Adaptive Estimation of Causal Periodic Autoregressive Model

This article deals with the adaptive estimation of a periodic autoregressive model, with unspecified innovation density satisfying only some general technical assumptions. We first establish, while verifying the adapted sufficient conditions of Swensen (1985 Swensen , A. R. ( 1985 ). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend . Journal of Multivariate Analysis 16 : 54 – 70 .[Crossref], [Web of Science ®] , [Google Scholar]) to our model, the Local Asymptotic Normality (LAN), the Local Asymptotic Quadratic (LAQ), and the Local Asymptotic properties satisfied by its central sequence. Secondly, the Locally Asymptotically Minimax (LAM) estimators are constructed. Using these results, we construct the adaptive estimators of the unknown autoregressive parameters. The performances of the established estimators are shown, via simulation studies.     

Adaptive Estimation of Periodic First-Order Threshold Autoregressive Model

This paper focuses on the adaptive estimation problem of a Periodic Self-Exciting Threshold Autoregressive (PSETAR) model. The adapted sufficient conditions of Swensen (1985 Swensen, A. R. 1985. The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. Journal of Multivariate Analysis, 16: 54–70. [Crossref], [Web of Science ®] , [Google Scholar]) to our model, are verified and then explored to establish the Local Asymptotic Normality (LAN), the Local Asymptotic Quadratic (LAQ) and the Local Asymptotic properties satisfied by its central sequence. Using these results, we construct adaptive estimators for the parameter model where the innovation density is unspecified but symmetric, while satisfying only some general conditions. The performances of these adaptive estimations are shown via simulation studies and an application on the modeling of the Fraser River data.     

Adaptive Test for Periodicity in Autoregressive Conditional Heteroskedastic Processes

This article is concerned with the periodicity testing problem in Autoregressive Conditional Heteroskedastic (ARCH) process. Adaptive locally asymptotically optimal test is derived, when the innovation density is unspecified but symmetric satisfying only some general technical assumptions, for the null hypothesis of classical ARCH process against an alternative of periodically correlated ARCH dependence. The main technical tool is LeCam's (1960 LeCam , L. ( 1960 ). Locally Asymptotically Normal Families of Distributions . University California Publ. Statistics 3:27–98 . [Google Scholar]) Local Asymptotic Normality (LAN) property. The LAN property of the central sequence is shown via the adapted sufficient Swensen's conditions (1985 Swensen , A. R. ( 1985 ). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend . Journal of Multivariate Analysis 16 : 54 – 70 .[Crossref], [Web of Science ®] , [Google Scholar]). The performance of the established test is shown via simulation studies.     

Test for periodicity in restrictive EXPAR models

Abstract

This article is devoted to study the problem of test of periodicity in the restricted exponential autoregressive (EXPAR) model. The local asymptotic normality property, of this model, is shown via the adapted sufficient conditions due to Swensen (1985 Swensen, A.R. (1985). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. J. Multivariate Anal. 16:54–70.[Crossref], [Web of Science ®] , [Google Scholar]). Using this result, in the case where the innovation density is specified, we obtain a parametric local asymptotic “most stringent” test.     

Second-Order Powers of a Class of Tests in the Presence of a Nuisance Parameter

The second-order local powers of a broad class of asymptotic chi-squared tests are considered in a composite case where both the parameter of interest and the nuisance parameter are possibly multidimensional for which no assumption has been made regarding global parametric orthogonality or curved exponentiality. The main result is that the second-order (point-by-point) local power identity holds if approximate third cumulants of a square-root version of the (modified) test statistic in the class vanish up to the second-order, which is an extension of Kakizawa (2010a Kakizawa , Y. ( 2010a ). Second-order power comparison of tests . Commun. Statist. Theor. Meth. 39 : 1424 – 1436 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in the absence of the nuisance parameter. It is also shown that in the presence of the nuisance parameter, such a third cumulant condition does not always imply the second-order local unbiasedness of the resulting test. Then, the adjusted likelihood ratio test by Mukerjee (1993b Mukerjee , R. ( 1993b ). An extension of the conditional likelihood ratio test to the general multiparameter case . Ann. Inst. Statist. Math. 45 : 759 – 771 .[Crossref], [Web of Science ®] , [Google Scholar]) can be interpreted as the second-order local unbiased modification after applying the third cumulant condition.     

