Testing for parameter restrictions in a stationary VAR model: A bootstrap alternative

This paper proposes the use of the bootstrap when the system Wald test is employed to test for linear restrictions in a stationary vector autoregressive (VAR) model. The bootstrap test is conducted using the estimated generalised least square estimator for VAR parameters, which considers contemporaneous correlations among the error terms. It is found that the bootstrap test shows little size distortion in small samples. In contrast, the asymptotic Wald test exhibits serious size distortion, severely over-rejecting the true null hypothesis in small samples. The bootstrap test also has desirable power properties, with its power particularly high when the model is near non-stationary and the error terms are highly correlated contemporaneously. As an application, the bootstrap Wald test is employed to test for the predictability of stock return from dividend yield using U.S. data.     

Bootstrap LM tests for higher-order spatial effects in spatial linear regression models

This paper first extends the methodology of Yang (J Econom 185:33–59, 2015) to allow for non-normality and/or unknown heteroskedasticity in obtaining asymptotically refined critical values for the LM-type tests through bootstrap. Bootstrap refinements in critical values require the LM test statistics to be asymptotically pivotal under the null hypothesis, and for this we provide a set of general methods for constructing LM and robust LM tests. We then give detailed treatments for two general higher-order spatial linear regression models: namely the \(\mathtt{SARAR}(p,q)\) model and the \(\mathtt{MESS}(p,q)\) model, by providing a complete set of non-normality robust LM and bootstrap LM tests for higher-order spatial effects, and a complete set of LM and bootstrap LM tests robust against both unknown heteroskedasticity and non-normality. Monte Carlo experiments are run, and results show an excellent performance of the bootstrap LM-type tests.     

Multivariate tests of the capm under heteroskedasticity: A note

This note extends the multivariate testing procedure to the case where heteroskedasticity is present. Previous tests of the CAPM relied on the market model. However, a substantial body of literature indicates that the error term in the market model is heteroskedastic. Failing to correct for heteroskedasticity can lead to biased estimates of the variance-covariance matrix and hence, incorrect statistics for hypothesis testing. In this note, a Wald test with a variance-covariance matrix corrected for heteroskedasticity is derived to test the CAPM. Using monthly data from 1926 to 1994, the adjusted test overwhelmingly rejects the zerobeta version of the CAPM for the 14 subperiods and for the entire sample period.     

Testing for joint significance in nonstationary binary choice model

This paper investigates the test of joint significance for binary choice model with multiple integrated explanatory variables. It is found that for the widely used logit and probit models, even though the estimators have a different convergence rate under null hypothesis compared with the case under alternative, the commonly used Wald statistic is still useful, and asymptotically chi-squared.     

Bootstrap methods for autocorrelation test with uncorrelated but not independent errors

By using bootstrap technique we investigate the properties of the Breusch [Breusch, T.S., 1978. Testing for autocorrelation in dynamic linear models. Australian Economic Papers 17, 334–355]–Godfrey [Godfrey, L.G., 1978. Testing for higher order serial correlation in regression equations when the regressors include lagged dependent variables. Econometrica 46, 1303–1310] autocorrelation tests in dynamic models with uncorrelated but not independent errors. In this paper we show that, under conditions when the errors are uncorrelated but not independent, even the best likelihood ratio test cannot achieve the asymptotic distribution under the null hypothesis of no autocorrelation. Standard bootstrap methods also fail to produce consistent results. To overcome this problem we applied several bootstrap testing methods for the same purpose and found the stationary bootstrap and Wild bootstrap with static model to perform adequately among the other bootstrap methods.     

Testing separate models with stochastic regressors

In this paper we generalize the R procedure to test a null linear regression model against a separate alternative in the context of generalized instrumental variable estimation, and thereby motivate the use of the standard F test. The relations between the F test and several one-degree-of-freedom separate tests are examined under the null, and the asymptotic distributions of the statistics are evaluated under local alternatives. It is shown that the one-degree-of-freedom tests can be more powerful than the F test under a Pesaran-type local alternative, and that the F test is more widely consistent than the one-degree-of-freedom tests.     

