Uncorrelated Innovations Research Articles

Spectral analysis for GARCH processes through a bilinear representation

In this paper, we deal with the frequency domain approach to describe some probabilistic properties of the processes using the associated transfer functions which seem more informative. So, in the first part, we propose an appropriate bilinear representation of model governed by a second-order Hermite polynomial with leading coefficient 1, and we give a necessary and sufficient condition for the causal and second-order stationary solution. In a second part, we establish the autocovariance and the associated spectral density function of such a representation, and for its squared version in terms of transfer functions. As a consequence, it is observed that the second-order properties are similar to an ARMA process with uncorrelated innovations and hence the resort to higher-order properties however becomes inevitable. Therefore, the explicit expressions of bispectral densities of the process and its squared version are given and several illustrative examples are emphasized. Our results are related to the work of Terdik (2000, §3) for the GARCH(1, 1) model.

A Simple Asymptotically F-Distributed Portmanteau Test for Diagnostic Checking of Time Series Models With Uncorrelated Innovations

We propose a simple asymptotically F-distributed portmanteau test for diagnostically checking whether the innovations in a parametric time series model are uncorrelated while allowing them to exhibit higher-order dependence of unknown forms. A transform of sample residual autocovariances removing the influence of parameter estimation uncertainty makes the test simple. Further, by employing the orthonormal series variance estimator, a special sample autocovariances estimator that is asymptotically invariant to parameter estimation uncertainty, we show that the proposed test statistic is asymptotically F-distributed under fixed-smoothing asymptotics. The asymptotic F-theory accounts for the estimation error of the variance estimator that the asymptotic chi-squared theory ignores. Moreover, an extensive Monte Carlo study demonstrates that the F-test has more accurate finite sample size than existing tests with virtually no power loss. An application to S&P 500 returns illustrates the merits of the proposed methodology.

Isometric operators on Hilbert spaces and Wold decomposition of stationary time series

Wold Theorem plays a fundamental role in the decomposition of weakly stationary time series. It provides a moving average representation of the process under consideration in terms of uncorrelated innovations, whatever the nature of the process is. From an empirical point of view, this result enables to identify orthogonal shocks, for instance in macroeconomic and financial time series. More theoretically, the decomposition of weakly stationary stochastic processes can be seen as a special case of the Abstract Wold Theorem, that allows to decompose Hilbert spaces by using isometric operators. In this work we explain this link in detail, employing the Hilbert space spanned by a weakly stationary time series and the lag operator as isometry. In particular, we characterize the innovation subspace by exploiting the adjoint operator. We also show that the isometry of the lag operator is equivalent to weak stationarity. Our methodology, fully based on operator theory, provides novel tools useful to discover new Wold-type decompositions of stochastic processes, in which the involved isometry is no more the lag operator. In such decompositions the orthogonality of innovations is ensured by construction since they are derived from the Abstract Wold Theorem.

Model identification for time series with dependent innovations

This paper investigates the impact of dependent but uncorrelated inno- vations (errors) on the traditional autoregressive moving average model (ARMA) order determination schemes such as autocorrelation function (ACF), partial auto- correlation function (PACF), extended autocorrelation function (EACF), and the unit-root test. The ARMA models with iid innovations have been studied exten- sively and are well-posed, but their properties with dependent but uncorrelated innovations are relatively less studied. In the presence of such innovations, we show that the ACF, PACF, and EACF are significantly impacted while the unit- root test is not affected. We also propose a new order determination scheme to address those impacts for analyzing time series with uncorrelated innovations.

Stochastic Volatility in General Equilibrium

The connections between stock market volatility and returns are studied within the context of a general equilibrium framework. The framework rules out it a priori any purely statistical relationship between volatility and returns by imposing uncorrelated innovations. The main model generates a two-factor structure for stock market volatility along with time-varying risk premiums on consumption and volatility risk. It also generates endogenously a dynamic leverage effect (volatility asymmetry), the sign of which depends upon the magnitudes of the risk aversion and the intertemporal elasticity of substitution parameters.

