Stochastic unit root processes: Maximum likelihood estimation,and new Lagrange multiplier and likelihood ratio tests

We show in this study that the maximum likelihood estimators of stochastic unit root (STUR) processes are consistent and asymptotically normally distributed. We also present two new tests for STUR. We first propose a Lagrange multiplier test and show that it has a standard χ² distribution asymptotically. We also propose a likelihood ratio test and show that it has an asymptotic distribution of 50–50 mixture of χ² and a point mass at 0. As an empirical example, we test the existence of STUR in the Canadian real exchange rate and explore the implication of STUR on the validity of purchasing power parity.     

ESTIMATING MULTICOUNTRY VAR MODELS*

This article presents a method to estimate the coefficients, to test specification hypotheses, and to conduct policy exercises in multicountry Vector Autoregressive (VAR) models with cross‐unit interdependencies, unit‐specific dynamics, and time variations in the coefficients. The framework of analysis is Bayesian: A prior flexibly reduces the dimensionality of the model and puts structure on the time variations, Markov chain Monte Carlo (MCMC) methods are used to obtain posterior distributions, and marginal likelihoods to check the fit of various specifications. Impulse responses and conditional forecasts are obtained with the output of an MCMC routine. The transmission of certain shocks across countries is analyzed.     

进口的经济增长效应——基于趋势突变的协整检验分析

当时间序列存在结构突变时,会降低单位根检验和协整检验的检验势.通过对进口对于经济增长的影响作用进行实证分析,可以得到进口没有发挥出应有的促进经济增长的作用;结构突变理论比较适合于研究转型期的中国经济问题的结论.     

Bootstrapping Smooth Functions of Slope Parameters and Innovation Variances in VAR(∞) Models*

It is common to conduct bootstrap inference in vector autoregressive (VAR) models based on the assumption that the underlying data‐generating process is of finite‐lag order. This assumption is implausible in practice. We establish the asymptotic validity of the residual‐based bootstrap method for smooth functions of VAR slope parameters and innovation variances under the alternative assumption that a sequence of finite‐lag order VAR models is fitted to data generated by a VAR process of possibly infinite order. This class of statistics includes measures of predictability and orthogonalized impulse responses and variance decompositions. Our approach provides an alternative to the use of the asymptotic normal approximation and can be used even in the absence of closed‐form solutions for the variance of the estimator. We illustrate the practical relevance of our findings for applied work, including the evaluation of macroeconomic models.     

Modified Quasi‐Likelihood Ratio Test for Regime Switching

In this paper we propose a modified quasi‐likelihood ratio test of the null hypothesis of one regime against the alternative of two regimes in Markov regime‐switching models. The asymptotic distribution of the proposed test statistic is a simple function of Gaussian random variables, and the inference is no more complicated than in the standard case. Our simulations show that the proposed test has good finite sample size and power that are comparable to the quasi‐likelihood ratio test of Cho and White. We apply our test to stock returns and Japanese policy functions.     

Specification via model selection in vector error correction models

This paper proposes a model selection approach for the specification of the cointegrating rank in the VECM representation of VAR models. Asymptotic properties of estimates are derived and their features compared with the traditional likelihood ratio based approach.     

THE EXTENT OF THE WORLD COFFEE MARKET

This paper employs a novel method to determine whether the prices of various types of coffee are co-integrated. In the spirit of Stigler's arbitrage definition of the market, an attempt is made to test whether all the prices co-integrate in pairs, implying that there is a single market for coffee. This test for co-integration is carried out using both linear and non-linear approaches. This finding demonstrates that the conclusions drawn from the linear and non-linear tests may be drastically different.     

