This paper considers the factor model Xt = ΛFt + et. Assuming a normal distribution for the idiosyncratic error et conditional on the factors {Ft}, conditional maximum likelihood estimators of the factor and factor-loading spaces are derived. These estimators are called generalized principal component estimators (GPCEs) without the normality assumption. This paper derives asymptotic distributions of the GPCEs of the factor and factor-loading spaces. It is shown that variance of the GPCE of the common component is smaller than that of the principal component estimator studied in Bai (2003, Econometrica 71, 135–172). The approximate variance of the forecasting error using the GPCE-based factor estimates is derived and shown to be smaller than that based on the principal component estimator. The feasible GPCE (FGPCE) of the factor space is shown to be asymptotically equivalent to the GPCE. The GPCE and FGPCE are shown to be more efficient than the principal component estimator in finite samples.

In this paper we derive the asymptotic distribution of the least absolute deviations (LAD) estimator of the autoregressive parameter under the unit root hypothesis, when the errors are assumed to have finite variances, and present LAD-based unit root tests, which, under heavy-tailed errors, are expected to be more powerful than tests based on least squares. The limiting distribution of the LAD estimator is that of a functional of a bivariate Brownian motion, similar to those encountered in cointegrating regressions. By appropriately correcting for serial correlation and other distributional parameters, the test statistics introduced here are found to have either conditional or unconditional normal limiting distributions. The results of the paper complement similar ones obtained by Knight (1991, Canadian Journal of Statistics 17, 261-278) for infinite variance errors. A simulation study is conducted to investigate the finite sample properties of our tests.

This article extends recent work on the Gaussian or quasi-maximum likelihood estimation of the parameters of a closed higher-order continuous time dynamic model by introducing exogenous variables into the model The method presented yields exact maximum likelihood estimates when the innovations are Gaussian and the exogenous variables are polynomials in time of degree not exceeding two, and it can be expected to yield very good estimates under more general conditions. It is applicable, in principle, to a system of any order with mixed stock and iow data. The precise formulas for its implementation are derived, in this article, for a second-order system in which both the endog-enous and exogenous variables are a mixture of stock and flow variables.

In this paper, we show that for panel AR(p) models, an instrumental variable (IV) estimator with instruments deviated from past means has the same asymptotic distribution as the infeasible optimal IV estimator when both N and T, the dimensions of the cross section and time series, are large. If we assume that the errors are normally distributed, the asymptotic variance of the proposed IV estimator is shown to attain the lower bound when both N and T are large. A simulation study is conducted to assess the estimator.

The use of probit and logit models has become quite common whenever the dependent variable in a regression is qualitative. These models have been used to explain either/or choices and decisions involving multiple alternatives. A two-dimensional graphical interpretation of these different models has been provided by Johnson [3]. The purpose of this paper is to provide a three-dimensional graphical exposition of the ordered probit model, which was first estimated by McKelvey and Zavoina [4] and is now built into computer packages, such as LIMDEP [1].

This paper discusses nonparametric models for panels of time series. There is already a substantial literature on nonlinear models and nonparametric methods in a regression and time series setting. But almost without exception these developments have been limited to univariate and multivariate models of moderate dimensions. Very little has been done for panels, where the dimension, often corresponding to a number of individuals, typically is very large but where the number of observations for each individual may be small or moderate. It is the aim of this paper to start a systematic theoretical treatment of nonparametric models for panels of time series, in particular on additive models. Extending existing methodology to the panel situation is by no means trivial because already for the parametric case many problems are unsolved. Our estimation approach is based on backfitting methods.

In this paper we consider a family of penalized likelihood criteria for determining the rank of the cointegration space, the number of lags, and the form of the intercept in vector autoregressions with possibly integrated processes. The paper provides a general consistency result for a class of model determination procedures in which the penalty depends on a simple parameter count only.

This paper considers a formulation of the extended constant or time-varying conditional correlation GARCH model that allows for volatility feedback of either the positive or negative sign. In the previous literature, negative volatility spillovers were ruled out by the assumption that all the parameters of the model are nonnegative, which is a sufficient condition for ensuring the positive definiteness of the conditional covariance matrix. In order to allow for negative feedback, we show that the positive definiteness of the conditional covariance matrix can be guaranteed even if some of the parameters are negative. Thus, we extend the results of Nelson and Cao (1992) and Tsai and Chan (2008) to a multivariate setting. For the bivariate case of order one, we look into the consequences of adopting these less severe restrictions and find that the flexibility of the process is substantially increased. Our results are helpful for the model-builder, who can consider the unrestricted formulation as a tool for testing various economic theories.

