Motivated by a nonparametric GARCH model we consider nonparametric additive autoregression models in the special case that the additive components are linked parametrically. We show that the parameter can be estimated with parametric rate and give the normal limit. Our procedure is based on two steps. In the first step nonparametric smoothers are used for the estimation of each additive component without taking into account the parametric link of the functions. In a second step the parameter is estimated by using the parametric restriction between the additive components. Interestingly, our method needs no undersmoothing in the first step.

The small sample bias of the least-squares coefficient estimator is examined in the dynamic multiple linear regression model with normally distributed whitenoise disturbances and an arbitrary number of regressors which are all exogenous except for the one-period lagged-dependent variable. We employ large sample (T → ∞) and small disturbance (σ → 0) asymptotic theory and derive and compare expressions to O(T−1) and to O(σ2), respectively, for the bias in the least-squares coefficient vector. In some simulations and for an empirical example, we examine the mean (squared) error of these expressions and of corrected estimation procedures that yield estimates that are unbiased to O(T−l) and to O(σ2), respectively. The large sample approach proves to be superior, easily applicable, and capable of generating more efficient and less biased estimators.

In this paper a new class of tests for parameter stability, the moving-estimates (ME) test, is proposed. It is shown that in the standard situation the ME test asymptotically equivalent to the maximal likelihood ratio test under the alternative of a temporary parameter shift. It is also shown that the asymptotic null distribution of the ME test is determined by the increments of a vector Brownian bridge and that under a broad class of alternatives the ME test is consistent and has nontrivial local power in general. Our simulations also demonstrate that the proposed test has power superior to other competing tests when parameters are temporarily instable.

Smoothed nonparametric estimates of the spectral density matrix at zero frequency have been widely used in econometric inference, because they can consistently estimate the covariance matrix of a partial sum of a possibly dependent vector process. When elements of the vector process exhibit long memory or antipersistence such estimates are inconsistent. We propose estimates which are still consistent in such circumstances, adapting automatically to memory parameters that can vary across the vector and be unknown.

We propose a closed-form estimator for the linear GARCH(1,1) model. The estimator has the advantage over the often used quasi-maximum likelihood estimator (QMLE) that it can be easily implemented and does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance process. We derive the asymptotic properties of the estimator, showing T(κ−1)/κ-consistency for some κ ∈ (1,2) when the fourth moment exists and -asymptotic normality when the eighth moment exists. We demonstrate that a finite number of Newton–Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the fourth moment exists. A simulation study confirms our theoretical results.The first author's research was supported by the Shoemaker Foundation. The second author's research was supported by the Economic and Social Science Research Council of the United Kingdom.

Baltagi [3] derived 2SLS and 3SLS analogues for a simultaneous equation model with error components. These were denoted by EC2SLS and EC3SLS. More recently, Balestra and Varadharajan-Krishnakumar [1] derived alternative 2SLS and 3SLS analogues; these were denoted by G2SLS and G3SLS. This note explains the relationship between these estimators and shows that the set of instruments employed by Balestra and Varadharajan-Krishnakumar is a subset of those used in Baltagi. In addition this paper shows that for the single equation case the extra set of instruments is redundant and both EC2SLS and G2SLS have the same asymptotic variance-covariance matrix. However, for the system of equations, it can be shown that EC3SLS is asymptotically more efficient than G3SLS.

In this paper, we analyze the influence of observed and unobserved initial values on the bias of the conditional maximum likelihood or conditional sum-of-squares (CSS, or least squares) estimator of the fractional parameter, d, in a nonstationary fractional time series model. The CSS estimator is popular in empirical work due, at least in part, to its simplicity and its feasibility, even in very complicated nonstationary models.

We consider a process, Xt, for which data exist from some point in time, which we call –N0 + 1, but we only start observing it at a later time, t = 1. The parameter (d, μ, σ2) is estimated by CSS based on the model ${\rm{\Delta }}_0^d \left( {X_t - \mu } \right) = \varepsilon _t ,t = N + 1, \ldots ,N + T$, conditional on X1,..., XN. We derive an expression for the second-order bias of $\hat d$ as a function of the initial values, Xt, t = – N0 + 1,..., N, and we investigate the effect on the bias of setting aside the first N observations as initial values. We compare $\hat d$ with an estimator, $\hat d_c $, derived similarly but by choosing μ = C. We find, both theoretically and using a data set on voting behavior, that in many cases, the estimation of the parameter μ picks up the effect of the initial values even for the choice N = 0.

If N0 = 0, we show that the second-order bias can be completely eliminated by a simple bias correction. If, on the other hand, N0 > 0, it can only be partly eliminated because the second-order bias term due to the initial values can only be diminished by increasing N.

It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory, producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient, and a limit normal distribution for the t-ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continue to hold in spite of the degeneracy in the limit theory and are therefore robust to initial conditions that extend to the infinite past.

