Let (X1, Y1), …, (Xn, Yn) be independent and identically distributed random variables and let l(x) be the unknown p-quantile regression curve of Y conditional on X. A quantile smoother ln(x) is a localized, nonlinear estimator of l(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {ln(x) – l(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1 |ln(x) − l(x)|. The derived result helps in the construction of a uniform confidence band for the quantile curve l(x). This confidence band can be applied as a econometric model check. An economic application considers the relation between age and earnings in the labor market by means of parametric model specification tests, which presents a new framework to describe trends in the entire wage distribution in a parsimonious way.

In this paper we propose a time-varying vector error correction model in which the cointegrating relationship varies smoothly over time. The Johansen setup is a special case of our model. A likelihood ratio test for time-invariant cointegration is defined and its asymptotic chi-square distribution is derived. We apply our test to the purchasing power parity hypothesis of international prices and nominal exchange rates, and we find evidence of time-varying cointegration.

We extend the framework of the fully modified, ordinary least squares (OLS) estimator introduced by Phillips and Hansen (1990) to the case of seasonally cointegrated processes at a given frequency. First we extend a weak convergence result of sample covariance matrices (Phillips, 1988) to the case of seasonal unit roots. Using a complex number framework, we then show that we can take into account the constraints that exist in a situation of seasonal cointegration as illustrated in Gregoir (1999a) and derive estimates of the cointegration vectors that allow for asymptotic normal inference. This allows us to propose a test whose null hypothesis is the existence of seasonal cointegration. A Monte Carlo exercise investigates the finite sample properties of this test procedure. The paper closes with the analysis of situations in which there exist more than one frequency at which seasonal cointegration can be observed.

This paper considers a nonparametric functional coefficient model with an unknown link function. The model gives flexibility to the standard interaction-variable model by allowing an arbitrary functional form of heterogeneous marginal effects. A local rank estimation procedure is proposed for the functional coefficients along with its asymptotic property.

We extend the representation theory of the autoregressive model in the fractional lag operator of Johansen (2008, Econometric Theory 24, 651–676). A recursive algorithm for the characterization of cofractional relations and the corresponding adjustment coefficients is given, and it is shown under which condition the solution of the model is fractional of order d and displays cofractional relations of order d − b and polynomial cofractional relations of order d − 2b,…, d − cb ≥ 0 for integer c; the cofractional relations and the corresponding moving average representation are characterized in terms of the autoregressive coefficients by the same algorithm. For c = 1 and c = 2 we find the results of Johansen (2008).

Most work in the area of nonlinear econometric modeling is based on a single equation and assumes exogeneity of the explanatory variables. Recently, work by Caner and Hansen (2004) and Psaradakis, Sola, and Spagnolo (2005) has considered the possibility of estimating nonlinear models by methods that take into account endogeneity but provide no tests for exogeneity. This paper examines the problem of testing for exogeneity in nonlinear threshold models. We suggest new Hausman-type tests and discuss the use of the bootstrap to improve the properties of asymptotic tests. The theoretical properties of the tests are discussed and an extensive Monte Carlo study is undertaken.

Recently, Shimotsu and Phillips (2005, Annals of Statistics 33, 1890–1933) developed a new semiparametric estimator, the exact local Whittle (ELW) estimator, of the memory parameter (d) in fractionally integrated processes. The ELW estimator has been shown to be consistent, and it has the same asymptotic distribution for all values of d, if the optimization covers an interval of width less than 9/2 and the mean of the process is known. With the intent to provide a semiparametric estimator suitable for economic data, we extend the ELW estimator so that it accommodates an unknown mean and a polynomial time trend. We show that the two-step ELW estimator, which is based on a modified ELW objective function using a tapered local Whittle estimator in the first stage, has an asymptotic distribution for (or when the data have a polynomial trend). Our simulation study illustrates that the two-step ELW estimator inherits the desirable properties of the ELW estimator.

We study a nonlinear panel data model in which the fixed effects are assumed to have finite support. The fixed effects estimator is known to have the incidental parameters problem. We contribute to the literature by making a qualitative observation that the incidental parameters problem in this model may not be not as severe as in the conventional case. Because fixed effects have finite support, the probability of correctly identifying the fixed effect converges to one even when the cross sectional dimension grows as fast as some exponential function of the time dimension. As a consequence, the finite sample bias of the fixed effects estimator is expected to be small.

A vector autoregressive model allowing for unit roots as well as an explosive characteristic root is developed. The Granger-Johansen representation shows that this results in processes with two common features: a random walk and an explosively growing process. Cointegrating and coexplosive vectors can be found that eliminate these common factors. The likelihood ratio test for a simple hypothesis on the coexplosive vectors is analyzed. The method is illustrated using data from the extreme Yugoslavian hyperinflation of the 1990s.

