Internal Market Efficiency, Market Co-movement, and Cross-Market Efficiency: The Case of Hong Kong and Shanghai Stock Markets

Abstract

We examine the effects of the 2008 financial crisis on the cross-market efficiency of the Hong Kong and Shanghai stock markets. Our results show a sharp decline in the cross-market efficiency during the financial crisis. We investigate whether this is due to lower internal market efficiency or higher market co-movement. The results show no evidence that the internal market efficiency dropped in Hong Kong or Shanghai during the crisis. In contrast, we document a strong increase in the market co-movement during the crisis. These results suggest that the decline in cross-market efficiency during the financial crisis is due to increased market co-movement and not a decline in internal market efficiency.

Appendix A

The wavelet transformation function is described by the following equation:

$$\begin{aligned} W\left( {\tau ,a} \right) =\int \mathop \int \nolimits _{+\infty }^{-\infty } f\left( t \right) \psi _{\tau ,a} \left( t \right) dt, \end{aligned}$$

where $W\left( {\tau ,a} \right)$ is the wavelet transform, $f\left( t \right)$ is the inner product of the function, and $\psi _{\tau ,a} \left( t \right)$ is the analyzing wavelet. The family of wavelet vectors is obtained by translation $\tau$ and dilatation a:

$$\begin{aligned} \psi _{\tau ,a} \left( t \right) =\frac{1}{\sqrt{a}}\psi _{\tau ,a} \left( {\frac{t-\tau }{a}} \right) . \end{aligned}$$

Energy normalization across the different scales is achieved with the factor$a^{-1/2}$. A common way to build wavelets of order n is to successively differentiate a smoothing function. A popular family of wavelets uses the Gaussian function:

$$\begin{aligned} g_n \left( t \right) =g^{\left( n \right) }\left( t \right) =\frac{d^{n}}{dt^{n}}e^{-\frac{t^{2}}{2}} \end{aligned}$$

The advantage of the Gaussian wavelet is that it can be shown that the analyzing wavelet $\psi$ is the Nth derivative of the Gaussian function, where the order of the differentiation is related to the scale a : 

$$\begin{aligned} \psi ^{\left( N \right) }\left( x \right) =\frac{d^{N}}{dx^{N}}e^{-\frac{x^{2}}{2}}. \end{aligned}$$

The wavelet transform is a well-adapted tool for studying scaling processes. The wavelet transform of a stochastic process with long-range dependence properties or self-similar processes satisfies the following: the wavelet coefficients at a fixed scale are stationary; the wavelet coefficients have short-range correlation. Therefore, the greater the number of wavelet vanishing moments, the shorter the correlations. Thus, when the number of vanishing moments is large enough, the wavelet coefficients are uncorrelated, and the $q-$order moment $M_q \left( a \right)$ of the wavelet coefficients at scale a reproduces the scaling property of the process.