# Testing the equality of two regression coefficients from same data but different frequency

I am using tick data of implied volatility and their underlying asset price and extract two data sets of hourly frequency and daily frequency. The two 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 the underlying asset. Please suggest how to test $$H_0: \beta_1=\beta_2$$.

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. $$SE_j$$ is the standard error of the respective coefficient.

• Can you explain a bit more what process it is that makes the two data sets dependent. Is the hourly data contained inside the daily data? Are $\beta_1$ and $\beta_2$ fitted parameters for different sampling methods from the same stream of data or for two different streams of data? What are you trying to achieve in comparing $\beta_1$ and $\beta_2$? Commented Aug 24, 2018 at 13:54
• Have you considered SUR? Or are your SEs too complicated for that be feasible? Commented Jul 31, 2020 at 16:35
• Isn't it a mathematical necessity that $\beta_1=\beta_2$? That would follow from the impressions you leave that each of these datasets is a differenced value of a common time series, only with different lags. I guess I'm assuming there's no daily seasonal component to either series, which leaves me wondering what assumptions we should be making.
– whuber
Commented Nov 21, 2023 at 17:00

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.

• Thanks for the reply. I did that but the model shows some dependency on data. For one set, I get the coefficients equal at 5% level, but if I swap the data sets, they are not. Can it be concluded that they are not equal? Commented Aug 28, 2018 at 6:47
• let me make sure I understand...if you run the hour model with data_set_1 and the day model with data_set_2, you get one result, and if you switch the data sets, you get a different result (based on the p-value dichotomy)...¿correct? Commented Aug 28, 2018 at 13:30
• one other thing...¿what would the units of the slope coefficient be? Commented Aug 28, 2018 at 13:31
• Thanks again for reply. Yes, you got it right. I use log difference for both returns and implied volatility, so the slope will have no units. Commented Aug 29, 2018 at 8:34
• For the inconsistent findings, my suggestion would be to go the "voice of the crowd" method...try other random-half splits (I'd recommend by whole days). See how often the test statistics you calculate agree and disagree. Then, the overall finding should suggest a reasonable interpretation. Commented Aug 29, 2018 at 14:12