Try to reproduce Robert E. McCulloch and Ruey S. Tsay’s paper Nonlinearity in High-Frequency Financial Data and Hierarchical Models with local market data.
the paper uses GLM to model high-frequency data, $D_i, N_i, \Delta t_i$, and $F_{i-1}$.
$D_i$ indicates the up jump or down jump of price movement. $N_i$ is the number of trade happened during the $\Delta t_i$ period. $\Delta t_i$ it the duration of the period in seconds. $F_{i-1}$ is the information in the $t_{i-1}$ time period. The model of $D_i$ is: $$D_i \vert(N_i, \Delta t_i, F_{i-1})=sign(\mu_i+\sigma_i \epsilon) $$ where $\epsilon \thicksim N(0,1)$, $$\mu_i=\omega_0+\omega_1 D_{i-1}+\omega_2ln(\Delta t_i)$$ $$ln(\sigma_i)=\beta^* \vert \sum_{j=1}^4 D_{i-j} \vert$$
$D_i$ could be either -1 or 1 for price up-jump or down-jump. From the model definition, it is clear that: $$ P(D_i=-1 \vert N_i, \Delta t_i, F_{i-1})=\Phi({- \mu_i \over \sigma_i})$$ $$ P(D_i=1 \vert N_i, \Delta t_i, F_{i-1})=1-\Phi({- \mu_i \over \sigma_i})$$ $$ E(D_i\vert N_i, \Delta t_i, F_{i-1})=1-2\Phi({- \mu_i \over \sigma_i})$$
if we map -1,1 to 1,0
$$ E(D_i\vert N_i, \Delta t_i, F_{i-1})=\Phi({- \mu_i \over \sigma_i})$$
Here $D_i$ follows a 0-1 distribution, definitely a distribution of exponential family. Intuitively the link function $g^{-1}(X^T \beta)$ is the CDF of normal distribution, the probit link function.
Here $$X=[1, D_i, ln(\Delta t_i), \vert \sum_{j=1}^4 D_{i-j} \vert] $$ The perameters to estimate is: $$\beta=[\omega_0, \omega_1, \omega_2, \beta^*] $$
My questions are:
The GLM need $X^t\beta$ in the link function to be a linear form, but ${- \mu_i \over \sigma_i}$ is not. Should I choose another link function or should I reform the $ {- \mu_i \over \sigma_i}$ (since the model proposed $ln(\sigma_i)$ form)?
For the computational reason, I mapped $D_i$ from -1,1 to 1,0. Do I need to map regressors as well? I guess not. Cause if I do, i. the calculation of $ \vert \sum_{j=1}^4 D_{i-j} \vert$will be different. ii. I am just using a different tag for the same 'truth'.
I have spent days searching and scratching this problem. I presume it could be a quite common modeling procedure since people normally want to model expectation and variance. It will be very appreciated if you may kindly provide me some guide.