threshold model with n thresholds

[sedki zn]

Dear all,

i hope you are doing good in this period of Covid.

i would like to ask you about threshold models when we have n threshold.
how do we write the model equation

for n=1 the equation is as follow :

and for n thresholds the following equation is it correct ??

Kind regards
SEDKI

[Clyde Schechter]

Yes, what you show is the correct n-thresholds analog of the 1-threshold model you gave. But there is a slight error in both models. As written, if the value of FC_it is exactly equal to one of the deltas, then it is not treated as part of any interval and, implicitly, the coefficient for such values is indeterminate. So you really should rewrite the models with a <=, rather than <, on one side of each interval. There, of course, may not be any observations in your real data affected by this, so as a practical matter it may not be important, but, in principle, it should be that way.

[sedki zn]

Dear Clyde Schechter,

Thank you very much for confirming and correcting me the models

kind regards
SEDKI

[sedki zn]

Dear Clyde Schechter

i hope you are doing good

please if i would add time fixed effect as well as industry fixed effect, my model becomes as follow? :

Kind regards
SEDKI

[Clyde Schechter]

Well assuming that firms do not change industry from one occasion to another, it would be Industry_i with no t-subscript. And the term for mu also would be mu_i.

[sedki zn]

Well assuming that firms do not change industry from one occasion to another, it would be Industry_i with no t-subscript. And the term for mu also would be mu_i.

thank you dear Clyde Schechter

concerning industry i got it,
concerning mu, i is not for individual effect and t for time effect ? if i would include time fixed effects why should not add the t ?

kind regards

[Clyde Schechter]

No. If you want to have both individual and time effects, they have to be separate. mu_i + nu_t. mu_it is not both individual and time effects: it is an effect for all possible combinations of i and t, and in that regard it is completely indistinguishable from epsilon_it, which you already have in the model. So you can't have both mu_it and epsilon_it in the model--it would be unidentifiable.

[sedki zn]

No. If you want to have both individual and time effects, they have to be separate. mu_i + nu_t. mu_it is not both individual and time effects: it is an effect for all possible combinations of i and t, and in that regard it is completely indistinguishable from epsilon_it, which you already have in the model. So you can't have both mu_it and epsilon_it in the model--it would be unidentifiable.

so the model must be as follow if i take into consideration both time and industry fixed effects :

is it correct ?

i am asking because in my articles they show results with time and industry fixed effects without mentioning them in the model.

[Clyde Schechter]

Well, I wouldn't use the same symbol, mu, for both the industry and time fixed effects. Generally with two fixed effects, they usually call one of them mu and the other alpha (though all that really matters is that the symbols be distinct from any other symbols used in the equation.)

i am asking because in my articles they show results with time and industry fixed effects without mentioning them in the model.

Well, I don't know what literature you are reading and different disciplines and different journals may have different conventions. When you include explicit fixed effects, particularly if there is more than one, you make the model equation longer and harder to read (as well as making it a more complete and accurate description of the model). Some people might feel that if you mention that the model is supposed to also include industry and time fixed effects that you can then omit those from the equation and it will look less intimidating and knowledgeable readers will understand that the mu_i and alpha_t terms are implicitly part of the model even though they aren't mentioned. It's not an unreasonable convention to use if the audience/readership can be counted on to understand it that way.

[sedki zn]

[QUOTE=Clyde Schechter;n1641730]Well, I wouldn't use the same symbol, mu, for both the industry and time fixed effects. Generally with two fixed effects, they usually call one of them mu and the other alpha (though all that really matters is that the symbols be distinct from any other symbols used in the equation.)

thank you very much dear Clyde Schechter


just one more question please when it is writing like this

it means we do not have individual fixed effects ( i mean it is specific effects ?)

[Clyde Schechter]

Correct. That equation impiles that all of the values of CAPEX_i,t are set to the exact same value--nothing specific for either industry or time.

[sedki zn]

Correct. That equation impiles that all of the values of CAPEX_i,t are set to the exact same value--nothing specific for either industry or time.

Thank you very much dear Professor

[sedki zn]

dear Clyde Schechter

i am back to you because i got confused again.
i'm studying an article and the author has given the following model :

it is a specific effect model (as mentioned by alpha 0 ); However in the results he has mentioned both time and industry fixed effects

how could this be possible ?

kind regards

[Clyde Schechter]

Assuming you are correctly representing what the author has done then the model would also include a mu_i term and another term for a year fixed effect as well. Now, as I mentioned in #9, sometimes people omit those terms from the models and just state elsewhere that those additional effects were included. This makes the model equation shorter and easier to read. While I personally think this is bad communication, if writing for an audience that is accustomed to this convention, it does no harm. It does, however, do harm, when the paper is given to readers like you who are not accustomed to the convention and are getting confused by it.

[sedki zn]

Assuming you are correctly representing what the author has done then the model would also include a mu_i term and another term for a year fixed effect as well. Now, as I mentioned in #9, sometimes people omit those terms from the models and just state elsewhere that those additional effects were included. This makes the model equation shorter and easier to read. While I personally think this is bad communication, if writing for an audience that is accustomed to this convention, it does no harm. It does, however, do harm, when the paper is given to readers like you who are not accustomed to the convention and are getting confused by it.

yes thank you professor

but instead of alpha 0 it should be Alpha i ( individual effects). yes ?
( as i know we can not include time fixed effects or industry fixed effects with alpha 0 ( specific effects) in the same model. please do correct me if i am wrong )