We consider linearity testing in a general class of nonlinear time series models of order one, involving a nonnegative nuisance parameter that (a) is not identified under the null hypothesis and (b) gives the linear model when equal to zero. This paper studies the asymptotic distribution of the likelihood ratio test and asymptotically equivalent supremum tests. The asymptotic distribution is described as a functional of chi-square processes and is obtained without imposing a positive lower bound for the nuisance parameter. The finite-sample properties of the sup-tests are studied by simulations.

This paper extends the Andrews (2002, Econometrica 71, 1661–1694) and Andrews and Kim (2006, Journal of Business & Economic Statistics 24, 379–394) ordinary least squares–based end-of-sample instability tests for linear regression models. The author proposes to quasi-difference the data first using a consistent estimate of the sum of the autoregressive coefficients of the error process and then test for the end-of-sample instability. For the cointegration model, the feasible quasi-generalized least squares (FQGLS) version of the Andrews and Kim (2006) P test is considered and is shown, by simulations, to be more robust to serial correlation in the error process and to have power no less than Andrews and Kim’s original test. For the linear time trend model, the FQGLS version of the Andrews (2002) S test is considered with the error process allowed to be nonstationary up to one unit root, and the new test is shown to be robust to potentially nonstationary serial correlation. A simulation study also shows that the finite-sample properties of the proposed test can be further improved when the Andrews (1993, Econometrica 61,139–165) or Andrews and Chen (1994, Journal of Business & Economic Statistics 12, 187–204) median unbiased estimate of the sum of the autoregressive coefficients is used.

We consider the stochastic process , t = 2, …, n, where sw(xt, xi) is a similarity function between the tth and the ith observations and {εt} is a random disturbance term. This process was originally axiomatized by Gilboa, Lieberman, and Schmeidler (2006, Review of Economics and Statistics 88, 433–444) as a way by which agents, or even nature, reason. In the present paper, consistency and the asymptotic distribution of the quasi-maximum likelihood estimator of the parameters of the model are established. Connections to other models and techniques are drawn. In its general form, the model does not fall within any class of nonstationary econometric models for which asymptotic theory is available. For this reason, the developments in this paper are new and nonstandard.

For long memory time series models with uncorrelated but dependent errors, we establish the asymptotic normality of the Whittle estimator under mild conditions. Our framework includes the widely used fractional autoregressive integrated moving average models with generalized autoregressive conditional heteroskedastic-type innovations. To cover nonstationary fractionally integrated processes, we extend the idea of Abadir, Distaso, and Giraitis (2007, Journal of Econometrics 141, 1353–1384) and develop the nonstationarity-extended Whittle estimation. The resulting estimator is shown to be asymptotically normal and is more efficient than the tapered Whittle estimator. Finally, the results from a small simulation study are presented to corroborate our theoretical findings.

An effective way to control for cross-section correlation when conducting a panel unit root test is to remove the common factors from the data. However, there remain many ways to use the defactored residuals to construct a test. In this paper, we use the panel analysis of nonstationarity in idiosyncratic and common components (PANIC) residuals to form two new tests. One estimates the pooled autoregressive coefficient, and one simply uses a sample moment. We establish their large-sample properties using a joint limit theory. We find that when the pooled autoregressive root is estimated using data detrended by least squares, the tests have no power. This result holds regardless of how the data are defactored. All PANIC-based pooled tests have nontrivial power because of the way the linear trend is removed.

We develop a nonparametric regression-based goodness-of-fit test for multifactor continuous-time Markov models using the conditional characteristic function, which often has a convenient closed form or can be approximated accurately for many popular continuous-time Markov models in economics and finance. An omnibus test fully utilizes the information in the joint conditional distribution of the underlying processes and hence has power against a vast class of continuous-time alternatives in the multifactor framework. A class of easy-to-interpret diagnostic procedures is also proposed to gauge possible sources of model misspecification. All the proposed test statistics have a convenient asymptotic N(0, 1) distribution under correct model specification, and all asymptotic results allow for some data-dependent bandwidth. Simulations show that in finite samples, our tests have reasonable size, thanks to the dimension reduction in nonparametric regression, and good power against a variety of alternatives, including misspecifications in the joint dynamics, but the dynamics of each individual component is correctly specified. This feature is not attainable by some existing tests. A parametric bootstrap improves the finite-sample performance of proposed tests but with a higher computational cost.

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.

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).

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.

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.