This paper considers nonstationary fractional autoregressive integrated moving-average (p,d,q) models with the fractionally differencing parameter d ∈ (− 1/2,1/2) and the autoregression function with roots on or outside the unit circle. Asymptotic inference is based on the conditional sum of squares (CSS) estimation. Under some suitable conditions, it is shown that CSS estimators exist and are consistent. The asymptotic distributions of CSS estimators are expressed as functions of stochastic integrals of usual Brownian motions. Unlike results available in the literature, the limiting distributions of various unit roots are independent of the parameter d over the entire range d ∈ (− 1/2,1/2). This allows the unit roots and d to be estimated and tested separately without loss of efficiency. Our results are quite different from the current asymptotic theories on nonstationary long memory time series. The finite sample properties are examined for two special cases through simulations.

This paper investigates the asymptotic properties of the t-statistic in spurious regressions when the bandwidth in the estimation of the heteroskedasticity and autocorrelation consistent (HAC) standard error is set proportional to the sample size. Using autocovariances of large lags, the so-defined HAC estimator is capable of capturing the high persistence of the regressor and regression residuals. It is shown that the resulting t-statistic converges to a nondegenerate limiting distribution for all cases of spurious regressions considered in the literature. This finding sheds some new light on the nature of spurious regressions.I am very grateful to Bruce Hansen and two anonymous referees for helpful comments and suggestions. All remaining errors and omissions are mine alone.

This paper compares models of fractional processes and associated weak convergence results based on moving average representations in the time domain with spectral representations. Both approaches have been applied in the literature on fractional processes. We point out that the conventional forms of these models are not equivalent, as is commonly assumed, even under a Gaussianity assumption. We show that it is necessary to distinguish between “two-sided” processes depending on both leads and lags from one-sided or “causal” processes, because in the case of fractional processes these models yield different limiting properties. We derive new representations of fractional Brownian motion and show how different results are obtained for, in particular, the distribution of stochastic integrals in the multivariate context. Our results have implications for valid statistical inference in fractional integration and cointegration models.We thank F. Hashimzade and two anonymous referees for their valuable comments.

The paper provides a careful, analytical account of Trygve Haavelmo’s use of the analogy between controlled experiments common in the natural sciences and econometric techniques. The experimental analogy forms the linchpin of the methodology for passive observation that he develops in his famous monograph, The Probability Approach in Econometrics (1944). Contrary to some recent interpretations of Haavelmo’s method, the experimental analogy does not commit Haavelmo to a strong apriorism in which econometrics can only test and reject theoretical hypotheses, rather it supports the acquisition of knowledge through a two-way exchange between theory and empirical evidence. Once the details of the analogy are systematically understood, the experimental analogy can be used to shed light on theory-consistent cointegrated vector autoregression (CVAR) scenario analyses. A CVAR scenario analysis can be interpreted as a clear example of Haavelmo’s ‘experimental’ approach; and, in turn, it can be shown to extend and develop Haavelmo’s methodology and to address issues that Haavelmo regarded as unresolved.

This paper proposes a consistent nonparametric test for testing for equality of unknown conditional quantile curves across subgroups within a survey data framework. Moreover, to improve the small-sample performance of the test, we propose a modified version of wild bootstrap procedure in a quantile context. Monte Carlo evidence shows that the performance of the test in small samples is adequate but is improved significantly when the bootstrap critical values are used.I thank the co-editor and two referees for helpful comments that improved the paper. Financial support from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.

We compare the distributional properties of the four predictors commonly used in practice. They are based on the maximum likelihood, two types of the least squared, and the Yule-Walker estimators. The asymptotic expansions of the distribution, bias, and mean-squared error for the four predictors are derived up to O(T−1), where T is the sample size. Examining the formulas of the asymptotic expansions, we find that except for the Yule-Walker type predictor, the other three predictors have the same distributional properties up to O(T−1).

This paper considers a series estimator of E[α(Y)|λ(X) = λ̄], (α,λ) ∈ 𝛢 × Λ, indexed by function spaces, and establishes the estimator's uniform convergence rate over λ̄ ∈ R, α ∈ 𝛢, and λ ∈ Λ, when 𝛢 and Λ have a finite integral bracketing entropy. The rate of convergence depends on the bracketing entropies of 𝛢 and Λ in general. In particular, we demonstrate that when each λ ∈ Λ is locally uniformly ℒ2-continuous in a parameter from a space of polynomial discrimination and the basis function vector pK in the series estimator keeps the smallest eigenvalue of E[pK(λ(X))pK(λ(X))‼] above zero uniformly over λ ∈ Λ, we can obtain the same convergence rate as that established by de Jong (2002, Journal of Econometrics 111, 1–9).