In this paper we introduce a new method of projection-type inference and describe it in the context of two stage least squares–based split-sample inference on subsets of structural coefficients in a linear instrumental variables regression model. The use of the new method not only guards against the uncontrolled overrejection of the true value of the parameters of interest but also reduces the conservativeness of the usual method of projection proposed by Dufour and his coauthors (Dufour, 1997, Econometrica 65, 1365–1388; Dufour and Jasiak, 2001, International Economic Review 41, 815–843; Dufour and Taamouti, 2005, discussion paper; Dufour and Taamouti, 2005, Econometrica 73, 1351–1365; Dufour and Taamouti, 2007, Journal of Econometrics 139, 133–153).

This paper considers the extension of the classical minimum distance approach for the pooling of estimates with various rates of convergence. Under a setting where relatively high rates of convergence can be attained, the minimum distance estimators are shown to be consistent and asymptotically normally distributed. The constrained estimates can be efficient relative to the unconstrained ones. The minimized distance function is shown to be asymptotically χ2-distributed, and can be used as a goodness-of-fit test for the constraints. As the extension is motivated by some social interactions models, which are of interest in their own right, we discuss this approach for the estimation and testing of a social interactions model.

We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes {(Yi, Xi)}. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging such estimators into other functionals where some control over higher order terms is required. We apply our results to the estimation of an additive M-regression model.

We consider nonparametric estimation of spectral densities of stationary processes, a fundamental problem in spectral analysis of time series. Under natural and easily verifiable conditions, we obtain consistency and asymptotic normality of spectral density estimates. Asymptotic distribution of maximum deviations of the spectral density estimates is also derived. The latter result sheds new light on the classical problem of tests of white noises.

The local linear method is popular in estimating nonparametric continuous-time diffusion models, but it may produce negative results for the diffusion (or volatility) functions and thus lead to insensible inference. We demonstrate this using U.S. interest rate data. We propose a new functional estimation method of the diffusion coefficient based on reweighting the conventional Nadaraya–Watson estimator. It preserves the appealing bias properties of the local linear estimator and is guaranteed to be nonnegative in finite samples. A limit theory is developed under mild requirements (recurrence) of the data generating mechanism without assuming stationarity or ergodicity.

Asymptotic likelihood analysis of cointegration in I (2) models (see Johansen, 1997, 2006; Boswijk, 2000; Paruolo, 2000) has shown that inference on most parameters is mixed normal, implying hypothesis test statistics with an asymptotic χ2 null distribution. The asymptotic distribution of the multicointegration parameter estimator so far has been characterized by a Brownian motion functional, which has been conjectured to have a mixed normal distribution, based on simulations. The present note proves this conjecture.

A well known result is that many of the tests used in econometrics, such as the Rao score (RS) test, may not be robust to misspecified alternatives, that is, when the alternative model does not correspond to the underlying data generating process. Under this scenario, these tests spuriously reject the null hypothesis too often. We generalize this result to generalized method of moments–based (GMM-based) tests. We also extend the method proposed in Bera and Yoon (1993, Econometric Theory 9, 649–658) for constructing RS tests that are robust to local misspecification to GMM-based tests. Finally, a further generalization for general estimating and testing functions is developed. This framework encompasses both likelihood and GMM-based results.

This note is concerned with the spectral properties of matrices associated with linear smoothers. We derive analytical results on the eigenvalues and eigenvectors of smoothing matrices by interpreting the latter as perturbations of matrices belonging to algebras with known spectral properties, such as the circulant and the generalized tau. These results are used to characterize the properties of a smoother in terms of an approximate eigen-decomposition of the associated smoothing matrix.

We derive a closed-form expression for the asymptotic distribution of the LIML estimator for the coefficients of both endogenous and exogenous variables in a partially identified linear structural equation. We extend previous results of Phillips (1989) and Choi and Phillips (1992), where the focus was on IV estimators. We show that partial failure of identification affects the LIML in that its moments do not exist even asymptotically.

Many regression models have two dimensions, say time (t = 1,…,T) and households (i = 1,…,N), as in panel data, error components, or spatial econometrics. In estimating such models we need to specify the structure of the error variance matrix Ω, which is of dimension T N × T N. If T N is large, then direct computation of the determinant and inverse of Ω is not practical. In this note we define structures of Ω that allow the computation of its determinant and inverse, only using matrices of orders T and N, and at the same time allowing for heteroskedasticity, for household- or station-specific autocorrelation, and for time-specific spatial correlation.

This paper establishes asymptotic properties of quasi-maximum likelihood estimators for spatial dynamic panel data with both time and individual fixed effects when the number of individuals n and the number of time periods T can be large. We propose a data transformation approach to eliminate the time effects. When n / T → 0, the estimators are consistent and asymptotically centered normal; when n is asymptotically proportional to T, they are consistent and asymptotically normal, but the limit distribution is not centered around 0; when n / T → ∞, the estimators are consistent with rate T and have a degenerate limit distribution. We also propose a bias correction for our estimators. When n1/3 / T → 0, the correction will asymptotically eliminate the bias and yield a centered confidence interval. The estimates from the transformation approach can be consistent when n is a fixed finite number.