How does Stata -predict- after logit?

[Karl Yesler]

Hi Statalist,

After running logit, how does stata predict the probability of outcome? More importantly and specifically, how do I reproduce the results manually? Here is an example using -predict- and using my attempt at manual calculation (which is somehow wrong?) produces 2 different results. I tried manual calculation after a linear regression (eg. substituting -reg- for -logit- here) and the results of -predict- and manual calculation are the same. What am I doing wrong with my manual calculation after -logit-?

**Load example dataset
sysuse auto, clear

**Regression
qui: logit foreign price

**Stata predicted probability
predict yhat

**Calc predicted probability manually
gen yhat2 = (price*_b[price] + _b[_cons])

sum yhat yhat2

Ultimately, I'm trying to calculate an elasticity manually and the results after running -logit- are not the same compared with using -margins, eyex- Again, the code below works for a linear regression but not logit. What am I missing? Do I need to somehow take into account the logit link? If so how?

///First by automated method (eyex)
**Load example dataset
sysuse auto, clear

**Regression
qui: logit foreign price

**Predict elasticities
margins, eyex(price) nose

///Second by manual method

**Load example dataset
sysuse auto, clear

**Regression
qui: logit foreign price

**Calc eyex
*Denominator
gen denom = (price*_b[price] + _b[_cons])

*Numerator
gen numer = (price*_b[price])


*Eyex
gen price_eyex = (numer/denom)

sum price_eyex

Thanks!

[Nick Cox]

Yes; you need to take account of the link. Otherwise logit would just be regression and predictions outside [0,1] would be possible. Documented, e.g. at http://www.stata.com/manuals14/rlogit.pdf

. sysuse auto, clear
(1978 Automobile Data)

. logit foreign price

Iteration 0:   log likelihood =  -45.03321
Iteration 1:   log likelihood = -44.947363
Iteration 2:   log likelihood =  -44.94724

Logistic regression                               Number of obs   =         74
                                                  LR chi2(1)      =       0.17
                                                  Prob > chi2     =     0.6784
Log likelihood =  -44.94724                       Pseudo R2       =     0.0019

------------------------------------------------------------------------------
     foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       price |   .0000353   .0000844     0.42   0.676    -.0001301    .0002006
       _cons |  -1.079792    .587834    -1.84   0.066    -2.231926    .0723411
------------------------------------------------------------------------------

. predict yhat
(option pr assumed; Pr(foreign))

. gen yhat2 = invlogit(_b[_cons] + _b[price] * price)

. su yhat*

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
        yhat |        74    .2972973    .0224245   .2761411   .3731745
       yhat2 |        74    .2972973    .0224245   .2761411   .3731745

.

[Karl Yesler]

Nick, thanks for your helpful comment. 2 follow up questions to get me where I am ultimately trying to go:

1) To calculate the -margins, eyex(price)- manually, the following code does not seem to produce the same results, even after accounting for the link. Again, the code works for a linear regression but not for the logit. There must be something I am misunderstanding?

///First by automated method (eyex)
**Load example dataset
sysuse auto, clear

**Regression
qui: logit foreign price 

**Predict elasticities
margins, eyex(price) nose

///Second by manual method

**Load example dataset
sysuse auto, clear

**Regression
qui: logit foreign price 

**Calc eyex
*Denominator
gen denom = invlogit(price*_b[price] + _b[_cons]) 

*Numerator
gen numer = invlogit(price*_b[price])


*Eyex
gen price_eyex = (numer/denom)

sum price_eyex

Secondly, aside from http://www.stata.com/manuals13/dfunctions.pdf, is there a resource which would even more clearly describe how I would take the link into account in the predictions using other link and distributions? For example log poisson, or probit?

///First by automated method (eyex)
sysuse auto, clear

glm rep78 price, family(poisson) link(log)

**Predict elasticities
margins, eyex(price) nose


///Second by manual method
sysuse auto, clear

glm rep78 price, family(poisson) link(log)


*Denominator
gen denom = ?????(price*_b[price] + _b[_cons]) 

*Numerator
gen numer = ?????(price*_b[price])


*Eyex
gen price_eyex = (numer/denom)

sum price_eyex

Many Thanks!