Obtaining any Wald statistic you want

While the Wald test can be used to test a non-linear hypothesis in a linear or non-linear regression model it is known that a particular hypothesis can be written in many ways when non-linear forms are permitted. This paper illustrates that it is possible to obtain virtually any value of the Wald statistic at different significance levels. It is also shown that in small samples the use of the χ² or F approximation for the distribution of the Wald statistic can be misleading for some forms of the non-linear Wald test.     

Wald,likelihood ratio,and infinite induced test statistics for joint one-sided hypothesis

The relationship between certain infinite induced test statistics and the Wald and likelihood ratio statistics for testing hypothesis with one-sided alternatives is considered. In addition, it is shown that one of these infinite induced statistics reduces to a simple finite induced test statistic when the Wald statistic is equal to zero.     

Structural Changes, Common Stochastic Trends, and Unit Roots in Panel Data

This paper studies the problem of unit root testing in the presence of multiple structural changes and common dynamic factors. Structural breaks represent infrequent regime shifts, while dynamic factors capture common shocks underlying the comovement of economic time series. We examine the modified Sargan–Bhargava (MSB) test in the panel data setting and propose ways to handle multiple structural changes and dynamic factors. Properties of the MSB test under these non-standard conditions are derived. For example, the test statistics are shown to be invariant, in the limit, to mean breaks. This invariance does not carry over to breaks in linear trends, where the test statistics will converge to functionals of weighted Brownian bridges. A simplified test statistic is then proposed, which is invariant to both mean and trend breaks. We further study pooled test statistic based on standardization and combination of p -values. Response surfaces for p -values of all test statistics are computed to facilitate the empirical implementation of the proposed methodology. The pooled tests are shown to have good finite sample performance.     

Immigration,unemployment and GDP in the host country: Bootstrap panel Granger causality analysis on OECD countries

This paper examines the causality relationship between immigration, unemployment and economic growth of the host country. We employ the panel Granger causality testing approach of Kònya (2006) that is based on SUR systems and Wald tests with country specific bootstrap critical values. This approach allows one to test for Granger-causality on each individual panel member separately by taking into account the contemporaneous correlation across countries. Using annual data over the 1980–2005 period for 22 OECD countries, we find that, only in Portugal, unemployment negatively causes immigration, while in any country, immigration does not cause unemployment. On the other hand, our results show that, in four countries (France, Iceland, Norway and the United Kingdom), growth positively causes immigration, whereas in any country, immigration does not cause growth.     

Heteroskedasticity testing through a comparison of Wald statistics

This paper shows that a test for heteroskedasticity within the context of classical linear regression can be based on the difference between Wald statistics in heteroskedasticity-robust and nonrobust forms. The test is asymptotically distributed under the null hypothesis of homoskedasticity as chi-squared with one degree of freedom. The power of the test is sensitive to the choice of parametric restriction used by the Wald statistics, so the supremum of a range of individual test statistics is proposed. Two versions of a supremum-based test are considered: the first version does not have a known asymptotic null distribution, so the bootstrap is employed to approximate its empirical distribution. The second version has a known asymptotic distribution and, in some cases, is asymptotically pivotal under the null. A simulation study illustrates the use and finite-sample performance of both versions of the test. In this study, the bootstrap is found to provide better size control than asymptotic critical values, namely with heavy-tailed, asymmetric distributions of the covariates. In addition, the use of well-known modifications of the heteroskedasticity consistent covariance matrix estimator of OLS coefficients is also found to benefit the tests’ overall behaviour.     

Financial development and economic growth nexus in the MENA countries: Bootstrap panel granger causality analysis

This paper investigates the direction of causality between financial development and economic growth in the Middle East and North African (MENA) countries. The panel causality testing approach, developed by Kónya (2006) [Kónya, L. (2006), exports and growth: Granger causality analysis on OECD countries with a panel data approach, Economic Modelling, 23, 978–992], based on the Seemingly Unrelated Regressions and Wald tests with the country specific bootstrap critical values, is applied to the panel of fifteen MENA countries for the period 1980–2007. In order to capture the different aspects of financial development, six different indicators are used. Empirical results show that there is no clear consensus on the direction of causality between financial development and economic growth for all measurements of financial development and it is also observed that the findings are country specific.     

An aspect of the wald test for linear restrictions in the seemingly unrelated regressions model

This paper shows that, in a SUR context, there is an exact relationship between the Wald statistic for testing linear restrictions and the ‘F statistic’ based upon an estimate of the covariance matrix that adjusts for degrees of freedom.     