This paper provides a simple estimation method for an error component regression model with general MA(q) remainder disturbances. The estimation method utilizes the transformation derived by Baltagi and Li [3] for an error component model with autoregressive remainder disturbances, and a standard orthogonalizing algorithm for the general MA(q) model. This estimation method is computationally simple utilizing only least-squares regressions. This is important for panel data regressions where brute force GLS is in many cases not feasible.This estimation method performs well relative to true GLS in Monte-Carlo experiments.

This paper considers nonparametric estimation of autoregressive panel data models with fixed effects. A within-group type series estimator is developed and its convergence rate and asymptotic normality are derived. It is found that the series estimator is asymptotically biased and the bias could reduce the mean-square convergence rate compared with the cross-section cases. A bias corrected nonparametric estimator is developed.

This paper deals with the problem of estimating the covariance matrix of the least-squares regression coefficients under heteroskedasticity and/or autocorrelation of unknown form. We consider an estimator proposed by White [17] and give a relatively simple proof of its consistency. Our proof is based on more easily verifiable conditions than those of White. An alternative estimator with improved small sample properties is also presented.

This paper develops a new test of orthogonality based on a zero restriction on the covariance between the dependent variable and the predictor. The test provides a useful alternative to regression-based tests when conditioning variables have roots close or equal to unity. In this case standard predictive regression tests can suffer from well-documented size distortion. Moreover, under the alternative hypothesis, they force the dependent variable to share the same order of integration as the predictor, whereas in practice the dependent variable often appears stationary and the predictor may be near-nonstationary. By contrast, the new test does not enforce the same orders of integration and is therefore capable of detecting a rich set of alternatives to orthogonality that are excluded by the standard predictive regression model. Moreover, the test statistic has a standard normal limit distribution for both unit root and local-to-unity conditioning variables, without prior knowledge of the local-to-unity parameter. If the conditioning variable is stationary, the test remains conservative and consistent. Simulations suggest good small-sample performance. As an empirical application, we test for the predictability of stock returns using two persistent predictors, the dividend-price ratio and short-term interest rate.

This article investigates model checks for a class of possibly nonlinear heteroskedastic time series models, including but not restricted to ARMA-GARCH models. We propose omnibus tests based on functionals of certain weighted standardized residual empirical processes. The new tests are asymptotically distribution-free, suitable when the conditioning set is infinite-dimensional, and consistent against a class of Pitman’s local alternatives converging at the parametric rate n−1/2, with n the sample size. A Monte Carlo study shows that the simulated level of the proposed tests is close to the asymptotic level already for moderate sample sizes and that tests have a satisfactory power performance. Finally, we illustrate our methodology with an application to the well-known S&P 500 daily stock index. The paper also contains an asymptotic uniform expansion for weighted residual empirical processes when initial conditions are considered, a result of independent interest.

This paper questions whether it is possible to derive consistency and asymptotic normality of the Gaussian quasi-maximum likelihood estimator (QMLE) for possibly the simplest multivariate GARCH model, namely, the multivariate ARCH(1) model of the Baba, Engle, Kraft, and Kroner form, under weak moment conditions similar to the univariate case. In contrast to the univariate specification, we show that the expectation of the log-likelihood function is unbounded, away from the true parameter value, if (and only if) the observable has unbounded second moment. Despite this nonstandard feature, consistency of the Gaussian QMLE is still warranted. The same moment condition proves to be necessary and sufficient for the stationarity of the score when evaluated at the true parameter value. This explains why high moment conditions, typically bounded sixth moment and above, have been used hitherto in the literature to establish the asymptotic normality of the QMLE in the multivariate framework.

