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.

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.

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.

The current article examines the limit distribution of the quasi-maximum likelihood estimator obtained from a directionally differentiable quasi-likelihood function and represents its limit distribution as a functional of a Gaussian stochastic process indexed by direction. In this way, the standard analysis that assumes a differentiable quasi-likelihood function is treated as a special case of our analysis. We also examine and redefine the standard quasi-likelihood ratio, Wald, and Lagrange multiplier test statistics so that their null limit behaviors are regular under our model framework.

In this paper we investigate the impact of persistent (nonstationary or near nonstationary) cycles on the asymptotic and finite-sample properties of standard unit root tests. Results are presented for the augmented Dickey–Fuller (ADF) normalized bias and t-ratio-based tests (Dickey and Fuller, 1979, Journal of the American Statistical Association 745, 427–431; Said and Dickey, 1984; Biometrika 71, 599–607). the variance ratio unit root test of Breitung (2002, Journal of Econometrics 108, 343–363), and the M class of unit-root tests introduced by Stock (1999, in Engle and White (eds.), A Festschrift in Honour of Clive W.J. Granger) and Perron and Ng (1996, Review of Economic Studies 63, 435–463). We show that although the ADF statistics remain asymptotically pivotal (provided the test regression is properly augmented) in the presence of persistent cycles, this is not the case for the other statistics considered and show numerically that the size properties of the tests based on these statistics are too unreliable to be used in practice. We also show that the t-ratios associated with lags of the dependent variable of order greater than two in the ADF regression are asymptotically normally distributed. This is an important result as it implies that extant sequential methods (see Hall, 1994, Journal of Business & Economic Statistics 17, 461–470; Ng and Perron, 1995, Journal of the American Statistical Association 90, 268–281) used to determine the order of augmentation in the ADF regression remain valid in the presence of persistent cycles.

We propose a new iterative estimation algorithm for use in semiparametric models where calculation of Z-estimators by conventional means is difficult or impossible. Unlike a Newton–Raphson approach, which makes use of the entire Hessian, this approach only uses curvature information associated with portions of the Hessian that are relatively easy to calculate. Consistency and asymptotic normality of estimators obtained from this algorithm are established under regularity conditions and an information dominance condition. Two specific examples, a quantile regression model with missing covariates and a GARCH-in-mean model with conditional mean of unknown functional form, demonstrate the applicability of the algorithm. This new approach can be interpreted as an extension of the maximization by parts estimation approach to semiparametric models.

In this article, we investigate the properties of heteroskedasticity and autocorrelation robust (HAR) test statistics in time series regression settings when observations are missing. We primarily focus on the nonrandom missing process case where we treat the missing locations to be fixed as T → ∞ by mapping the missing and observed cutoff dates into points on [0,1] based on the proportion of time periods in the sample that occur up to those cutoff dates. We consider two models, the amplitude modulated series (Parzen, 1963) regression model, which amounts to plugging in zeros for missing observations, and the equal space regression model, which simply ignores the missing observations. When the amplitude modulated series regression model is used, the fixed-b limits of the HAR test statistics depend on the locations of missing observations but are otherwise pivotal. When the equal space regression model is used, the fixed-b limits of the HAR test statistics have the standard fixed-b limits as in Kiefer and Vogelsang (2005). We discuss methods for obtaining fixed-b critical values with a focus on bootstrap methods and find the naive i.i.d. bootstrap with missing dates fixed to be an effective and practical way to obtain the fixed-b critical values.

Given observations on a stationary economic vector time series process we show that the best h-step ahead forecast (best in the sense of having minimal mean square forecast error) of one of the variables can be consistently estimated by nonparametric regression on an ARMA memory index. Our approach is based on a combination of the ARMA memory index modeling approach of Bierens [7] with a modification to time series of the nonparametric kernel regression approach of Devroye and Wagner [16]. This approach is truly model-free, as no explicit specification of the distribution of the data generating process is needed.

