This paper considers regression models for cross-section data that exhibit cross-section dependence due to common shocks, such as macroeconomic shocks. The paper analyzes the properties of least squares (LS) and instrumental variables (IV) estimators in this context. The results of the paper allow for any form of cross-section dependence and heterogeneity across population units. The probability limits of the LS and IV estimators are determined and necessary and sufficient conditions are given for consistency. The asymptotic distributions of the estimators are found to be mixed normal after re-centering and scaling. t, Wald, and F statistics are found to have asymptotic standard normal, χ2, and scaled χ2 distributions, respectively, under the null hypothesis when the conditions required for consistency of the parameter under test hold. But, the absolute values of t statistics and Wald and F statistics are found to diverge to infinity under the null hypothesis when these conditions fail. Confidence intervals exhibit similarly dichotomous behavior. Hence, common shocks are found to be innocuous in some circumstances, but quite problematic in others.
Models with factor structures for errors, regressors, and IV’s are considered. Using the general results, conditions are determined under which consistency of the LS and IV estimators holds and fails in models with factor structures. The results are extended to cover heterogeneous and functional factor structures in which common factors have different impacts on different population units.
Extensions to generalized method of moments estimators are discussed.
Keywords: Asymptotics, Common shocks, Dependence, Exchangeability, Factor model, Inconsistency, regression
This paper introduces tests for cointegration breakdown that may occur over a relatively short time period, such as at the end of the sample. The breakdown may be due to a shift in the cointegrating vector or due to a shift in the errors from being I(0) to being I(1). Tests are introduced based on the post-breakdown sum of squared residuals and the post-breakdown sum of squared reverse partial sums of residuals. Critical values are provided using a parametric subsampling method.
The regressors in the model are taken to be arbitrary linear combinations of deterministic, stationary, and integrated random variables. The tests are asymptotically valid when the number of observations in the breakdown period, m, is fixed and finite as the total sample size, T + m, goes to infinity. The tests are asymptotically valid under weak conditions.
Simulation results indicate that the tests work well in the scenarios considered.
Use of the tests is illustrated by testing for interest rate parity breakdown during the Asian financial crisis of 1997.
Keywords: Cointegration, least squares estimator, model breakdown, parameter change test, structural change
The local Whittle (or Gaussian semiparametric) estimator of long range dependence, proposed by Künsch (1987) and analyzed by Robinson (1995a), has a relatively slow rate of convergence and a finite sample bias that can be large. In this paper, we generalize the local Whittle estimator to circumvent these problems. Instead of approximating the short-run component of the spectrum, φ(λ), by a constant in a shrinking neighborhood of frequency zero, we approximate its logarithm by a polynomial. This leads to a “local polynomial Whittle” (LPW) estimator. We specify a data-dependent adaptive procedure that adjusts the degree of the polynomial to the smoothness of φ(λ) at zero and selects the bandwidth. The resulting “adaptive LPW” estimator is shown to achieve the optimal rate of convergence, which depends on the smoothness of φ(λ) at zero, up to a logarithmic factor.
Keywords: Adaptive estimator, Asymptotic bias, Asymptotic normality, Bias reduction, Local polynomial, Long memory, Minimax rate, Optimal bandwidth, Whittle likelihood
This paper determines coverage probability errors of both delta method and parametric bootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memory Gaussian time series. CIs for the long-memory parameter d0 are included. The results establish that the bootstrap provides higher-order improvements over the delta method. Analogous results are given for tests. The CIs and tests are based on one or other of two approximate maximum likelihood estimators. The first estimator solves the first-order conditions with respect to the covariance parameters of a “plug-in” log-likelihood function that has the unknown mean replaced by the sample mean. The second estimator does likewise for a plug-in Whittle log-likelihood.
The magnitudes of the coverage probability errors for one-sided bootstrap CIs for covariance parameters for long-memory time series are shown to be essentially the same as they are with iid data. This occurs even though the mean of the time series cannot be estimated at the usual n1/2 rate.