In the linear instrumental variables model with possibly weak instruments we derive the asymptotic size of testing procedures when instruments locally violate the exogeneity assumption. We study the tests by Anderson and Rubin (1949, The Annals of Mathematical Statistics 20, 46–63), Moreira (2003, Econometrica 71, 1027–1048), and Kleibergen (2005, Econometrica 73, 1103–1123) and their generalized empirical likelihood versions. These tests have asymptotic size equal to nominal size when the instruments are exogenous but are size distorted otherwise. While in just-identified models all the tests that we consider are equally size-distorted asymptotically, the Anderson-Rubin type tests are less size-distorted than the tests of Moreira (2003) and Kleibergen in over-identified situations. On the other hand, we also show that there are parameter sequences under which the former test asymptotically overrejects more frequently. Given that strict exogeneity of instruments is often a questionable assumption, our findings should be important to applied researchers who are concerned about the degree of size distortion of their inference procedure. We suggest robustness of asymptotic size under local model violations as a new alternative measure to choose among competing testing procedures. We also investigate the subsampling and hybrid tests introduced in Andrews and Guggenberger (2010a, Journal of Econometrics 158, 285–305) and show that they do not offer any improvement in terms of size-distortion reduction over the Anderson-Rubin type tests.

This paper considers the solution of multivariate linear rational expectations models. It is described how all possible classes of solutions (namely, the unique stable solution, multiple stable solutions, and the case where no stable solution exists) of such models can be characterized using the quadratic determinantal equation (QDE) method of Binder and Pesaran (1995, in M.H. Pesaran & M. Wickens [eds.], Handbook of Applied Econometrics: Macroeconomics, pp. 139–187. Oxford: Basil Blackwell). To this end, some further theoretical results regarding the QDE method expanding on previous work are presented. In addition, numerical techniques are discussed allowing reasonably fast determination of the dimension of the solution set of the model under consideration using the QDE method. The paper also proposes a new, fully recursive solution method for models involving lagged dependent variables and current and future expectations. This new method is entirely straightforward to implement, fast, and applicable also to high-dimensional problems possibly involving coefficient matrices with a high degree of singularity.

This paper considers a nonparametric time series regression model with a nonstationary regressor. We construct a nonparametric test for whether the regression is of a known parametric form indexed by a vector of unknown parameters. We establish the asymptotic distribution of the proposed test statistic. Both the setting and the results differ from earlier work on nonparametric time series regression with stationarity. In addition, we develop a bootstrap simulation scheme for the selection of suitable bandwidth parameters involved in the kernel test as well as the choice of simulated critical values. An example of implementation is given to show that the proposed test works in practice.

Standard methods for analyzing linear-latent variable models rely on the assumption that the observed variables are normally distributed. Normality allows statistical inferences to be carried out based solely on the first-and second-order moments. In general, inferences for nonnormally distributed data require the estimates of matrices of third-and fourth-order moments. In the present paper, we show that inferences based on normal theory retain validity and asymptotic efficiency under general assumptions that allow for considerable departure from normality. In particular, we obtain conditions under which correct asymptotic inferences are attained when replacing a matrix of higher order moments by a matrix that depends only on cross-product moments of the data.

The main uniform convergence results of Hansen (2008, Econometric Theory 24, 726–748) are generalized in two directions: Data are allowed to (a) be heterogeneously dependent and (b) depend on a (possibly unbounded) parameter. These results are useful in semiparametric estimation problems involving time-inhomogeneous models and/or sampling of continuous-time processes. The usefulness of these results is demonstrated by two applications: kernel regression estimation of a time-varying AR(1) model and the kernel density estimation of a Markov chain that has not been initialized at its stationary distribution.

Most nonparametric identification results for the mixed proportional hazards model for single spell duration data depend crucially on the proportional hazards assumption. Here, it is shown that variation in covariates over time, combined with variation across observations, is sufficient to ensure identification without the proportional hazards assumption. The required variation over time is minimal, and the mixed hazards model is identified without the proportional hazards assumption in the presence of standard time-varying covariates.Thanks to Kåre Bævre, Zhiyang Jia, Tore Schweder, Rolf Aaberge, and John K. Dagsvik for useful comments.