This paper investigates a simple dynamic linear panel regression model with both fixed effects and time effects. Using “large n and large T” asymptotics, we approximate the distribution of the fixed effect estimator of the autoregressive parameter in the dynamic linear panel model and derive its asymptotic bias. We find that the same higher order bias correction approach proposed by Hahn and Kuersteiner (2002, Econometrica 70, 1639–1659) can be applied to the dynamic linear panel model even when time specific effects are present.We thank Peter Phillips and three anonymous referees for helpful comments. The first author gratefully acknowledges financial support from NSF grant SES-0313651. The second author appreciates the Faculty Development Awards of USC for research support.

Impulse response functions from time series models are standard tools for analyzing the relationship between economic variables. The asymptotic distribution of orthogonalized impulse responses is derived under the assumption that finite order vector autoregressive (VAR) models are fitted to time series generated by possibly infinite order processes. The resulting asymptotic distributions of forecast error variance decompositions are also given.

Volatility component models have received considerable attention recently, not only because of their ability to capture complex dynamics via a parsimonious parameter structure, but also because it is believed that they can handle well structural breaks or nonstationarities in asset price volatility. This paper revisits component volatility models from a statistical perspective and attempts to explore the stationarity of the underlying processes. There is a clear need for such an analysis, since any discussion about nonstationarity presumes we know when component models are stationary. As it turns out, this is not the case and the purpose of the paper is to rectify this. We also look into the sampling behavior of the maximum likelihood estimates of recently proposed volatility component models and establish their consistency and asymptotic normality.

This paper investigates selection and averaging of linear regressions with a possible structural break. Our main contribution is the construction of a Mallows criterion for the structural break model. We show that the correct penalty term is nonstandard and depends on unknown parameters, but it can be approximated by an average of limiting cases to yield a feasible penalty with good performance. Following Hansen (2007, Econometrica 75, 1175–1189) we recommend averaging the structural break estimates with the no-break estimates where the weight is selected to minimize the Mallows criterion. This estimator is simple to compute, as the weights are a simple function of the ratio of the penalty to the Andrews SupF test statistic.

To assess performance we focus on asymptotic mean-squared error (AMSE) in a local asymptotic framework. We show that the AMSE of the estimators depends exclusively on the parameter variation function. Numerical comparisons show that the unrestricted least-squares and pretest estimators have very large AMSE for certain regions of the parameter space, whereas our averaging estimator has AMSE close to the infeasible optimum.

A Bayesian reference analysis of the cointegrated vector autoregression is presented based on a new prior distribution. Among other properties, it is shown that this prior distribution distributes its probability mass uniformly over all cointegration spaces for a given cointegration rank and is invariant to the choice of normalizing variables for the cointegration vectors. Several methods for computing the posterior distribution of the number of cointegrating relations and distribution of the model parameters for a given number of relations are proposed, including an efficient Gibbs sampling approach where all inferences are determined from the same posterior sample. Simulated data are used to illustrate the procedures and for discussing the well-known issue of local nonidentification.The author thanks Luc Bauwens, Anant Kshirsagar, Peter Phillips, Herman van Dijk, four anonymous referees, and especially Daniel Thorburn for helpful comments. Financial support from the Swedish Council of Research in Humanities and Social Sciences (HSFR) grant F0582/1999 and Swedish Research Council (Vetenskapsrådet) grant 412-2002-1007 is gratefully acknowledged. The views expressed in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Executive Board of Sveriges Riksbank.

Testing for white noise has been well studied in the literature of econometrics and statistics. For most of the proposed test statistics, such as the well-known Box–Pierce test statistic with fixed lag truncation number, the asymptotic null distributions are obtained under independent and identically distributed assumptions and may not be valid for dependent white noise. Because of recent popularity of conditional heteroskedastic models (e.g., generalized autoregressive conditional heteroskedastic [GARCH] models), which imply nonlinear dependence with zero autocorrelation, there is a need to understand the asymptotic properties of the existing test statistics under unknown dependence. In this paper, we show that the asymptotic null distribution of the Box–Pierce test statistic with general weights still holds under unknown weak dependence as long as the lag truncation number grows at an appropriate rate with increasing sample size. Further applications to diagnostic checking of the autoregressive moving average (ARMA) and fractional autoregressive integrated moving average (FARIMA) models with dependent white noise errors are also addressed. Our results go beyond earlier ones by allowing non-Gaussian and conditional heteroskedastic errors in the ARMA and FARIMA models and provide theoretical support for some empirical findings reported in the literature.

This paper takes up Bayesian inference in a general trend stationary model for macroeconomic time series with independent Student-t disturbances. The model is linear in the data, but nonlinear in parameters. An informative but nonconjugate family of prior distributions for the parameters is introduced, indexed by a single parameter that can be readily elicited. The main technical contribution is the construction of posterior moments, densities, and odd ratios by using a six-step Gibbs sampler. Mappings from the index parameter of the family of prior distribution to posterior moments, densities, and odds ratios are developed for several of the Nelson–Plosser time series. These mappings show that the posterior distribution is not even approximately Gaussian, and they indicate the sensitivity of the posterior odds ratio in favor of difference stationarity to the choice of the prior distribution.