This paper studies the identification and estimation of weighted average derivatives of conditional location functionals including conditional mean and conditional quantiles in settings where either the outcome variable or a regressor is interval-valued. Building on Manski and Tamer (2002, Econometrica 70(2), 519–546) who study nonparametric bounds for mean regression with interval data, we characterize the identified set of weighted average derivatives of regression functions. Since the weighted average derivatives do not rely on parametric specifications for the regression functions, the identified set is well-defined without any functional-form assumptions. Under general conditions, the identified set is compact and convex and hence admits characterization by its support function. Using this characterization, we derive the semiparametric efficiency bound of the support function when the outcome variable is interval-valued. Using mean regression as an example, we further demonstrate that the support function can be estimated in a regular manner by a computationally simple estimator and that the efficiency bound can be achieved.

We extend the notion of cointegration for time series taking values in a potentially infinite dimensional Banach space. Examples of such time series include stochastic processes in $C[0,1]$ equipped with the supremum distance and those in a finite dimensional vector space equipped with a non-Euclidean distance. We then develop versions of the Granger–Johansen representation theorems for I(1) and I(2) autoregressive (AR) processes taking values in such a space. To achieve this goal, we first note that an AR(p) law of motion can be characterized by a linear operator pencil (an operator-valued map with certain properties) via the companion form representation, and then study the spectral properties of a linear operator pencil to obtain a necessary and sufficient condition for a given AR(p) law of motion to admit I(1) or I(2) solutions. These operator-theoretic results form a fundamental basis for our representation theorems. Furthermore, it is shown that our operator-theoretic approach is in fact a closely related extension of the conventional approach taken in a Euclidean space setting. Our theoretical results may be especially relevant in a recently growing literature on functional time series analysis in Banach spaces.

This note proves that the estimator of panel data censored regression models proposed in Honoré [2] is median unbiased when only one parameter is estimated. This result is obtained without parametric assumptions about the distribution of the error terms.

A multivariate quantile regression model with a factor structure is proposed to study data with multivariate responses with covariates. The factor structure is allowed to vary with the quantile levels, which is more flexible than the classical factor models. Assuming the number of factors is small, and the number of responses and the input variables are growing with the sample size, the model is estimated with the nuclear norm regularization. The incurred optimization problem can only be efficiently solved in an approximate manner by off-the-shelf optimization methods. Such a scenario is often seen when the empirical loss is nonsmooth or the numerical procedure involves expensive subroutines, for example, singular value decomposition. To show that the approximate estimator is still statistically accurate, we establish a nonasymptotic bound on the Frobenius risk and prediction risk. For implementation, a numerical procedure that provably marginalizes the approximation error is proposed. The merits of our model and the proposed numerical procedures are demonstrated through the Monte Carlo simulation and an application to finance involving a large pool of asset returns.

For regression models with general unstable regressors having characteristic roots on the unit circle and general stationary errors independent of the regressors, sufficient conditions are investigated under which the ordinary least squares estimator (OLSE) is asymptotically efficient in that it has the same limiting distribution as the generalized least squares estimator (GLSE) under the same normalization. A key condition for the asymptotic efficiency of the OLSE is that one multiplicity of a characteristic root of the regressor process is strictly greater than the multiplicities of the other roots. Under this condition, the covariance matrix Γ of the errors and the regressor matrix X are shown to satisfy a relationship (ΓX = XC + V for some matrix C) for V asymptotically dominated by X, which is analogous to the condition (ΓX = XC for some matrix C) for numerical equivalence of the OLSE and the GLSE.

Because the ARMA–GARCH model can generate data with some important properties such as skewness, heavy tails, and volatility persistence, it has become a benchmark model in analyzing financial and economic data. The commonly employed quasi maximum likelihood estimation (QMLE) requires a finite fourth moment for both errors and the sequence itself to ensure a normal limit. The self-weighted quasi maximum exponential likelihood estimation (SWQMELE) reduces the moment constraints by assuming that the errors and their absolute values have median zero and mean one, respectively. Therefore, it is necessary to test zero median of errors before applying the SWQMELE, as changing zero mean to zero median destroys the ARMA–GARCH structure. This paper develops an efficient empirical likelihood test without estimating the GARCH model but using the GARCH structure to reduce the moment effect. A simulation study confirms the effectiveness of the proposed test. The data analysis shows that some financial returns do not have zero median of errors, which cautions the use of the SWQMELE.

