Quasi-maximum likelihood estimation of multivariate diffusions
-
Xiao Huang
Abstract
This paper introduces quasi-maximum likelihood estimator for multivariate diffusions based on discrete observations. A numerical solution to the stochastic differential equation is obtained by higher order Wagner-Platen approximation and it is used to derive the first two conditional moments. Monte Carlo simulation shows that the proposed method has good finite sample property for both normal and non-normal diffusions. In an application of estimating stochastic volatility models, we find evidence of closeness between the CEV model and the GARCH stochastic volatility model. This finding supports the discrete time GARCH modeling of market volatility.
Appendix A
A.1 Proof of Theorem 1
Consider the order γ strong Wanger-Platen expansion for 
in equation (5.5.3) in Kloeden and Platen (1999)
and the order γ strong Wanger-Platen approximation for in equation (11)
Based on (32) and (33), we have
where 
Using Lemma 10.8.1 in Kloeden and Platen (1999) and tq−tq-1=Δ, we obtain
For the domain (−∞,+∞)or [0,+ ∞) we discuss two cases: case 1 when |xi|≥1 and case 2 when 0≤|xi|<1.
Case 1: |xi|≥1
Without loss of generality, we assume 
Under the rth order polynomial growth assumption in (12) and the result that Euclidean norm is less than or equal to the l1-norm, we have
where the second inequality follows Theorem 4.5.4 in Kloeden and Platen (1999) and the fact that t0=0 and K5=2r(2r+1)K2, where K is the constant in Assumption 2.
Given the description of approximation order γ below (11), we can verify that 2l(α)≥2γ+2>2γ+1 when l(α)=n(α) and l(α)+n(α)≥2γ+1 when l(α)≠n(α). Hence results in (36) can more compactly written as
For a fixed γ, K6 ( 
) is a constant, and K5 is also a constant under the assumption in (11). Since the approximation 
is derived for a given Xtq–1, tq–1 is fixed in the approximation, and 
as Δ→0 Even in the case when tq-1 grows as the number of observation increases, still holds as long as Δ→0 fast enough.
Case 2: 0≤|xi|<1 (excluding 0 if either a(x;θ) or b(x;θ) is not differentiable at the point 0).
Similar to case 1, we assume 
Under the rth order polynomial growth assumption in (13), when 0≤xi<1 (35) can be written as
For the expectation 
let the lower bound of integration be c and c→0+ and we have
provided that p(x1) is a square integrable density function for x1. If we do not impose the assumption of E((x1)–2r)<+∞, the r.h.s. of (39) will grow to infinity as c→0+. However, we will still have in (38) as long as Δ→0 fast enough. Alternatively, we may impose an assumption on the rate at which the diffusion process approaches boundary point and require that Δ→0 at a faster rate. Combining this analysis with the result in (37), we have as Δ→0. Consequently, Ztq→0 in (34) and the result in (14) holds.
The result in (15) can be obtained using Chebychev’s inequality. As Δ→0, we obtain 
which implies 
in probability in (15).
A.2 Approximation expressions
Consider the SDE in (7) which nests SDEs in (27) and (28). When l(a)= 3 the strong Wagner-Platen approximations for 
and 
are given, respectively, by
Expressions for l(α)=4 are available from the author upon request.
I thank the editor Bruce Mizrach and two referees for many constructive comments. I also thank Yong Bao, Robert Kimmel, Stefano Mazzotta, and seminar participants at the 2009 Meetings of the Midwest Econometrics Group for helpful comments. I am also grateful to Eckhard Platen for his valuable comments and suggestions. All remaining errors are mine. Financial support from the Coles College of Business at Kennesaw State University is greatly acknowledged
- 1
Wagner-Platen expansion and approximation are also called Itô-Taylor expansion and approximation in Kloeden and Platen (1999).
- 2
Let β(i,j) denote the ith derivative of β w.r.t. x1 and jth derivative w.r.t. x2. Let ξ′, ξ″ and ξ(r) denote the 1st, 2nd, and rth derivative of ξ w.r.t. x1 with r≥3. Similar definitions apply to ζ and ϕ in (8).
- 3
I would like to thank Peter E. Kloeden for clarifying some notation in Theorem 10.6.3 of Kloeden and Platen (1999).
- 4
To be consistent with Assumption 1, we exclude all boundaries (0 or ±∞) for x
- 5
A more detailed description of VIX can be found at: http://www.cboe.com/micro/VIX/vixwhite.pdf.
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Articles in the same Issue
- Stochastically weighted average conditional moment tests of functional form
- Do Latin American Central Bankers Behave Non-Linearly? The Experiences of Brazil, Chile, Colombia and Mexico
- Empirical analysis of ARMA-GARCH models in market risk estimation on high-frequency US data
- Quasi-maximum likelihood estimation of multivariate diffusions
- Time-varying cointegration, identification, and cointegration spaces
- Noncausality and asset pricing
- State space Markov switching models using wavelets
Articles in the same Issue
- Stochastically weighted average conditional moment tests of functional form
- Do Latin American Central Bankers Behave Non-Linearly? The Experiences of Brazil, Chile, Colombia and Mexico
- Empirical analysis of ARMA-GARCH models in market risk estimation on high-frequency US data
- Quasi-maximum likelihood estimation of multivariate diffusions
- Time-varying cointegration, identification, and cointegration spaces
- Noncausality and asset pricing
- State space Markov switching models using wavelets
