Infinite order vector autoregressive (VAR) models have been used in a number of applications ranging from spectral density estimation, impulse response analysis, and tests for cointegration and unit roots, to forecasting. For estimation of such models it is necessary to approximate the infinite order lag structure by finite order VARs. In practice, the order of approximation is often selected by information criteria or by general-to-specific specification tests. Unlike in the finite order VAR case these selection rules are not consistent in the usual sense, and the asymptotic properties of parameter estimates of the infinite order VAR do not follow as easily as in the finite order case. In this paper it is shown that the parameter estimates of the infinite order VAR are asymptotically normal with zero mean when the model is approximated by a finite order VAR with a data dependent lag length. The requirement for the result to hold is that the selected lag length satisfies certain rate conditions with probability tending to one. Two examples of selection rules satisfying these requirements are discussed. Uniform rates of convergence for the parameters of the infinite order VAR are also established.Very helpful comments by the editor and two referees led to a substantial improvement of the manuscript. I am particularly indebted to one of the referees for pointing out an error in the proofs. All remaining errors are my own. Financial support from NSF grant SES−0095132 is gratefully acknowledged.

The system two stage least squares estimator for the linear panel data model is shown to have different characterizations depending on the choice of instrument matrix. The more general estimator, where, in effect, separate reduced form linear projections are estimated for each time period, also has the advantage of being applicable when the number of instruments changes across time periods. The issue of efficient estimation is also treated.

A simple regression approach to HAC and LRV estimation is suggested. The method exploits the fact that the quantities of interest relate to only one point of the spectrum (the origin). The new estimator is simply the explained sum of squares in a linear regression whose regressors are a set of trend basis functions. Positive definiteness in the estimate is therefore automatically enforced, and the technique can be implemented with standard regression packages. No kernel choice is needed in practical implementation, but basis functions need to be chosen and a smoothing parameter corresponding to the number of basis functions needs to be selected. An automated approach to making this selection based on optimizing the asymptotic mean squared error is derived. The limit theory of the new estimator shows that its properties, including the convergence rate, are comparable to those of conventional HAC estimates constructed from quadratic kernels.My thanks go to Bruce Hansen, Guido Kuersteiner, and two referees for comments on an earlier version of the paper. NSF research support under grant SES 00-92509 is acknowledged.

On May 18, 2005, Francesc Marmol died of cancer in Madrid, Spain, at the age of 38. Francesc (Paco) was born in Barcelona to a family of modest means. He graduated from the Universitat Autònoma de Barcelona with a B.Sc. in Economics. He later went on to obtain a Ph.D. in the International Doctorate in Economic Analysis, where he defended a thesis on asymptotic theory for integrated processes in 1996. After University, Paco took a position at the Universidad Carlos III de Madrid, where he was Professor at the Economics Department.

In this paper we consider the estimation of the expectation vector Xβ under the general linear model {y,Xβ,σ2V}. We introduce a new handy representation for the rank of the difference of the covariance matrices of the ordinary least squares estimator OLSE(Xβ) (= Hy, say) and the best linear unbiased estimator BLUE(Xβ) (= Gy, say). From this formula some well-known conditions for the equality between Hy and Gy follow at once. We recall that the equality between Hy and Gy can be characterized by the rank-subtractivity ordering between the covariance matrices of y and Hy. This rank characterization suggests a particular presentation for the rank of the difference of the covariance matrices of Hy and Gy. We show, however, that this presentation is valid if and only if the model is connected.

This note gives a proof of the equivalence between two expressions of the moving-average impact matrix; one is given by Johansen (1995, Likelihood-Based Inference in Cointegrated Vector Autoregressive Models) and Paruolo (1997, Econometric Theory 13, 79–118), and the other appears in Phillips (1998, Journal of Econometrics 83, 21–56) as the limit of the impulse response function. Because these expressions are shown to be parametrically equivalent, we can make the same statistical inference about the impact matrix based on either Johansen (1995) and Paruolo (1997) or Phillips (1998).

In the statistics literature, asymptotic properties of the Maximum Product of Spacings estimator are derived from first principles. We propose an alternative derivation based on the comparison between its objective function and that of the Maximum Likelihood estimator.We thank the co-editor Paolo Paruolo for his patience and an anonymous referee for providing references to the statistics literature.

Let x be a random variable whose first three moments exist. If the density of x is unimodal and positively skewed, then counterexamples are provided which show that the inequality mode ≤ median ≤ mean does not necessarily hold.I thank Andrey Vasnev for help with the graphs and Jan Magnus for various helpful discussions. I also thank Martin Bland, Paolo Paruolo, Peter Phillips, Michael Rockinger, and a referee for their comments. ESRC grant R000239538 is gratefully acknowledged.

Let α* be any basis of col(Π) and β* be any basis of col(Π′); we note col(α⊥) = col(α*⊥), col(β⊥) = col(β*⊥) so no distinction is made between α⊥ and α*⊥, β⊥ and β*⊥ in the following discussion.

