Estimation and asymptotic covariance matrix for stochastic volatility models

Abstract

In this paper we compute the asymptotic variance-covariance matrix of the method of moments estimators for the canonical Stochastic Volatility model. Our procedure is based on a linearization of the initial process via the log-squared transformation of Breidt and Carriquiry (Modelling and prediction, honoring Seymour Geisel. Springer, Berlin, 1996). Knowledge of the asymptotic variance-covariance matrix of the method of moments estimators offers a concrete possibility for the use of the classical testing procedures. The resulting asymptotic standard errors are then compared with those proposed in the literature applying different parameter estimates. Applications on simulated data support our results. Finally, we present empirical applications on the daily returns of Euro-US dollar and Yen-US dollar exchange rates.

Appendices

Appendix 1: Computation of $V_{\eta }$

The elements of $\widehat{\eta }$ are the sample mean $\widehat{m}_X$, the sample variance $\widehat{\gamma }_X (0)$ and the sample first-order autocovariance $\widehat{\gamma }_X (1)$ of $x_t = \log y^2_t$. Thus the asymptotic variance-covariance matrix is

$$\begin{aligned} V_{\eta } = \lim _{T \rightarrow + \infty } {\text {var}} [\sqrt{T} (\widehat{\eta } - \eta )] = \sum _{j = - \infty }^{+ \infty } {\text {cov}} (\epsilon _t, \epsilon _{t - j}) \end{aligned}$$

where

$$\begin{aligned} \epsilon _t = \left( x_t^{*} \quad x_t^{* 2} \quad x_t^{*} x^{*}_{t-1} \right) ^{'} \end{aligned}$$

and

$$\begin{aligned} x^{*}_t = x_t - \alpha - \mu (1 - \rho )^{-1} = \log y^2_t - \alpha - \mu (1 - \rho )^{-1}. \end{aligned}$$

Write $x_t^{*}$ as $h_t^{*} + e_t$, where $h^{*}_t = h_t - \mu (1 - \rho )^{-1}$ and $e_t = \log u^2_t - c_1$. Then the transition equation in (2.1) becomes $h_t^{*} = \rho h^{*}_{t - 1} + v_t$. Now $e_t$ and $h^{*}_t$ have zero mean and are independent. Furthermore, we have

$$\begin{aligned} \begin{aligned} {\text {cov}} \left( h^{*}_t , h^{*}_{t-i} \right)&= \frac{\rho ^{|i|}}{1 - \rho ^2 } \sigma ^2_v \\ {\text {cov}} \left( h^{*}_t h^{*}_{t-i} , h^{*}_{t - j} \right)&= 0 \\ {\text {cov}} \left( h^{*}_t h^{*}_{t-i}, h^{*}_{t - j } h^{*}_{t- \ell }\right)&= \frac{\rho ^{|j| + |i - \ell |} + \rho ^{ |\ell | + | i - j |}}{( 1 - \rho ^2 )^2 } \sigma ^4_v \end{aligned} \end{aligned}$$

for any integers i, j and $\ell$. Using these formulae, the elements of $V_{\eta }$ are derived as follows:

