Time-varying autoregressive conditional duration model

The main goal of this work is to generalize the autoregressive conditional duration (ACD) model applied to times between trades to the case of time-varying parameters. The use of wavelets allows that parameters vary through time and makes possible the modeling of non-stationary processes without preliminary data transformations. The time-varying ACD model estimation was done by maximum-likelihood with standard exponential distributed errors. The properties of the estimators were assessed via bootstrap. We present a simulation exercise for a non-stationary process and an empirical application to a real series, namely the TELEMAR stock. Diagnostic and goodness of fit analysis suggest that the time-varying ACD model simultaneously modeled the dependence between durations, intra-day seasonality and volatility.     

Mixture inverse Gaussian for unobserved heterogeneity in the autoregressive conditional duration model

In this paper, we assume that the duration of a process has two different intrinsic components or phases which are independent. The first is the time it takes for a trade to be initiated in the market (for example, the time during which agents obtain knowledge about the market in which they are operating and accumulate information, which is coherent with Brownian motion) and the second is the subsequent time required for the trade to develop into a complete duration. Of course, if the first time is zero then the trade is initiated immediately and no initial knowledge is required. If we assume a specific compound Bernoulli distribution for the first time and an inverse Gaussian distribution for the second, the resulting convolution model has a mixture of an inverse Gaussian distribution with its reciprocal, which allows us to specify and test the unobserved heterogeneity in the autoregressive conditional duration (ACD) model.

Our proposals make it possible not only to capture various density shapes of the durations but also easily to accommodate the behaviour of the tail of the distribution and the non monotonic hazard function. The proposed model is easy to fit and characterizes the behaviour of the conditional durations reasonably well in terms of statistical criteria based on point and density forecasts.     

Efficient estimation for the conditional autoregressive model

Recently, spatial regression models have been attracting a great deal of attention in areas ranging from effect of traffic congestion on accident rates to the analysis of trends in gastric cancer mortality. In this paper, we propose efficient estimators for the regression coefficients of the spatial conditional autoregressive model, when uncertain auxiliary information is available about these coefficients. We provide efficiency comparisons of the proposed estimators based on asymptotic risk analysis and Monte Carlo simulations. We apply the proposed methods to real data on Boston housing prices and illustrate how a bootstrapping approach can be employed to compute prediction errors of the estimators.     

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.     

A likelihood based estimator for vector autoregressive processes

A onestep estimator, which is an approximation to the unconditional maximum likelihood estimator (MLE) of the coefficient matrices of a Gaussian vector autoregressive process is presented. The onestep estimator is easy to compute and can be computed using standard software. Unlike the computation of the unconditional MLE, the computation of the onestep estimator does not require any iterative optimization and the computation is numerically stable. In finite samples the onestep estimator generally has smaller mean square error than the ordinary least squares estimator. In a simple model, where the unconditional MLE can be computed, numerical investigation shows that the onestep estimator is slightly worse than the unconditional MLE in terms of mean square error but superior to the ordinary least squares estimator. The limiting distribution of the onestep estimator for processes with some unit roots is derived.     

On diagnostic checking of the autoregressive conditional intensity model

The autoregressive conditional intensity model proposed by Russell (1998) is a promising option for fitting multivariate high frequency irregularly spaced data. The authors acknowledge the validity of this model by showing the independence of its generalized residuals, a crucial assumption of the model formulation not readily recognized by researchers. The authors derive the large‐sample distribution of the autocorrelations of the generalized residual series and use it to construct a goodness‐of‐fit test for the model. Empirical results compare the performance of their test with other off‐the‐shelf tests such as the Ljung–Box test. They illustrate the use of their test with transaction records of the HSBC stock.     

Trimmed least squares estimator as best trimmed linear conditional estimator for linear regression model

A class of trimmed linear conditional estimators based on regression quantiles for the linear regression model is introduced. This class serves as a robust analogue of non-robust linear unbiased estimators. Asymptotic analysis then shows that the trimmed least squares estimator based on regression quantiles ( Koenker and Bassett ( 1978 ) ) is the best in this estimator class in terms of asymptotic covariance matrices. The class of trimmed linear conditional estimators contains the Mallows-type bounded influence trimmed means ( see De Jongh et al ( 1988 ) ) and trimmed instrumental variables estimators. A large sample methodology based on trimmed instrumental variables estimator for confidence ellipsoids and hypothesis testing is also provided.     

A spatially adaptive conditional autoregressive prior for area health data

Different priors have been suggested to reflect spatial dependence in area health outcomes or in spatial regression residuals. However, to assume that residuals demonstrate spatial clustering only is a strong prior belief and alternatives have been suggested. A scheme suggested by Leroux et al. [B. Leroux, X. Lei, N. Breslow, Estimation of disease rates in small areas: A new mixed model for spatial dependence, in: M. Halloran, D. Berry (Eds.), Statistical Models in Epidemiology, the Environment and Clinical Trials, Springer-Verlag, New York, 1999, pp. 135–178] involves a single set of random effects and a spatial correlation parameter with extreme values corresponding to pure spatial and pure unstructured residual variation. This paper considers a spatially adaptive extension of that prior to reflect the fact that the appropriate mix between local and global smoothing may not be constant across the region being studied. Local smoothing will not be indicated when an area is disparate from its neighbours (e.g. in terms of social or environmental risk factors for the health outcome being considered). The prior for varying spatial correlation parameters may be based on a regression structure which includes possible observed sources of disparity between neighbours. A case study considers probabilities of long term illness in 133 small areas in NE London, with disparities based on a measure of socio-economic deprivation.     