This paper is concerned with the practical problem of conducting inference in a vector time series setting when the data are unbalanced or incomplete. In this case, one can work with only the common sample, to which a standard HAC/bootstrap theory applies, but at the expense of throwing away data and perhaps losing efficiency. An alternative is to use some sort of imputation method, but this requires additional modeling assumptions, which we would rather avoid. We show how the sampling theory changes and how to modify the resampling algorithms to accommodate the problem of missing data. We also discuss efficiency and power. Unbalanced data of the type we consider are quite common in financial panel data; see, for example, Connor and Korajczyk (1993, Journal of Finance 48, 1263–1291). These data also occur in cross-country studies.I thank Greg Connor, Esfandiar Maasoumi, Peter Phillips, Peter Robinson, Yoon Whang, and two referees for helpful comments.

(a) Let , where Q is the fixed effects (FE) transformation given by Q = INT − P with P = IN [otimes ] ιTιT′/T.

This paper proposes a new class of heteroskedastic and autocorrelation consistent (HAC) covariance matrix estimators. The standard HAC estimation method reweights estimators of the autocovariances. Here we initially smooth the data observations themselves using kernel function–based weights. The resultant HAC covariance matrix estimator is the normalized outer product of the smoothed random vectors and is therefore automatically positive semidefinite. A corresponding efficient GMM criterion may also be defined as a quadratic form in the smoothed moment indicators whose normalized minimand provides a test statistic for the overidentifying moment conditions.

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.

This solution offers additional insights into the theory of block-tridiagonal Toeplitz matrices. Block Toeplitz matrices have constant blocks on each block diagonal parallel to the block main diagonal. A block partitioned matrix is said to be block-tridiagonal if the nonzero blocks occur only on the block subdiagonal, the block main diagonal, and the block superdiagonal. Block-tridiagonal Toeplitz matrices are particularly nice in that they are inexpensive to investigate. Our first observation on such particular block Toeplitz matrices is easy to check, and its proof is therefore left to the reader.

This paper provides a survey on a class of so-called subspace methods whose main proponent is CCA proposed by Larimore (1983, Proceedings of the 1983 American Control Conference 2). Because they are based on regressions these methods for the estimation of ARMAX systems are attractive as a result of their conceptual simplicity and their numerical advantages as compared to traditional estimators based on criterion optimization. Under the assumption of correct specification the methods provide consistent and asymptotically normal estimates for stationary ARMAX processes where the innovations may be conditionally heteroskedastic and the exogenous inputs are strictly stationary of sufficiently short memory. For stationary autoregressive moving average (ARMA) processes with independent and identically distributed (i.i.d.) Gaussian innovations the estimates are even asymptotically efficient. For I(1) ARMA processes the estimates of both the long-run and the short-run dynamics are consistent without using the knowledge that the data are integrated in the algorithm. Additionally the algorithms provide easily accessible information on the appropriateness of the chosen model complexity. The algorithms include a number of design parameters that have to be set by the user. These include the order of the estimated system. This paper collects up-to-date knowledge on the effects of these design parameters, leading to a number of suggested automated choices to obtain a fully automated estimation procedure.

Let P in be a projector with PH its conjugate transpose.

This paper considers the problems facing decision makers using econometric models in real time. It identifies the key stages involved and highlights the role of automated systems in reducing the effect of data snooping. It sets out many choices that researchers face in construction of automated systems and discusses some of the possible ways advanced in the literature for dealing with them. The role of feedbacks from the decision maker's actions to the data generating process is also discussed and highlighted through an example.

I am delighted to announce the following Econometric Theory Awards for 2005.

This paper uses the specific-to-general methodological approach that is widely used in science, in which problems with existing theories are resolved as the need arises, to illustrate a number of important developments in the modeling of univariate and multivariate financial volatility. Some of the difficulties in analyzing time-varying univariate and multivariate conditional volatility and stochastic volatility include the number of parameters to be estimated and the computational complexities associated with multivariate conditional volatility models and both univariate and multivariate stochastic volatility models. For these reasons, among others, automated inference in its present state requires modifications and extensions for modeling in empirical financial econometrics. As a contribution to the development of automated inference in modeling volatility, 20 important issues in the specification, estimation, and testing of conditional and stochastic volatility models are discussed. A “potential for automation rating” (PAR) index and recommendations regarding the possibilities for automated inference in modeling financial volatility are given in each case.

In Perez-Amaral, Gallo, and White (2003, Oxford Bulletin of Economics and Statstics 65, 821–838), the authors proposed an automatic predictive modeling tool called relevant transformation of the inputs network approach (RETINA). It is designed to embody flexibility (using nonlinear transformations of the predictors of interest), selective search within the range of possible models, control of collinearity, out-of-sample forecasting ability, and computational simplicity. In this paper we compare the characteristics of RETINA with PcGets, a well-known automatic modeling method proposed by David Hendry. We point out similarities, differences, and complementarities of the two methods. In an example using U.S. telecommunications demand data we find that RETINA can improve both in- and out-of-sample over the usual linear regression model and over some models suggested by PcGets. Thus, both methods are useful components of the modern applied econometrician's automated modeling tool chest.