The bayesmh command now includes an asymmetric Laplace distribution (ALD) as a new likelihood function. You can use ALD to model nonnormal outcomes with pronounced skewness and kurtosis.

Let's see it work

Consider the mathscores dataset, which contains math scores of pupils in the third year (math3) and fifth year (math5) from different schools (school) in Inner London (Mortimore et al. 1988).

. webuse mathscores

. describe

Contains data from https://www.stata-press.com/data/r19/mathscores.dta
 Observations:           887
    Variables:             3                  9 May 2022 23:31


Variable      Storage   Display    Value                                     
    name         type    format    label      Variable label

school          float   %9.0g                 School ID
math3           float   %9.0g                 Year 3 math score
math5           float   %9.0g                 Year 5 math score
Sorted by:

We use ALD to examine the skewness (asymmetry) of the math5 variable. For this purpose, we use asymlaplaceq() with a random quantile parameter {tau}. A negative skewness is indicated by {tau} greater than 0.5; positive skewness is indicated by {tau} less than 0.5; and {tau} of 0.5 indicates no skewness, a perfect symmetry. We apply a flat uniform(0, 1) prior for {tau}. We specify a regression model with only a constant term for math5, which corresponds to the location parameter of ALD, and a scale parameter {scale} as in the previous examples.

. bayesmh math5, likelihood(asymlaplaceq({scale}, {tau}))
     prior({math5:}, normal(0,10000)) 
     prior({scale},  igamma(0.01,0.01))
     prior({tau},    uniform(0,1))
     block({math5:} {scale}{tau}, split)
     initial({tau} 0.5) rseed(19)

Burn-in ...
Simulation ...

Model summary

 
Likelihood:
  math5 ~ asymlaplaceq({math5:_cons},{scale},{tau})

Priors:
  {math5:_cons} ~ normal(0,10000)
        {scale} ~ igamma(0.01,0.01)
          {tau} ~ uniform(0,1)
 

Bayesian asymm. Laplace (quantile) regression    MCMC iterations  =     12,500
Random-walk Metropolis–Hastings sampling         Burn-in          =      2,500
                                                 MCMC sample size =     10,000
                                                 Number of obs    =        887
                                                 Acceptance rate  =       .424
                                                 Efficiency:  min =     .03815
                                                              avg =     .05242
Log marginal-likelihood = -2846.2182                          max =     .07833




             
 
                                                Equal-tailed    
             
 
      Mean   Std. dev.     MCSE     Median  [95% cred. interval]
 
 
 
math5        
 
                                                                
       _cons 
 
  37.00486    .045309   .001619   37.00222    36.9161   37.10699

       scale 
 
  1.073886   .0655504   .003246   1.071063   .9511403   1.211991
         tau 
 
  .8602184   .0085491   .000438   .8601294    .843211   .8767714

We see that the estimated 95% credible interval of [0.84, 0.88] suggests that {tau} is greater than 0.5, indicating a strong negative skewness. A more formal test for negative skewness can be conducted by using the bayestest interval command.

. bayestest interval {tau}, lower(0.5)

Interval tests     MCMC sample size =    10,000

       prob1 : {tau} > 0.5
 


             
 
      Mean    Std. dev.      MCSE
 
 
 
       prob1 
 
         1     0.00000          0

Indeed, the posterior probability of {tau} greater than 0.5 is 1.