Exact moments of the sample autocorrelations from series generated by general arima processes of order (p, d, q), d=0 or 1

Formulae for the numerical computation of the first four exact moments of the sample autocorrelations, given a time series realisation from a general autoregressive moving average process of order (p, d, q) with d=0 or 1, are presented. The exact mean and variance of the sample autocorrelations are computed for various sample sizes and several time series models. The evaluated results are compared with those obtained from approximate formulae for the mean and variance of the sample autocorrelations. A specification of the numerical accuracy of the first two exact moments is included.     

Asymptotics for Autocovariances and Integrated Periodograms for Linear Processes Observed at Lower Frequencies

One of the most frequently used class of processes in time series analysis is the one of linear processes. For many statistical quantities, among them sample autocovariances and sample autocorrelations, central limit theorems are available in the literature. We investigate classical linear processes under a nonstandard observation pattern; namely, we assume that we are only able to observe the linear process at a lower frequency. It is shown that such observation pattern destroys the linear structure of the observations and leads to substantially different asymptotic results for standard statistical quantities. Central limit theorems are given for sample autocovariances and sample autocorrelations as well as more general integrated periodograms and ratio statistics. Moreover, for specific autoregressive processes, the possibilities to estimate the parameters of the underlying autoregression from lower frequency observations are addressed. Finally, we suggest for autoregressions of order 2 a valid bootstrap procedure. A small simulation study demonstrates the performance of the bootstrap proposal for finite sample size.     

Distributional properties of portfolio weights

In this paper, we prove several distributional properties for optimal portfolio weights. The weights are estimated by replacing the parameters with the sample counterparts. All results for finite samples are made assuming normally distributed returns. We calculate the exact covariances for the weights obtained by the expected quadratic utility. Additionally we derive the multivariate density function of the global minimum variance portfolio and the univariate density of the tangency portfolio. We obtain the conditional density for the Sharpe ratio optimal weights and show that the expectations of the Sharpe ratio optimal weights do not exist. Moreover, we determine the asymptotic distributions of the estimated weights assuming that the returns follow a multivariate stationary Gaussian process.     

Nonstationary nonlinear heteroskedasticity

In this paper, we consider time series with the conditional heteroskedasticities that are given by nonlinear functions of integrated processes. Such time series are said to have nonlinear nonstationary heteroskedasticity (NNH), and the functions generating conditional heterogeneity are called heterogeneity generating functions (HGF's). Various statistical properties of time series with NNH are investigated for a wide class of HGF's. For NNH models with a variety of HGF's, volatility clustering and leptokurtosis, which are common features of ARCH type models, are manifest. In particular, it is shown that the sample autocorrelations of their squared processes vanish only very slowly, or do not even vanish at all, in the limit. Volatility clustering is therefore well expected. The NNH models with certain types of HGF's indeed have sample characteristics that are very similar to those of ARCH type models. Moreover, the sample kurtosis of the NNH model either diverges or has a stable limiting distribution with support truncated on the left by the kurtosis of the innovations. This would well explain the presence of leptokurtosis in many observed time series data. To illustrate the empirical relevancy of our model, we analyze the spreads between the forward and spot rates of USD/DM exchange rates. It is found that the conditional variances of the spreads can be well modelled as a nonlinear function of the levels of the spot rates.     

Consistent estimation of a regression with errors in the variables

Summary As is well known, least squares estimates of regression coefficients are inconsistent if the variables are measured with random errors. In the classical case of known variances and covariances for these error variables, consistent estimates can be derived. It is shown that these estimators generally have a joint asymptotic normal distribution, the covariance matrix of which is derived. No use is made of normality assumptions, but knowledge of the third and fourth moments of error variables is utilized.     

Classification performance of mathematical programming techniques in discriminant analysis: Results for small and medium sample sizes

The performance on small and medium-size samples of several techniques to solve the classification problem in discriminant analysis is investigated. The techniques considered are two widely used parametric statistical techniques (Fisher's linear discriminant function and Smith's quadratic function), and a class of recently proposed nonparametric estimation techniques based on mathematical programming (linear and mixed-integer programming). A simulation study is performed, analyzing the relative performance of the above techniques in the two-group case, for various small sample sizes, moderate group overlap and across six different data conditions. Training samples as well as validation samples are used to assess the classificatory performance of the techniques. The degree of group overlap and sample sizes selected for analysis in this paper are of interest in practice because they closely reflect conditions of many real data sets. The results of the experiment show that Smith's nonlinear quadratic function tends to be superior on the training samples and validation samples when the variances–covariances across groups are heterogeneous, while the mixed-integer technique performs best on the training samples when the variances–covariances are equal, and on validation samples with equal variances and discrete uniform independent variables. The mixed-integer technique and the quadratic discriminant function are also found to be more sensitive than the other techniques to the sample size, giving disproportionally inaccurate results on small samples.     

