Self-weighted least absolute deviation estimation for infinite variance autoregressive models

Summary.  How to undertake statistical inference for infinite variance autoregressive models has been a long-standing open problem. To solve this problem, we propose a self-weighted least absolute deviation estimator and show that this estimator is asymptotically normal if the density of errors and its derivative are uniformly bounded. Furthermore, a Wald test statistic is developed for the linear restriction on the parameters, and it is shown to have non-trivial local power. Simulation experiments are carried out to assess the performance of the theory and method in finite samples and a real data example is given. The results are entirely different from other published results and should provide new insights for future research on heavy-tailed time series.     

Adaptive estimation in partially linear autoregressive models

The authors consider a partially linear autoregressive model and construct kernel‐based estimates for both the parametric and nonparametric components. They propose an estimation procedure for the model and illustrate it through simulated and real data. Their work shows that the proposed estimation procedure not only has good asymptotic properties but also works well numerically. It also suggests that a partially linear autoregression is more appropriate than a completely nonparametric autoregression for some sets of data.     

On periodic autoregressive stochastic volatility models: structure and estimation

To capture both the volatility evolution and the periodicity feature in the autocorrelation structure exhibited by many nonlinear time series, a Periodic AutoRegressive Stochastic Volatility (PAR-SV ) model is proposed. Some probabilistic properties, namely the strict and second-order periodic stationarity, are provided. Furthermore, conditions for the existence of higher-order moments are established. The autocovariance structure of the squares and higher order powers of the PAR-SV process is studied. Its dynamic properties are shown to be consistent with financial time series empirical findings. Ways in which the model may be estimated are discussed. Finally, a simulation study of the performance of the proposed estimation methods is provided and the PAR-SV is applied to model the spot rates of the euro and US dollar both against the Algerian dinar. The empirical analysis shows that the proposed PAR-SV model can be considered as a viable alternative to the periodic generalized autoregressive conditionally heteroscedastic (PGARCH) model.     

Ridge estimation in regression problems with autocorrelated errors: A monte carlo study

This paper presents the results of a Monte Carlo study of OLS and GLS based adaptive ridge estimators for regression problems in which the independent variables are collinear and the errors are autocorrelated. It studies the effects of degree of collinearity, magnitude of error variance, orientation of the parameter vector and serial correlation of the independent variables on the mean squared error performance of these estimators. Results suggest that such estimators produce greatly improved performance in favorable portions of the parameter space. The GLS based methods are best when the independent variables are also serially correlated.     

Linear model estimation errors with estimated design parameters

The problem of error estimation of parameters b in a linear model,Y = Xb+ e, is considered when the elements of the design matrix X are functions of an unknown ‘design’ parameter vector c. An estimated value c is substituted in X to obtain a derived design matrix [Xtilde]. Even though the usual linear model conditions are not satisfied with [Xtilde], there are situations in physical applications where the least squares solution to the parameters is used without concern for the magnitude of the resulting error. Such a solution can suffer from serious errors.

This paper examines bias and covariance errors of such estimators. Using a first-order Taylor series expansion, we derive approximations to the bias and covariance matrix of the estimated parameters. The bias approximation is a sum of two terms:One is due to the dependence between ? and Y; the other is due to the estimation errors of ? and is proportional to b, the parameter being estimated. The covariance matrix approximation, on the other hand, is composed of three omponents:One component is due to the dependence between ? and Y; the second is the covariance matrix ∑_b corresponding to the minimum variance unbiased b, as if the design parameters were known without error; and the third is an additional component due to the errors in the design parameters. It is shown that the third error component is directly proportional to bb'. Thus, estimation of large parameters with wrong design matrix [Xtilde] will have larger errors of estimation. The results are illustrated with a simple linear example.     

Bootstrapping time series models

This paper surveys recent development in bootstrap methods and the modifications needed for their applicability in time series models. The paper discusses some guidelines for empirical researchers in econometric analysis of time series. Different sampling schemes for bootstrap data generation and different forms of bootstrap test statistics are discussed. The paper also discusses the applicability of direct bootstrapping of data in dynamic models and cointegrating regression models. It is argued that bootstrapping residuals is the preferable approach. The bootstrap procedures covered include the recursive bootstrap, the moving block bootstrap and the stationary bootstrap.     

Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean

Linear vector autoregressive (VAR) models where the innovations could be unconditionally heteroscedastic are considered. The volatility structure is deterministic and quite general, including breaks or trending variances as special cases. In this framework we propose ordinary least squares (OLS), generalized least squares (GLS) and adaptive least squares (ALS) procedures. The GLS estimator requires the knowledge of the time-varying variance structure while in the ALS approach the unknown variance is estimated by kernel smoothing with the outer product of the OLS residual vectors. Different bandwidths for the different cells of the time-varying variance matrix are also allowed. We derive the asymptotic distribution of the proposed estimators for the VAR model coefficients and compare their properties. In particular we show that the ALS estimator is asymptotically equivalent to the infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) in a given range and hence justifies data-driven bandwidth rules. Using these results we build Wald tests for the linear Granger causality in mean which are adapted to VAR processes driven by errors with a nonstationary volatility. It is also shown that the commonly used standard Wald test for the linear Granger causality in mean is potentially unreliable in our framework (incorrect level and lower asymptotic power). Monte Carlo experiments illustrate the use of the different estimation approaches for the analysis of VAR models with time-varying variance innovations.     

Bootstrap prediction intervals for autoregressive conditional duration models

In the recent past, the autoregressive conditional duration (ACD) models have gained popularity in modelling the durations between successive events. The aim of this paper is to propose a simple and distribution free re-sampling procedure for developing the forecast intervals of linear ACD Models. We use the conditional least squares method to estimate the parameters of the ACD Model instead of the conditional Maximum Likelihood Estimation or Quasi-Maximum Likelihood Estimation and show that they are consistent for large samples. The properties of the proposed procedure are illustrated by a simulation study and an application to two real data sets.     

Missing data in linear models with correlated errors

One common method for analyzing data in experimental designs when observations are missing was devised by Yates (1933), who developed his procedure based upon a suggestion by R. A. Fisher. Considering a linear model with independent, equi-variate errors, Yates substituted algebraic values for the missing data and then minimized the error sum of squares with respect to both the unknown parameters and the algebraic values. Yates showed that this procedure yielded the correct error sum of squares and a positively biased hypothesis sum of squares.

Others have elaborated on this technique. Chakrabarti (1962) gave a formal proof of Fisher's rule that produced a way to simplify the calculations of the auxiliary values to be used in place of the missing observations. Kshirsagar (1971) proved that the hypothesis sum of squares based on these values was biased, and developed an easy way to compute that bias. Sclove     

Application of shrinkage estimation in linear regression models with autoregressive errors

In this paper, we consider the shrinkage and penalty estimation procedures in the linear regression model with autoregressive errors of order p when it is conjectured that some of the regression parameters are inactive. We develop the statistical properties of the shrinkage estimation method including asymptotic distributional biases and risks. We show that the shrinkage estimators have a significantly higher relative efficiency than the classical estimator. Furthermore, we consider the two penalty estimators: least absolute shrinkage and selection operator (LASSO) and adaptive LASSO estimators, and numerically compare their relative performance with that of the shrinkage estimators. A Monte Carlo simulation experiment is conducted for different combinations of inactive predictors and the performance of each estimator is evaluated in terms of the simulated mean-squared error. This study shows that the shrinkage estimators are comparable to the penalty estimators when the number of inactive predictors in the model is relatively large. The shrinkage and penalty methods are applied to a real data set to illustrate the usefulness of the procedures in practice.     

The specification of vector autoregressive moving average models

In this paper we propose a new identification method based on the residual white noise autoregressive criterion (Pukkila et al., 1990) to select the order of VARMA structures. Results from extensive simulation experiments based on different model structures with varying number of observations and number of component series are used to demonstrate the performance of this new procedure. We also use economic and business data to compare the model structures selected by this order selection method with those identified in other published studies.     

Parameter estimation for a new distribution for the strength of brittle fibers: a simulation study

Parameter estimates of a new distribution for the strength of brittle fibers and composite materials are considered. An algorithm for generating random numbers from the distribution is suggested. Two parameter estimation methods, one based on a simple least squares procedure and the other based on the maximum likelihood principle, are studied using Monte Carlo simulation. In most cases, the maximum likelihood estimators were found to have somewhat smaller root mean squared error and bias than the least squares estimators. However, the least squares estimates are generally good and provide useful initial values for the numerical iteration used to find the maximum likelihood estimates.     

