Bayesian inference and forecasting in the stationary bilinear model

A stationary bilinear (SB) model can be used to describe processes with a time-varying degree of persistence that depends on past shocks. This study develops methods for Bayesian inference, model comparison, and forecasting in the SB model. Using monthly U.K. inflation data, we find that the SB model outperforms the random walk, first-order autoregressive AR(1), and autoregressive moving average ARMA(1,1) models in terms of root mean squared forecast errors. In addition, the SB model is superior to these three models in terms of predictive likelihood for the majority of forecast observations.     

A Bayesian panel data framework for examining the economic growth convergence hypothesis: do the G7 countries converge?

In this paper, we suggest a Bayesian panel (longitudinal) data approach to test for the economic growth convergence hypothesis. This approach can control for possible effects of initial income conditions, observed covariates and cross-sectional correlation of unobserved common error terms on inference procedures about the unit root hypothesis based on panel data dynamic models. Ignoring these effects can lead to spurious evidence supporting economic growth divergence. The application of our suggested approach to real gross domestic product panel data of the G7 countries indicates that the economic growth convergence hypothesis is supported by the data. Our empirical analysis shows that evidence of economic growth divergence for the G7 countries can be attributed to not accounting for the presence of exogenous covariates in the model.     

Unit Root Testing in Heteroscedastic Panels Using the Cauchy Estimator

The Cauchy estimator of an autoregressive root uses the sign of the first lag as instrumental variable. The resulting IV t-type statistic follows a standard normal limiting distribution under a unit root case even under unconditional heteroscedasticity, if the series to be tested has no deterministic trends. The standard normality of the Cauchy test is exploited to obtain a standard normal panel unit root test under cross-sectional dependence and time-varying volatility with an orthogonalization procedure. The article’s analysis of the joint N, T asymptotics of the test suggests that (1) N should be smaller than T and (2) its local power is competitive with other popular tests. To render the test applicable when N is comparable with, or larger than, T, shrinkage estimators of the involved covariance matrix are used. The finite-sample performance of the discussed procedures is found to be satisfactory.     

Tests for Unit-Root versus Threshold Specification With an Application to the Purchasing Power Parity Relationship

We consider modeling the real exchange rate by a stationary three-regime self-exciting threshold autoregressive (SETAR) model with possibly a unit root in the middle regime. This representation is consistent with purchasing power parity in the presence of trading costs. Our main contribution is to provide statistical tools for testing unit root versus a SETAR. First, we show that a SETAR with a unit root in the middle regime is stationary and mixing under reasonable assumptions. Second, we derive analytically the asymptotic distribution of our unit-root test under the null. Using monthly real exchange rate data, our test rejects the null of unit-root against a threshold process for five European series.     

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].     

Bayesian analysis of panel data using an MTAR model

Bayesian analysis of panel data using a class of momentum threshold autoregressive (MTAR) models is considered. Posterior estimation of parameters of the MTAR models is done by using a simple Markov Chain Monte Carlo (MCMC) algorithm. Selection of appropriate differenced variables, test for asymmetry and unit roots are recast as model selections and a simple way of computing posterior probabilities of the candidate models is proposed. The proposed method is applied to the yearly unemployment rates of 51 US states and the results show strong evidence of stationarity and asymmetry.     

房地产价格区域间联动与泡沫的空间扩散 ——基于2000－2005年中国35个大中城市面板数据的实证检验

 利用中国35个大中城市2000－2005年间的面板数据，本文首先对房地产价格中的泡沫成分进行测度；然后构建泡沫自回归模型，并对其残差进行CSD检验，结果显示不同城市间房地产泡沫的演化过程相互影响。与前人研究不同，以房地产泡沫为研究对象，不仅可以确认中国不同城市间房地产价格存在联动性，而且可以证明消费者的适应性预期是其中重要的传导机制之一。     

Bayesian tests for unit root and multiple breaks

A Bayesian approach is considered for identifying sources of nonstationarity for models with a unit root and breaks. Different types of multiple breaks are allowed through crash models, changing growth models, and mixed models. All possible nonstationary models are represented by combinations of zero or nonzero parameters associated with time trends, dummy for breaks, or previous levels, for which Bayesian posterior probabilities are computed. Multiple tests based on Markov chain Monte Carlo procedures are implemented. The proposed method is applied to a real data set, the Korean GDP data set, showing a strong evidence for two breaks rather than the usual unit root or one break.     