Analytic moments for GJR-GARCH (1, 1) processes

For a GJR-GARCH(1, 1) specification with a generic innovation distribution we derive analytic expressions for the first four conditional moments of the forward and aggregated returns and variances. Moments for the most commonly used GARCH models are stated as special cases. We also derive the limits of these moments as the time horizon increases, establishing regularity conditions for the moments of aggregated returns to converge to normal moments. A simulation study using these analytic moments produces approximate predictive distributions which are free from the bias affecting simulations. An empirical study using almost 30 years of daily equity index, exchange rate and interest rate data applies Johnson SU and Edgeworth expansion distribution fitting to our closed-form formulae for higher moments of returns.     

Multivariate residual‐based finite‐sample tests for serial dependence and ARCH effects with applications to asset pricing models

In this paper, we propose several finite‐sample specification tests for multivariate linear regressions (MLR). We focus on tests for serial dependence and ARCH effects with possibly non‐Gaussian errors. The tests are based on properly standardized multivariate residuals to ensure invariance to error covariances. The procedures proposed provide: (i) exact variants of standard multivariate portmanteau tests for serial correlation as well as ARCH effects, and (ii) exact versions of the diagnostics presented by Shanken ( 1990 ) which are based on combining univariate specification tests. Specifically, we combine tests across equations using a Monte Carlo (MC) test method so that Bonferroni‐type bounds can be avoided. The procedures considered are evaluated in a simulation experiment: the latter shows that standard asymptotic procedures suffer from serious size problems, while the MC tests suggested display excellent size and power properties, even when the sample size is small relative to the number of equations, with normal or Student‐t errors. The tests proposed are applied to the Fama–French three‐factor model. Our findings suggest that the i.i.d. error assumption provides an acceptable working framework once we allow for non‐Gaussian errors within 5‐year sub‐periods, whereas temporal instabilities clearly plague the full‐sample dataset. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonparametric discrimination of time series data

A normality assumption is usually made for the discrimination between two stationary time series processes. A nonparametric approach is desirable whenever there is doubt concerning the validity of this normality assumption. In this paper a nonparametric approach is suggested based on kernel density estimation firstly on (p+1) sample autocorrelations and secondly on (p+1) consecutive observations. A numerical comparison is made between Fishers linear discrimination based on sample autocorrelations and kernel density discrimination for AR and MA processes with and without Gaussian noise. The methods are applied to some seismological data.     

Maximum likelihood methods for fitting the burr type XII distribution to multiply (progressively) censored life test data

Summary This paper develops mathematical and computational methodology for fitting, by the method of maximum likelihood (ML), the Burr Type XII distribution to multiply (or progressively) censored life test data. Mathematical expressions are given for approximating the asymptotic variances and covariances of the ML estimates (MLEs) of the parameters of the Burr Type XII distribution. A rigorous mathematical analysis is undertaken to investigate the existence and uniqueness of the MLEs for arbitrary sample data. The methodology of this paper is applied to progressively censored sample data arising in a life test experiment.     

A new fluctuation test for constant variances with applications to finance

We present a test to determine whether variances of time series are constant over time. The test statistic is a suitably standardized maximum of cumulative first and second moments. We apply the test to time series of various assets and find that the test performs well in applications. Moreover, we propose a portfolio strategy based on our test which hedges against potential financial crises and show that it works in practice.     

Confidence intervals for batch properties based on both sampling and analytical variance

Summary If different random samples are taken from a heterogeneous batch and each sample is analysed once or in replicate, the average result is a point estimation of the true but unknown mean of the batch property concerned. In this paper, formulae are given for the numbers of degrees of freedom, which are necessary for interval estimations based on both the sampling and the analytical variance. It is essential to distinguish between situations in which estimates of the variances are available and in which they are not.     

Record values of the Lomax distribution

In this paper, firstly an introduction to the idea of record values for a sequence of independent and identically (Lomax or Pareto II) distributed random variables is given. Some of the distributional properties of these record values and moments up to second order are derived. These moments clearly depend on the location, scale and shape parameter of the Lomax distribution and two types of estlmators of these parameters, based on a series of observed record values are presented.     