A Persistence Based Decomposition of Macroeconomic and Financial Time Series

The Classical Wold Decomposition Theorem allows to split a weakly stationary time series x into a non-deterministic component, driven by uncorrelated innovations, and a deterministic term. This decomposition is a special case of the Abstract Wold Theorem, which deals with isometric operators defined on Hilbert spaces. As the lag operator is isometric on the Hilbert space H_t(x) spanned by the sequence {x_{t-k}_k}, the Classical Wold Decomposition for time series obtains. Moreover, the \emph{scaling operator} is isometric on the Hilbert space H_t(e), spanned by the classical Wold innovations of x, and it provides an Extended Wold Decomposition. Thus, the process x may be seen as a sum, across scales, of uncorrelated components that explain different layers of persistence, from temporary fluctuations to low-frequency shocks. Multiscale impulse response functions are, then, defined. Conversely, the sum of suitable uncorrelated components delivers a weakly stationary process. This decomposition fruitfully applies to ARMA and fractional ARIMA processes.

Stochastic Volatility in General Equilibrium

The connections between stock market volatility and returns are studied within the context of a general equilibrium framework. The framework rules out a priori any purely statistical relationship between volatility and returns by imposing uncorrelated innovations. The main model generates a two-factor structure for stock market volatility along with time-varying risk premiums on consumption and volatility risk. It also generates endogenously a dynamic leverage effect (volatility asymmetry), the sign of which depends upon the magnitudes of the risk aversion and the intertemporal elasticity of substitution parameters.

The Phillips unit root tests for polynomials of integrated processes

In this paper, we derive the limiting distributions of the first order serial correlation coefficient and its t-statistic, which are the basis for the non-parametric unit root tests of Phillips (1987), for polynomials of integrated processes. The resulting limiting distributions depend upon nuisance parameters and in general the modification proposed by Phillips (1987), to achieve a nuisance parameter free limiting distribution, is not feasible for polynomials of integrated processes. For the special case of serially uncorrelated innovations, the limiting distributions are nuisance parameter free and are simulated. The distributions shift to the left with increasing variance for increasing polynomial orders.

Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence

This paper considers the effect of least squares procedures for nearly unstable linear time series with strongly dependent innovations. Under a general framework and appropriate scaling, it is shown that ordinary least squares procedures converge to functionals of fractional Ornstein–Uhlenbeck processes. We use fractional integrated noise as an example to illustrate the important ideas. In this case, the functionals bear only formal analogy to those in the classical framework with uncorrelated innovations, with Wiener processes being replaced by fractional Brownian motions. It is also shown that limit theorems for the functionals involve nonstandard scaling and nonstandard limiting distributions. Results of this paper shed light on the asymptotic behavior of nearly unstable long-memory processes.

Predictive Regressions: Method of Least Autocorrelation

Conventional predictive regressions produce biased and inefficient coefficient estimates in small samples when the predicting variable is Gaussian first order persistent and its innovations are highly correlated with the error series of the return. We propose a new estimation method (the method of least autocorrelation) conditional on the assumption of serially uncorrelated innovations to the dependent variable. The new estimator minimizes the weighted sum of squared autocorrelations of residuals obtained from fitting a return series through estimated parameters. The asymptotic properties of the estimator and a refined weighting matrix are discussed. Finite sample behavior of the estimator is compared with existing estimation methods in the simulation study. The new method produces more accurate and efficient point estimate of the true parameter and improves performance in the power and size tests. Our empirical results provide some fresh evidence on the predictability of stock returns during the post-war time period.

STOCHASTIC VOLATILITY

Hull and White [1] have priced a European call option for the case in which the volatility of the underlying asset is a lognormally distributed random variable. They have obtained their formula under the assumption of uncorrelated innovations in security price and volatility. Although the option pricing formula has a power series representation, the question of convergence has been left unanswered. This paper presents an iterative method for calculating all the higher order moments of volatility necessary for the process of proving convergence theoretically. Moreover, simulation results are given that show the practical convergence of the series. These results have been obtained by using a displaced geometric Brownian motion as a volatility process.

Inference for Autocorrelations under Weak Assumptions

Abstract In this article we consider the large-sample behavior of estimates of autocorrelations and autoregressive moving average (ARMA) coefficients, as well as their distributions, under weak conditions. Specifically, the usual text book formulas for variances of these estimates are based on strong assumptions and should not be routinely applied without careful consideration. Such is the case when the time series follows an ARMA process with uncorrelated innovations that may not be assumed to be independent and identically distributed. As a specific case, it is well known that if the process is independent and identically distributed, then the sample autocorrelation estimates, scaled by the square root of the sample size, are asymptotically standard normal. This result is used extensively as a diagnostic check on the residuals of a fitted model, or as an initial test on the observed time series to determine whether further model fitting is warranted. In this article we show that this result can be quite...