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 bayesian analysis of trend determination in economic time series

In this paper we provide a comprehensive Bayesian posterior analysis of trend determination in general autoregressive models. Multiple lag autoregressive models with fitted drifts and time trends as well as models that allow for certain types of structural change in the deterministic components are considered. We utilize a modified information matrix-based prior that accommodates stochastic nonstationarity, takes into account the interactions between long-run and short-run dynamics and controls the degree of stochastic nonstationarity permitted. We derive analytic posterior densities for all of the trend determining parameters via the Laplace approximation to multivariate integrals. We also address the sampling properties of our posteriors under alternative data generating processes by simulation methods. We apply our Bayesian techniques to the Nelson-Plosser macroeconomic data and various stock price and dividend data. Contrary to DeJong and Whiteman (1989a,b,c), we do not find that the data overwhelmingly favor the existence of deterministic trends over stochastic trends. In addition, we find evidence supporting Perron's (1989) view that some of the Nelson and Plosser data are best construed as trend stationary with a change in the trend function occurring at 1929.     

A bayesian analysis of trend determination in economic time series

In this paper we provide a comprehensive Bayesian posterior analysis of trend determination in general autoregressive models. Multiple lag autoregressive models with fitted drifts and time trends as well as models that allow for certain types of structural change in the deterministic components are considered. We utilize a modified information matrix-based prior that accommodates stochastic nonstationarity, takes into account the interactions between long-run and short-run dynamics and controls the degree of stochastic nonstationarity permitted. We derive analytic posterior densities for all of the trend determining parameters via the Laplace approximation to multivariate integrals. We also address the sampling properties of our posteriors under alternative data generating processes by simulation methods. We apply our Bayesian techniques to the Nelson-Plosser macroeconomic data and various stock price and dividend data. Contrary to DeJong and Whiteman (1989a,b,c), we do not find that the data overwhelmingly favor the existence of deterministic trends over stochastic trends. In addition, we find evidence supporting Perron's (1989) view that some of the Nelson and Plosser data are best construed as trend stationary with a change in the trend function occurring at 1929.     

Bayesian inference for random coefficient dynamic panel data models

We develop a hierarchical Bayesian approach for inference in random coefficient dynamic panel data models. Our approach allows for the initial values of each unit's process to be correlated with the unit-specific coefficients. We impose a stationarity assumption for each unit's process by assuming that the unit-specific autoregressive coefficient is drawn from a logitnormal distribution. Our method is shown to have favorable properties compared to the mean group estimator in a Monte Carlo study. We apply our approach to analyze energy and protein intakes among individuals from the Philippines.     

Testing for Unit Roots in Dynamic Panels in the Presence of a Deterministic Trend: Re-examining the Unit Root Hypothesis for Real Stock Prices and Dividends

In this paper, we suggest a similar unit root test statistic for dynamic panel data with fixed effects. The test is based on the LM, or score, principle and is derived under the assumption that the time dimension of the panel is fixed, which is typical in many panel data studies. It is shown that the limiting distribution of the test statistic is standard normal. The similarity of the test with respect to both the initial conditions of the panel and the fixed effects is achieved by allowing for a trend in the model using a parameterisation that has the same interpretation under both the null and alternative hypotheses. This parameterisation can be expected to increase the power of the test statistic. Simulation evidence suggests that the proposed test has empirical size that is very close to the nominal level and considerably more power than other panel unit root tests that assume that the time dimension of the panel is large. As an application of the test, we re-examine the stationarity of real stock prices and dividends using disaggregated panel data over a relatively short period of time. Our results suggest that while real stock prices contain a unit root, real dividends are trend stationary.     

