Phillips-Perron-type unit root tests in the nonlinear ESTAR framework

Summary In this paper, we propose Phillips-Perron type, semi-parametric testing procedures to distinguish a unit root process from a mean-reverting exponential smooth transition autoregressive one. The limiting nonstandard distributions are derived under very general conditions and simulation evidence shows that the tests perform better than the standard Phillips-Perron or Dickey-Fuller tests in the region of the null. We would like to thank conference participants of the Pfingsttagung 2005 in Münster for their helpful comments.     

Size distortion of asymmetric unit root tests in the presence of level shifts

The finite-sample size properties of momentum-threshold autoregressive (MTAR) asymmetric unit root tests are examined in the presence of level shifts under the null hypothesis. The original MTAR test using a fixed threshold is found to exhibit severe size distortion when a break in level occurs early in the sample period, leading to an increased probability of an incorrect inference of asymmetric stationarity. For later breaks the test is also shown to suffer from undersizing. In contrast, the use of consistent-threshold estimation results in a test which is relatively robust to level shifts.     

A new modified Jackknifed estimator for the Poisson regression model

The Poisson regression is very popular in applied researches when analyzing the count data. However, multicollinearity problem arises for the Poisson regression model when the independent variables are highly intercorrelated. Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators and some methods for estimating the ridge parameter k in the Poisson regression have been proposed. It has been found that some estimators are better than the commonly used maximum-likelihood (ML) estimator and some other RR estimators. In this study, the modified Jackknifed Poisson ridge regression (MJPR) estimator is proposed to remedy the multicollinearity. A simulation study and a real data example are provided to evaluate the performance of estimators. Both mean-squared error and the percentage relative error are considered as the performance criteria. The simulation study and the real data example results show that the proposed MJPR method outperforms the Poisson ridge regression, Jackknifed Poisson ridge regression and the ML in all of the different situations evaluated in this paper.     

The successive raising estimator and its relation with the ridge estimator

The raised estimators are used to reduce collinearity in linear regression models by raising a column in the experimental data matrix which may be nearly linear with the other columns. The raising procedure has two components, namely stretching and rotating, which we can analyze separately. We give the relationship between the raised estimators and the classical ridge estimators. Using a case study, we show how to determine the perturbation parameter for the raised estimators by controlling the amount of precision to be retained in the original data.     

A consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function when the sample observations are contaminated with random noise is studied. A particular estimator f?_n(x) is proposed which uses kernel-density and deconvolution techniques. The estimator f?_n(x) is shown to be uniformly consistent, and its appearance and properties are affected by constants M_n and h_n which the user may choose. The optimal choices of M_n and h_n depend on the sample size n, the noise distribution, and the true distribution which is being estimated. Particular selections for M_n and h_n which minimize upper-bound functions of the mean squared error for f?_n(x) are recommended.     

A pre-test like estimator dominating the least-squares method

We develop a pre-test type estimator of a deterministic parameter vector β$\mathit{\beta }$ in a linear Gaussian regression model. In contrast to conventional pre-test strategies, that do not dominate the least-squares (LS) method in terms of mean-squared error (MSE), our technique is shown to dominate LS when the effective dimension is greater than or equal to 4. Our estimator is based on a simple and intuitive approach in which we first determine the linear minimum MSE (MMSE) estimate that minimizes the MSE. Since the unknown vector β$\mathit{\beta }$ is deterministic, the MSE, and consequently the MMSE solution, will depend in general on β$\mathit{\beta }$ and therefore cannot be implemented. Instead, we propose applying the linear MMSE strategy with the LS substituted for the true value of β$\mathit{\beta }$ to obtain a new estimate. We then use the current estimate in conjunction with the linear MMSE solution to generate another estimate and continue iterating until convergence. As we show, the limit is a pre-test type method which is zero when the norm of the data is small, and is otherwise a non-linear shrinkage of LS.     

A new heteroskedasticity-consistent covariance matrix estimator and inference under heteroskedasticity

To solve the heteroscedastic problem in linear regression, many different heteroskedasticity-consistent covariance matrix estimators have been proposed, including HC0 estimator and its variants, such as HC1, HC2, HC3, HC4, HC5 and HC4m. Each variant of the HC0 estimator aims at correcting the tendency of underestimating the true variances. In this paper, a new variant of HC0 estimator, HC5m, which is a combination of HC5 and HC4m, is proposed. Both the numerical analysis and the empirical analysis show that the quasi-t inference based on HC5m is typically more reliable than inferences based on other covariance matrix estimators, regardless of the existence of high leverage points.     

