How can I get theta for Inverse hyperbolic sine transformation by mle?

[Bing Zhang]

Hi,everyone
I want to use mle to estimate theta for Inverse hyperbolic sine transformation , I have written the likelihood function as follows:

Because the y in myresearch is profit ,and it has negative values.So I want to use mle to estimate the theta and some mistakes arised. it shows "= is not a valid command name" after `y'=asinh(`theta' * $ML_y) / `theta'.Here is my code:

----------------------- copy starting from the next line -----------------------

sysuse auto.dta

cap program drop mylf
program define mylf       
    args  lnf mu sigma  theta
    `y'=asinh(`theta' * $ML_y) / `theta'
    quietly replace `lnf' = ln(normalden((`y' - `mu')/`sigma')) - ln(`sigma') - (1/2)*ln(1+(($ML_y)^2) * ((`theta')^2))
end

reg price mpg rep78 headroom trunk
matrix b=e(b)
matrix b=b,0

ml model lf mylf ( price = mpg rep78 headroom trunk)(sigma:)(theta:)
ml check
ml init b,copy skip
ml max

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I wonder if you can figure out mistakes in my code,many thanks in advance

[Stephen Jenkins]

A couple of quick remarks: In

`y'=asinh(`theta' * $ML_y) / `theta'

Should this be

local y   asinh(`theta' * $ML_y) / `theta'

The following code runs:

sysuse auto.dta

cap program drop mylf
program define mylf       
    args  lnf mu sigma  theta
    local y   asinh(`theta' * $ML_y) / `theta'
    quietly replace `lnf' = ln(normalden((`y' - `mu')/`sigma')) - ln(`sigma') - (1/2)*ln(1+(($ML_y)^2) * ((`theta')^2))
end

reg price mpg rep78 headroom trunk
matrix b=e(b)
matrix b=b,0

ml model lf mylf ( price = mpg rep78 headroom trunk)(sigma:)(theta:)
ml check
* ml init b,copy skip
ml max

The ml init line is commented out because leaving it in led to an error

. ml init b,copy skip
initial vector: matrix must be dimension 7
r(503);

The results of running the code (with the line commented out):

initial:       log likelihood = -843.43942
rescale:       log likelihood = -843.43942
rescale eq:    log likelihood =  -754.6528
Iteration 0:   log likelihood =  -754.6528  (not concave)
Iteration 1:   log likelihood = -684.47831  (not concave)
Iteration 2:   log likelihood = -637.72835  (not concave)
Iteration 3:   log likelihood = -634.69412  (not concave)
Iteration 4:   log likelihood = -632.40265  (not concave)
Iteration 5:   log likelihood = -630.39139  
Iteration 6:   log likelihood = -619.16139  
Iteration 7:   log likelihood = -614.18974  (not concave)
Iteration 8:   log likelihood = -613.68869  
Iteration 9:   log likelihood = -613.61111  
Iteration 10:  log likelihood =   -613.611  
Iteration 11:  log likelihood =   -613.611  

                                                Number of obs     =         69
                                                Wald chi2(4)      =       0.01
Log likelihood =   -613.611                     Prob > chi2       =     1.0000

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
eq1          |
         mpg |  -.0231133   .2348256    -0.10   0.922    -.4833631    .4371364
       rep78 |   .0699242    .710734     0.10   0.922    -1.323089    1.462937
    headroom |  -.0674762   .6864197    -0.10   0.922    -1.412834    1.277882
       trunk |   .0147577    .150134     0.10   0.922    -.2794995    .3090149
       _cons |   6.579106   60.24232     0.11   0.913    -111.4937    124.6519
-------------+----------------------------------------------------------------
sigma        |
       _cons |   .2020261   2.052022     0.10   0.922    -3.819863    4.223916
-------------+----------------------------------------------------------------
theta        |
       _cons |   1.543306   15.67519     0.10   0.922     -29.1795    32.26611
------------------------------------------------------------------------------

I have not checked whether your likelihood expression is correct for your model specification.

[Bing Zhang]

Mank thanks ,I will check again.

[Parinay Gupta]

Hello, Bing, I am using a similar approach for my dataset, how did you derive your likelihood function? I am referring to Pence (2006) and it is slightly different