Feeds:
Posts

## USING AUTOCORRELATION FOR AUTOREGRESSIVE COEFFICIENTS WITH EXOGENOUS MARKET VOLATILITY

We are interested in models of the form
$v^{individual stock}_t = \sum_{j=1}^N a_j v^{individual stock}_{t-j} + \sum_{j=1}^N b_j v^{aggregate volatility}_{t-j} + e_t$

So following standard procedure (see this note for example) we multiply this equation with $v_{t-d}$ take expectations and divide by the variance of $v_t$ which we assume constant assuming $v_t$ is stationary.

Let $x_t = v^{individual stock}_t$ and $z_t =v^{aggregate volatility}_t$, and $\sigma_x = E[x_t^2]$ and $\sigma_z = E[z_t^2]$.  Let $A = toeplitz( 1,r_1,\dots,r_{N-1} )$ and $B=toeplitz( s_1,\dots,s_N)$ where $s_k = corr(z_t,x_{t-k})$. We have to solve:

$[A ,\frac{\sigma_z}{\sigma_x} B] (a_1,\dots,a_N,b_1,\dots,b_N)^T = (r_1,\dots,r_N)^T$

So besides the autocorrelation of $z_t$ we need these cross correlations $s_k = corr(z_t,x_{t-k})$.