# Dynamic model and serial correlation

I'm interested in estimating the effect of the $$event_t$$ variable in the price volatility of a financial instrument with the following model $$V_t = \beta_0 + \beta_1 \ |\Delta p_t| + \beta_2 \ event_t + u_t$$ where $$t = \{0, 14, 28, 42, 70, \dots ,364\}$$ represents the first day of every odd week; $$\Delta p_t = p_t - p_{t-1}$$ is the price difference between the current day and the day (not two weeks) before; $$V_t = \frac{1}{5}\sqrt{\sum_{i=t+2}^{t + 6} \Delta p_i^2}$$ is the volatility in the days (not weeks) after the event; and $$u_t$$ is the error term.

I can safely assume that the market is efficient in the sense that two-week-old prices don't help to predict the price of the current week. i.e. $$E(p_t| p_{t-14}, p_{t-28}, \dots) = E(p_t)$$. However, this assumption doesn't guarantee there in no serial correlation between $$p_t$$ and $$p_{t-1}$$.

If the case was with $$t= \{0, 1, 2, 3, 4, \dots ,365\}$$, then test for serial correlation would make sense to decide if I estimate the model using methods such as OLS or Cochrane-Orcutt. Since I'm taking a biweekly series with the dependent variable lagged by a day, I'm not sure if I should test for serial correlation. And if so, how?

Edit Following @markleeds advise, a more sensible model is do a daily regression and add a dummy variable when the event occurs:

$$V_t = \beta_0 + \beta_1 \ |\Delta p_t| + \beta_2 \ d\_event_t + \beta_3 \ d\_event_t \cdot event_t + u_t,$$

with $$t = \{1, 2, 3, \dots ,365\}$$, $$d\_event_t$$ is a dummy that takes one when the event occurs and zero otherwise; $$event_t$$ is a continuous variable; $$\Delta p_t = p_t - p_{t-1}$$ is the price difference between the current day and the day before; $$V_t = \frac{1}{5}\sqrt{\sum_{i=t+2}^{t + 6} \Delta p_i^2}$$ is the volatility; and $$u_t$$ is the error term.

• Its a little weird that you are using leads rather than lags but that shouldn't be a technical issue for a test of autocorrelation. You rightfully point out that the biweekly spacing is uncommon. I would suggest setting up a simulation for the test where you first measure the data daily and run the test and then see how it does when you remove until you get to 2 week, and see how that affects the test results. My sense is that if the model is correct then this shouldn't be a big issue... but its a big assumption that the model is correct. Might other day lags matter as well? – Andrew M Jun 28 at 15:41
• @user3889486: I'm still not so clear (sorry for confusion ) but it seems to me like you are trying to account for the fact that the $V_t$ is effected by changes in price ( in addition to the event ) so don't you want correlation between the two variables, $V_{t}$ and $\triangle p_{t}$ ? Yes, differences of log prices ( returns ) tend to be auto-correlated ( depends on what horizon ) but since you're modelling that directly, I'm still not sure what's auto-correlated ? Thanks. – mark leeds Jun 28 at 23:23
• Also, in regard to Andrew's suggestion, I think he makes a good point. You could include all of the data along with an indicator variable denoting when the event occurs and when it doesn't. Note that if you just have data where the event occurs, then there's no baseline from which to detect the effect. I don't have time to look now but there's probably a regression example on the net where a dummy variable is used in a regression setting in order to test for the effect of some legislation say ( or whatever ) etc so you could use that sort of technique and include Andrew's suggestion. – mark leeds Jun 28 at 23:30
• One more thing, since it's good to get other perspectives. Your should take a look the koyck distributed lag which is dynamic andl used to capture when an effect is over time rather than all at once. There is so much literature on koyck that I'm not sure what to link to. Let me look at what I have and then send a couple of suggestions. Also, note that, as it stands, you're model really isn't dynamic in the sense that there's no feedback into the next period. ( by Koyck DL , I'm thinking of something along the lines of $V_t = \rho *V_{t-1} + \beta \times \triangle p_{t} + \epsilon_t$. ) – mark leeds Jun 29 at 7:43
• There's about thirty others that I have in my folder that are also okay but I think that this one provides a nice balance of theory with examples. www-personal.umich.edu/~franzese/… – mark leeds Jun 29 at 7:47