$$\begin{aligned}&\begin{aligned} V_{\eta } (1, 1)&= \sum _{j = - \infty }^{+ \infty } {\text {cov}} \left( h_t^{*} + e_t , h^{*}_{ t - j} + e_{ t - j}\right) \\&= \sum _{j = - \infty }^{+ \infty } \left[ {\text {cov}} \left( h_t^{*} , h^{*}_{ t - j} \right) + {\text {cov}} \left( e_t , e_{ t - j} \right) \right] \\&= \frac{1}{(1 - \rho )^2} \sigma ^2_v + c_2 \end{aligned} \\&\begin{aligned} V_{\eta } (2, 2)&= \sum _{j = - \infty }^{+ \infty } {\text {cov}} \left[ \left( h^{*}_t + e_t \right) ^2 , \left( h^{*}_{t - j} + e_{t - j} \right) ^2 \right] \\&= \sum _{j = - \infty }^{+ \infty } \left[ {\text {cov}} \left( h^{* 2}_t , h^{* 2}_{t - j} \right) + 4 {\text {cov}} \left( h^{*}_t e_t , h^{*}_{t - j} e_{t - j}\right) + {\text {cov}} \left( e_t^2 , e_{t - j}^2 \right) \right] \\&= \frac{2 (1 + \rho ^2 )}{(1 - \rho ^2 )^3 } \sigma ^4_v + \frac{4 c_2}{1 - \rho ^2 } \sigma ^2_v + c_4 - c_2^2 \end{aligned}\\&\begin{aligned} V_{\eta } (3, 3)&= \sum _{j = - \infty }^{+ \infty } {\text {cov}} \left[ \left( h^{*}_t + e_t \right) \left( h^{*}_{t - 1} + e_{t-1}\right) , \left( h^{*}_{t - j} + e_{t - j} \right) \left( h^{*}_{t-j-1} + e_{t - j -1}\right) \right] \\&= \sum _{j = - \infty }^{+ \infty } \left[ {\text {cov}} \left( h^{*}_t h^{*}_{t - 1} , h^{*}_{t - j} h^{*}_{t - j - 1} \right) + {\text {cov}} \left( h^{*}_t e_{t - 1} , h^{*}_{t - j} e_{t - j - 1} \right) \right. \\&\qquad \left. + {\text {cov}} \left( h^{*}_t e_{t - 1} , e_{t - j} h^{*}_{t - j - 1} \right) + {\text {cov}} \left( e_t h^{*}_{t - 1} , h^{*}_{t - j} e_{t - j - 1} \right) \right. \\&\qquad \left. + {\text {cov}} \left( e_t h^{*}_{t - 1} , e_{t - j} h^{*}_{t - j - 1} \right) + {\text {cov}} \left( e_t e_{t - 1} , e_{t - j} e_{t - j - 1} \right) \right] \\&= \frac{1 - \rho ^4 + 4 \rho ^2}{ (1 - \rho ^2 )^3} \sigma ^4_v + \frac{2 c_2 (1 + \rho ^2 )}{1 - \rho ^2 } \sigma ^2_v + c_2^2 \end{aligned} \end{aligned}$$

$$\begin{aligned} V_{\eta } (2,1)= & {} \sum _{j = - \infty }^{+ \infty } {\text {cov}} \left[ \left( h^{*}_t + e_t \right) ^2 , h^{*}_{t - j} + e_{t - j} \right] = \sum _{j = - \infty }^{+ \infty } {\text {cov}} \left( e_t^2, e_{t-j}\right) \\= & {} E\left( e^3_t\right) = c_3\\&V_{\eta } (3,1) = \sum _{j = - \infty }^{+ \infty } {\text {cov}} \left[ \left( h^{*}_t + e_t \right) \left( h^{*}_{t-1} + e_{t-1}\right) , h^{*}_{t - j} + e_{t - j} \right] = 0\\ V_{\eta } (3,2)= & {} \sum _{j = - \infty }^{+ \infty } {\text {cov}} \left[ \left( h^{*}_t + e_t \right) \left( h^{*}_{t-1} + e_{t-1} \right) , \left( h^{*}_{t - j} + e_{t - j}\right) ^2 \right] \\= & {} \sum _{j = - \infty }^{+ \infty } \left[ {\text {cov}} \left( h^{*}_t h^{*}_{t-1} , h^{* 2}_{t - j}\right) + 2 {\text {cov}} \left( h^{*}_t e_{t-1} , h^{*}_{t - j} e_{t - j} \right) \right. \\&\left. + 2 {\text {cov}} \left( h^{*}_{t -1} e_t , h^{*}_{t - j} e_{t - j} \right) \right] \\= & {} \sum _{j = - \infty }^{+ \infty } \frac{2 \, \rho ^{|j| + |j-1|}}{(1 - \rho ^2 )^2} \, \sigma ^4_v + 4 \, {\text {cov}} \left( h^{*}_t , h^{*}_{t - j} \right) \, c_2 \\= & {} \left[ \frac{2 \, \rho }{(1 - \rho ^2 )^2} + 2 \sum _{j = 1}^{+ \infty } \, \frac{\rho ^{2 j - 1}}{(1 - \rho ^2 )^2} + 2 \, \sum _{j = - \infty }^{- 1} \, \frac{ \rho ^{- (2 j - 1)}}{(1 - \rho ^2 )^2} \right] \, \sigma ^4_v + \frac{4 \, c_2 \, \rho }{ (1 - \rho ^2)} \, \sigma ^2_v \\= & {} \left[ \frac{2 \, \rho }{(1 - \rho ^2 )^2} + 2 \sum _{i = 0}^{+ \infty } \, \frac{\rho ^{2 i + 1}}{(1 - \rho ^2 )^2} + 2 \, \sum _{i = 0}^{+ \infty } \, \frac{ \rho ^{ 2 i + 3}}{(1 - \rho ^2 )^2} \right] \, \sigma ^4_v + \frac{4 \, c_2 \, \rho }{ (1 - \rho ^2)} \, \sigma ^2_v \\= & {} \left[ \frac{2 \, \rho }{(1 - \rho ^2 )^2} + \frac{2 \, \rho }{(1 - \rho ^2 )^3} + \frac{2 \, \rho ^3 }{(1 - \rho ^2 )^3} \right] \, \sigma ^4_v \, + \frac{4 \, c_2 \, \rho }{ (1 - \rho ^2)} \, \sigma ^2_v \\= & {} \frac{4\, \rho }{(1 - \rho ^2 )^3} \, \sigma ^4_v + \frac{4 \, c_2 \, \rho }{(1 - \rho ^2)} \, \sigma ^2_v \end{aligned}$$

