This paper develops a generalized autoregressive conditional correlation (GARCC) model when the standardized residuals follow a random coefficient vector autoregressive process. As a multivariate generalization of the Tsay (1987, Journal of the American Statistical Association 82, 590–604) random coefficient autoregressive (RCA) model, the GARCC model provides a motivation for the conditional correlations to be time varying. GARCC is also more general than the Engle (2002, Journal of Business & Economic Statistics 20, 339–350) dynamic conditional correlation (DCC) and the Tse and Tsui (2002, Journal of Business & Economic Statistics 20, 351–362) varying conditional correlation (VCC) models and does not impose unduly restrictive conditions on the parameters of the DCC model. The structural properties of the GARCC model, specifically, the analytical forms of the regularity conditions, are derived, and the asymptotic theory is established. The Baba, Engle, Kraft, and Kroner (BEKK) model of Engle and Kroner (1995, Econometric Theory 11, 122–150) is demonstrated to be a special case of a multivariate RCA process. A likelihood ratio test is proposed for several special cases of GARCC. The empirical usefulness of GARCC and the practicality of the likelihood ratio test are demonstrated for the daily returns of the Standard and Poor's 500, Nikkei, and Hang Seng indexes.

We provide a theoretical framework in which to study the accuracy of bootstrap P values, which may be based on a parametric or nonparametric bootstrap. In the parametric case, the accuracy of a bootstrap test will depend on the shape of what we call the critical value function. We show that, in many circumstances, the error in rejection probability of a bootstrap test will be one whole order of magnitude smaller than that of the corresponding asymptotic test. We also propose a simulation method for estimating this error that requires the calculation of only two test statistics per replication.

This paper considers estimation of multiplicative, unobserved components panel data models without imposing a strict exogeneity assumption on the conditioning variables. The method of moments estimators proposed have significant robustness properties. They require only a conditional mean assumption and apply to models with lagged dependent variables and to finite distributed lag models with arbitrary feedback from the explained to future values of the explanatory variables. The model is particularly suited to nonnegative explained variables, including count variables, continuously distributed nonnegative outcomes, and even binary variables. The general model can also be applied to certain nonlinear Euler equations.

It is well known that most of the standard specification tests are not robust when the alternative is misspecified. Using the asymptotic distributions of standard Lagrange multiplier (LM) test under local misspecification, we suggest a robust specification test. This test essentially adjusts the mean and covariance matrix of the usual LM statistic. We show that for local misspecification the adjusted test is asymptotically equivalent to Neyman's C(α) test, and therefore, shares the optimality properties of the C(α) test. The main advantage of the new test is that, compared to the C(α) test, it is much simpler to compute. Our procedure does require full specification of the model and there might be some loss of asymptotic power relative to the unadjusted test if the model is indeed correctly specified.

Expansion and collapse are two key features of a financial asset bubble. Bubble expansion may be modeled using a mildly explosive process. Bubble implosion may take several different forms depending on the nature of the collapse and therefore requires some flexibility in modeling. This paper first strengthens the theoretical foundation of the real time bubble monitoring strategy proposed in Phillips, Shi and Yu (2015a,b, PSY) by developing analytics and studying the performance characteristics of the testing algorithm under alternative forms of bubble implosion which capture various return paths to market normalcy. Second, we propose a new reverse sample use of the PSY procedure for detecting crises and estimating the date of market recovery. Consistency of the dating estimators is established and the limit theory addresses new complications arising from the alternative forms of bubble implosion and the endogeneity effects present in the reverse regression. A real-time version of the strategy is provided that is suited for practical implementation. Simulations explore the finite sample performance of the strategy for dating market recovery. The use of the PSY strategy for bubble monitoring and the new procedure for crisis detection are illustrated with an application to the Nasdaq stock market.

Asymptotic theory for heteroskedasticity autocorrelation consistent (HAC) covariance matrix estimators requires the truncation lag, or bandwidth, to increase more slowly than the sample size. This paper considers an alternative approach covering the case with the asymptotic covariance matrix estimated by kernel methods with truncation lag equal to sample size. Although such estimators are inconsistent, valid tests (asymptotically pivotal) for regression parameters can be constructed. The limiting distributions explicitly capture the truncation lag and choice of kernel. A local asymptotic power analysis shows that the Bartlett kernel delivers the highest power within a group of popular kernels. Finite sample simulations suggest that, regardless of the kernel chosen, the null asymptotic approximation of the new tests is often more accurate than that for conventional HAC estimators and asymptotics. Finite sample results on power show that the new approach is competitive.