This paper investigates the asymptotic size properties of a two-stage test in the linear instrumental variables model when in the first stage a Hausman (1978) specification test is used as a pretest of exogeneity of a regressor. In the second stage, a simple hypothesis about a component of the structural parameter vector is tested, using a t-statistic that is based on either the ordinary least squares (OLS) or the two-stage least squares estimator (2SLS), depending on the outcome of the Hausman pretest. The asymptotic size of the two-stage test is derived in a model where weak instruments are ruled out by imposing a positive lower bound on the strength of the instruments. The asymptotic size equals 1 for empirically relevant choices of the parameter space. The size distortion is caused by a discontinuity of the asymptotic distribution of the test statistic in the correlation parameter between the structural and reduced form error terms. The Hausman pretest does not have sufficient power against correlations that are local to zero while the OLS-based t-statistic takes on large values for such nonzero correlations. Instead of using the two-stage procedure, the recommendation then is to use a t-statistic based on the 2SLS estimator or, if weak instruments are a concern, the conditional likelihood ratio test by Moreira (2003).

This paper proposes model selection criteria (MSC) for unconditional moment models using generalized empirical likelihood (GEL) statistics. The use of GEL-statistics in lieu of J-statistics (in the spirit of Andrews, 1999, Econometrica 67, 543–564; and Andrews and Lu, 2001, Journal of Econometrics 101, 123–164) leads to an alternative interpretation of the MSCs that emphasizes the common information-theoretic rationale underlying model selection procedures for both parametric and semiparametric models. The result of this paper also provides a GEL-based model selection alternative to the information criteria–based nonnested tests for generalized method of moments models considered in Kitamura (2000, University of Wisconsin). The results of a Monte Carlo experiment are reported to illustrate the finite-sample performance of the selection criteria and their impact on parameter estimation.The authors gratefully acknowledge support from the NSF (Hong: SES-0079495, Shum: SES-0003352) and the Fellowship of Woodrow Wilson Scholars (Preston). We thank the co-editor Don Andrews, Xiaohong Chen, John Geweke, Bo Honore, Yuichi Kitamura, Serena Ng, Harry Paarsch, Gautam Tripathi, and two anonymous referees for insightful suggestions and helpful comments.

We conduct Monte Carlo simulations to evaluate the use of information criteria (Akaike information criterion [AIC] and Schwarz information criterion [SC]) as an alternative to various probability-based tests for determining cointegrating rank in multivariate analysis. First, information criteria are used to determine cointegrating rank given the lag order in a levels vector autoregression. Second, information criteria are used to determine the lag order and cointegrating rank simultaneously. Results show that AIC has an advantage over trace tests for cointegrated or stationary processes in small samples. AIC does not perform well in large samples. The performance of SC is close to that of the trace test. SC shows better large sample results than AIC and the trace test, even if the series are close to nonstationary series or they contain large negative moving average components. We also find evidence that supports the joint estimation of lag order and cointegrating rank by the SC criterion. We conclude that information criteria can complement traditional parametric tests.We are grateful to Peter C.B. Phillips and an anonymous referee for their comments, which significantly improved the paper.

The nonparametric estimation of a regression function from conditional moment restrictions involving instrumental variables is considered. The rate of convergence of penalized estimators is studied in the case where the regression function is not identified from the conditional moment restriction. We also study the gain of modifying the penalty in the estimation, considering derivatives in the penalty. We analyze the effect of this modification on the identification of the regression function and the rate of convergence of its estimator.

This paper studies the estimation of a semi-strong GARCH(1,1) model when it does not have a stationary solution, where semi-strong means that we do not require the errors to be independent over time. We establish necessary and sufficient conditions for a semi-strong GARCH(1,1) process to have a unique stationary solution. For the nonstationary semi-strong GARCH(1,1) model, we prove that a local minimizer of the least absolute deviations (LAD) criterion converges at the rate to a normal distribution under very mild moment conditions for the errors. Furthermore, when the distributions of the errors are in the domain of attraction of a stable law with the exponent κ ∈ (1, 2), it is shown that the asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is non-Gaussian but is some stable law with the exponent κ ∈ (0, 2). The asymptotic distribution is difficult to estimate using standard parametric methods. Therefore, we propose a percentile-t subsampling bootstrap method to do inference when the errors are independent and identically distributed, as in Hall and Yao (2003). Our result implies that the least absolute deviations estimator (LADE) is always asymptotically normal regardless of whether there exists a stationary solution or not, even when the errors are heavy-tailed. So the LADE is more appealing when the errors are heavy-tailed. Numerical results lend further support to our theoretical results.