A Copula Nonlinear Granger Causality

We propose a new copula nonlinear Granger causality test that is more robust than the current available linear and nonlinear Granger causality tests when there exists an asymmetric and nonlinear directional dependence. To perform the statistical test of the copula nonlinear causality, the Gaussian Copula Marginal Regression (GCMR) model and copula directional dependence (Kim and Hwang, 2017) are employed in this paper. By using GCMR and two-sample permutation test with rank sum statistic for the copula nonlinear Granger causality, we can confirm that the result of the proposed copula nonlinear Granger causality test is a reliable test through the simulated data and real data both for small and large sample sizes.     

Consumer prices,wholesale prices,and causality

This paper uses the Wald variant of the Granger direct causality test recently prescribed by Geweke, Meese, and Dent (1983) to assess the causal nature of the pairwise relationship between U.S. consumer and wholesale (producer) prices. For the sample period, January 1947 to December 1983, as well as for the two sample subperiods, January 1947 to June 1971 and May 1974 to December 1983, the test reveal that bidirectional causality, or feedback, exists between monthly observations on seasonally unadjusted consumer and wholesale price changes. The finding of bidirectional causality corroborates other empirical evidence suggesthing that a simultaneous equation approach represents the appropriate way to estimate a bivariate model that consists of consumer and wholesale prices.     

Testing for Granger causality in heterogeneous mixed panels

In this paper, we propose a simple Granger causality procedure based on Meta analysis in heterogeneous mixed panels. Firstly, we examine the finite sample properties of the causality test through Monte Carlo experiments for panels characterized by both cross-section independency and cross-section dependency. Then, we apply the procedure for investigating the export led growth hypothesis in a panel data of twenty OECD countries.     

Inequalities for LR,W, and LM statistics

Sufficient conditions for inequalities involving the Hessian-based Wald and Lagrange multiplier statistics and the likelihood ratio statistic are presented. A condition for non-existence of strong inequality results is also given, as well as some examples.     

财政收支、货币供给与经济增长关系及变化趋势

运用向量误差修正模型、Wald系数约束检验,对我国财政收支、货币供给与经济增长关系进行实证分析。研究发现:在短期内,财政收支、货币供给是经济增长的Granger原因,财政收支、经济增长是货币供给增加的Granger原因;但从长期看,财政收支、货币供给与经济增长的因果关系不显著。因此,结合当前我国人民币升值压力和通胀预期的实际,应将财政政策与货币供给政策配合使用,即数量型工具和价格型工具要搭配,短期政策和长期政策要相搭配。     

Economic growth,exports and foreign direct investment in Least Developed Countries: A panel Granger causality analysis

This paper investigates potential Granger causality among the real GDP, real exports and inward FDI in Least Developed Countries for the period between 1970 and 2009. A new panel-data approach developed in Kónya (2006) [Kónya (2006), Exports and growth: Granger causality analysis on OECD countries with a panel data approach, Economic Modelling, 23, 978–992] which is based on SUR systems and Wald tests with country specific bootstrap critical values has been employed. The results indicate direct, one-period-ahead, unidirectional causality from exports to GDP in Haiti, Rwanda and Sierra Leone, and from GDP to exports in Angola, Chad and Zambia. Considering the FDI–Growth nexus, there is evidence of FDI Granger-causing GDP in Benin and Togo, and GDP Granger-causing FDI in Burkina Faso, Gambia, Madagascar and Malawi. While studying EXP–FDI relations, this paper finds that the causality is from FDI to real exports in Benin, Chad, Haiti, Mauritania, Niger, Togo and Yemen, and from real exports to FDI in Haiti, Madagascar, Mauritania, Malawi, Rwanda, Senegal and Zambia.     

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.     

TESTING FOR STATIONARITY IN HETEROGENEOUS PANEL DATA IN THE CASE OF MODEL MISSPECIFICATION

This paper investigates the performance of the tests proposed by Hadri and by Hadri and Larsson for testing for stationarity in heterogeneous panel data under model misspecification. The panel tests are based on the well known KPSS test (cf. Kwiatkowski et al.) which considers two models: stationarity around a deterministic level and stationarity around a deterministic trend. There is no study, as far as we know, on the statistical properties of the test when the wrong model is used. We also consider the case of the simultaneous presence of the two types of models in a panel. We employ two asymptotics: joint asymptotic, T, N →∞ simultaneously, and T fixed and N allowed to grow indefinitely. We use Monte Carlo experiments to investigate the effects of misspecification in sample sizes usually used in practice. The results indicate that the assumption that T is fixed rather than asymptotic leads to tests that have less size distortions, particularly for relatively small T with large N panels (micro‐panels) than the tests derived under the joint asymptotics. We also find that choosing a deterministic trend when a deterministic level is true does not significantly affect the properties of the test. But, choosing a deterministic level when a deterministic trend is true leads to extreme over‐rejections. Therefore, when unsure about which model has generated the data, it is suggested to use the model with a trend. We also propose a new statistic for testing for stationarity in mixed panel data where the mixture is known. The performance of this new test is very good for both cases of T asymptotic and T fixed. The statistic for T asymptotic is slightly undersized when T is very small (≤10).     