A Note on Unit Root Tests and GARCH Errors: A Simulation Experiment

This article re-examines the Monte Carlo experiments in Seo (1999 Seo , B. ( 1999 ). Distribution theory for unit root tests with conditional heteroskedasticity . J. Econometrics 91 : 113 – 144 .[Crossref], [Web of Science ®] , [Google Scholar]) for unit root tests with GARCH errors. We report a Monte Carlo study with data generated from various GARCH(1, 1) processes where 0.8 ≤ α + β < 1 and β > α. In this case, the Dickey–Fuller test works better than the Seo test.     

A finite-sample sensitivity analysis of the Dickey–Fuller test under local-to-unity detrending

In recent research, Elliott et al. (1996) Elliott, G. 1996. Efficient tests for an autoregressive unit root. Econometrica, 64: 813–836. [Crossref], [Web of Science ®] [Google Scholar] have shown the use of local-to-unity detrending via generalized least squares (GLS) to substantially increase the power of the Dickey–Fuller (1979) unit root test. In this paper the relationship between the extent of detrending undertaken, determined by the detrending parameter &art1;, and the power of the resulting GLS-based Dickey–Fuller (DF-GLS) test is examined. Using Monte Carlo simulation it is shown that the values of &art1; suggested by Elliott et al. (1996) Elliott, G. 1996. Efficient tests for an autoregressive unit root. Econometrica, 64: 813–836. [Crossref], [Web of Science ®] [Google Scholar] on the basis of a limiting power function seldom maximize the power of the DF-GLS test for the finite samples encountered in applied research. This result is found to hold for the DF-GLS test including either an intercept or an intercept and a trend term. An empirical examination of the order of integration of the UK household savings ratio illustrates these findings, with the unit root hypothesis rejected using values of &art1; other than that proposed by Elliott et al. (1996) Elliott, G. 1996. Efficient tests for an autoregressive unit root. Econometrica, 64: 813–836. [Crossref], [Web of Science ®] [Google Scholar].     

On Testing Sample Selection Bias Under the Multicollinearity Problem

ABSTRACT

This paper reviews and extends the literature on the finite sample behavior of tests for sample selection bias. Monte Carlo results show that, when the “multicollinearity problem” identified by Nawata (1993 Nawata , K. ( 1993 ). A note on the estimation of models with sample-selection biases . Economics Letters 42 : 15 – 24 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) is severe, (i) the t-test based on the Heckman–Greene variance estimator can be unreliable, (ii) the Likelihood Ratio test remains powerful, and (iii) nonnormality can be interpreted as severe sample selection bias by Maximum Likelihood methods, leading to negative Wald statistics. We also confirm previous findings (Leung and Yu, 1996 Leung , S. F. , Yu , S. ( 1996 ). On the choice between sample selection and two-part models . Journal of Econometrics 72 : 197 – 229 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) that the standard regression-based t-test (Heckman, 1979 Heckman , J. J. ( 1979 ). Sample selection bias as a specification error . Econometrica 47 : 153 – 161 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and the asymptotically efficient Lagrange Multiplier test (Melino, 1982 Melino , A. ( 1982 ). Testing for sample selection bias . Review of Economic Studies 49 : 151 – 153 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), are robust to nonnormality but have very little power.     

On Asymptotic Normality of the Local Polynomial Regression Estimator with Stochastic Bandwidths

Nonparametric density and regression estimators commonly depend on a bandwidth. The asymptotic properties of these estimators have been widely studied when bandwidths are non stochastic. In practice, however, in order to improve finite sample performance of these estimators, bandwidths are selected by data driven methods, such as cross-validation or plug-in procedures. As a result, nonparametric estimators are usually constructed using stochastic bandwidths. In this article, we establish the asymptotic equivalence in probability of local polynomial regression estimators under stochastic and nonstochastic bandwidths. Our result extends previous work by Boente and Fraiman (1995 Boente , G. , Fraiman , R. ( 1995 ). Asymptotic distribution of data-driven smoothers in density and regression estimation under dependence . Can. J. Statist. 23 : 383 – 397 .[Crossref], [Web of Science ®] , [Google Scholar]) and Ziegler (2004 Ziegler , K. ( 2004 ). Adaptive kernel estimation of the mode in nonparametric random design regression model . Probab. Mathemat. Statist. 24 : 213 – 235 . [Google Scholar]).     

