method of estimating functions to study the class of autoregressive conditional heteroscedasticity (ARCH) models. We derive the optimal estimating functions by combining linear and quadratic estimating functions. The resultant estimators are more efficient than the quasi-maximum likelihood estimator. If the assumption of conditional normality is imposed, the estimator obtained by using the theory of estimating functions is identical to that obtained by using the maximum likelihood method in finite samples. The relative efficiencies of the estimating function (EF) approach in comparison with the quasi-maximum likelihood estimator are developed. We illustrate the EF approach using a univariate GARCH(1,1) model with conditional normal, Student-t, and gamma distributions. The efficiency benefits of the EF approach relative to the quasi-maximum likelihood approach are substantial for the gamma distribution with large skewness. Simulation analysis shows that the finite-sample properties of the estimators from the EF approach are attractive. EF estimators tend to display less bias and root mean squared error than the quasi-maximum likelihood estimator. The efficiency gains are substantial for highly nonnormal distributions. An example demonstrates that implementation of the method is straightforward.     

Bootstrap adaptive estimation: The trimmed-mean example

We consider the problem of choosing among a class of possible estimators by selecting the estimator with the smallest bootstrap estimate of finite sample variance. This is an alternative to using cross-validation to choose an estimator adaptively. The problem of a confidence interval based on such an adaptive estimator is considered. We illustrate the ideas by applying the method to the problem of choosing the trimming proportion of an adaptive trimmed mean. It is shown that a bootstrap adaptive trimmed mean is asymptotically normal with an asymptotic variance equal to the smallest among trimmed means. The asymptotic coverage probability of a bootstrap confidence interval based on such adaptive estimators is shown to have the nominal level. The intervals based on the asymptotic normality of the estimator share the same asymptotic result, but have poor small-sample properties compared to the bootstrap intervals. A small-sample simulation demonstrates that bootstrap adaptive trimmed means adapt themselves rather well even for samples of size 10.     

Overnight and Daytime Stock-Return Dynamites on the London Stock Exchange: The Impacts of “Big Bang” and the 1987 Stock-Market Crash

We explore the time series properties of stock returns on the London Stock Exchange around the 1986 market restructuring (Big Bang) and the 1987 stock-market crash using a modified generalized autoregressive conditional heteroscedasticity model. Using this general dynamic model, which allows (a) intradaily returns to have different impacts and persistence on stock-return volatility, (b) return effects on volatility to be asymmetric, and (c) intradaily returns to follow conditional distributions with different fourth moments, we uncover important changes in return dynamics and conditional fourth moments following Big Bang and the 1987 crash not reported before.     

Beta seasonal autoregressive moving average models

In this paper, we introduce the class of beta seasonal autoregressive moving average (βSARMA) models for modelling and forecasting time series data that assume values in the standard unit interval. It generalizes the class of beta autoregressive moving average models [Rocha AV and Cribari-Neto F. Beta autoregressive moving average models. Test. 2009;18(3):529–545] by incorporating seasonal dynamics to the model dynamic structure. Besides introducing the new class of models, we develop parameter estimation, hypothesis testing inference, and diagnostic analysis tools. We also discuss out-of-sample forecasting. In particular, we provide closed-form expressions for the conditional score vector and for the conditional Fisher information matrix. We also evaluate the finite sample performances of conditional maximum likelihood estimators and white noise tests using Monte Carlo simulations. An empirical application is presented and discussed.     

Likelihood-based estimation for Gaussian MRFs

Markov random fields (MRFs) express spatial dependence through conditional distributions, although their stochastic behavior is defined by their joint distribution. These joint distributions are typically difficult to obtain in closed form, the problem being a normalizing constant that is a function of unknown parameters. The Gaussian MRF (or conditional autoregressive model) is one case where the normalizing constant is available in closed form; however, when sample sizes are moderate to large (thousands to tens of thousands), and beyond, its computation can be problematic. Because the conditional autoregressive (CAR) model is often used for spatial-data modeling, we develop likelihood-inference methodology for this model in situations where the sample size is too large for its normalizing constant to be computed directly. In particular, we use simulation methodology to obtain maximum likelihood estimators of mean, variance, and spatial-depencence parameters (including their asymptotic variances and covariances) of CAR models.