Many estimators and tests are of the form of a ratio of quadraticforms in normal variables. Excepting a few very special cases littleis known about the density or distribution of these ratios,particularly if we allow for noncentrality in the quadratic forms.This paper assumes this generality and derives saddlepointapproximations for this class of statistics. We first derive andprove the existence of an exact inversion based on the jointcharacteristic function. Then the saddlepoint algorithm is appliedand the leading term found, and analytic justification of theasymptotic nature of the approximation is given. As an illustrationwe consider the calculation of sizes and powers ofF-tests, where a new exact result is found.

Many unit root and cointegration tests require an estimate of thespectral density function at frequency zero of some process.Commonly used are kernel estimators based on weighted sums ofautocovariances constructed using estimated residuals from an AR(1)regression. However, it is known that with substantially correlatederrors, the OLS estimate of the AR(1) parameter is severely biased.In this paper, we first show that this least-squares bias induces asignificant increase in the bias and mean-squared error (MSE) ofkernel-based estimators. We then consider a variant of theautoregressive spectral density estimator that does not share theseshortcomings because it bypasses the use of the estimate from theAR(1) regression. Simulations and local asymptotic analyses show itsbias and MSE to be much smaller than those of a kernel-basedestimator when there is strong negative serial correlation. We alsoinclude a discussion about the appropriate choice of the truncationlag.

In this paper, we study the bias-corrected test developed in Fan(1994). It is based on the integrated squared difference between akernel estimator of the unknown density function of a random vectorand a kernel smoothed estimator of the parametric density functionto be tested under the null hypothesis. We provide an alternativeasymptotic approximation of the finite-sample distribution of thistest by fixing the smoothing parameter. In contrast to the normalapproximation obtained in Fan (1994) in which the smoothingparameter shrinks to zero as the sample size grows to infinity, weobtain a non-normal asymptotic distribution for the bias-correctedtest. A parametric bootstrap procedure is proposed to approximatethe critical values of this test. We show both analytically and bysimulation that the proposed bootstrap procedure works. Consistencyand local power properties of the bias-corrected test with a fixedsmoothing parameter are also discussed.

In this article we present a new method for computing the theoreticalautocovariance function of an autoregressive moving average model.The importance of our theorem is that it yields two interestingresults: First, a closed-form solution is derived in terms of theroots of the autoregressive polynomial and the parameters of themoving average part. Second, a sufficient condition for the lack ofmodel redundancy is obtained.

In this paper, we study the asymptotics of nonstationary fractionalintegrated time series, the long memory time series withd ≥ ½, with special attention focused on thecases when d = (2p + 1)/2 forinteger n no less than 0. There is considerableempirical evidence showing long memory of this magnitude in manyeconomic time series including the inflation rate and the stockmarket volatility. A study of the large-sample property is thereforeboth needed and useful. Also, we found the asymptotics ofnonstationary fractional integrated time series useful in the studyof the large-sample theory of the Kwiatkowski–Phillips–Schmidt–Shintest (1992, Journal of Econometrics 54,159–178).

When the disturbances of a regression model follow anI(1) process there is a tendency to estimate abreak point in the middle of the sample, even though a break pointdoes not actually exist. In this note, we provide a mathematicalproof for this phenomenon.

The abundance of high-frequency financial data and the rapiddevelopment of computer hardware have combined to transformfinancial economics into, arguably, the most empirically orientedfield within the social sciences. At the same time, as a result ofthe difficulty of conducting genuine market experiments, empiricalfinance remains firmly grounded in the tradition of model-drivenstatistical inference that is characteristic of economics. Even so,the richness of data has often spurred a practical orientation thatis more familiar in the natural sciences. The combination has provedfertile, leading to the classification of a set of loosely connectedempirical topics as a distinct entity, financialeconometrics.