A Note on the Convergence Rate of Strong Law of Large Numbers for Positively Associated Sequences

By the inequalities established in this article, we obtain the convergence rate of strong law of large numbers for positively associated sequences. The results derived extend and improve the corresponding ones in Vronskii (1999 Vronskii , M. A. ( 1999 ). Rate of convergence in the slln for associated sequences and fields . Theory Probab. Appl. 43 ( 3 ): 449 – 462 .[Crossref], [Web of Science ®] , [Google Scholar]).     

The Block-Block Bootstrap for Time Series

This article proves that the block-block bootstrap of Andrews (2004 Andrews , D. W. K. ( 2004 ). The block-block bootstrap: improved asymptotic refinements . Econometrica 72 ( 3 ): 673 – 700 .[Crossref], [Web of Science ®] , [Google Scholar]) can be helpful to provide asymptotic refinements for the GMM estimator when autocorrelation structures of moment functions are unknown (i.e., incorporating the HAC covariance matrix) and when we allow for statistics that are inefficient. The asymptotic refinements of this block-block bootstrap in the time series context are shown to exist with the use of less restricted kernels than in the block bootstrap in Inoue and Shintani (2006 Inoue , A. , Shintani , M. ( 2006 ). Bootstrapping GMM estimators for time series . J. Econometrics 113 : 531 – 555 .[Crossref] , [Google Scholar]), since they do not require to have a characteristic exponent larger than 2. The procedure allows to apply in practice kernels that guarantee that the HAC covariance matrix estimator is positive semidefinite, and to get asymptotic refinements at the same time.     

Approximate Asymptotic Variance-Covariance Matrix for the Whittle Estimators of GAR(1) Parameters

Generalized Autoregressive (GAR) processes have been considered to model some features in time series. The Whittle's estimates have been investigated for the GAR(1) process by a simulation study by Shitan and Peiris (2008 Shitan , M. , Peiris , S. ( 2008 ). Generalised autoregressive (GAR) model: a comparison of maximum likelihood and whittle estimation procedures using a simulation study . Commun. Statist. Simul. Computat. 37 ( 3 ): 560 – 570 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). This article derives approximate theoretical expressions for the enteries of the asymptotic variance-covariance matrix for those estimates of GAR(1) parameters. These results are supported by a simulation study.     

Testing for Unit Roots and the Impact of Quadratic Trends,with an Application to Relative Primary Commodity Prices

In practice a degree of uncertainty will always exist concerning what specification to adopt for the deterministic trend function when running unit root tests. While most macroeconomic time series appear to display an underlying trend, it is often far from clear whether this component is best modeled as a simple linear trend (so that long-run growth rates are constant) or by a more complicated nonlinear trend function which may, for instance, allow the deterministic trend component to evolve gradually over time. In this article, we consider the effects on unit root testing of allowing for a local quadratic trend, a simple yet very flexible example of the latter. Where a local quadratic trend is present but not modeled, we show that the quasi-differenced detrended Dickey–Fuller-type test of Elliott et al. (1996 Elliott , G. , Rothenberg , T. J. , Stock , J. H. ( 1996 ). Efficient tests for an autoregressive unit root . Econometrica 64 : 813 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]) has both size and power which tend to zero asymptotically. An extension of the Elliott et al. (1996 Elliott , G. , Rothenberg , T. J. , Stock , J. H. ( 1996 ). Efficient tests for an autoregressive unit root . Econometrica 64 : 813 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]) approach to allow for a quadratic trend resolves this problem but is shown to result in large power losses relative to the standard detrended test when no quadratic trend is present. We consequently propose a simple and practical approach to dealing with this form of uncertainty based on a union of rejections-based decision rule whereby the unit root is rejected whenever either of the detrended or quadratic detrended unit root tests rejects. A modification of this basic strategy is also suggested which further improves on the properties of the procedure. An application to relative primary commodity price data highlights the empirical relevance of the methods outlined in this article. A by-product of our analysis is the development of a test for the presence of a quadratic trend which is robust to whether the data admit a unit root.     