A Unit Root Test With Multiple Trend Breaks: A Theory and an Application to US and Japanese Macroeconomic Time-Series

This study develops a unit root test that allows for an alternative hypothesis with multiple trend breaks of unknown dates, and proves that tests with fewer than the true number of breaks fail, incorrectly, to reject the null asymptotically. The asymptotic distributions of appropriate t -statistics under the unit root null are determined. In empirical applications, this test rejects the null for the Nelson–Plosser velocity of the USA, the postwar quarterly real output series of the USA and Japan, and Japanese real consumption, but not for real consumption of the USA. The finite-sample properties are examined with Monte Carlo experiments.
JEL Classification Numbers: C32, C12, E32.     

Assessing the role of components of life satisfaction

Factors affecting subjective well-being and their gross effect measurement have been widely studied. However, how people derive utility from these factors has not been fully explored. This article provides a way to decompose the gross effect of major determinants on life satisfaction into direct and indirect effects and make inference for the latter. Because the indirect effect is nonlinear in parameters associated with different models in an equation system, it creates a problem in estimating the standard error. Besides, the categorical nature in survey data further introduces bias to the covariance estimation even asymptotically. To address these issues without knowing the form of heteroskedasticity under the null hypothesis, we propose to extend the wild bootstrap procedure in this particular context. Its robustness against various data properties is validated via several simulation experiments. Using three recent waves of World Values Survey, we find that the relative importance of life control has significantly increased recently, and endowing citizens with the right to enjoy their freedoms and liberties is a more effective policy in raising life satisfaction than promoting national income.     

Testing causality among five fertilizer prices

This article identifies the causal relationships among the prices of five fertilizers using both asymptotic Wald test and resampling (bootstrap) techniques. Monthly data for urea, muriate of potash, rock phosphate, diammonium phosphate and triple super phosphate between 2008 (March) and 2016 (March) are used for the analysis. Results show that the urea price Granger causes all other fertilizer prices. The results of the study are significant in understanding the price dynamics and identify the reference fertilizer price that is useful for the fertilizer industry, farmers and government.     

Testing for unit roots with income distribution data

In this paper I test the unit root hypothesis for US log GNP using the information available in income distribution data. The percentile data of an income distribution are shown to follow the same autoregressive pattern as does mean income. Under the null hypothesis of a unit root log GNP is cointegrated with the percentile data. A sequence of augmented HEGY-Tests, however, presents strong evidence against the unit root hypothesis for the distribution data and hence for log GNP. Using a full information estimation procedure for the percentiles under the alternative yields an estimate of the autoregressive coefficient which is in principle testable by an approximate Dickey-Hasza-Fuller test. The appropriate critical values are found by bootstrap methods. Again, inference is clearly unfavorable for the unit root hypothesis.An earlier version of this paper was presented at the ESEM 1993 in Uppsala. I thank Prof. Jürgen Wolters, seminar participants at the Free University of Berlin and an anonymous referee of this journal for valuable comments. The usual disclaimer applies.     

Bootstrapping covariate stationarity tests for inflation rates

This paper investigates the stationarity properties of international inflation rates by bootstrapping two stationarity tests with covariates in Jansson (2004). When the asymptotic critical values are used, the two powerful tests are found to reject the null hypothesis less in the presence of a large negative moving-average (MA) error in inflation. To cope with this problem, a parametric bootstrap scheme is developed and then is investigated by a Monte Carlo study. The simulation results demonstrate that the bootstrap tests display a better control over the empirical rejection rates at finite samples. Furthermore, after applying these tests to the inflation in G-10 countries, we find that one of the two tests using bootstrap critical values yields inferences that differ from when using asymptotic ones, and as a whole, the bootstrap tests consistently provide strong evidence in support of mean reversion in inflation in most countries of the G-10.     

Conflict among testing procedures?

This paper demonstrates that under certain conditions the Lagrange Multiplier test provides a better approximation to the asymptotic X² distribution than the more familiar Wald and Likelihood Ratio tests.     