The two-sample version of the celebrated Pearson goodness-of-fit problem has been a topic of extensive research, and several tests like the Kolmogorov-Smirnov and Cramér-von Mises have been suggested. Although these tests perform fairly well as omnibus tests for comparing two probability density functions (PDFs), they may have poor power against specific departures such as in location, scale, skewness, and kurtosis. We propose a new test for the equality of two PDFs based on a modified version of the Neyman smooth test using empirical distribution functions minimizing size distortion in finite samples. The suggested test can detect the specific directions of departure from the null hypothesis. Specifically, it can identify deviations in the directions of mean, variance, skewness, or tail behavior. In a finite sample, the actual probability of type-I error depends on the relative sizes of the two samples. We propose two different approaches to deal with this problem and show that, under appropriate conditions, the proposed tests are asymptotically distributed as chi-squared. We also study the finite sample size and power properties of our proposed test. As an application of our procedure, we compare the age distributions of employees with small employers in New York and Pennsylvania with group insurance before and after the enactment of the “community rating” legislation in New York. It has been conventional wisdom that if community rating is enforced (where the group health insurance premium does not depend on age or any other physical characteristics of the insured), then the insurance market will collapse, since only older or less healthy patients would prefer group insurance. We find that there are significant changes in the age distribution in the population in New York owing mainly to a shift in location and scale.

This paper considers a nonparametric conditional moment test of stability of an econometric model against the alternative of instability. The alternative hypothesis allows for more than one structural change, although in this case it has to be fairly smooth. This complements existing results for stability in a parametric setting. Also, it is shown that the test is always consistent, unlike the available “parametric” tests, which normally rely on the assumption of a correct specification of the model, at least under the null hypothesis of no structural instability. Moreover, we show that the test has local power comparable to the parametric ones; that is, its asymptotic efficiency is greater than zero. A Monte Carlo experiment about the performance of our test is described.

This paper develops a test of autocorrelation in the presence of heteroskedasticity of unknown form in the nonlinear regression model. The test statistic is based on the sample autocovariance of the residuals standardized by a nonparametric kernel estimate of the unknown heteroskedasticity function. Under appropriate conditions, the test statistic is shown to have a limiting chi-square distribution. Local power and consistency results for the test are also established. Monte Carlo experiments show that the test has reasonable size performance and generally dominates some of the existing tests in terms of finite-sample power.

In this comment, I first want to offer a few general remarks on the problems caused by nonuniformity in the convergence of coverage probabilities as pointed out in Kabaila's (1995) very stimulating paper. Second, I also argue that within the context of set estimation after model selection one typically will be more interested in conditional coverage probabilities than in unconditional ones. Concentrating on the last example of Section 2 in Kabaila, I then illustrate how a confidence region can be constructed such that the conditional coverage probabilities do not suffer from the nonuniformity problem discussed in Kabaila. As a by-product, I show that the confidence region considered in this example in Kabaila is not a very natural one and that it has undesirable conditional coverage properties, which I find actually more troublesome than the lack of uniformity in the unconditional coverage probability pointed out by Kabaila.

We give a straightforward condition sufficient for determining the minimum asymptotic variance estimator in certain classes of estimators relevant to econometrics. These classes are relatively broad, as they include extremum estimation with smooth or nonsmooth objective functions; also, the rate of convergence to the asymptotic distribution is not required to be n−½. We present examples illustrating the content of our result. In particular, we apply our result to a class of weighted Huber estimators, and obtain, among other things, analogs of the generalized least-squares estimator for least Lp-estimation, 1 ≤ p < ∞.

This paper considers the distribution of the Dickey-Fuller test in a model with non-zero initial value and drift and trend. We show how stochastic integral representations for the limiting distribution can be derived either from the local to unity approach with local drift and trend or from the continuous record asymptotic results of Sørensen [29]. We also show how the stochastic integral representations can be utilized as the basis for finding the corresponding characteristic functions via the Fredholm approach of Nabeya and Tanaka [16,17], This “link” between those two approaches may be of general interest. We further tabulate the asymptotic distribution by inverting the characteristic function. Using the same methods, we also find the characteristic function for the asymptotic distribution for the Schmidt-Phillips [26] unit root test. Our results show very clearly the dependence of the various tests on the initial value of the time series.

We derive the semiparametric efficiency bound in dynamic models of conditional quantiles under a sole strong mixing assumption. We also provide an expression of Stein’s (1956) least favorable parametric submodel. Our approach is as follows: First, we construct a fully parametric submodel of the semiparametric model defined by the conditional quantile restriction that contains the data generating process. We then compare the asymptotic covariance matrix of the MLE obtained in this submodel with those of the M-estimators for the conditional quantile parameter that are consistent and asymptotically normal. Finally, we show that the minimum asymptotic covariance matrix of this class of M-estimators equals the asymptotic covariance matrix of the parametric submodel MLE. Thus, (i) this parametric submodel is a least favorable one, and (ii) the expression of the semiparametric efficiency bound for the conditional quantile parameter follows.