Data-Driven Tests for Trend

This article proposes a new nonparametric test for the ordered alternatives problem in the k-sample setting for null hypothesis of lack of trend. This article further elaborates upon and extends the results of Ledwina and Wy?upek (2012a Ledwina , T. , Wy?upek , G. ( 2012a ). Two-sample test against one-sided alternatives . Scand. J. Statist. 39 : 358 – 381 .[Crossref], [Web of Science ®] , [Google Scholar]) obtained for k = 2. Simulations show that the new test has high and stable power and is able to control the Type I error to satisfactory extent, thus solving the problem posed in Terpstra and Magel (2003 Terpstra , J. T. , Magel , R. C. ( 2003 ). A new nonparametric test for the ordered alternative problem . J. Nonparametr. Statist. 15 : 289 – 301 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Our theoretical results say that asymptotic errors of both kinds do not exceed significance level, thus implying that the test is asymptotically unbiased.     

Best Spatial Two‐Stage Least Squares Estimators for a Spatial Autoregressive Model with Autoregressive Disturbances

Abstract

Estimation of a cross‐sectional spatial model containing both a spatial lag of the dependent variable and spatially autoregressive disturbances are considered. [Kelejian and Prucha (1998)] Kelejian, H. H. and Prucha, I. R. 1998. A generalized spatial two‐stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances. J. Real Estate Financ. and Economics, 17: 99–121. [Crossref], [Web of Science ®] , [Google Scholar]described a generalized two‐stage least squares procedure for estimating such a spatial model. Their estimator is, however, not asymptotically optimal. We propose best spatial 2SLS estimators that are asymptotically optimal instrumental variable (IV) estimators. An associated goodness‐of‐fit (or over identification) test is available. We suggest computationally simple and tractable numerical procedures for constructing the optimal instruments.     

A Broad Class of Partially Specified Autoregressions on Multi-Casting Data

Extending the bifurcating autoregressive (BAR) process (cf. Cowan and Staudte, 1986 Cowan , R. , Staudte , R. G. ( 1986 ). The bifurcating autoregression model in cell lineage studies . Biometrics 42 : 769 – 783 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to multi-casting (multi-splitting) data, Hwang and Choi (2009 Hwang , S. Y. , Choi , M. S. ( 2009 ). Modeling and large sample estimation for multi-casting autoregression . Statist. Prob. Lett. 79 : 1943 – 1950 .[Crossref], [Web of Science ®] , [Google Scholar]) introduced multi-casting autoregression (MCAR, for short) defined on multi-casting tree structured data. This article is concerned with the case when the MCAR model is partially specified only through conditional mean and variance without directly imposing autoregressive (AR) structure. The resulting class of models will be referred to as P-MCAR (partially specified MCAR). The P-MCAR considerably enlarges the class of multi-casting models including (as special cases) MCAR, random coefficient MCAR, conditionally heteroscedastic multi-casting models and binomial-thinning processes. Moment structures for this broad P-MCAR class are investigated. Least squares (LS) estimation method is discussed and asymptotic relative efficiency (ARE) of the generalized-LS over ordinary-LS is obtained in a closed form. A simulation study is conducted to illustrate results.     

Peaks-Over-Threshold Modeling Under Random Censoring

Recently, the topic of extreme value under random censoring has been considered. Different estimators for the index have been proposed (see Beirlant et al., 2007 Beirlant , J. , Guillou , A. , Dierckx , G. , Fils-Villetard , A. ( 2007 ). Estimation of the extreme value index and extreme quantiles under random censoring . Extremes 10 : 151 – 174 .[Crossref] , [Google Scholar]). All of them are constructed as the classical estimators (without censoring) divided by the proportion of non censored observations above a certain threshold. Their asymptotic normality was established by Einmahl et al. (2008 Einmahl , J. H. J. , Fils-Villetard , A. , Guillou , A. ( 2008 ). Statistics of extremes under random censoring . Bernoulli 14 ( 1 ): 207 – 227 . [Google Scholar]). An alternative approach consists of using the Peaks-Over-Threshold method (Balkema and de Haan, 1974 Balkema , A. , de Haan , L. ( 1974 ). Residual life at great age . Ann. Probab. 2 : 792 – 804 .[Crossref], [Web of Science ®] , [Google Scholar]; Smith, 1987 Smith , R. L. ( 1987 ). Estimating tails of probability distributions . Ann. Statist. 15 : 1174 – 1207 .[Crossref], [Web of Science ®] , [Google Scholar]) and to adapt the likelihood to the context of censoring. This leads to ML-estimators whose asymptotic properties are still unknown. The aim of this article is to propose one-step approximations, based on the Newton-Raphson algorithm. Based on a small simulation study, the one-step estimators are shown to be close approximations to the ML-estimators. Also, the asymptotic normality of the one-step estimators has been established, whereas in case of the ML-estimators it is still an open problem. The proof of our result, whose approach is new in the Peaks-Over-Threshold context, is in the spirit of Lehmann's theory (1991 Lehmann , E. L. ( 1991 ). Theory of Point Estimation . Pacific Grove , CA : Wadsworth & Brooks/Cole Advanced Books & Software .[Crossref] , [Google Scholar]).     