This paper addresses the problem of inference on the moving average impact matrix and on its row and column spaces in cointegrated 1(1) VAR processes. The choice of bases (i.e., the identification) of these spaces, which is of interest in the definition of the common trend structure of the system, is discussed. Maximum likelihood estimators and their asymptotic distributions are derived, making use of a relation between properly normalized bases of orthogonal spaces, a result that may be of separate interest. Finally, Wald-type tests are given, and their use in connection with existing likelihood ratio tests is discussed.

We consider the limiting distributions of M-estimates of an “autoregressive” parameter when the observations come from an integrated linear process with infinite variance innovations. It is shown that M-estimates are, asymptotically, infinitely more efficient than the least-squares estimator (in the sense that they have a faster rate of convergence) and are conditionally asymptotically normal.

This paper investigates the bias and the weak Bahadur representation of a local polynomial estimator of the conditional quantile function and its derivatives. The bias and Bahadur remainder term are studied uniformly with respect to the quantile level, the covariates, and the smoothing parameter. The order of the local polynomial estimator can be higher than the differentiability order of the conditional quantile function. Applications of the results deal with global optimal consistency rates of the local polynomial quantile estimator, performance of random bandwidths, and estimation of the conditional quantile density function. The latter allows us to obtain a simple estimator of the conditional quantile function of the private values in a first-price sealed bids auction under the independent private values paradigm and risk neutrality.

Haavelmo’s seminal 1943 and 1944 papers are the first rigorous treatment of causality. In them, he distinguished the definition of causal parameters from their identification. He showed that causal parameters are defined using hypothetical models that assign variation to some of the inputs determining outcomes while holding all other inputs fixed. He thus formalized and made operational Marshall’s (1890) ceteris paribus analysis. We embed Haavelmo’s framework into the recursive framework of Directed Acyclic Graphs (DAGs) commonly used in the literature of causality (Pearl, 2000) and Bayesian nets (Lauritzen, 1996). We compare the analysis of causality based on a methodology inspired by Haavelmo’s ideas with other approaches used in the causal literature of DAGs. We discuss the limitations of methods that solely use the information expressed in DAGs for the identification of economic models. We extend our framework to consider models for simultaneous causality, a central contribution of Haavelmo.

Estimation of cointegrated systems via autoregressive approximation is considered in the framework developed by Saikkonen (1992, Econometric Theory 8, 1-27). The asymptotic properties of the estimated coefficients of the autoregressive error correction model (ECM) and the pure vector autoregressive (VAR) representations are derived under the assumption that the autoregressive order goes to infinity with the sample size. These coefficients are often used for analyzing the relationships between the variables; therefore, they are important for applied work. Tests for linear restrictions on the coefficients of both the ECM and the pure VAR representation are considered under the present assumptions. It is found that they have limiting x2 distributions. Tests are also derived under the assumption that the number of restrictions goes to infinity with the sample size.

This paper develops tests for the null hypothesis of cointegration in the nonlinear regression model with I(1) variables. The test statistics we use in this paper are Kwiatkowski, Phillips, Schmidt, and Shin’s (1992; KPSS hereafter) tests for the null of stationarity, though using other kinds of tests is also possible. The tests are shown to depend on the limiting distributions of the estimators and parameters of the nonlinear model when they use full-sample residuals from the nonlinear least squares and nonlinear leads-and-lags regressions. This feature makes it difficult to use them in practice. As a remedy, this paper develops tests using subsamples of the regression residuals. For these tests, first, the nonlinear least squares and nonlinear leads-and-lags regressions are run and residuals are calculated. Second, the KPSS tests are applied using subresiduals of size b. As long as b/T → 0 as T → ∞, where T is the sample size, the tests using the subresiduals have limiting distributions that are not affected by the limiting distributions of the full-sample estimators and the parameters of the model. Third, the Bonferroni procedure is used for a selected number of the subresidual-based tests. Monte Carlo simulation shows that the tests work reasonably well in finite samples for polynomial and smooth transition regression models when the block size is chosen by the minimum volatility rule. In particular, the subresidual-based tests using the leads-and-lags regression residuals appear to be promising for empirical work. An empirical example studying the U.S. money demand equation illustrates the use of the tests.