Keywords: Asymptotics, confidence intervals, delta method, Edgeworth expansion, Gaussian process, long memory, maximum likelihood estimator, parametric bootstrap, t statistic, Whittle likelihood
This paper considers tests for structural instability of short duration, such as at the end of the sample. The key feature of the testing problem is that the number, m, of observations in the period of potential change is relatively small — possibly as small as one. The well-known F test of Chow (1960) for this problem only applies in a linear regression model with normally distributed iid errors and strictly exogenous regressors, even when the total number of observations, n + m, is large.
We generalize the F test to cover regression models with much more general error processes, regressors that are not strictly exogenous, and estimation by instrumental variables as well as least squares. In addition, we extend the F test to nonlinear models estimated by generalized method of moments and maximum likelihood.
Asymptotic critical values that are valid as n approaches infinity with m fixed are provided using a subsampling-like method. The results apply quite generally to processes that are strictly stationary and ergodic under the null hypothesis of no structural instability.
In this paper, we prove the validity of an Edgeworth expansion to the distribution of the Whittle maximum likelihood estimator for stationary long-memory Gaussian models with unknown parameter . The error of the (s-2)-order expansion is shown to be o(n(s-2)/2) – the usual iid rate — for a wide range of models, including the popular ARFIMA(p,d,q) models. The expansion is valid under mild assumptions on the behavior of spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a Theorem by Fox and Taqqu (1987) concerning the asymptotic behavior of Toeplitz matrices.
Lieberman, Rousseau, and Zucker (2002) (LRZ) establish a valid Edgeworth expansion for the maximum likelihood estimator for stationary long-memory Gaussian models. For a significant class of models, their expansion is shown to have an error of o(n-1). The results given here improve upon those of LRZ in that the results provide an Edgeworth expansion for an asymptotically efficient estimator, as LRZ do, but the error of the expansion is shown to be o(n-(s-2)/2), not o(n-1), for a broad range of models.
Keywords: ARFIMA, Edgeworth expansion, Long Memory, Whittle estimator
This paper provides bounds on the errors in coverage probabilities of maximum likelihood-based, percentile-t, parametric bootstrap confidence intervals for Markov time series processes. These bounds show that the parametric bootstrap for Markov time series provides higher-order improvements (over confidence intervals based on first order asymptotics) that are comparable to those obtained by the parametric and nonparametric bootstrap for iid data and are better than those obtained by the block bootstrap for time series. Additional results are given for Wald-based confidence regions.
The paper also shows that k-step parametric bootstrap confidence intervals achieve the same higher-order improvements as the standard parametric bootstrap for Markov processes. The k-step bootstrap confidence intervals are computationally attractive. They circumvent the need to compute a nonlinear optimization for each simulated bootstrap sample. The latter is necessary to implement the standard parametric bootstrap when the maximum likelihood estimator solves a nonlinear optimization problem.
Keywords: Asymptotics, Edgeworth expansion, Gauss-Newton, k-step bootstrap, maximum likelihood estimator, Newton-Raphson, parametric bootstrap, t statistic
The local Whittle (or Gaussian semiparametric) estimator of long range dependence, proposed by Künsch (1987) and analyzed by Robinson (1995a), has a relatively slow rate of convergence and a finite sample bias that can be large. In this paper, we generalize the local Whittle estimator to circumvent those problems. Instead of approximating the short-run component of the spectrum, φ(λ), by a constant in a shrinking neighborhood of frequency zero, we approximate its logarithm by a polynomial. This leads to a “local polynomial Whittle” (LPW) estimator.
Following the work of Robinson (1995a), we establish the asymptotic bias, variance, mean-squared error (MSE), and normality of the LPW estimator. We determine the asymptotically MSE-optimal bandwidth, and specify a plug-in selection method for its practical implementation. When φ(λ) is smooth enough near the origin, we find that the bias of the LPW estimator goes to zero at a faster rate than that of the local Whittle estimator, and its variance is only inflated by a multiplicative constant. In consequence, the rate of convergence of the LPW estimator is faster than that of the local Whittle estimator, given an appropriate choice of the bandwidth m.