A Generalization of the Skew-Normal Distribution: The Beta Skew-Normal

We consider a new generalization of the skew-normal distribution introduced by Azzalini (1985 Azzalini , A. ( 1985 ). A class of distributions which includes the normal ones . Scand. J. Statis. 12 ( 2 ): 171 – 178 .[Web of Science ®] , [Google Scholar]). We denote this distribution Beta skew-normal (BSN) since it is a special case of the Beta generated distribution (Jones, 2004 Jones , M. C. ( 2004 ). Families of distributions of order statistics . Test 13 ( 1 ): 1 – 43 .[Crossref], [Web of Science ®] , [Google Scholar]). Some properties of the BSN are studied. We pay attention to some generalizations of the skew-normal distribution (Bahrami et al., 2009 Bahrami , W. , Agahi , H. , Rangin , H. ( 2009 ). A two-parameter Balakrishnan skew-normal distribution . J. Statist. Res. Iran 6 : 231 – 242 . [Google Scholar]; Sharafi and Behboodian, 2008 Sharafi , M. , Behboodian , J. ( 2008 ). The Balakrishnan skew-normal density . Statist. Pap. 49 : 769 – 778 .[Crossref], [Web of Science ®] , [Google Scholar]; Yadegari et al., 2008 Yadegari , I. , Gerami , A. , Khaledi , M. J. ( 2008 ). A generalization of the Balakrishnan skew-normal distribution . Statist. Probab. Lett. 78 : 1165 – 1167 .[Crossref], [Web of Science ®] , [Google Scholar]) and to their relations with the BSN.     

Asymptotics of improved generalized moment estimators for spatial autoregressive error models

Abstract

This article considers linear models with a spatial autoregressive error structure. Extending Arnold and Wied (2010) Arnold, M., Wied, D. (2010). Improved GMM estimation of the spatial autoregressive error model. Econ. Lett. 108:65–68.[Crossref], [Web of Science ®] , [Google Scholar], who develop an improved generalized method of moment (GMM) estimator for the parameters of the disturbance process to reduce the bias of existing estimation approaches, we establish the asymptotic normality of a new weighted version of this improved estimator and derive the efficient weighting matrix. We also show that this efficiently weighted GMM estimator is feasible as long as the regression matrix of the underlying linear model is non stochastic and illustrate the performance of the new estimator by a Monte Carlo simulation and an application to real data.     

Tests for Multiple Outliers in an Exponential Sample

By applying the recursion of Huffer (1988 Huffer, F. 1988. Divided differences and the joint distribution of linear combinations of spacings. Journal of Applied Probability, 25: 346–354. [Crossref], [Web of Science ®] , [Google Scholar]) repeatedly, we propose an algorithm for evaluating the null joint distribution of Dixon-type test statistics for testing discordancy of k upper outliers in exponential samples. By using the critical values of Dixon-type test statistics determined from the proposed algorithm and those of Cochran-type test statistics presented earlier by Lin and Balakrishnan (2009 Lin, C. T. and Balakrishnan, N. 2009. Exact computation of the null distribution of a test for multiple outliers in an exponential sample. Computational Statistics & Data Analysis, 53: 3281–3290. [Crossref], [Web of Science ®] , [Google Scholar]), we carry out an extensive Monte Carlo study to investigate the powers and the error probabilities for the effects of masking and swamping when the number of outliers k = 2 and 3. Based on our empirical findings, we recommend Rosner’s (1975 Rosner, B. 1975. On the detection of many outliers. Technometrics, 17: 221–227. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) sequential test procedure based on Dixon-type test statistics for testing multiple outliers from an exponential distribution.