I am using tick data of implied volatility and their underlying asset price and extract 2 data sets of hourly frequency and daily frequency. The 2 data sets are formed from the same tick data. I run OLS regression for each data set.
$$Y_{t,hourly} = \beta_1 X_{hourly} + \epsilon_t$$
$$Y_{t,daily} = \beta_2 X_{daily} + \epsilon_t$$
$Y$ is change in implied volatility and $X$ is log return of underlying asset.
Please suggest how to test the equality. The data is fundamentally dependent
so the $Z$ test, $$Z = \frac{\beta_1 – \beta_2}{\sqrt{}{SE_1^2 +SE_2^2}} $$ cannot be used as it assumes data to be independent. $SE_1$ is the standard error of the respective coefficient.
Thank you.
Best Answer
If you have enough data to assess daily vs. hourly rates, I am going to assume you have enough data to split the data in half. Because linear regression assumes a constant slope (which I assume you believe to be true), then you should obtain similar slopes for different regions of the line (top or bottom) or for different steps along the line. Of course, this introduces another source of chance variability, but as a "quick" solution, this may be reasonable.
Step 1: break the data into two sets: even days and odd days
Step 2: calculate the regression coefficients for hour from one set & day from the other
Step 3: use the 2nd formula you've presented above
This will (1) introduce more variability because you are using less data to make your inferences, but (2) it will break the inter-dependency violation.