In this article, starting from continuous-time local level unobserved components models for stock and flow data we derive locally best invariant (LBI) stationarity tests for data available at potentially irregularly spaced points in time. We demonstrate that the form of the LBI test differs between stock and flow variables. In cases where the data are observed at regular intervals throughout the sample we show that the LBI tests for stock and flow data both reduce to the form of the standard stationarity test in the discrete-time local level model. Here we also show that the asymptotic local power of the LBI test increases with the sampling frequency in the case of stock, but not flow, variables. Moreover, for a fixed time span we show that the LBI test for stock (flow) variables is (is not) consistent against a fixed alternative as the sampling frequency increases to infinity. We also consider the case of mixed frequency data in some detail, providing asymptotic critical values for the LBI tests for both stock and flow variables, together with a finite-sample power study. Our results suggest that tests that ignore the infraperiod aspect of the data involve rather small losses in efficiency relative to the LBI test in the case of flow variables but can result in significant losses of efficiency when analyzing stock variables.We are grateful to co-editor Jeff Wooldrige, two anonymous referees, Marcus Chambers, Andrew Harvey, Rod McCrorie, Neil Shephard, and seminar participants at Nuffield College, Oxford, for their helpful comments and suggestions on earlier versions of this paper. We are especially indebted to Marcus Chambers, who suggested the formulation of the observation equations used in our continuous-time unobserved components models for stock and flow variables. All errors are ours.

We give a brief introduction to the vector autoregressive model for cointegrated I(2) variables and show how some plausible economic relations can be formulated in the I(2) framework in such a way that likelihood ratio tests for their validity are asymptotically χ2 distributed.The authors thank Paolo Paruolo for helpful comments and the ESF for financial support in the framework of the EMM network.

This paper introduces a novel Itô diffusion process to model high-frequency financial data that can accommodate low-frequency volatility dynamics by embedding the discrete-time nonlinear exponential generalized autoregressive conditional heteroskedasticity (GARCH) structure with log-integrated volatility in a continuous instantaneous volatility process. The key feature of the proposed model is that, unlike existing GARCH-Itô models, the instantaneous volatility process has a nonlinear structure, which ensures that the log-integrated volatilities have the realized GARCH structure. We call this the exponential realized GARCH-Itô model. Given the autoregressive structure of the log-integrated volatility, we propose a quasi-likelihood estimation procedure for parameter estimation and establish its asymptotic properties. We conduct a simulation study to check the finite-sample performance of the proposed model and an empirical study with 50 assets among the S&P 500 compositions. Numerical studies show the advantages of the proposed model.

This paper deals with the estimation of linear dynamic models of the autoregressive moving average type for the conditional mean for stationary time series with conditionally heteroskedastic innovation process. Estimation is performed using a particular class of subspace methods that are known to have computational advantages as compared to estimation based on criterion minimization. These advantages are especially strong for high-dimensional time series. Conditions to ensure consistency and asymptotic normality of the subspace estimators are derived in this paper. Moreover asymptotic equivalence to quasi maximum likelihood estimators based on the Gaussian likelihood in terms of the asymptotic distribution is proved under mild assumptions on the innovations. Furthermore order estimation techniques are proposed and analyzed.

Recently, Cavaliere, Georgiev, and Taylor (2018, Econometric Theory 34, 302–348) (CGT) considered the augmented Dickey–Fuller (ADF) test for a unit-root model with linear noise driven by i.i.d. infinite variance innovations and showed that ordinary least squares (OLS)-based ADF statistics have the same distribution as in Chan and Tran (1989, Econometric Theory 5, 354–362) for i.i.d. infinite variance noise. They also proposed an interesting question to extend their results to the case with infinite variance GARCH innovations as considered in Zhang, Sin, and Ling (2015, Stochastic Processes and their Applications 125, 482–512). This paper addresses this question. In particular, the limit distributions of the ADF for random walk models with short-memory linear noise driven by infinite variance GARCH innovations are studied. We show that when the tail index $\alpha <2$ , the limit distributions are completely different from that of CGT and the estimator of the parameters of the lag terms used in the ADF regression is not consistent. This paper provides a broad treatment of unit-root models with linear GARCH noises, which encompasses the commonly entertained unit-root IGARCH model as a special case.

We provide new asymptotic theory for kernel density estimators, when these are applied to autoregressive processes exhibiting moderate deviations from a unit root. This fills a gap in the existing literature, which has to date considered only nearly integrated and stationary autoregressive processes. These results have applications to nonparametric predictive regression models. In particular, we show that the null rejection probability of a nonparametric t test is controlled uniformly in the degree of persistence of the regressor. This provides a rigorous justification for the validity of the usual nonparametric inferential procedures, even in cases where regressors may be highly persistent.

This article studies the asymptotic properties of empirical nonparametric regressions that partially misspecify the relationships between nonstationary variables. In particular, we analyze nonparametric kernel regressions in which a potential nonlinear cointegrating regression is misspecified through the use of a proxy regressor in place of the true regressor. Such models occur in linear and nonlinear regressions where the regressor suffers from measurement error or where the true regressor is a latent or filtered variable as in mixed-data-sampling. The treatment allows for endogenous regressors as the latent variable and proxy variables that cointegrate asymptotically with the true latent variable, including correctly specified as well as misspecified systems, and is therefore intermediate between nonlinear nonparametric cointegrating regression and completely spurious nonparametric nonstationary regression. The results relate to recent work on dynamic misspecification in nonparametric nonstationary systems and the limit theory accommodates regressor variables with autoregressive roots that are local to unity and whose errors are driven by long memory and short memory innovations, thereby encompassing applications with a wide range of economic and financial time series. Some implications for forecasting under misspecification are also examined.

