Indices of financial returns typically display sample kurtosis that declines towards the Gaussian value 3 as the sampling interval increases. This paper uses stochastic unit root (STUR) and continuous time analysis to explain the phenomenon. Limit theory for the sample kurtosis reveals that STUR specifications provide two sources of excess kurtosis, both of which decline with the sampling interval. Limiting kurtosis is shown to be random and is a functional of the limiting price process. Using a continuous time version of the model under no-drift, local drift, and drift inclusions, we suggest a new continuous time kurtosis measure for financial returns that assists in reconciling these models with the empirical kurtosis characteristics of returns. Simulations are reported and applications to several financial indices demonstrate the usefulness of this approach.
Two approaches have dominated formulations designed to capture small departures from unit root autoregressions. The first involves deterministic departures that include local-to-unity (LUR) and mildly (or moderately) integrated (MI) specifications where departures shrink to zero as the sample size n→∞. The second approach allows for stochastic departures from unity, leading to stochastic unit root (STUR) specifications. This paper introduces a hybrid local stochastic unit root (LSTUR) specification that has both LUR and STUR components and allows for endogeneity in the time varying coefficient that introduces structural elements to the autoregression. This hybrid model generates trajectories that, upon normalization, have non-linear diffusion limit processes that link closely to models that have been studied in mathematical finance, particularly with respect to option pricing. It is shown that some LSTUR parameterizations have a mean and variance which are the same as a random walk process but with a kurtosis exceeding 3, a feature which is consistent with much financial data. We develop limit theory and asymptotic expansions for the process and document how inference in LUR and STUR autoregressions is affected asymptotically by ignoring one or the other component in the more general hybrid generating mechanism. In particular, we show how confidence belts constructed from the LUR model are affected by the presence of a STUR component in the generating mechanism. The import of these findings for empirical research are explored in an application to the spreads on US investment grade corporate debt.
Lieberman and Phillips (2016; Journal of Econometrics; LP) introduced a multivariate stochastic unit root (STUR) model, which allows for random, time varying local departures from a unit root (UR) model, where nonlinear least squares (NLLS) may be used for estimation and inference on the STUR coefficient. In a structural version of this model where the driver variables of the STUR coefficient are endogenous, the NLLS estimate of the STUR parameter is inconsistent, as are the corresponding estimates of the associated covariance parameters. This paper develops a nonlinear instrumental variable (NLIV) as well as GMM estimators of the STUR parameter which conveniently addresses endogeneity. We derive the asymptotic distributions of the NLIV and GMM estimators and establish consistency under similar orthogonality and relevance conditions to those used in the linear model. An overidentification test and its asymptotic distribution are also developed. The results enable inference about structural STUR models and a mechanism for testing the local STUR model against a simple UR null, which complements usual UR tests. Simulations reveal that the asymptotic distributions of the the NLIV and GMM estimators of the STUR parameter as well as the test for overidentifying restrictions perform well in small samples and that the distribution of the NLIV estimator is heavily leptokurtic with a limit theory which has Cauchy-like tails. Comparisons of STUR coefficient and a standard UR coefficient test show that the one-sided UR test performs poorly against the one-sided STUR coefficient test both as the sample size and departures from the null rise.
This paper extends recent findings of Lieberman and Phillips (2014) on stochastic unit root (SUR) models to a multivariate case including a comprehensive asymptotic theory for estimation of the model’s parameters. The extensions are useful because they lead to a generalization of the Black-Scholes formula for derivative pricing. In place of the standard assumption that the price process follows a geometric Brownian motion, we derive a new form of the Black-Scholes equation that allows for a multivariate time varying coefficient element in the price equation. The corresponding formula for the value of a European-type call option is obtained and shown to extend the existing option price formula in a manner that embodies the effect of a stochastic departure from a unit root. An empirical application reveals that the new model is consistent with excess skewness and kurtosis in the price distribution relative to a lognormal distribution.
