Robustifying multivariate trend tests to nonstationary volatility

This article studies inference of multivariate trend model when the volatility process is nonstationary. Within a quite general framework we analyze four classes of tests based on least squares estimation, one of which is robust to both weak serial correlation and nonstationary volatility. The existing multivariate trend tests, which either use non-robust standard errors or rely on non-standard distribution theory, are generally non-pivotal involving the unknown time-varying volatility function in the limit. Two-step residual-based i.i.d. bootstrap and wild bootstrap procedures are proposed for the robust tests and are shown to be asymptotically valid. Simulations demonstrate the effects of nonstationary volatility on the trend tests and the good behavior of the robust tests in finite samples.     

Statistical estimation of multivariate Ornstein–Uhlenbeck processes and applications to co-integration

Ornstein–Uhlenbeck models are continuous-time processes which have broad applications in finance as, e.g., volatility processes in stochastic volatility models or spread models in spread options and pairs trading. The paper presents a least squares estimator for the model parameter in a multivariate Ornstein–Uhlenbeck model driven by a multivariate regularly varying Lévy process with infinite variance. We show that the estimator is consistent. Moreover, we derive its asymptotic behavior and test statistics. The results are compared to the finite variance case. For the proof we require some new results on multivariate regular variation of products of random vectors and central limit theorems. Furthermore, we embed this model in the setup of a co-integrated model in continuous time.     

Time series properties of ARCH processes with persistent covariates

We investigate the time series properties of a volatility model, whose conditional variance is specified as in ARCH with an additional persistent covariate. The included covariate is assumed to be an integrated or nearly integrated process, with its effect on volatility given by a wide class of nonlinear volatility functions. In the paper, such a model is shown to generate many important characteristics that are commonly observed in financial time series. In particular, the model yields persistence in volatility, and also well predicts leptokurtosis. This is true for any type of volatility functions considered in the paper, as long as the covariate is integrated or nearly integrated. Stationary covariates cannot produce important characteristics observed in many financial time series. We present two empirical applications of the model, which show that the default premium (the yield spread between Baa and Aaa corporate bonds) affects stock return volatility and the interest rate differential between two countries accounts for exchange rate return volatility. The forecast evaluation shows that the model generally outperforms GARCH and FIGARCH at relatively lower frequencies.     

The uncertainty in extreme risk forecasts from covariate-augmented volatility models

For a GARCH-type volatility model with covariates, we derive asymptotically valid forecast intervals for risk measures, such as the Value-at-Risk or Expected Shortfall. To forecast these, we use estimators from extreme value theory. In the volatility model, we allow for leverage effects and the inclusion of exogenous variables, e.g., volatility indices or high-frequency volatility measures. In simulations, we find coverage of the forecast intervals to be adequate for sufficiently extreme risk levels and sufficiently large samples, which is consistent with theory. Finally, we investigate if covariate information from volatility indices or high-frequency data improves risk forecasts for major US stock indices. While—in our framework—volatility indices appear to be helpful in this regard, intra-day data are not.     

Testing the assumptions behind importance sampling

Importance sampling is used in many areas of modern econometrics to approximate unsolvable integrals. Its reliable use requires the sampler to possess a variance, for this guarantees a square root speed of convergence and asymptotic normality of the estimator of the integral. However, this assumption is seldom checked. In this paper we use extreme value theory to empirically assess the appropriateness of this assumption. Our main application is the stochastic volatility model, where importance sampling is commonly used for maximum likelihood estimation of the parameters of the model.     

Estimation of semiparametric locally stationary diffusion models

This paper proposes a class of locally stationary diffusion processes. The model has a time varying but locally linear drift and a volatility coefficient that is allowed to vary over time and space. The model is semiparametric because we allow these functions to be unknown and the innovation process is parametrically specified, indeed completely known. We propose estimators of all the unknown quantities based on long span data. Our estimation method makes use of the property of local stationarity. We establish asymptotic theory for the proposed estimators as the time span increases, so we do not rely on infill asymptotics. We apply this method to interest rate data to illustrate the validity of our model. Finally, we present a simulation study to provide the finite-sample performance of the proposed estimators.     

Jump-robust volatility estimation using nearest neighbor truncation

We propose two new jump-robust estimators of integrated variance that allow for an asymptotic limit theory in the presence of jumps. Specifically, our MedRV estimator has better efficiency properties than the tripower variation measure and displays better finite-sample robustness to jumps and small (“zero”) returns. We stress the benefits of local volatility measures using short return blocks, as this greatly alleviates the downward biases stemming from rapid fluctuations in volatility, including diurnal (intraday) U-shape patterns. An empirical investigation of the Dow Jones 30 stocks and extensive simulations corroborate the robustness and efficiency properties of our nearest neighbor truncation estimators.     