Nonstationary panel data analysis: an overview of some recent developments

This paper overviews some recent developments in panel data asymptotics, concentrating on the nonstationary panel case and gives a new result for models with individual effects. Underlying recent theory are asymptotics for multi-indexed processes in which both indexes may pass to infinity. We review some of the new limit theory that has been developed, show how it can be applied and give a new interpretation of individual effects in nonstationary panel data. Fundamental to the interpretation of much of the asymptotics is the concept of a panel regression coefficient which measures the long run average relation across a section of the panel. This concept is analogous to the statistical interpretation of the coefficient in a classical regression relation. A variety of nonstationary panel data models are discussed and the paper reviews the asymptotic properties of estimators in these various models. Some recent developments in panel unit root tests and stationary dynamic panel regression models are also reviewed.     

Nonstationary panel data analysis: an overview of some recent developments

This paper overviews some recent developments in panel data asymptotics, concentrating on the nonstationary panel case and gives a new result for models with individual effects. Underlying recent theory are asymptotics for multi-indexed processes in which both indexes may pass to infinity. We review some of the new limit theory that has been developed, show how it can be applied and give a new interpretation of individual effects in nonstationary panel data. Fundamental to the interpretation of much of the asymptotics is the concept of a panel regression coefficient which measures the long run average relation across a section of the panel. This concept is analogous to the statistical interpretation of the coefficient in a classical regression relation. A variety of nonstationary panel data models are discussed and the paper reviews the asymptotic properties of estimators in these various models. Some recent developments in panel unit root tests and stationary dynamic panel regression models are also reviewed.     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.     

Combining p-Values in Non-Stationary Panels

This article summarizes and discusses the existing p-value pooling approaches and compares their performances in the context of panel unit root tests. When the data are free of contemporaneous correlation, most tests achieve very high power. However, in the presence of contemporaneous correlation, most tests suffer from moderate to severe size distortions. When the panel contains both stationary and nonstationary series, the power of tests increases as the cross-sectional units grows. Among all the tests under study, the mean-of-Z test yields the highest power for the benchmark model, while the Fisher test is most robust for complicated model structures.     

Using the Residual White Noise Autoregressive Order Determination Criterion to Identify Unit Roots in Arima Models

We present a simplified form of a univariate identification approach for time series models based on the residual white noise autoregressive order determination criterion and linear estimation methods. We also show how the procedure can be used to identify the degree of differencing necessary to induce stationarity in data. The performance of this approach is also contrasted with Portmanteau tests for detection of white noise residuals and with Dickey-Fuller and Bayesian procedures for detection of unit roots. Simulated and economic data are used to demonstrate the capabilities of the modified approach.     

Testing for coefficient stability of AR(1) model when the null is an integrated or a stationary process

In this paper, we propose a new test for coefficient stability of an AR(1) model against the random coefficient autoregressive model of order 1 neither assuming a stationary nor a non-stationary process under the null hypothesis of a constant coefficient. The proposed test is obtained as a modification of the locally best invariant (LBI) test by Lee [(1998). Coefficient constancy test in a random coefficient autoregressive model. J. Statist. Plann. Inference 74, 93–101]. We examine finite sample properties of the proposed test by Monte Carlo experiments comparing with other existing tests, in particular, the LBI test by McCabe and Tremayne [(1995). Testing a time series for difference stationary. Ann. Statist. 23 (3), 1015–1028], which is for the null of a unit root process against the alternative of a stochastic unit root process.     

具有GARCH（1，1）-正态误差项的非对称单位根检验研究

摘　　要:理论研究表明许多经济变量呈现出非对称的门限自回归（TAR）或动态门限自回归（M-TAR）数据生成机制,因而非对称单位根检验就成为该领域的主要研究方向之一。本文对非对称单位根检验Enders-Granger方法在GARCH（1,1）-正态误差项下的检验水平与检验势作了系统的仿真研究。研究表明：GARCH（1,1）-正态误差项的TAR或M-TAR模型会对该方法的检验水平和检验势产生重要影响。