Probit Regression [Parameter Stability]

[harshil59]

In short, I'm running a probit regression on whether a team was given a penalty kick decision and using it to investigate whether social pressures such as the audience have an effect on referee's in the awarding of penalty kicks.

I want to test for parameter stability since the I've collected data for 3 seasons worth and want to know whether I need to run the regression for the whole sampel or based on the fact of parameter instability, include a regression for each season.

From, what I've researched with probit regressions, the simple chow test cannot be used (still not 100% sure why).

My question is then how could I test for parameter stability?

My supervisor for my work told me I needed to run 4 regressions (one for the first season, one for the second, one for the third, and one using data for all seasons) and then decide. But I'm not sure how this would work either.

Any help is very much appreciated, I've been stuck for days now and would be forever in debt if anyone could help me on this. If you need more information, about the problem, feel free to ask.

Thanks, in advance!

[trubador]

You cannot use Chow test here, because the result of F-test is no longer valid in this context. What your supervisor suggests is called Likelihood Ratio (LR) test. In the first step, you fit the restricted model to data from all three seasons and get the log likelihood (e.g. Rlogl). Next, estimate the model separately for each season and compute the log likelihood for the unrestricted model by summing the log likelihoods of three seasons (i.e. Ulogl = logl1 + logl2 + logl3). You can then carry out the LR test in EViews as follows:

Code: Select all

scalar dof = (3-1)*number of variables including constant scalar lr = 2*(ulogl-rlogl) scalar lrprob = 1-@cchisq(lr, dof)

[harshil59]

You cannot use Chow test here, because the result of F-test is no longer valid in this context. What your supervisor suggests is called Likelihood Ratio (LR) test. In the first step, you fit the restricted model to data from all three seasons and get the log likelihood (e.g. Rlogl). Next, estimate the model separately for each season and compute the log likelihood for the unrestricted model by summing the log likelihoods of three seasons (i.e. Ulogl = logl1 + logl2 + logl3). You can then carry out the LR test in EViews as follows:

Code: Select all

scalar dof = (3-1)*number of variables including constant scalar lr = 2*(ulogl-rlogl) scalar lrprob = 1-@cchisq(lr, dof)

How would the value be interupted? (I got a value of 0.106121965091932 for the lrprob)

[RLOG=-436.7475]
ULOG= (-128.0900-148.9918-135.0864)= (-412.1682)

Edit- did a bit of reading online and found the log likelhihood follows approxitmiately a chi squared distribution. Where the degrees of freedom are extra number of variables included in the Rlog model (all three seasons). In this case it is 2 for me, and this means the critical value is 0.010. Given the value of 0.106 > 0.010 we can conclude that the three seasons should be modelled seperetely? (I have no idea if what I said is correct-so my apologies if I'm talking utter nonsense)

[trubador]

No, that is valid for a single model where you restrict the coefficients of variables of interest. In your case, you have 3 models with m variables. So, as I clearly indicated in the code, the degrees of freedom should be (3-1)*m. You already compute the probability, so you do not need the critical value. This is your significance level, and now it is up to you (and you only) to decide whether to model them jointly or separately. If you like, you can do some reading on Hypothesis Testing.

[harshil59]

No, that is valid for a single model where you restrict the coefficients of variables of interest. In your case, you have 3 models with m variables. So, as I clearly indicated in the code, the degrees of freedom should be (3-1)*m. You already compute the probability, so you do not need the critical value. This is your significance level, and now it is up to you (and you only) to decide whether to model them jointly or separately. If you like, you can do some reading on Hypothesis Testing.

Thank you for taking time out to reply. I understand Hypothesis testing but I just want to know what the value I've calculated is refering to. Is it Akin to a p value and if so what is the null in this case?

[harshil59]

Bump, still can't find the Null of a LR test in my case...

[trubador]

There is nothing specific to your case in this context.

Yes, this is a simple hypothesis testing and the calculated statistic has a p-value, which is comparable to that of your choice of significance level.

Null hypothesis is that there is no significant difference between the restricted and unrestricted models. If you cannot reject the null, it means that you can pool the data and build a single model. Otherwise (i.e. if you reject the null), it means that seasons have different characteristics, so you should model them separately to estimate the parameters of interest.

[harshil59]

Thank you ever so much for replying that clears things up but just one thing. Shouldn't the restricted model be when we model each season sepretely and then add up the likilhoods (this is because in these models we are imposing that the season dummy variables are=0). From this, the unrestricted model is the data set that runs a single model with data from all three season.


From your post earlier, you stated that it's the opposite way around where the restricted is data from all three seasons and the unrestricted is found by summing each likelihood from each season. This appears to be the inverse of what I'm saying above.

Thanks so much again, once I have this piece of information I can practically finish my project!!!

[trubador]

No, the restricted model is the one with pooled data since you reduce the number of coefficients in order to represent each season with the same parameter values.

[harshil59]

FIRSTLY- thank you for replying I promise I'll try not to bother you, but this actually the last thing I need help with. Your reply and help is very useful and appreciated!!!

Forigive me for my stupidity but is what you are trying to say we have for example 20 variables for the pooled regression (see img) and for the seperate regressions we have 18 variables (since we don't include the season dummy's) but we have 3 different regressions for each variable. Therefore we effectively have 54 variables (18*3 seasons) to represent the variables over the course of the same period whereas in the pooled regression we have less coefficients (20) to represent the same data span?

http://puu.sh/i2pSg/259d9f7a4a.png - Pooled regression

http://puu.sh/i2pTt/13df54e762.png- Season 1 (for example regression) -

[trubador]

No, I am not saying that. Those are different than that of the ones I mentioned, because seasonal dummies capture only the impact on the constant. You should include all crossproducts in order to model the changes in other coefficients as well.

[harshil59]

Okay so what you're saying is the pooled regression is the restricted one because we re imposing the restriction that the coefficient for each variable is the same across each season. Whereas, in the unresticted one it is because we are allowing the parameters to vary across the seasons?

[harshil59]

On more thing trubador (sorry to rebump old thread):


My restricted model which has data from all three seasons has 16 parameters whereas each season modelled separately has 14 (since we drop the 2 season dummies when modelling each season separately).

Therefore on the first line of code ( scalar dof = (3-1)*number of variables including constant)

Would the number of variables including the constant be under the restricted or unrestricted model i.e 16 or 14?


Thanks!