The asymptotic equivalence between the iterated improved 2sls estimator and the 3sls estimator

In this paper we show that the 3SLS estimator of a system of equations is asymptotically equivalent to an iterative 2SLS estimator applied to each equation, augmented with the residuals from the other equations. This result is a natural extension of Telser (1964).     

The asymptotic equivalence between the iterated improved 2sls estimator and the 3sls estimator

In this paper we show that the 3SLS estimator of a system of equations is asymptotically equivalent to an iterative 2SLS estimator applied to each equation, augmented with the residuals from the other equations. This result is a natural extension of Telser (1964).     

A comparison of alternative unit root tests

In this paper we evaluate the performance of three methods for testing the existence of a unit root in a time series, when the models under consideration in the null hypothesis do not display autocorrelation in the error term. In such cases, simple versions of the Dickey-Fuller test should be used as the most appropriate ones instead of the known augmented Dickey-Fuller or Phillips-Perron tests. Through Monte Carlo simulations we show that, apart from a few cases, testing the existence of a unit root we obtain actual type I error and power very close to their nominal levels. Additionally, when the random walk null hypothesis is true, by gradually increasing the sample size, we observe that p-values for the drift in the unrestricted model fluctuate at low levels with small variance and the Durbin-Watson (DW) statistic is approaching 2 in both the unrestricted and restricted models. If, however, the null hypothesis of a random walk is false, taking a larger sample, the DW statistic in the restricted model starts to deviate from 2 while in the unrestricted model it continues to approach 2. It is also shown that the probability not to reject that the errors are uncorrelated, when they are indeed not correlated, is higher when the DW test is applied at 1% nominal level of significance.     

On the cohen-sackrowitz estimator of a common mean

The paper reconsider certain estimators proposed by COHENand SACKROWITZ[Ann.Statist.(1974)2,1274-1282,Ann.Statist.4,1294]for the common mean of two normal distributions on the basis of independent samples of equal size from the two populations. It derives the ncecessary and sufficient condition for improvement over the first sample mean, under squared error loss, for any member of a class containing these. It shows that the estimator proposded by them for simultaneous improvement over botyh sample means has the desired property if and only if the common size of the samples is at least nine. The requirement is milder than that for any other estimator at the present state of knolwledge and may be constrasted with their result which implies the desired property of the estimator only if the common size of the samples is at least fifteen. Upper bounds for variances if the estimators derived by them are also improved     

Testing for a unit root in a stationary ESTAR process

This article develops a statistic for testing the null of a linear unit root process against the alternative of a stationary exponential smooth transition autoregressive model. The asymptotic distribution of the test is shown to be nonstandard but nuisance parameter-free and hence critical values are obtained by simulations. Simulations show that the proposed statistic has considerable power under various data generating scenarios. Applications to real exchange rates also illustrate the ability of our test to reject null of unit root when some of the alternative tests do not.     

A Bayes-type estimator for the Bradley-Terry model for paired comparison

A Bayes-type estimator is proposed for the worth parameter π_i and for the treatment effect parameter ln π_i in the Bradley-Terry Model for paired comparison. In contrast to current Bayes estimators which require iterative numberical calculations, this estimator has a closed form expression. This estimation technique is also extended to obtain estimators for the Luce Multiple Comparison Model. An application of this technique to a 2³ factorial experiment with paired comparisons is presented.     

On the asymptotic distribution of a simple unit root test for trending and breaking series

In this paper, we derive the asymptotic distribution of Popp's (2008) innovational outlier unit root test for trending series with a break. The results of Zivot and Andrews (1992) are applied to provide the limiting results of these new test statistics. We tabulate their asymptotic and finite sample critical values, and illustrate the use of the new statistics with an application to the unemployment rate series for 23 OECD countries.     