We show that the LPW estimator attains the optimal rate of convergence for a class of spectra containing those for which φ(λ) is smooth of order s > 1 near zero. When φ(λ) is infinitely smooth near zero, the rate of convergence of the LPW estimator based on a polynomial of high degree is arbitrarily close to n-1/2.
It is well known that a one-step scoring estimator that starts from any N1/2-consistent estimator has the same first-order asymptotic efficiency as the maximum likelihood estimator. This paper extends this result to k-step estimators and test statistics for k > 1, higher-order asymptotic efficiency, and general extremum estimators and test statistics.
The paper shows that a k-step estimator has the same higher-order asymptotic efficiency, to any given order, as the extremum estimator towards which it is stepping, provided (i) k is sufficiently large, (ii) some smoothness and moment conditions hold, and (iii) a condition on the initial estimator holds.
For example, for the Newton-Raphson k-step estimator, we obtain asymptotic equivalence to integer order s providedk > s + 1. Thus, for k = 1, 2, and 3, one obtains asymptotic equivalence to first, third, and seventh orders respectively. This means that the maximum differences between the probabilities that the (N1/2-normalized) k-step and extremum estimators lie in any convex set are o(1), o(N-3/2), and o(N-3) respectively.
The widely used log-periodogram regression estimator of the long-memory parameter d proposed by Geweke and Porter-Hudak (1983) (GPH) has been criticized because of its finite-sample bias, see Agiakloglou, Newbold, and Wohar (1993). In this paper, we propose a simple bias-reduced log-periodogram regression estimator, ^dr, that eliminates the first- and higher-order biases of the GPH estimator. The bias-reduced estimator is the same as the GPH estimator except that one includes frequencies to the power 2k for k = 1,…,r, for some positive integer r, as additional regressors in the pseudo-regression model that yields the GPH estimator. The reduction in bias is obtained using assumptions on the spectrum only in a neighborhood of the zero frequency, which is consistent with the semiparametric nature of the long-memory model under consideration.
Following the work of Robinson (1995b) and Hurvich, Deo, and Brodsky (1998), we establish the asymptotic bias, variance, and mean-squared error (MSE) of ^dr, determine the MSE optimal choice of the number of frequencies, m, to include in the regression, and establish the asymptotic normality of ^dr. These results show that the bias of ^dr goes to zero at a faster rate than that of the GPH estimator when the normalized spectrum at zero is sufficiently smooth, but that its variance only is increased by a multiplicative constant. In consequence, the optimal rate of convergence to zero of the MSE of ^dr is faster than that of the GPH estimator.
We establish the optimal rate of convergence of a minimax risk criterion for estimators of d when the normalized spectral density is in a class that includes those that are smooth of order s > 1 at zero. We show that the bias-reduced estimator ^dr attains this rate when r > (s-2)/2 and m is chosen appropriately. For s > 2, the GPH estimator does not attain this rate. The proof of these results uses results of Giraitis, Robinson, and Samarov (1997).
Some Monte Carlo simulation results for stationary Gaussian ARFIMA(1,d,1) models show that the bias-reduced estimators perform well relative to the standard log-periodogram estimator.
Keywords: Asymptotic bias, asymptotic normality, bias reduction, frequency domain, long-range dependence, optimal rate, rate of convergence, strongly dependent time series
This paper considers the problem of choosing the number bootstrap repetitions B to use with the BCa bootstrap confidence intervals introduced by Efron (1987). Because the simulated random variables are ancillary, we seek a choice of B that yields a confidence interval that is close to the ideal bootstrap confidence interval for which B = ∞. We specifiy a three-step method of choosing B that ensures that the lower and upper lengths of the confidence interval deviate from those of the ideal bootstrap confidence interval by at most a small percentage with high probability.