In this article it is shown how the cointegration or joint trending behavior of economic time series helps to alleviate the errors in variables problem.

Tail approximation of the asymptotic distribution of the log likelihood ratio test for cointegration in a vector autoregressive process is studied. In dimension 2, an approximation of weighted χ2 type is derived by applying multivariate saddlepoint approximation techniques to a Fourier inversion integral.

This paper proposes Pearson-type statistics based on implied probabilities to detect structural change. The class of generalized empirical likelihood estimators (see Smith 1997, The Economic Journal107, 503–519) assigns a set of implied probabilities to each observation such that moment conditions are satisfied. The proposed test statistics for structural change are based on the information content in these implied probabilities. We consider cases of structural change with unknown breakpoint that can occur in the parameters of interest or in the overidentifying restrictions used to estimate these parameters. We also propose a structural change test based on implied probabilities that is robust to weak identification or cases in which parameters are completely unidentified. The test statistics considered here have competitive size and power properties. Moreover, they are computed in a single step, which eliminates the need to compute the weighting matrix required for generalized method of moments estimation.

A model is suggested to forecast economic time series. This model incorporates some innovative ideas of Harrison and Stevens [20] for building into the forecasting process important external shocks to the systems. Thus the occurrence of possibly significant real-world events may cause a fundamental change in the time series in question. The Jeffreys-Savage (JS) Bayesian theory of hypothesis testing is used to test the hypothesis that a particular event has been such as to free the series from its immediate past behavior. When the event frees the series in this way, then we model the sequence of observations following such an event (until the next such event) as an exchangeable sequence. In the simplest case of 0–1 valued data, such as in recording the ups and downs of the value of a particular commodity or stock, our alternative hypothesis is a Pólya process, and the null hypothesis is a simple random walk (unit roots model) with p = .50. Any exchangeable sequence is strictly stationary, and the observations in the Polya process are positively correlated, which can give rise to “explosive” behavior of the series at isolated time points. We then use the JS theory to predict future observations by taking a weighted average of the optimal predictions for each model, with weights given by the posterior probabilities of the hypotheses. Results of simulation studies are presented which compare the predictive performance of the fully Bayesian method based upon the JS theory with those based upon the “p-value” or pre-test method. The de Finetti method for scoring predictions is used to assess their empirical performance. A theoretical methodology, which extends the “evaluation game” of Hill [28,37], is developed for comparing predictors.

We show boundedness in probability uniformly in sample size of a general M-estimator for multiple linear regression in time series. The positive criterion function for the M-estimator is assumed lower semicontinuous and sufficiently large for large argument. Particular cases are the Huber-skip and quantile regression. Boundedness requires an assumption on the frequency of small regressors. We show that this is satisfied for a variety of deterministic and stochastic regressors, including stationary and random walks regressors. The results are obtained using a detailed analysis of the condition on the regressors combined with some recent martingale results.

In Wooldridge (2005), I posed a problem whose solution involved showing that the so-called ignorability of treatment assumption, commonly used in the treatment effects literature, was necessarily violated. In particular, in a setting with multiple treatment effects and without distributional or functional form assumptions, I assumed that treatment was randomized with respect to the counterfactual outcomes but not with respect to the covariates. I applied the law of iterated expectations to provide a simple proof that ignorability (given the covariates) necessarily fails.

We consider the Gaussian ARFIMA(j,d,l) model, with spectral density and an unknown mean . For this class of models, the n−1-normalized information matrix of the full parameter vector, (μ,θ), is asymptotically degenerate. To estimate θ, Dahlhaus (1989, Annals of Statistics 17, 1749–1766) suggested using the maximizer of the plug-in log-likelihood, , where is any n(1−2d)/2-consistent estimator of μ. The resulting estimator is a plug-in maximum likelihood estimator (PMLE). This estimator is asymptotically normal, efficient, and consistent, but in finite samples it has some serious drawbacks. Primarily, none of the Bartlett identities associated with are satisfied for fixed n. Cheung and Diebold (1994, Journal of Econometrics 62, 301–316) conducted a Monte Carlo simulation study and reported that the bias of the PMLE is about 3–4 times the bias of the regular maximum likelihood estimator (MLE). In this paper, we derive asymptotic expansions for the PMLE and show that its second-order bias is contaminated by an additional term, which does not exist in regular cases. This term arises because of the failure of the first Bartlett identity to hold and seems to explain Cheung and Diebold's simulated results. We derive similar expansions for the Whittle MLE, which is another estimator tacitly using the plug-in principle. An application to the ARFIMA(0,d,0) shows that the additional bias terms are considerable.Research on this topic commenced during 2000–2002, while the author was visiting the Cowles Foundation for Research in Economics at Yale University. The author is most grateful to the Cowles Foundation for their generous hospitality and to Donald Andrews and Peter Phillips for numerous helpful comments.

In this note, determinants are explicitly calculated for the covariance matrices of differenced and double-differenced AR(1) variables.The author thanks Peter C.B. Phillips for introducing the author to Grenander and Szegö's book on Toeplitz matrices and giving useful comments. The author also thanks two anonymous referees for helpful comments on earlier drafts of the note.