|
on Econometric Time Series |
| By: | Roger Lord (Faculty of Economics and Business Economics, Erasmus Universiteit Rotterdam); Remmert Koekkoek (Robeco Alternative Investments); Dick van Dijk (Faculty of Economics and Business Economics, Erasmus Universiteit Rotterdam) |
| Abstract: | When using an Euler discretisation to simulate a mean-reverting square root process, one runs into the problem that while the process itself is guaranteed to be nonnegative, the discretisation is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the Heston stochastic volatility model, where the variance is modelled as a square root process. Consequently, when using an Euler discretisation, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimise the upward bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to a recent quasi-second order scheme of Kahl and Jäckel and the exact scheme of Broadie and Kaya. The choice of fix is found to be extremely important. The full truncation scheme by far outperforms all biased schemes in terms of bias, root-mean-squared error, and hence should be the preferred discretisation method for simulation of the Heston model and extensions thereof. |
| Keywords: | Stochastic volatility; Heston; square root process; Euler-Maruyama; discretisation; strong convergence; weak convergence; boundary behaviour |
| JEL: | C63 G13 |
| Date: | 2006–05–18 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:uvatin:20060046&r=ets |
| By: | Heiss, Florian |
| Abstract: | In applied microeconometric panel data analyses, time-constant random effects and first-order Markov chains are the most prevalent structures to account for intertemporal correlations in limited dependent variable models. An example from health economics shows that the addition of a simple autoregressive error terms leads to a more plausible and parsimonious model which also captures the dynamic features better. The computational problems encountered in the estimation of such models - and a broader class formulated in the framework of nonlinear state space models - hampers their widespread use. This paper discusses the application of different nonlinear filtering approaches developed in the time-series literature to these models and suggests that a straightforward algorithm based on sequential Gaussian quadrature can be expected to perform well in this setting. This conjecture is impressively confirmed by an extensive analysis of the example application. |
| Keywords: | LDV models; panel data; state space; numerical integration; health |
| JEL: | C15 C23 C35 I10 |
| Date: | 2006–06 |
| URL: | http://d.repec.org/n?u=RePEc:lmu:muenec:1157&r=ets |
| By: | James H. Stock; Mark W. Watson |
| Abstract: | The conventional heteroskedasticity-robust (HR) variance matrix estimator for cross-sectional regression (with or without a degrees of freedom adjustment), applied to the fixed effects estimator for panel data with serially uncorrelated errors, is inconsistent if the number of time periods T is fixed (and greater than two) as the number of entities n increases. We provide a bias-adjusted HR estimator that is (nT)1/2 -consistent under any sequences (n, T) in which n and/or T increase to ∞.The conventional heteroskedasticity-robust (HR) variance matrix estimator for cross-sectional regression (with or without a degrees of freedom adjustment), applied to the fixed effects estimator for panel data with serially uncorrelated errors, is inconsistent if the number of time periods T is fixed (and greater than two) as the number of entities n increases. We provide a bias-adjusted HR estimator that is (nT)1/2 -consistent under any sequences (n, T) in which n and/or T increase to ∞. |
| JEL: | C23 C12 |
| Date: | 2006–06 |
| URL: | http://d.repec.org/n?u=RePEc:nbr:nberte:0323&r=ets |
| By: | Juan Carlos Escanciano (Universidad de Navarra); Carlos Velasco (Universidad Carlos III) |
| Abstract: | This paper proposes an omnibus test for testing a generalized version of the martingale difference hypothesis (MDH). This generalized hypothesis includes the usual MDH, testing for conditional moments constancy such as conditional homoscedasticity (ARCH effects) or testing for directional predictability. Here we propose a unified approach for dealing with all of these testing problems. These hypotheses are long standing problems in econometric time series analysis, and typically have been tested using the sample autocorrelations or in the spectral domain using the periodogram. Since these hypotheses cover also nonlinear predictability, tests based on those second order statistics are inconsistent against uncorrelated processes in the alternative hypothesis. To circumvent this problem we introduce the pairwise integrated regression functions as measures of linear and nonlinear dependence. With our test there is no need to choose a lag order depending on sample size, to smooth the data or to formulate a parametric alternative model. Moreover, our test is robust to higher order dependence, in particular to conditional heteroskedasticity. Under general dependence the asymptotic null distribution depends on the data generating process, so a bootstrap procedure is considered and a Monte Carlo study examines its finite sample performance. Then we investigate the martingale and conditional heteroskedasticity properties of the Pound/Dollar exchange rate. |
| JEL: | C12 |
| URL: | http://d.repec.org/n?u=RePEc:una:unccee:wp0606&r=ets |
| By: | Javier Hualde (Universidad de Navarra); Peter Robinson (London School of Economics) |
| Abstract: | A semiparametric bivariate fractionally cointegrated system is considered, integration orders possibly being unknown and I(0) unobservable inputs having nonparametric spectral density. Two kinds of estimate of the cointegrating parameter ν are considered, one involving inverse spectral weighting and the other, unweighted statistics with a spectral estimate at frequency zero. We establish under quite general conditions the asymptotic distributional properties of the estimates of ν, both in case of "strong cointegration" (when the difference between integration orders of observables and cointegrating errors exceeds 1/2) and in case of "weak cointegration" (when that difference is less than 1/2), which includes the case of (asymptotically) stationary observables. Across both cases, the same Wald test statistic has the same standard null χ² limit distribution, irrespective of whether integration orders are known or estimated. The regularity conditions include unprimitive ones on the integration orders and spectral density estimates, but we check these under more primitive conditions on particular estimates. Finite-sample properties are examined in a Monte Carlo study.A semiparametric bivariate fractionally cointegrated system is considered, integration orders possibly being unknown and I(0) unobservable inputs having nonparametric spectral density. Two kinds of estimate of the cointegrating parameter ν are considered, one involving inverse spectral weighting and the other, unweighted statistics with a spectral estimate at frequency zero. We establish under quite general conditions the asymptotic distributional properties of the estimates of ν, both in case of "strong cointegration" (when the difference between integration orders of observables and cointegrating errors exceeds 1/2) and in case of "weak cointegration" (when that difference is less than 1/2), which includes the case of (asymptotically) stationary observables. Across both cases, the same Wald test statistic has the same standard null χ² limit distribution, irrespective of whether integration orders are known or estimated. The regularity conditions include unprimitive ones on the integration orders and spectral density estimates, but we check these under more primitive conditions on particular estimates. Finite-sample properties are examined in a Monte Carlo study. |
| JEL: | C32 |
| URL: | http://d.repec.org/n?u=RePEc:una:unccee:wp0706&r=ets |
| By: | Javier Hualde (Universidad de Navarra); Carlos Velasco (Universidad Carlos III) |
| Abstract: | We propose tests of the null of spurious relationship against the alternative of fractional cointegration among the components of a vector of fractionally integrated time series. Our test statistics have an asymptotic chi-square distribution under the null and rely on GLS-type of corrections which control for the short run correlation of the weak dependent components of the fractionally integrated processes. We emphasize corrections based on nonparametric modelization of the innovations' autocorrelation, relaxing important conditions which are standard in the literature, and, in particular, being able to consider simultaneously (asymptotically) stationary or nonstationary processes. Relatively weak conditions on the corresponding short run and memory parameter estimates are assumed. The new tests are consistent with a divergence rate that, in most of the cases, as we show in a simple situation, depends on the cointegration degree. Finite-sample properties of the tests are analysed by means of a Monte Carlo experiment. |
| JEL: | C12 C13 C22 |
| URL: | http://d.repec.org/n?u=RePEc:una:unccee:wp0806&r=ets |