Appendix 2: Proof of Theorem A

Set $\theta = \left( \mu \, \, \rho \, \, \sigma _{v}^2\right) ^{'}$ and $\eta =(m_X \, \, \gamma _{X}(0) \, \, \gamma _{X}(1))^{'}$. Let $\Theta = \mathbb {R} \times (- 1, 1) \times (0, + \infty )$ be the open space of the parameter vector $\theta$. Let $\eta = \eta (\theta )$ be the map from $\Theta$ into $\mathbb {R}^3$ defined by Eqs. (2.3), (2.5) and (2.6). This map is injective and formed by rational functions, hence it is differentiable. The Jacobian matrix $J = \frac{\partial \, \eta }{\partial \, \theta ^{'}}$ has full rank as $\det J = - (1 + \rho ) \, (1 - \rho ^2)^{-3}\, \sigma _{v}^{2} \ne 0$ for $|\rho | < 1$. The inverse map $\theta = \theta (\eta )$ from the image of $\eta$ into $\Theta$ is defined by the equations

$$\begin{aligned} \begin{aligned} \mu =&\frac{\gamma _X (0) - \gamma _X (1) - c_2}{\gamma _X (0) - c_2} (m_X - c_1) \qquad \qquad \rho = \frac{\gamma _X (1)}{\gamma _X (0) - c_2 } \\&\qquad \qquad \sigma ^2_v = \frac{[\gamma _X (0) - c_2 ]^2 - [\gamma _X (1)]^2 }{\gamma _X (0) - c_2}. \end{aligned} \end{aligned}$$

(6.1)

where $\gamma _X (0) \ne c_2 = \sigma _{e}^2$ by (2.5). Let $\widehat{\theta }$ be the MM estimator of $\theta$ in Model (1.1). As T grows larger, $\widehat{\mu }$, $\widehat{\rho }$, and $\widehat{\sigma }^2_v$ converge to their population counterparts (or probability limits). Let ${\widehat{\eta }}_T = ({{\widehat{m}}}_X \, \, {\widehat{\gamma }}_X (0)\, \, {\widehat{\gamma }}_X (1))^{'}$ be the sample estimator of $\eta$. Then ${\widehat{\eta }}_T \rightarrow \eta$ as $T \rightarrow \infty$. Substituting $\widehat{m}_X$, $\widehat{\gamma }_X (0)$ and $\widehat{\gamma }_X (1)$ on the right–hand sides of (6.1) yields a consistent estimator of $\theta$ that is asymptotically equivalent to $\widehat{\theta }$. More precisely, the sequence $\{ {\widehat{\eta }}_T \}$ satisfies $\sqrt{T} ({\widehat{\eta }}_T - \eta ) \rightarrow \mathcal{N}(0, V_{\eta })$ in distribution. Applying the Delta Method gives $\sqrt{T} ({\widehat{\theta }}_T - \theta ) = \sqrt{T} (\theta ({\widehat{\eta }}_T) - \theta (\eta )) \rightarrow \mathcal{N}(0, V_{\theta })$ in distribution, where the asymptotic variance-covariance matrix $V_{\theta }$ is given by $\frac{\partial \, \theta }{\partial \, \eta ^{'}} \, V_{\eta }\, \frac{\partial \, \theta ^{'}}{\partial \, \eta } = J^{-1} \, V_{\eta } \, (J^{'})^{-1}$. This proves the statement.