Power issues when testing the Markov switching model with the sup likelihood ratio test using U.S. output

The likelihood ratio (LR) test statistic for the test of a linear AR(1) model against the alternative of a Markov switching model does not possess the standard χ² distribution. Garcia (1998) derives the asymptotic distribution of the Sup LR test statistic under these non-standard conditions allowing the researcher to easily compare the two models. This paper examines the power properties of this test statistic using Monte Carlo experiments calibrated to U.S. output growth data. The results suggest a test of reasonable power. When the experiments are calibrated to annual data, power is 82% at 200 observations. When the experiments are calibrated to quarterly data power is 57% for the same sample size. First Version Received: March 2000/Final Version Received: March 2001     

Finite-sample distribution of the augmented Dickey–Fuller test with lag optimization

This article presents a response surface analysis for the distribution of the augmented Dickey–Fuller (ADF) test with optimized lag order in the autoregression using frequently applied data-dependent methods. Simulation results demonstrate that finite-sample distribution of the test depends critically on the lag-order determination rule. The univariate test and its panel counterparts exhibit size distortion when the lag order in the autoregression is optimized but inappropriate distribution is employed in test implementation. Response surface coefficients reported for the finite-sample distribution of the test with lag optimization are therefore useful tools for practitioners in applied research. The simulation evidence and practical use of the response surface coefficients are substantiated with empirical illustrations.     

A THREE‐STEP PROCEDURE FOR ESTIMATING AND TESTING COINTEGRATED ARMAX MODELS*

To deal with a variety of inferential problems on non‐stationary cointegrated time series, this paper proposes a computationally feasible method based on the Whittle likelihood and examines its performance. For the empirical application of our method, the paper investigates three sets of Japanese and US monetary and financial time‐series data. To evaluate the p‐value of the likelihood ratio statistic, we propose an approximation procedure based on the gamma distribution and the accompanying Laguerre expansion for reducing the computational burden. We also provide a numerical procedure for the asymptotic covariance matrix of the Whittle estimator.     

中国进出口额与利率变动的关系——基于交易费用的一个解释

文章运用ADF检验、协整关系检验等对我国的出口额占进口额的比例(简称出口占进口比)和利率的关系进行实证检验,结果表明我国的出口占进口比和名叉利率闻存在着长期协整关系,并且是负相关关系.随后本文测算了每单位实际利率变动对出口占进口比的影响,结果表明大多数样本的影响值小于0,这个结果显然和传统理论的解释相异,为此文章建立了一个存在交易费用的两阶段开放经济模型解释我国的情况,最后提出一点启示.     

Statistical inference of partially linear varying coefficient spatial autoregressive models

This paper proposes a semiparametric partially linear varying coefficient spatial autoregressive model, which is a generalization of standard spatial autoregressive model and partially linear spatial autoregressive model. To estimate the unknown spatial lag parameter, constant coefficients and coefficient functions, a profile quasi-maximum likelihood approach based on the local-linear method is introduced. To test the existence of the spatial effects, a generalized likelihood ratio test statistic is proposed, and a residual-based bootstrap procedure is used to derive the p-value of the test. Some simulations are conducted to examine the performance of our proposed procedures and the results are satisfactory. Furthermore, a real-world example is given to demonstrate the application of the proposed procedures.     

Testing for Unit Roots in Small Panels with Short-run and Long-run Cross-sectional Dependencies

An IV approach, using as instruments non-linear transformations of the lagged levels, is explored to test for unit roots in panels with general dependency and heterogeneity across cross-sectional units. We allow not only for the cross-sectional dependencies of innovations, but also for the presence of co-integration across cross-sectional levels. Unbalanced panels and panels with differing individual short-run dynamics and cross-sectionally related dynamics are also permitted. We also more carefully formulate the unit root hypotheses in panels. In particular, using order statistics, we make it possible to test for and against the presence of unit roots in some of the individual units for a given panel. The individual IV t -ratios, which are the bases of our tests, are asymptotically and normally distributed and cross-sectionally independent. Therefore, the critical values of the order statistics as well as the usual average statistic can be easily obtained from simple elementary probability computations. We show via a set of simulations that our tests work well, whereas other existing tests fail to perform properly. As an illustration, we apply our tests to the panels of real exchange rates, and find no evidence for the purchasing power parity hypothesis, which is in sharp contrast with the previous studies.     