Double unit root tests for cross-sectionally dependent panel data

This paper proposes various double unit root tests for cross-sectionally dependent panel data. The cross-sectional correlation is handled by the projection method [P.C.B. Phillips and D. Sul, Dynamic panel estimation and homogeneity testing under cross section dependence, Econom. J. 6 (2003), pp. 217–259; H.R. Moon and B. Perron, Testing for a unit root in panels with dynamic factors, J. Econom. 122 (2004), pp. 81–126] or the subtraction method [J. Bai and S. Ng, A PANIC attack on unit roots and cointegration, Econometrica 72 (2004), pp. 1127–1177]. Pooling or averaging is applied to combine results from different panel units. Also, to estimate autoregressive parameters the ordinary least squares estimation [D.P. Hasza and W.A. Fuller, Estimation for autoregressive processes with unit roots, Ann. Stat. 7 (1979), pp. 1106–1120] or the symmetric estimation [D.L. Sen and D.A. Dickey, Symmetric test for second differencing in univariate time series, J. Bus. Econ. Stat. 5 (1987), pp. 463–473] are used, and to adjust mean functions the ordinary mean adjustment or the recursive mean adjustment are used. Combinations of different methods in defactoring to eliminate the cross-sectional dependency, integrating results from panel units, estimating the parameters, and adjusting mean functions yields various available tests for double unit roots in panel data. Simple asymptotic distributions of the proposed test statistics are derived, which can be used to find critical values of the test statistics.

We perform a Monte Carlo experiment to compare the performance of these tests and to suggest optimal tests for a given panel data. Application of the proposed tests to a real data, the yearly export panel data sets of several Latin–American countries for the past 50 years, illustrates the usefulness of the proposed tests for panel data, in that they reveal stronger evidence of double unit roots than the componentwise double unit root tests of Hasza and Fuller [Estimation for autoregressive processes with unit roots, Ann. Stat. 7 (1979), pp. 1106–1120] or Sen and Dickey [Symmetric test for second differencing in univariate time series, J. Bus. Econ. Stat. 5 (1987), pp. 463–473].     

A new estimator of finite population mean based on the dual use of the auxiliary information

When a sufficient correlation between the study variable and the auxiliary variable exists, the ranks of the auxiliary variable are also correlated with the study variable, and thus, these ranks can be used as an effective tool in increasing the precision of an estimator. In this paper, we propose a new improved estimator of the finite population mean that incorporates the supplementary information in forms of: (i) the auxiliary variable and (ii) ranks of the auxiliary variable. Mathematical expressions for the bias and the mean-squared error of the proposed estimator are derived under the first order of approximation. The theoretical and empirical studies reveal that the proposed estimator always performs better than the usual mean, ratio, product, exponential-ratio and -product, classical regression estimators, and Rao (1991 Rao, T.J. (1991). On certail methods of improving ration and regression estimators. Commun. Stat. Theory Methods 20(10):3325–3340.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Singh et al. (2009 Singh, R., Chauhan, P., Sawan, N., Smarandache, F. (2009). Improvement in estimating the population mean using exponential estimator in simple random sampling. Int. J. Stat. Econ. 3(A09):13–18. [Google Scholar]), Shabbir and Gupta (2010 Shabbir, J., Gupta, S. (2010). On estimating finite population mean in simple and stratified random sampling. Commun. Stat. Theory Methods 40(2):199–212.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Grover and Kaur (2011 Grover, L.K., Kaur, P. (2011). An improved estimator of the finite population mean in simple random sampling. Model Assisted Stat. Appl. 6(1):47–55. [Google Scholar], 2014) estimators.     

Symmetric Test for Second Differencing in Univariate Time Series

A test for the null hypothesis that a time series has characteristic equations with two unit roots is presented. The test, based on a standard regression computation, is shown to have good power properties when compared to previously existing tests.     

Bounded integrated processes and unit root tests

In the framework of integrated processes, the problem of testing the presence of unknown boundaries which constrain the process to move within a closed interval is considered. To analyze this problem, the concept of bounded integrated process is introduced, thus allowing to formally define boundary conditions for I(1) processes. A new class of tests, which are based on the rescaled range of the process, is introduced in order to test the null hypothesis of no boundary conditions. The limit distribution of the test statistics involved can be expressed in terms of the distribution of the range of Brownian functionals, while the power properties are obtained by deriving some asymptotic results for I(1) processes with boundary conditions. Both theoretical and simulation investigations show that range-based tests outperform standard unit root tests significantly when used to detect the presence of boundary conditions. A previous draft of the paper (Cavaliere, 2000) was presented at the 8th World Congress of the Econometric Society, Seattle, 11–16 August 2000. I wish sincerely to thank: Martin Jacobsen for his patience in discussing weak convergence to regulated Brownian motions and his valuable suggestions; the Department of Theoretical Statistics of the University of Copenhagen whose hospitality is gratefully acknowledged; Tommaso Proietti for important suggestions; Silvano Bordignon and partecipants at the CIdE seminar, University of Padua, June 2000; two anonymous referees. Partial financial support from 60% M.U.R.S.T. research grants is acknowledged.