Wavelet analysis of change-points in a non-parametric regression with heteroscedastic variance

In this paper we develop wavelet methods for detecting and estimating jumps and cusps in the mean function of a non-parametric regression model. An important characteristic of the model considered here is that it allows for conditional heteroscedastic variance, a feature frequently encountered with economic and financial data. Wavelet analysis of change-points in this model has been considered in a limited way in a recent study by Chen et al. (2008) with a focus on jumps only. One problem with the aforementioned paper is that the test statistic developed there has an extreme value null limit distribution. The results of other studies have shown that the rate of convergence to the extreme value distribution is usually very slow, and critical values derived from this distribution tend to be much larger than the true ones. Here, we develop a new test and show that the test statistic has a convenient null limit N(0,1) distribution. This feature gives the proposed approach an appealing advantage over the existing approach. Another attractive feature of our results is that the asymptotic theory developed here holds for both jumps and cusps. Implementation of the proposed method for multiple jumps and cusps is also examined. The results from a simulation study show that the new test has excellent power and the estimators developed also yield very accurate estimates of the positions of the discontinuities.     

Edgeworth expansions for realized volatility and related estimators

This paper shows that the asymptotic normal approximation is often insufficiently accurate for volatility estimators based on high frequency data. To remedy this, we derive Edgeworth expansions for such estimators. The expansions are developed in the framework of small-noise asymptotics. The results have application to Cornish–Fisher inversion and help setting intervals more accurately than those relying on normal distribution.     

Asymptotic normality of narrow-band least squares in the stationary fractional cointegration model and volatility forecasting

We consider semiparametric frequency domain analysis of cointegration between long memory processes, i.e. fractional cointegration, allowing derivation of useful long-run relations even among stationary processes. The approach is due to Robinson (1994b. Annals of Statistics 22, 515–539) and uses a degenerating part of the periodogram near the origin to form a narrow-band frequency domain least squares (FDLS) estimator of the cointegrating relation, which is consistent for arbitrary short-run dynamics. We derive the asymptotic distribution theory for the FDLS estimator of the cointegration vector in the stationary long memory case, thus complementing Robinson's consistency result. An application to the relation between the volatility realized in the stock market and the associated implicit volatility derived from option prices is offered.     

The Effects of Additive Outliers and Measurement Errors when Testing for Structural Breaks in Variance*

This article discusses the asymptotic and finite‐sample properties of the CUSUM tests for detecting structural breaks in volatility when the data are perturbed with (additive) outliers and/or measurement errors. The special focus is on the parametric and non‐parametric tests in Inclán and Tiao (1994) and Kokoszka and Leipus (2000) . Whereas the asymptotic distribution of the former can be largely affected, the distribution of the latter remains invariant and renders consistent break‐point estimates. In small samples, however, large additive outliers are able to generate sizeable distortions in both tests, which explains some of the contradictory findings in previous literature.     

Spectral density and spectral distribution inference for long memory time series via fixed-b asymptotics

This paper studies taper-based estimates of the spectral density utilizing a fixed bandwidth ratio asymptotic framework, and makes several theoretical contributions: (i) we treat multiple frequencies jointly, (ii) we allow for long-range dependence or anti-persistence at differing frequencies, (iii) we allow for tapers that are only piecewise smooth or discontinuous, including flat-top and truncation tapers, (iv) we study higher-order accuracy through the limit distribution’s Laplace Transform, (v) we develop a taper-based estimation theory for the spectral distribution, and show how confidence bands can be constructed. Simulation results produce quantiles and document the finite-sample size properties of the estimators, and a few empirical applications demonstrate the utility of the new methods.     

Testing the parametric form of the volatility in continuous time diffusion models—a stochastic process approach

We present new tests for the form of the volatility function which are based on stochastic processes of the integrated volatility. We prove weak convergence of these processes to centered processes whose conditional distributions are Gaussian. In the case of testing for a constant volatility the limiting process are standard Brownian bridges. As a consequence an asymptotic distribution free test and bootstrap tests (for testing of a general parametric form) can easily be implemented. It is demonstrated that the new tests are more than the currently available procedures. The new approach is also demonstrated by means of a simulation study.     

Bootstrapping realized multivariate volatility measures

We propose a bootstrap method for statistics that are a function of multivariate high frequency returns such as realized regression, covariance and correlation coefficients. We show that the finite sample performance of the bootstrap is superior to the existing first-order asymptotic theory. Nevertheless, and contrary to the existing results in the bootstrap literature for regression models subject to error heteroskedasticity, the Edgeworth expansion for the pairs bootstrap that we develop here shows that this method is not second-order accurate. We argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility.     