THE POLITICAL ECONOMY OF BUDGET DEFICITS

In Leachman et al. (2005) we use the multicointegration approach to test for sustainable fiscal budgeting processes in a stochastic setting in 15 industrialized countries. In this paper, we extend the analysis in order to rank these same countries as well as an additional three, according to the degree to which their budget processes are sustainable. Rankings are related to theories regarding the political economy of budget deficits. Evidence clearly indicates that fiscal performance is better where fiscal budgeting institutions are strong. Additionally, we find that in conjunction with fiscal strength, greater degrees of federalism positively affect intertemporal budget management.     

Tests for cointegration rank and the initial condition

Many economic events involve initial observations that substantially deviate from long-run steady state. Such initial conditions are known to affect the power of univariate unit root tests diversely, whereas their impact on multivariate tests is largely unknown. This paper investigates the impact of the initial condition on the power of tests for cointegration rank, such as Johansen??s widely used likelihood ratio test, tests with prior adjustment for deterministic terms, and a test based on the eigenvalues of the companion matrix. We find that the power of the likelihood ratio test is increasing in the magnitude of the initial condition, whereas the power of the other tests is generally decreasing. We exploit these findings in an application to price convergence.     

A system of demand equations for Austria

This paper presents results of parameter estimations of a small system of demand equations for Austria. The functional form of the equations follows the log-linear specification well known as the “Rotterdam”-System. Using annual data from 1954 to 1977 we estimate the absolute price version for a rather aggregated system consisting of four sectors of consumption expenditures. Aitken estimation with and without linear restrictions is the adopted estimation method. Tests for the validity of the general linear restrictions axe performed employing the usual criteria. Relations among the test statistics are discussed. Taking into accountBeaton's [1972] argument of the appropriate use of likelihood ratio tests we present results also after iterating on the restricted error-covariance matrix. The question of negative semidefiniteness of the matrix of price coefficients is examined by inspection of its characteristic roots and the calculation of their approximated asymptotic covariance matrix. Finally, our results are confronted with such of other comparable studies.     

汇率波动、货币政策传导渠道及有效性——兼论"不可能三角"在我国的适用性

本文尝试把人民币汇率波动纳入货币政策中介目标监控体系,采用1996年1季度至2008年2季度的季度数据进行实证检验。研究表明:(1)我国货币政策是通过货币渠道来影响实际经济总量的,并且狭义货币供应量更能实现对实际产出目标的传导和调控;(2)人民币汇率波动的扩大会减少货币供应量;(3)不可能三角在我国是成立的;(4)将人民币汇率波动纳入货币政策中介目标监控体系能提高货币政策的有效性。     

Tests of restrictions in large demand systems

This paper compares some asymptotic tests of linear restrictions, using exact results and Monte Carlo methods, in systems which contain the same regressors in each equation. It is found that under a variety of conditions, a statistic suggested by Deaton provides a better approximation to the true distribution in small sample situations than the Wald, likelihood ratio or Lagrange multiplier tests.     

Remittances and GDP in Jamaica: An ARDL Bounds Testing Approach to Cointegration

Similar to other developing nations, Jamaica’s remittances, specifically inflows, are an important source of income support and foreign exchange earnings. Anecdotally, much has been said about the relationship between remittances and GDP in this country. Yet, less has been established using rigorous statistical inference. We test for unit roots with structural breaks and use the autoregressive distributed lag (ARDL) approach to cointegration to help fill this lacuna in the literature on Jamaica. Using annual data for the 1976–2014 period, we examine the relationship between GDP and remittances, both measured in constant 2010 US dollar terms, as we control for the common determinants of economic growth. The main finding is that GDP and remittances are cointegrated relationship wherein they both reinforce each other positively. This finding is statistically robust as the ARDL models have well-behaved errors and parameters that are generally stable over the period. We discuss policy implications of this finding.     

Modelling the US real GNP with fractionally integrated techniques

The US real GNP is analysed by means of fractionally integrated techniques. LM tests proposed by Robinson for testing unit roots and other fractionally integrated hypotheses are applied to the quarterly GNP series and to its log-transformation. The maximum likelihood estimation procedure of Sowell for estimating ARFIMA models is implemented. The results indicate that the order of integration of the US real output is much higher than one, and thus, the standard approach of taking first differences may still produce series with long memory behaviour.