We consider an additive model with second-order interaction terms. Both marginal integration estimators and a combined backfitting-integration estimator are proposed for all components of the model and their derivatives. The corresponding asymptotic distributions are derived. Moreover, two test statistics for testing the presence of interactions are proposed. Asymptotics for the test functions and local power results are obtained. Because direct implementation of the test procedure based on the asymptotics would produce inaccurate results unless the number of observations is very large, a bootstrap procedure is provided, which is applicable for small or moderate sample sizes. Further, based on these methods a general test for additivity is developed. Estimation and testing methods are shown to work well in simulation studies. Finally, our methods are illustrated on a five-dimensional production function for a set of Wisconsin farm data. In particular, the separability hypothesis for the production function is discussed.

We consider several issues related to Durbin-Wu-Hausman tests; that is, tests based on the comparison of two sets of parameter estimates. We first review a number of results about these tests in linear regression models, discuss what determines their power, and propose a simple way to improve power in certain cases. We then show how in a general nonlinear setting they may be computed as “score” tests by means of slightly modified versions of any artificial linear regression that can be used to calculate Lagrange multiplier tests, and explore some of the implications of this result. In particular, we show how to create a variant of the information matrix test that tests for parameter consistency. We examine the conventional information matrix test and our new version in the context of binary-choice models, and provide a simple way to compute both tests using artificial regressions.

This paper considers tests for the rank of a matrix for which a root-T consistent estimator is available. However, in contrast to tests associated with the minimum chi-square and asymptotic least squares principles, the estimator's asymptotic variance matrix is not required to be either full or of known rank. Test statistics based on certain estimated characteristic roots are proposed whose limiting distributions are a weighted sum of independent chi-squared variables. These weights may be simply estimated, yielding convenient estimators for the limiting distributions of the proposed statistics. A sequential testing procedure is presented that yields a consistent estimator for the rank of a matrix. A simulation experiment is conducted comparing the characteristic root statistics advocated in this paper with statistics based on the Wald and asymptotic least squares principles.

In this paper we provide an asymptotic analysis of generalized bipower measures of the variation of price processes in financial economics. These measures encompass the usual quadratic variation, power variation, and bipower variations that have been highlighted in recent years in financial econometrics. The analysis is carried out under some rather general Brownian semimartingale assumptions, which allow for standard leverage effects.Ole E. Barndorff-Nielsen's work is supported by the Centre for Analytical Finance (CAF), which is funded by the Danish Social Science Research Council. Neil Shephard's research is supported by the UK's ESRC through the grant “High frequency financial econometrics based upon power variation.” We thank the editor, Peter Phillips, and the referees for their stimulating comments on an earlier version.

We analyze linear panel regression models with interactive fixed effects and predetermined regressors, for example lagged-dependent variables. The first-order asymptotic theory of the least squares (LS) estimator of the regression coefficients is worked out in the limit where both the cross-sectional dimension and the number of time periods become large. We find two sources of asymptotic bias of the LS estimator: bias due to correlation or heteroscedasticity of the idiosyncratic error term, and bias due to predetermined (as opposed to strictly exogenous) regressors. We provide a bias-corrected LS estimator. We also present bias-corrected versions of the three classical test statistics (Wald, LR, and LM test) and show their asymptotic distribution is a χ2-distribution. Monte Carlo simulations show the bias correction of the LS estimator and of the test statistics also work well for finite sample sizes.

We consider general, and possibly nonparametric, GARCH(1,1) processes. First we give conditions for the existence and the uniqueness of stationary ergodic solutions. Then we identify additional conditions for geometric ergodicity. These conditions consist of mild restrictions on the distribution of the latent independent process. No moment assumption is made on the generalized autoregressive conditionally heteroskedastic (GARCH) process. Applications to the asymptotic behavior of sample autocorrelations and to unit-root tests are proposed.This work was supported by INTAS (research project 03-51-3714). The authors gratefully acknowledge the quick and careful reading of the manuscript by Bruce Hansen and three referees. Their detailed comments led to a greatly improved presentation.

This paper proposes the least absolute shrinkage and selection operator–type (Lasso-type) generalized method of moments (GMM) estimator. This Lasso-type estimator is formed by the GMM objective function with the addition of a penalty term. The exponent of the penalty term in the regular Lasso estimator is equal to one. However, the exponent of the penalty term in the Lasso-type estimator is less than one in the analysis here. The magnitude of the exponent is reduced to avoid the asymptotic bias. This estimator selects the correct model and estimates it simultaneously. In other words, this method estimates the redundant parameters as zero in the large samples and provides the standard GMM limit distribution for the estimates of the nonzero parameters in the model. The asymptotic theory for our estimator is nonstandard. We conduct a simulation study that shows that the Lasso-type GMM correctly selects the true model much more often than the Bayesian information Criterion (BIC) and another model selection procedure based on the GMM objective function.