Functional data analysis for volatility

We introduce a functional volatility process for modeling volatility trajectories for high frequency observations in financial markets and describe functional representations and data-based recovery of the process from repeated observations. A study of its asymptotic properties, as the frequency of observed trades increases, is complemented by simulations and an application to the analysis of intra-day volatility patterns of the S&P 500 index. The proposed volatility model is found to be useful to identify recurring patterns of volatility and for successful prediction of future volatility, through the application of functional regression and prediction techniques.     

Adaptive consistent unit-root tests based on autoregressive threshold model

This paper proposes SupWald tests from a threshold autoregressive model computed with an adaptive set of thresholds. Simple examples of adaptive threshold sets are given. A second contribution of the paper is a general asymptotic null limit theory when the threshold variable is a level variable. We obtain a pivotal null limiting distribution under some simple conditions for bounded or asymptotically unbounded thresholds. Our general approach is flexible enough to allow a choice of the auxiliary threshold model or of the threshold set involved in the test specifically designed for nonlinear stationary alternatives relevant for macroeconomic and financial topics involving arbitrage in presence of transaction costs. A Monte-Carlo study and an application to the interest rates spread for French, German, New-Zealander and US post-1980 monthly data illustrate the ability of the adaptive SupWald tests to reject unit-root when the ADF does not.     

Central limit theorems and uniform laws of large numbers for arrays of random fields

Over the last decades, spatial-interaction models have been increasingly used in economics. However, the development of a sufficiently general asymptotic theory for nonlinear spatial models has been hampered by a lack of relevant central limit theorems (CLTs), uniform laws of large numbers (ULLNs) and pointwise laws of large numbers (LLNs). These limit theorems form the essential building blocks towards developing the asymptotic theory of M-estimators, including maximum likelihood and generalized method of moments estimators. The paper establishes a CLT, ULLN, and LLN for spatial processes or random fields that should be applicable to a broad range of data processes.     

A General Approach to Testing Volatility Models in Time Series

Volatility models have been playing important roles in economics and finance. Using a generalized spectral second order derivative approach, we propose a new class of generally applicable omnibus tests for the adequacy of linear and nonlinear volatility models. Our tests have a convenient asymptotic null N(0,1) distribution, and can detect a wide range of misspecifications for volatility dynamics, including both neglected linear and nonlinear volatility dynamics. Distinct from the existing diagnostic tests for volatility models, our tests are robust to time-varying higher order moments of unknown form (e.g., time-varying skewness and kurtosis). They check a large number of lags and are therefore expected to be powerful against neglected volatility dynamics that occurs at higher order lags or display long memory properties. Despite using a large number of lags, our tests do not suffer much from the loss of a large number of degrees of freedom, because our approach naturally discounts higher order lags, which is consistent with the stylized fact that economic or financial markets are affected more by the recent past events than by the remote past events. No specific estimation method is required, and parameter estimation uncertainty has no impact on the convenient limit N(0,1) distribution of the test statistics. Moreover, there is no need to formulate an alternative volatility model, and only estimated standardized residuals are needed to implement our tests. We do not have to calculate tedious and model-specific score functions or derivatives of volatility models with respect to estimated parameters, which are required in some existing popular diagnostic tests for volatility models. We examine the finite sample performance of the proposed tests. It is documented that the new tests are rather powerful in detecting neglected nonlinear volatility dynamics which the existing tests can easily miss. They are useful diagnostic tools for practitioners when modelling volatility dynamics.     

Efficient estimation of a multivariate multiplicative volatility model

We propose a multivariate generalization of the multiplicative volatility model of Engle and Rangel (2008), which has a nonparametric long run component and a unit multivariate GARCH short run dynamic component. We suggest various kernel-based estimation procedures for the parametric and nonparametric components, and derive the asymptotic properties thereof. For the parametric part of the model, we obtain the semiparametric efficiency bound. Our method is applied to a bivariate stock index series. We find that the univariate model of Engle and Rangel (2008) appears to be violated in the data whereas our multivariate model is more consistent with the data.     

A two-stage realized volatility approach to estimation of diffusion processes with discrete data

This paper motivates and introduces a two-stage method of estimating diffusion processes based on discretely sampled observations. In the first stage we make use of the feasible central limit theory for realized volatility, as developed in [Jacod, J., 1994. Limit of random measures associated with the increments of a Brownian semiartingal. Working paper, Laboratoire de Probabilities, Universite Pierre et Marie Curie, Paris] and [Barndorff-Nielsen, O., Shephard, N., 2002. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society. Series B, 64, 253–280], to provide a regression model for estimating the parameters in the diffusion function. In the second stage, the in-fill likelihood function is derived by means of the Girsanov theorem and then used to estimate the parameters in the drift function. Consistency and asymptotic distribution theory for these estimates are established in various contexts. The finite sample performance of the proposed method is compared with that of the approximate maximum likelihood method of [Aït-Sahalia, Y., 2002. Maximum likelihood estimation of discretely sampled diffusion: A closed-form approximation approach. Econometrica. 70, 223–262].