Test for dispersion constancy in stochastic differential equation models

Abstract

In this paper, we propose a constancy test for volatility in It Oo processes based on discretely sampled data. The test statistic constitutes an integration of the Ljung-Box test statistic and the kurtosis statistic in the Jarque-Bera test. It is shown that under regularity conditions, the proposed test asymptotically follows a chi-square distribution under the null hypothesis of constant volatility. To evaluate the test, empirical sizes and powers were examined through a simulation study. Analysis of real data including ultra-high frequency transaction data and interest rates was also conducted for illustration.

Change Point Test for Dispersion Parameter Based on Discretely Observed Sample From Sde Models

“…Thus, the test is free from any drift changes. In fact, Lee and Guo [6] considered the constancy test for the dispersion parameter in the situation that b(·) is constant under the null hypothesis. They proposed a test, which is a hybrid of the Jarque-Bera's normality test and the Ljung-Box test, and figured out that the test is unaffected by the drift parameter estimation.…”

Monitoring change point for diffusion parameter based on discretely observed sample from stochastic differential equation models

“…In the literature, statistical goodness of fit tests for SDE models have been proposed by many authors. For example, Aït‐Sahalia and Chen et al , considered a discrepancy measure regarding the transitional distribution of the process, and Lee and Guo considered a test for dispersion constancy. Here, owing to its simplicity and effectiveness, we employ the test of Lee and Guo (abbreviated as LG test) to perform a model check test.…”

“…Consider the SDE where ( θ , σ ) is a p + 1‐dimensional unknown parameter, a is a known real valued function, and is a standard Brownian motion. As mentioned earlier, many well‐known SDE models can be transformed to using the Lamperti transform (see Iacus and Lee and Guo ). A typical example is the BS model: Ito's rule ensuresthat X t = log S t has a form in : d X t =( μ − σ 2 /2) d t + σ d W t .…”

“…For instance, for the Itô process a suitable Lamperti transform is (cf. Iacus and Lee and Guo ): the SDE of f ( Z t )satisfies Model . Choose the time interval [0, T ]that has a constant dispersion. The test of Lee and Guo is applied to search a period with constant volatility by testing for the dispersion constancy in a given time interval.…”

“…Iacus and Lee and Guo ): the SDE of f ( Z t )satisfies Model . Choose the time interval [0, T ]that has a constant dispersion. The test of Lee and Guo is applied to search a period with constant volatility by testing for the dispersion constancy in a given time interval. We will illustrate the test procedure of Lee and Guo by the empirical example of Section 5. Determine the inter‐inspection time length ρ T , the sampling frequency and the sampling size n k = sup{ n : n h n ≤ k ρ T } in the inspection time interval [0, k ρ T ], where 0 < ρ , ζ < 1. Estimate the drift parameters of the SDE model using the mle's, and then estimate the diffusion parameter σ 2 for the in‐control period by , where . Construct the standardized control chart as described in Section 4.1. (S2) Prospective online monitoring in phase II: Perform the test at the inspection time k ρ T and estimate the dispersion parameter recursively by for k = k T +1, k T +2,…. (S3) Out‐of‐control stage: We perform the (S2) procedure to detect the change in the diffusion parameter.…”