On the distribution of the sample autocorrelation coefficients

Sample autocorrelation coefficients are widely used to test the randomness of a time series. Despite its unsatisfactory performance, the asymptotic normal distribution is often used to approximate the distribution of the sample autocorrelation coefficients. This is mainly due to the lack of an efficient approach in obtaining the exact distribution of sample autocorrelation coefficients. In this paper, we provide an efficient algorithm for evaluating the exact distribution of the sample autocorrelation coefficients. Under the multivariate elliptical distribution assumption, the exact distribution as well as exact moments and joint moments of sample autocorrelation coefficients are presented. In addition, the exact mean and variance of various autocorrelation-based tests are provided. Actual size properties of the Box–Pierce and Ljung–Box tests are investigated, and they are shown to be poor when the number of lags is moderately large relative to the sample size. Using the exact mean and variance of the Box–Pierce test statistic, we propose an adjusted Box–Pierce test that has a far superior size property than the traditional Box–Pierce and Ljung–Box tests.     

Multidimensional strong large deviation results

We establish strong large deviation results for an arbitrary sequence of random vectors under some assumptions on the normalized cumulant generating function. In other words, we give asymptotic approximations for a multivariate tail probability of the same kind as the one obtained by Bahadur and Rao (Ann Math Stat 31:1015–1027, 1960) for the sample mean (in the one-dimensional case). The proof of our results follows the same lines as in Chaganty and Sethuraman (J Stat Plan Inference, 55:265–280, 1996). We also present three statistical applications to illustrate our results, the first one dealing with a vector of independent sample variances, the second one with a Gaussian multiple linear regression model and the third one with the multivariate Nadaraya–Watson estimator. Some numerical results are also presented for the first two applications.     

DCC- and DECO-HEAVY: Multivariate GARCH models based on realized variances and correlations

This paper introduces the scalar DCC-HEAVY and DECO-HEAVY models for conditional variances and correlations of daily returns based on measures of realized variances and correlations built from intraday data. Formulas for multi-step forecasts of conditional variances and correlations are provided. Asymmetric versions of the models are developed. An empirical study shows that in terms of forecasts the scalar HEAVY models outperform the scalar BEKK-HEAVY model based on realized covariances and the scalar BEKK, DCC, and DECO multivariate GARCH models based exclusively on daily data.     

On estimation of some abbreviated social welfare measures

Point and interval estimations of two abbreviated social welfare functions are studied based on Schutz and Gini coefficients. The limit distributions of the estimators are derived in terms of sample L-moments and mean absolute deviation. A simulation study is performed to assess the bias in the estimators and the coverage probability of the confidence intervals. Unlike the previous studies our results do not require the grouping of population into a fixed number of quantiles. The proposed methods use distribution-free unbiased estimators and exact sample L-moment variances. An application is given that illustrates the benefits of the proposed methods.     

Gains in efficiency from joint estimation of systems of autoregressive-moving average processes

The paper examines gains in efficiency from joint estimation of systems of ARMA processes where cross-correlation is due to contemporaneous correlation among disturbances. The asymptotic variance of joint estimates is derived and it involves only variances and covariances among purely AR processes corresponding to the AR and MA parts of the constituent processes. Small sample gains are evaluated by Monte Carlo methods. Application of joint estimation to two short-term interest rates is shown to result in more accurate post-sample predictions relative to both univariate models and the FMP econometric model.     

Nonlinearity,nonstationarity, and thick tails: How they interact to generate persistence in memory

We consider nonlinear functions of random walks driven by thick-tailed innovations. Nonlinearity, nonstationarity, and thick tails interact to generate a spectrum of autocorrelation patterns consistent with the observed persistence in memory of many economic and financial time series. Depending upon the type of transformation considered and whether it is observed with noise, the autocorrelations are given by unity, random constants, or hyperbolically decaying deterministic functions, possibly with some independent noise, and thus may decay slowly or even not at all. Along with other sample characteristics, such patterns suggest that these three ingredients may generate the ubiquitous evidence for long memory.     

Analytic derivatives and the computation of GARCH estimates

In the context of univariate GARCH models we show how analytic first and second derivatives of the log-likelihood can be successfully employed for estimation purposes. Maximum likelihood GARCH estimation usually relies on the numerical approximation to the log-likelihood derivatives, on the grounds that an exact analytic differentiation is much too burdensome. We argue that this is not the case and that the computational benefit of using the analytic derivatives (first and second) may be substantial. Furthermore, we make a comparison of various gradient algorithms that are used for the maximization of the GARCH Gaussian likelihood. We suggest the implementation of a globally efficient computation algorithm that is obtained by suitably combining the use of the estimated information matrix with that of the exact Hessian during the maximization process. As this would appear a straightforward extension, we then study the finite sample performance of the exact Hessian and its approximations (that is, the estimated information, outer products and misspecification robust matrices) in inference.