This paper develops consistent model and moment selection criteria for GMM estimation. The criteria select the correct model specification and all correct moment conditions asymptotically. The selection criteria resemble the widely used likelihood-based selection criteria BIC, HQIC, and AIC. (The latter is not consistent.) The GMM selection criteria are based on the J statistic for testing over-identifying restrictions. Bonus terms reward the use of fewer parameters for a given number of moment conditions and the use of more moment conditions for a given number of parameters.
The paper applies the model and moment selection criteria to dynamic panel data models with unobserved individual effects. The paper shows how to apply the selection criteria to select the lag length for lagged dependent variables, to detect the number and locations of structural breaks, to determine the exogeneity of regressors, and/or to determine the existence of correlation between some regressors and the individual effect.
To illustrate the finite sample performance of the selection criteria and their impact on parameter estimation, the paper reports the results of a Monte Carlo experiment on a dynamic panel data model.
Keywords: Akaike information criterion, Bayesian information criterion, consistent selection procedure, generalized method of moments estimator, instrumental variables estimator, model selection, moment selection, panel data model, test of over-identifying restrictions.
This paper considers testing problems where several of the standard regularity conditions fail to hold. We consider the case where (i) parameter vectors in the null hypothesis may lie on the boundary of the maintained hypothesis and (ii) there may be a nuisance parameter that appears under the alternative hypothesis, but not under the null. The paper establishes the asymptotic null and local alternative distributions of quasi-likelihood ratio, rescaled quasi-likelihood ratio, Wald, and score tests in this case. The results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models.
Examples treated in the paper are: (1) tests of the null hypothesis of no conditional heteroskedasticity in a GARCH(1, 1) regression model and (2) tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with (possibly) correlated random coefficients.
Keywords: Asymptotic distribution, boundary, conditional heteroskedasticity, extremum estimator, GARCH model, inequality restrictions, likelihood ratio test, local power, maximum likelihood estimator, parameter restrictions, random coefficients regression, quasi-maximum likelihood estimator, quasi-likelihood ratio test, restricted estimator, score test, Wald test
This paper establishes the higher-order equivalence of the k-step bootstrap, introduced recently by Davidson and MacKinnon (1999a), and the standard bootstrap. The k-step bootstrap is a very attractive alternative computationally to the standard bootstrap for statistics based on nonlinear extremum estimators, such as generalized method of moment and maximum likelihood estimators. The paper also extends results of Hall and Horowitz (1996) to provide new results regarding the higher-order improvements of the standard bootstrap and the k-step bootstrap for extremum estimators (compared to procedures based on first-order asymptotics).
The results of the paper apply to Newton-Raphson (NR), default NR, line-search NR, and Gauss-Newton k-step bootstrap procedures. The results apply to the nonparametric iid bootstrap, non-overlapping and overlapping block bootstraps, and restricted and unrestricted parametric bootstraps. The results cover symmetric and equal-tailed two-sided t tests and confidence intervals, one-sided t tests and confidence intervals, Wald tests and confidence regions, and J tests of over-identifying restrictions.
The optimal block length for the accuracy of tests and confidence intervals is shown to be proportional to N1/4 for both non-overlapping and overlapping block bootstraps in the context considered.
In addition, the paper provides some results that establish the equivalence of the higher-order efficiency of non-bootstrap k-step statistics and extremum statistics. These results extend results of Pfanzagl (1974), Robinson (1988), and others.
Keywords: Asymptotics, block bootstrap, Edgeworth expansion, extremum estimator, Gauss-Newton, generalized method of moments estimator, higher-order efficiency, k-step bootstrap, maximum likelihood estimator, Newton-Raphson, parametric bootstrap, t statistic, test of over-identifying restrictions
. The error of the (s-2)-order expansion is shown to be o(n(s-2)/2) – the usual iid rate — for a wide range of models, including the popular ARFIMA(p,d,q) models. The expansion is valid under mild assumptions on the behavior of spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a Theorem by Fox and Taqqu (1987) concerning the asymptotic behavior of Toeplitz matrices.