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

<p>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. <span class="math-container">$E(p_t| p_{t-14}, p_{t-28}, \dots) = E(p_t)$</span>. However, this assumption doesn't guarantee there in no serial correlation between <span class="math-container">$p_t$</span> and <span class="math-container">$p_{t-1}$</span>.</p>

<p>If the case was with <span class="math-container">$t= \{0, 1, 2, 3, 4, \dots ,365\}$</span>, 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?</p>

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

<p><span class="math-container">$$V_t = \beta_0 + \beta_1 \ |\Delta p_t| + \beta_2 \ d\_event_t + \beta_3 \ d\_event_t \cdot event_t + u_t,$$</span> </p>

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

CC BY-SA 4.0
2019-06-28T05:57:57.653Z
Hi: are you saying that $V_t$ is the "later" volatility in the current week ? so, the volatility IMMEDIATELY AFTER the price difference on the left hand side  ? Thanks.
mark leeds
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2019-06-28T14:52:28.803Z
That's correct.
user3889486
CC BY-SA 4.0
2019-06-28T14:55:41.767Z
Not clear on what you think is auto-correlated. you're taking the difference between $p_t$ and $p_{t-1}$ to calculate the dependent variable ( note that if $p_{t}$ is log price then the difference is the same as one day return ) so what is possibly auto-correlated ?
mark leeds
CC BY-SA 4.0
2019-06-28T15:27:44.603Z
In the RHS, $\Delta p_t$ is a function of $p_t$, and, in the LHS, $V_t$ is a function of $p_{t+1}$. So my doubt is that I might have a concealed dynamic model that is a source of autocorrelation.
user3889486
CC BY-SA 4.0
2019-06-28T15:41:45.793Z
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
CC BY-SA 4.0
2019-06-28T16:31:39.720Z
@AndrewM When you mentioned run the test daily, are you referring to the model via something like The Durbins h test statistic? If I test the full model, I can only run them biweekly as the space between events is two weeks.
I can only test daily if I remove the event out of the equation. In this case, I would be testing an incomplete model. Not to mention that now exogeneity is violated since the dependent and independent variables overlap in time.
user3889486
CC BY-SA 4.0
2019-06-28T17:32:55.447Z
I meant to simulate some data and see how the test works on simulated data when you go from daily to biweekly data even though have daily lags/leads. Then you can see if the test is valid for your actual data...
Andrew M
CC BY-SA 4.0
2019-06-28T23:23:22.987Z
@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
CC BY-SA 4.0
2019-06-28T23:30:23.253Z
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
CC BY-SA 4.0
2019-06-29T01:32:40.233Z
@markleeds Thanks for pointing out the dummy variable! This makes much more sense.
I can consider the the event is 0 when it doesn't occur and its value when it does. I'm adding the new model considering your suggestion.
user3889486
CC BY-SA 4.0
2019-06-29T03:34:47.623Z
Just as a general piece of advice, before you change the model, I would look at the simpler model where you don't even include event in the model. This way, you can see if reasonable things are going on. For example, if you expect coefficient on price change to be positive, is it ? Is it effected by the time window you consider ? Doing regressions ( particularly in finance-econometrics and dynamic ) is tricky business so it's best to start out simpler and work  towards something more complex. I'm glad that dummy variable suggestion may have helped. It was really Andew's suggestion anyway.
mark leeds
CC BY-SA 4.0
2019-06-29T07:43:35.400Z
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
CC BY-SA 4.0
2019-06-29T07:47:41.730Z
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. http://www-personal.umich.edu/~franzese/DeBoefKeele.2008.TakingTimeSeriously.pdf
mark leeds
CC BY-SA 4.0
2019-06-29T15:17:33.470Z
Last things ( two ): 1) Note that, in the paper, they use the term "partial adjustment" for the koyck disributed lag. 2) As far as a textbook for this type of "lag" material, I recommend Andrew Harvey's "The Economic Analysis of Time Series". He has a terse writing style which means that each page takes time but that book is really great.
mark leeds
CC BY-SA 4.0
2019-06-29T17:00:48.190Z
@markleeds Thank you, I'll take a look at them!
user3889486
CC BY-SA 4.0
2019-06-30T22:27:40.740Z
good luck with it. sounds like an interesting econometric problem.
mark leeds
<econometrics>
Dynamic model and serial correlation
user3889486
# Dynamic model and serial correlation

(Asked by: username_0 on 2019-06-28)

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.

**Tags:** econometrics
> **Comments:**
> **username_1**: Hi: are you saying that $V_t$ is the "later" volatility in the current week ? so, the volatility IMMEDIATELY AFTER the price difference on the left hand side  ? Thanks.  
> **username_0**: That's correct.  
> **username_1**: Not clear on what you think is auto-correlated. you're taking the difference between $p_t$ and $p_{t-1}$ to calculate the dependent variable ( note that if $p_{t}$ is log price then the difference is the same as one day return ) so what is possibly auto-correlated ?  
> **username_0**: In the RHS, $\Delta p_t$ is a function of $p_t$, and, in the LHS, $V_t$ is a function of $p_{t+1}$. So my doubt is that I might have a concealed dynamic model that is a source of autocorrelation.  
> **username_2**: 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?  
> **username_0**: @AndrewM When you mentioned run the test daily, are you referring to the model via something like The Durbins h test statistic? If I test the full model, I can only run them biweekly as the space between events is two weeks. I can only test daily if I remove the event out of the equation. In this case, I would be testing an incomplete model. Not to mention that now exogeneity is violated since the dependent and independent variables overlap in time.  
> **username_2**: I meant to simulate some data and see how the test works on simulated data when you go from daily to biweekly data even though have daily lags/leads. Then you can see if the test is valid for your actual data...  
> **username_1**: @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.  
> **username_1**: 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.  
> **username_0**: @markleeds Thanks for pointing out the dummy variable! This makes much more sense. I can consider the the event is 0 when it doesn't occur and its value when it does. I'm adding the new model considering your suggestion.  
> **username_1**: Just as a general piece of advice, before you change the model, I would look at the simpler model where you don't even include event in the model. This way, you can see if reasonable things are going on. For example, if you expect coefficient on price change to be positive, is it ? Is it effected by the time window you consider ? Doing regressions ( particularly in finance-econometrics and dynamic ) is tricky business so it's best to start out simpler and work  towards something more complex. I'm glad that dummy variable suggestion may have helped. It was really Andew's suggestion anyway.  
> **username_1**: 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 $. )  
> **username_1**: 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. http://www-personal.umich.edu/~franzese/DeBoefKeele.2008.TakingTimeSeriously.pdf  
> **username_1**: Last things ( two ): 1) Note that, in the paper, they use the term "partial adjustment" for the koyck disributed lag. 2) As far as a textbook for this type of "lag" material, I recommend Andrew Harvey's "The Economic Analysis of Time Series". He has a terse writing style which means that each page takes time but that book is really great.  
> **username_0**: @markleeds Thank you, I'll take a look at them!  
> **username_1**: good luck with it. sounds like an interesting econometric problem.  

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## 0 answers