Appendix 3: Proof of Theorem B

Set $\theta _{*} = (\xi \, \, \beta \, \, \sigma _{w}^2)^{'}$ and $\theta = (\mu \, \, \rho \, \, \sigma _{v}^2)^{'}$. Let $\Theta _{*} = \mathbb {R} \times \{(- 1, 0) \cup (0, 1) \} \times (0, + \infty )$ be the open space of the parameter vector $\theta _{*}$. Let $\theta = \theta (\theta _{*})$ be the map from $\Theta _{*}$ into $\mathbb {R}^3$ defined by the following equations which are derived from Sect. 3:

$$\begin{aligned} \begin{aligned} \mu&= \xi \, - \, \left( 1 + \sigma _{w}^2 \, \sigma _{e}^{- 2}\, \beta \right) \alpha \\ \rho&= - \sigma _{w}^2 \, \sigma _{e}^{- 2}\, \beta \\ \sigma _{v}^2&= (1 + \beta ^2) \, \sigma _{w}^2 \, - \, \left( 1 + \sigma _{w}^4 \, \sigma _{e}^{- 4}\, \beta ^2\right) \, \sigma _{e}^2 \end{aligned} \end{aligned}$$

(6.2)

This map is differentiable. The Jacobian matrix $D = \frac{\partial \, \theta }{\partial \theta _{*}^{'}}$ has full rank as $\det D = \sigma _{w}^2 \, \sigma _{e}^{- 2}\, (\beta ^2-1) \ne 0$ for $|\beta | <1$. The map $\theta = \theta (\theta _{*})$ is injective and the identification problem is completely solved as the transformation $\theta _{*} = \theta _{*} (\theta )$ from the image of $\theta$ into $\Theta _{*}$ is derived as follows. Substituting (3.5) into (3.4) and then multiplying by $\beta$, we get

$$\begin{aligned} \rho \sigma ^2_e \beta ^2 + \left[ \sigma ^2_v + (1 + \rho ^2 ) \sigma ^2_e \right] \beta + \rho \sigma ^2_e = 0 \end{aligned}$$

(6.3)

which gives

$$\begin{aligned} \beta = \frac{- \sigma ^2_v - (1 + \rho ^2 ) \sigma ^2_e + \sqrt{\Delta }}{ 2 \rho \sigma ^2_e } \end{aligned}$$

(6.4)

where

$$\begin{aligned} \Delta = \left[ \sigma ^2_v + (1 + \rho ^2 ) \sigma ^2_e \right] ^2 - 4 \rho ^2 \sigma ^4_e = \left[ \sigma ^2_v + (1 - \rho )^2 \, \sigma ^2_e \right] \, \left[ \sigma ^2_v + (1 + \rho )^2 \, \sigma ^2_e \right] > 0 . \end{aligned}$$

Note that $- \sqrt{\Delta }$ in (6.4) is not acceptable since $| \beta | < 1$ (and $\beta \ne 0$). This condition ensures the invertibility of the process. In particular, if $\rho >0$, then we have $-1< \beta <0$. From (6.4) and (3.5) we obtain

$$\begin{aligned} \sigma ^2_w = \frac{\sigma ^2_v + (1 + \rho ^2 ) \sigma ^2_e + \sqrt{\Delta }}{2}. \end{aligned}$$

(6.5)

Now the equations of the map $\theta _{*} = \theta _{*} (\theta )$ can be easily derived from (3.2), (6.4) and (6.5). From Theorem A we have ${\widehat{\theta }}_T \rightarrow \theta$ as T goes to $\infty$ and $\sqrt{T} ({\widehat{\theta }}_T - \theta ) \rightarrow \mathcal{N}(0, V_{\theta })$ in distribution. Applying the Delta Method gives $\sqrt{T} ({\widehat{\theta }}_{* T} - \theta _{*}) = \sqrt{T} (\theta _{*}({\widehat{\theta }}_T) - \theta _{*}(\theta )) \rightarrow \mathcal{N}(0, V_{\theta _{*}})$ in distribution, where

$$\begin{aligned} V_{\theta _{*}} = \frac{\partial \, \theta _{*}}{\partial \theta ^{'}} \, V_{\theta } \, \frac{\partial \, \theta _{*}^{'}}{\partial \, \theta } = D^{-1} \, V_{\theta } \, (D^{'})^{- 1} = D^{- 1}\, J^{-1} \, V_{\eta } \, (J^{'})^{-1} \, (D^{'})^{- 1}. \end{aligned}$$

This proves the statement.