Many estimators and tests are of the form of a ratio of quadratic forms in normal variables. Excepting a few very special cases little is 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 saddlepoint approximations for this class of statistics. We first derive and prove the existence of an exact inversion based on the joint characteristic function. Then the saddlepoint algorithm is applied and the leading term found, and analytic justification of the asymptotic nature of the approximation is given. As an illustration we consider the calculation of sizes and powers of F-tests, where a new exact result is found.

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

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

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

In this paper, we study the asymptotics of nonstationary fractional integrated time series, the long memory time series with d ≥ ½, with special attention focused on the cases when d = (2p + 1)/2 for integer n no less than 0. There is considerable empirical evidence showing long memory of this magnitude in many economic time series including the inflation rate and the stock market volatility. A study of the large-sample property is therefore both needed and useful. Also, we found the asymptotics of nonstationary fractional integrated time series useful in the study of the large-sample theory of the Kwiatkowski–Phillips–Schmidt–Shin test (1992, Journal of Econometrics 54, 159–178).

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

The abundance of high-frequency financial data and the rapid development of computer hardware have combined to transform financial economics into, arguably, the most empirically oriented field within the social sciences. At the same time, as a result of the difficulty of conducting genuine market experiments, empirical finance remains firmly grounded in the tradition of model-driven statistical inference that is characteristic of economics. Even so, the richness of data has often spurred a practical orientation that is more familiar in the natural sciences. The combination has proved fertile, leading to the classification of a set of loosely connected empirical topics as a distinct entity, financial econometrics.