Berry-Esseen-type bounds for the kernel estimator of conditional distribution and conditional quantiles

Berry-Esseen bounds of order O(n^−1/2) have been obtained for several classes of statistics. In this paper, the rates of convergence in central limit theorem for conditional empirical functions and conditional sample quantiles based on kernel estimators are studied for both conditional and unconditional distributions.     

Bias of the lse estimator of the first order autoregressive model under tukey contamination

Results of an exhaustive study of the bias of the least square estimator (LSE) of an first order autoregression coefficient α in a contaminated Gaussian model are presented. The model describes the following situation. The process is defined as X_t = α X_t-1 + Y_t . Until a specified time T, Y_t are iid normal N(0, 1). At the moment T we start our observations and since then the distribution of Y_t, t≥T, is a Tukey mixture T(εσ) = (1 – ε)N(0,1) + εN(0, σ²). Bias of LSE as a function of α and ε, and σ² is considered. A rather unexpected fact is revealed: given α and ε, the bias does not change montonically with σ (“the magnitude of the contaminant”), and similarly, given α and σ, the bias is not growing with ε (“the amount of contaminants”).     

Performance of control charts for autoregressive conditional heteroscedastic processes

This paper examines the robustness of control schemes to data conditional heteroscedasticity. Overall, the results show that the control schemes which do not account for heteroscedasticity fail in providing reliable information on the status of the process. Consequently, incorrect conclusions will be drawn by applying these procedures in the presence of data conditional heteroscedasticity. Control charts with time-varying control limits are shown to be useful in that context.     

Dimension reduction for the conditional kth moment in regression

The idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model. Central subspaces are designed to capture all the information for the regression and to provide a population structure for dimension reduction. Here, we introduce the central k th-moment subspace to capture information from the mean, variance and so on up to the k th conditional moment of the regression. New methods are studied for estimating these subspaces. Connections with sliced inverse regression are established, and examples illustrating the theory are presented.     

Lag order selection for an optimal autoregressive covariance matrix estimator

A good parametric spectral estimator requires an accurate estimate of the sum of AR coefficients, however a criterion which minimizes the innovation variance not necessarily yields the best spectral estimate. This paper develops an alternative information criterion considering the bias in the sum of the parameters for the autoregressive estimator of the spectral density at frequency zero.     

Estimating the conditional tail index by integrating a kernel conditional quantile estimator

This paper deals with the estimation of the tail index of a heavy-tailed distribution in the presence of covariates. A class of estimators is proposed in this context and its asymptotic normality established under mild regularity conditions. These estimators are functions of a kernel conditional quantile estimator depending on some tuning parameters. The finite sample properties of our estimators are illustrated on a small simulation study.     

A copula‐graphic estimator for the conditional survival function under dependent censoring

Rivest Wells (2001) showed that in situations where the dependence between a lifetime and a censoring variable can be modeled by a given Archimedean copula, the copula‐graphic estimator of Zheng Klein (1995) has an explicit form. The authors extend this work to the fixed design regression case. They show that the copula‐graphic estimator then has an asymptotic representation and a Gaussian limit. They also assess the influence of a misspecified copula function on the performance of the estimator. Their developments are illustrated with data on the survival of the Atlantic halibut.     

Numerical computation of exact moments of the least squares estimator in a first-order stationary autoregressive model

This paper examines the least squares estimator of the autoregressive coefficient in a first-order sta¬tionary autoregressive model. Exact lower-order moments are computed by numerical integrations. From the study of moment values, it is found that the exact distribution of the least squares estimator may be well approximated by a beta distribution.     

Alternative moment approximations for berkson's minimum logit chi-squared estimator

Expressions are derived for the bias to order J^-1 , the variance to order J^-2 and the mean squared error to order J^-2 of Berkson's minimum logit chi-squared estimator where J is the number of distinct design points. These moment approximations are numerically compared to Monte Carlo estimates of the true moments and the moment approximations of Amemiya (1980) which are appropriate when the “average” number of observations per design point is large. They are used to compare the mean squared error of the minimum logit chi-squared estimator to that of the maximum likelihood estimator and to investigate the effect of bias on confidence intenrals constructed using the minimum logit chi-squared estimator.     

A non-stationary integer-valued autoregressive model

It is frequent to encounter a time series of counts which are small in value and show a trend having relatively large fluctuation. To handle such a non-stationary integer-valued time series with a large dispersion, we introduce a new process called integer-valued autoregressive process of order p with signed binomial thinning (INARS(p)). This INARS(p) uniquely exists and is stationary under the same stationary condition as in the AR(p) process. We provide the properties of the INARS(p) as well as the asymptotic normality of the estimates of the model parameters. This new process includes previous integer-valued autoregressive processes as special cases. To preserve integer-valued nature of the INARS(p) and to avoid difficulty in deriving the distributional properties of the forecasts, we propose a bootstrap approach for deriving forecasts and confidence intervals. We apply the INARS(p) to the frequency of new patients diagnosed with acquired immunodeficiency syndrome (AIDS) in Baltimore, Maryland, U.S. during the period of 108 months from January 1993 to December 2001.     

A conditional perspective of weighted variance estimation of the optimal regression estimator

The estimation of the variance for the GREG (general regression) estimator by weighted residuals is widely accepted as a method which yields estimators with good conditional properties. Since the optimal (regression) estimator shares the properties of GREG estimators which are used in the construction of weighted variance estimators, we introduce the weighting procedure also for estimating the variance of the optimal estimator. This method of variance estimation was originally presented in a seemingly ad hoc manner, and we shall discuss it from a conditional point of view and also look at an alternative way of utilizing the weights. Examples that stress conditional behaviour of estimators are then given for elementary sampling designs such as simple random sampling, stratified simple random sampling and Poisson sampling, where for the latter design we have conducted a small simulation study.