The nonlinear dynamic relationship of exchange rates: Parametric and nonparametric causality testing

The present study investigates the linear and nonlinear causal linkages among six currencies denoted relative to United States dollar (USD), namely Euro (EUR), Great Britain Pound (GBP), Japanese Yen (JPY), Swiss Frank (CHF), Australian Dollar (AUD) and Canadian Dollar (CAD). The data spans two periods between 3/20/1991 and 3/20/2007. We apply a new nonparametric test for Granger non-causality by Diks and Panchenko [Diks, C., Panchenko, V., 2005. A note on the Hiemstra–Jones test for Granger noncausality. Studies in Nonlinear Dynamics and Econometrics 9 (art. 4); Diks, C., Panchenko, V., 2006. A new statistic and practical guidelines for nonparametric Granger causality testing. Journal of Economic Dynamics & Control 30, 1647–1669] and the linear Granger test on the return time series. To detect strictly nonlinear causality, we examine the pairwise VAR-filtered residuals as well as in a six-variate formulation. We find remaining significant bi- and uni-directional causal nonlinear relationships in the series. Finally, we investigate causality after controlling for conditional heteroskedasticity using a GARCH–BEKK model. Whilst the nonparametric test statistics are smaller in some cases, significant nonlinear causal linkages persisted even after GARCH filtering during both periods. This indicates that currency returns may exhibit asymmetries and statistically significant higher-order moments.     

Exchange rates and fundamentals: A bootstrap panel data analysis

This study attempts to re-examine the Granger non-causality from exchange rates to observed fundamentals based on the present value model of Engel and West (2005). To this end, we employ the bootstrap panel Granger non-causality analysis, which allows us to untangle the causal nexus between exchange rates and fundamentals in panel data. Among the main results, it is found that the null hypothesis of no cross-sectional dependence across the members of the panel is strongly rejected, indicating that the bootstrap critical value is required in conducting the panel Granger non-causality test. The null hypothesis of Granger non-causality running from the fundamentals to exchange rates is significantly rejected, implying that the monetary approach of exchange rate determination is a useful benchmark to understand the evolution of the exchange rate. Empirical evidences also show that exchange rates Granger-case the fundamentals, supporting the view that exchange rates are determined as the present value that depends in part on observed fundamentals.     

Critical values for the augmented efficient Wald test for fractional unit roots

This paper presents response surface estimates of finite sample critical values of the Efficient Wald test for Fractional Unit Roots of Lobato and Velasco (Econometrica 75:575–590, 2007) in the presence of deterministic components. Lag-adjusted critical values of the augmented versions of the tests illustrate that as in the context of traditional unit root and stationarity tests, incorporating adjustments for serial correlation affects the finite sample distributions of the test statistics.     

Wald,likelihood ratio,and Lagrange multiplier tests as an F test

This expository note contains, in the case of a multiple regression model, a derivation of the Wald, Lagrange, and Likelihood Ratio tests as a function of the small sample F test. This also allows us to readily establish the well-known small sample inequalities among the first three large sample tests.     

基于不确定性视角的PPI与CPI传导关系真相研究

通货膨胀背景下,有关PPI和CPI之间的价格传导关系成为备受关注的热点问题,但现有文献均局限于PPI和CPI均值意义上的格兰杰因果性讨论。事实上,无论从宏观理论出发还是从计量理论出发,PPI和CPI衡量的通胀率的不确定性(inflation uncertainty)及不确定性间的相关性更应引起学术界以及货币当局的关注。鉴于此,我们借鉴Hafner and Herwartz(2004)提出的二阶矩意义的格兰杰因果检验方法,构建Wald形式的统计量对CPI和PPI不确定性因果关系进行检验,发现从通胀不确定性视角来看,PPI和CPI是相互传导的,从传导模式来看,无论以PPIMG还是以CGP"I充当"PPI,均只能在短期影响CPI,而CPI对它们的影响在长短期都是一贯的。由此推断,CPI到PPI方向构成了二者不确定性传导的"系统性"路径,通胀治理须从需求方考虑。