[Joao Santos Silva]

Dear Javier Rua Montes,

That is an excellent question. Loosely speaking, as you increase the number of powers in the RESET test, you broaden the range of alternatives against which the test has power, but reduce the power of the test against some alternatives. So, it is not surprising to see that the model passes the test when you add the cubes. The question is what to do in situations like that. There is not much you can do to change the specification of the model, but you can try to add regressors and possibly more data (gravity models often fail the RESET when we use just a subset of countries).

Best wishes,

Joao

[Eric de Souza]

@João Santos Silva: interesting coincidence. Just yesterday I was trying to find some theoretical literature on the RESET precisely in connection with the number of powers and how it affects the test. Do you have any reference by chance? Thanks

[Joao Santos Silva]

Dear Eric de Souza,

I do not have a reference, but the idea is as follows. The power of the test depends on the non-centrality parameter and the critical value. As we add more powers the critical value goes up because of the number of degrees of freedom increases; however the non-centrality parameter may not increase enough to offset the higher critical value. So, there is a number of powers that maximizes the power against some alternative, going beyond that number of powers reduces the power of the test.

Best wishes,

Joao

[Eric de Souza]

Thanks, João.

[Jeff Wooldridge]

To add to João's very helpful response, you can think of an easier situation. Suppose the true model is E(y|x) = b₀ + b₁*x + b₂*x². The most powerful test for the null of a linear conditional mean is to simply add x². If you add x² and x³ then the test will still have pretty good power, but it will have less power than the most powerful test because it includes the cubed term. Of course, if the true mean is more complicated than the quadratic, adding the cube is often a good idea.

[Joao Santos Silva]

Jeff's nice example reminded me of something else: in the case of the probit it is always good to include squares and cubes as the two terms have a clear interpretation: the significance of the quadratic term is an indication of skewness and the significance of the cubic term is an indication of non-normal kurtosis. In general, in binary models, I find it useful to include both terms; I do not think I have ever found an example where including more terms (as in Stata's command) is useful.

[Eric de Souza]

@Jeff and @ João: your comments are very helpful. Thanks.

[Eric de Souza]

On second thoughts, wouldn't the cube be an indication of skewness and the fourth power an indication of excess kurtosis?

[Joao Santos Silva]

Eric de Souza,

Nope, what I said is correct :-)

Best wishes,

Joao

[Eric de Souza]

OK. Will work through it .

[Edward Gardner]

Jeff's nice example reminded me of something else: in the case of the probit it is always good to include squares and cubes as the two terms have a clear interpretation: the significance of the quadratic term is an indication of skewness and the significance of the cubic term is an indication of non-normal kurtosis. In general, in binary models, I find it useful to include both terms; I do not think I have ever found an example where including more terms (as in Stata's command) is useful.

Hi Joao. Would you happen to know a reference where I can do further reading about this?

[Joao Santos Silva]

Dear Edward Gardner,

I know about that result from a very old paper (mid/late 80s?) by Chris Orme; I believe I have a copy of it in my office. However, the same result can probably be found in Paul Ruud's PhD thesis and is referred in

Newey, W.K. (1985), "Maximum Likelihood Specification Testing and Conditional Moment Tests," Econometrica, Vol. 53, No. 5, pp. 1047-1070.

Best wishes,

Joao

[Edward Gardner]

Dear Edward Gardner,

I know about that result from a very old paper (mid/late 80s?) by Chris Orme; I believe I have a copy of it in my office. However, the same result can probably be found in Paul Ruud's PhD thesis and is referred in

Newey, W.K. (1985), "Maximum Likelihood Specification Testing and Conditional Moment Tests," Econometrica, Vol. 53, No. 5, pp. 1047-1070.

Best wishes,

Joao

Thank you very much Joao, your response is greatly appreciated. Best wishes, Edward.