Semi-parametric estimation of disequilibrium models

This paper presents an easy-to-compute semi-parametric (SP) method to estimate a simple disequilibrium model proposed by Fair and Jaffee (1972). The proposed approach is based on a non-parametric interpretation of the EM (Expectation and Maximization) principle (Dempster et al; 1977) and the least squares method. The simple disequilibrium model includes the demand equation, the supply equation, and the condition that only the minimum of quantity demanded and quantity supplied is observed. The method used here allows one to consistently estimate the disequilibrium model without fully specifying the distribution of error terms in both demand and supply equations. Our Monte Carlo study suggests that the proposedestimator is better than the normal maximum likelihood estimator under asymmetric error distributions. and comparable to the nlaximunl likelihood estimator under synirnetric error distributions in finite samples. Aggregate U.S. labor market data from Quandt and Rosen (1988) is used to illustrate the procedure.     

Semiparametric estimation of employment duration models

There are a variety of economic areas, such as studies of employment duration and of the durability of capital goods, in which data on important variables typically are censored. The standard techinques for estimating a model from censored data require the distributions of unobservable random components of the model to be specified a priori up to a finite set of parameters, and misspecification of these distributions usually leads to inconsistent parameter estimates. However, economic theory rarely gives guidance about distributions and the standard estimation techniques do not provide convenient methods for identifying distributions from censored data. Recently, several distribution-free or semiparametric methods for estimating censored regression models have been developed. This paper presents the results of using two such methods to estimate a model of employment duration. The paper reports the operating characteristics of the semiparametric estimators and compares the semiparametric estimates with those obtained from a standard parametric model.     

Robust estimation for the coefficient of a first order autoregressive process

Nonparametric methods, Theil's method and Hussain's method have been applied to simple linear regression problems for estimating the slope of the regression line.We extend these methods and propose a robust estimator to estimate the coefficient of a first order autoregressive process under various distribution shapes, A simulation study to compare Theil's estimator, Hus-sain's estimator, the least squares estimator, and the proposed estimator is also presented.     

Improved autoregressive forecasts in the presence of non-normal errors

This paper is concerned with obtaining more accurate point forecasts in the presence of non-normal errors. Specifically, we apply the residual augmented least-squares (RALS) estimator to autoregressive models to utilize the additional moment restrictions embodied in non-normal errors. Monte Carlo experiments are performed to compare our RALS forecasts to forecasts based on the ordinary least-squares estimator and the least absolute deviations (LAD) estimator. We find that the RALS approach provides superior forecasts when the data are skewed. Compared to the LAD forecast, the RALS forecast has smaller mean squared prediction errors in the baseline case with normal errors.     

On the estimates of the parameters of the first order autoregressive process

Two methods of estimation for the parameters of an AR(1) process which are based on a non-linear least-squares approach are presented. On the basis of some simulation results they are compared with two maximum likelihood estimates and their relative merits are discussed.     

A simulation study of l1:estimation of a seasonal moving average time series model

A detailed simulation study is reported on the application of l₁:estimations to a seasonal moving average model. It is found that the asymptotic normal distribution is a nonapproximation to the finite sample distribution. However, the expected benefits of l₁:estimation relative to l₂:are partially realised for nonnormal innovative distributions.     

A note on the limiting properties of the least squares estimation for the random coefficient autoregressive model

This note is concerned with the limiting properties of the least squares estimation for the random coefficient autoregressive model. In contrast with existing results, ours is applicable to a wide range of models under more general assumptions.     

A study on geographically weighted spatial autoregression models with spatial autoregressive disturbances

Abstract

Spatial heterogeneity and correlation are both considered in the geographical weighted spatial autoregressive model. At present, this kind of model has aroused the attention of some scholars. For the estimation of the model, the existing research is based on the assumption that the error terms are independent and identically distributed. In this article we use a computationally simple procedure for estimating the model with spatially autoregressive disturbance terms, both the estimates of constant coefficients and variable coefficients are obtained. Finally, we give the large sample properties of the estimators under some ordinary conditions. In addition, application study of the estimation methods involved will be further explored in a separate study.