A time-varying autoregression is considered with a similarity-based coefficient and possible drift. It is shown that the random walk model has a natural interpretation as the leading term in a small-sigma expansion of a similarity model with an exponential similarity function as its autoregressive coefficient. Consistency of the quasi-maximum likelihood estimator of the parameters in this model is established, the behaviors of the score and Hessian functions are analyzed and test statistics are suggested. A complete list is provided of the normalization rates required for the consistency proof and for the score and Hessian functions standardization. A large family of unit root models with stationary and explosive alternatives are characterized within the similarity class through the asymptotic negligibility of a certain quadratic form that appears in the score function. A variant of the stochastic unit root model within the class is studied and a large sample limit theory provided which leads to a new nonlinear diffusion process limit showing the form of the drift and conditional volatility induced by this model. Some simulations and a brief empirical application to data on an Australian Exchange Traded Fund are included.
An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d in (-1/2,1/2). The class of spectral densities considered includes as a special case the stationary and invertible ARFIMA(p,d,q) model. The leading term of the expansion is of the order O(1/k1-2d), where k is the autocovariance order, consistent with the well known power law decay for such processes, and is shown to be accurate to an error of O(1/k3-2d). The derivation uses Erdélyi’s (1956) expansion for Fourier-type integrals when there are critical points at the boundaries of the range of integration - here the frequencies {0,2}. Numerical evaluations show that the expansion is accurate even for small k in cases where the autocovariance sequence decays monotonically, and in other cases for moderate to large k. The approximations are easy to compute across a variety of parameter values and models.
Keywords: Autocovariance, Asymptotic expansion, Critical point, Fourier integral, Long memory
This paper establishes error orders for integral limit approximations to traces of powers to the pth order) of products of Toeplitz matrices. Such products arise frequently in the analysis of stationary time series and in the development of asymptotic expansions. The elements of the matrices are Fourier transforms of functions which we allow to be bounded, unbounded, or even to vanish on [-π,π], thereby including important cases such as the spectral functions of fractional processes. Error rates are also given in the case in which the matrix product involves inverse matrices. The rates are sharp up to an arbitrarily small ε > 0. The results improve on the o(1) rates obtained in earlier work for analogous products. For the p = 1 case, an explicit second order asymptotic expansion is found for a quadratic functional of the autocovariance sequences of stationary long memory time series. The order of magnitude of the second term in this expansion is shown to depend on the long memory parameters. It is demonstrated that the pole in the first order approximation is removed by the second order term, which provides a substantially improved approximation to the original functional.
We establish the validity of an Edgeworth expansion to the distribution of the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. The result covers ARFIMA type models, including fractional Gaussian noise. The method of proof consists of three main ingredients: (i) verification of a suitably modified version of Durbin’s (1980) general conditions for the validity of the Edgeworth expansion to the joint density of the log-likelihood derivatives; (ii) appeal to a simple result of Skovgaard (1986) to obtain from this an Edgeworth expansion for the joint distribution of the log-likelihood derivatives; (iii) appeal to and extension of arguments of Bhattacharya and Ghosh (1978) to accomplish the passage from the result on the log-likelihood derivatives to the result for the maximum likelihood estimators. We develop and make extensive use of a uniform version of Dahlhaus’s (1989) Theorem~5.1 on products of Toeplitz matrices; the extension of Dahlhaus’s result is of interest in its own right. A small numerical study of the efficacy of the Edgeworth expansion is presented for the case of fractional Gaussian noise.
We apply and extend Firth’s (1993) modified score estimator to deal with a class of stationary Gaussian long-memory processes. Our estimator removes the first order bias of the maximum likelihood estimator. A small simulation study reveals the reduction in the bias is considerable, while it does not inflate the corresponding mean squared error.
The maximum likelihood estimator (MLE) of the fractional difference parameter in the Gaussian ARFIMA(0,d,0) model is well known to be asymptotically N(0, 6/π2 ). This paper develops a second order asymptotic expansion to the distribution of this statistic. The correction term for the density is shown to be independent of d, so that the MLE is second order pivotal for d. This feature of the MLE is unusual, at least in time series contexts. Simulations show that the normal approximation is poor and that the expansions make significant improvements in accuracy.