Tests for additive heteroskedasticity: Goldfeld and Quandt revisited

A number of new tests for heteroskedasticity have recently become available. Using Monte Carlo methods this paper explores the small sample properties of some of these tests in the context of additive heteroskedasticity. Lagrange multiplier and Wald tests (and variants thereof) are found to be inferior to the likelihood ratio and Goldfeld and QuandtF tests. This is a reconfirmation of the conclusions obtained byGoldfeld/Quandt [1972] in their study of additive heteroskedasticity. The paper also contains some new results onAmemiya's GLS estimator of the additive heteroskedastic structure.     

Robust Inference by Sub-sampling

This paper provides a simple technique of carrying out inference robust to serial correlation, heteroskedasticity and spatial correlation on the estimators which follow an asymptotic normal distribution. The idea is based on the fact that the estimates from a larger sample tend to have a smaller variance which can be expressed as a function of the variance of the estimator from smaller subsamples. The major advantage of the technique other than the ease of application and simplicity is its finite sample performance both in terms of the empirical null rejection probability as well as the power of the test. It does not restrict the data in terms of structure in any way and works pretty well for any kind of heteroskedasticity, autocorrelation and spatial correlation in a finite sample. Furthermore, unlike theoretical HAC robust techniques available in the existing literature, it does not require any kernel estimation and hence eliminates the discretion of the analyst to choose a specific kernel and bandwidth. The technique outperforms the Ibragimov and Müller (2010) approach in terms of null rejection probability as well as the local asymptotic power of the test.

     

Testing for Granger non-causality in heterogeneous panels

This paper proposes a very simple test of Granger (1969) non-causality for heterogeneous panel data models. Our test statistic is based on the individual Wald statistics of Granger non causality averaged across the cross-section units. First, this statistic is shown to converge sequentially to a standard normal distribution. Second, the semi-asymptotic distribution of the average statistic is characterized for a fixed T sample. A standardized statistic based on an approximation of the moments of Wald statistics is hence proposed. Third, Monte Carlo experiments show that our standardized panel statistics have very good small sample properties, even in the presence of cross-sectional dependence.     

Improving the Performance of a Long‐Run Variance Ratio Test for a Unit Root

Cai and Shintani (2006, Econometric Theory, 22, 347–372) considered the impact of introducing an inconsistent long‐run variance estimator when constructing a class of kernel‐based ratio tests for testing non‐stationarity in the series. They found that the quotient of two estimators with different rates of convergence under the null and the alternative hypotheses may lead to a test having an interesting size and power trade‐off. This paper develops modified versions of this test, presents new asymptotic results and tabulates critical values. The finite sample performance is explored through Monte Carlo simulations. The results show that the modifications proposed lead to more powerful unit root tests.     

Tests for the Poolability of Panel Data in the Presence of Unknown Heteroskedasticity: an Asymptotic Extension of ANOVA

This paper presents asymptotic tests for the poolability of panel data in the presence of heteroskedasticity. Also, the test statistic is compared with the well-known F -test statistic using simulation studies. The test statistic is regarded as an asymptotic extension of ANOVA in the sense that the statistic measures the distance between the null and the alternative models using the idea of ANOVA. Simulation studies show that the real size of the test statistic is in the neighbourhood of the nominal size, though that of the F -statistic is quite different. Even when the error term is independent and identically and normally distributed, its real size is near the nominal size and the power is not much inferior to that of the F -statistic when the sample size is large.
JEL Classification Numbers: C12, C23.     

Bias correction and bootstrapping of error component models for panel data: Theory and applications

The wellknown Wallace-Hussain estimator is applied in pooled models with random individual effects, and the magnitude of the bias caused by the estimator is estimated by bootstrap methods. Furthermore, the significance of the bias is tested using an asymptotic test based on the bootstrap results.A preliminary version of this paper was presented at the Econometric Society European Meeting ESEM '88 in Bologna. The authors thank an anonymous referee for helpful comments. The data for one of the applications (the earnings function) are from the German Socio-Economic Panel and we thank the DIW and the Sfb 3 for the permission to use these data.     

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.     

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.