This paper investigates through Monte Carlo simulation the finite sample properties of likelihood ratio tests for cointegrating ranks that were proposed by Johansen (1991, Econometrica 59, 1551–1580). We transform the model into a canonical form so that the experiment is well controlled without loss of generality and then conduct a comprehensive simulation study. As expected, the test performance is very sensitive to the value of the stationary root(s) of the process. We also find that the test performance depends crucially on the correlation between the innovations that drive the stationary and the nonstationary components of the process. We conclude that 100 observations are not sufficient to ensure reasonably good performance uniformly over the values of the nuisance parameters that affect the distributions of the test statistics.

We consider estimation of parameters in a regression model with endogenous regressors. The endogenous regressors along with a large number of other endogenous variables are driven by a small number of unobservable exogenous common factors. We show that the estimated common factors can be used as instrumental variables and they are more efficient than the observed variables in our framework. Whereas standard optimal generalized method of moments estimator using a large number of instruments is biased and can be inconsistent, the factor instrumental variable estimator (FIV) is shown to be consistent and asymptotically normal, even if the number of instruments exceeds the sample size. Furthermore, FIV remains consistent even if the observed variables are invalid instruments as long as the unobserved common components are valid instruments. We also consider estimating panel data models in which all regressors are endogenous but share exogenous common factors. We show that valid instruments can be constructed from the endogenous regressors. Although single equation FIV requires no bias correction, the faster convergence rate of the panel estimator is such that a bias correction is necessary to obtain a zero-centered normal distribution.

We consider semiparametric estimation of the memory parameter in a long memory stochastic volatility model. We study the estimator based on a log periodogram regression as originally proposed by Geweke and Porter-Hudak (1983, Journal of Time Series Analysis 4, 221–238). Expressions for the asymptotic bias and variance of the estimator are obtained, and the asymptotic distribution is shown to be the same as that obtained in recent literature for a Gaussian long memory series. The theoretical result does not require omission of a block of frequencies near the origin. We show that this ability to use the lowest frequencies is particularly desirable in the context of the long memory stochastic volatility model.

We suggest using a class of semiparametric dynamic panel data models to capture individual variations in panel data. The model assumes linearity in some continuous/discrete variables that can be exogenous/endogenous and allows for nonlinearity in other weakly exogenous variables. We propose a nonparametric generalized method of moments (NPGMM) procedure to estimate the functional coefficients, and we establish the consistency and asymptotic normality of the resulting estimators.

Semiparametric estimates of long memory seem useful in the analysis of long financial time series because they are consistent under much broader conditions than parametric estimates. However, recent large sample theory for semiparametric estimates forbids conditional heteroskedasticity. We show that a leading semiparametric estimate, the Gaussian or local Whittle one, can be consistent and have the same limiting distribution under conditional heteroskedasticity as under the conditional homoskedasticity assumed by Robinson (1995, Annals of Statistics 23, 1630–61). Indeed, noting that long memory has been observed in the squares of financial time series, we allow, under regularity conditions, for conditional heteroskedasticity of the general form introduced by Robinson (1991, Journal of Econometrics 47, 67–84), which may include long memory behavior for the squares, such as the fractional noise and autoregressive fractionally integrated moving average form, and also standard short memory ARCH and GARCH specifications.

We construct efficient estimators of the identifiable parameters in a regression model when the errors follow a stationary parametric ARCH(P) process. We do not assume a functional form for the conditional density of the errors, but do require that it be symmetric about zero. The estimators of the mean parameters are adaptive in the sense of Bickel [2]. The ARCH parameters are not jointly identifiable with the error density. We consider a reparameterization of the variance process and show that the identifiable parameters of this process are adaptively estimable.

We note that likelihood inference can be based on an unbiased simulation-based estimator of the likelihood when it is used inside a Metropolis–Hastings algorithm. This result has recently been introduced in statistics literature by Andrieu, Doucet, and Holenstein (2010, Journal of the Royal Statistical Society, Series B, 72, 269–342) and is perhaps surprising given the results on maximum simulated likelihood estimation. Bayesian inference based on simulated likelihood can be widely applied in microeconomics, macroeconomics, and financial econometrics. One way of generating unbiased estimates of the likelihood is through a particle filter. We illustrate these methods on four problems, producing rather generic methods. Taken together, these methods imply that if we can simulate from an economic model, we can carry out likelihood–based inference using its simulations.