The central limit theorem for student's distribution—solution.

Dynamic economic theories usually have implications on and only on the conditional mean dynamics of economic processes. Using a generalized spectral derivative approach, Hong and Lee (2005, Review of Economic Studies 72, 499–541) recently proposed a new class of omnibus nonparametric specification tests for linear and nonlinear time series conditional mean models, where the dimension of the conditioning information set may be infinite. The tests can detect a wide range of model misspecifications in mean while being robust to conditional heteroskedasticity and time-varying higher order moments of unknown form. They enjoy an asymptotic “nuisance parameter–free” property in the sense that parameter estimation uncertainty has no impact on the asymptotic N(0,1) distribution of the test statistics. As a result, only the estimated residuals from the null parametric model are needed to implement the tests, and no specific estimation is required.

Although parameter estimation uncertainty has no impact on the asymptotic distribution of the tests, it may have significant impact on the finite-sample distribution, and such an impact may become more substantial as the number of estimated parameters increases. In this paper, we adopt the Wooldridge (1990, Econometric Theory 6, 17–43) device for parametric m-tests to the Hong and Lee (2005) nonparametric tests to reduce the impact of parameter estimation uncertainty. Asymptotic size and power properties of the modified tests are investigated, and simulation studies show that the modified tests generally have better sizes in finite samples and are robust to parameter estimation uncertainty. In the meantime, the size improvement does not cause loss of power against a wide range of alternatives when using the empirical critical values for the tests. These results suggest that the modified generalized spectral derivative tests can be a useful tool in time series conditional mean modeling.We thank Bruce Hansen (the co-editor) and three referees for insightful and helpful comments and suggestions. We also are grateful for financial support from the National Science Foundation Grant SES–0111769 and the Cheung Kong Visiting Scholarship by the Chinese Ministry of Education and Xiamen University, China.

Jan Kmenta is the author of the internationally respected text The Elements of Econometrics [1] and co-editor of several books related to econometric model building. His published research over 40 years relates to many facets of econometric theory and practice, including the estimation of production functions, the evaluation of structural econometric models, and estimation in the face of missing data. This work has appeared in the leading journals of the profession. A 1966 paper [11] is of historical interest for being the second econometric model constructed of the Australian economy. In his work, through both constructive contributions and methodological critique, he has always sought to highlight and advance the evidently close connection between economics and econometrics.

Kmenta was born on January 3, 1928, in Prague, the Czech Republic. He was educated at the Jirasek State Gymnasium in Prague (1939–47) and the Czech University of Technology in Prague (1947–49). At the University of Sydney (1952–55), Australia, he graduated with a Bachelor of Economics degree (First Class Honors). He obtained a Master of Arts degree (1959) and a Doctor of Philosophy degree (1964) from Stanford University with a doctoral dissertation entitled “Australian Postwar Immigration: An Econometric Study.” After teaching stints in Australia, Stanford, and Wisconsin, he taught at Michigan State from 1965 to 1973 and at the University of Michigan from 1973 to 1993. He is currently Professor Emeritus of Economics and Statistics, University of Michigan, and Visiting Professor, CERGE-EI, Charles University, Prague.

His various academic awards and prizes include the Alexander von Humboldt Foundation Award for Senior U.S. Scientists, the Michigan Economic Society Best Professor Award, the Royal Economic Society Lecturer at the University of Leicester, and the Karel Englis Medal of the Academy of Sciences of the Czech Republic. He was awarded an honorary doctorate from the University of Saarland, Germany, and served as associate editor, Journal of the American Statistical Association (1973–1979 and 1985–1992), associate editor, Review of Economics and Statistics (1975–1992), and associate editor, Metrika (1981–1985). Kmenta was listed as 40th among all economists ranked by the total number of citations (Medoff 1989).

This is an edited transcript of two tape-recorded interviews conducted with Professor Kmenta in Sydney on March 12, 2004, as he visited the University of New South Wales to present the seminar Econometrics: A Failed Science? I am indebted to one of Kmenta's former students, Eric Sowey, for suggesting this project and for his enthusiasm and support throughout.