Box-Cox Transformation
Goal: Simultaneously obtain Normal and Homoskedatic errors (not always possible) through a power
transformation on Y where all Yi > 0. Typically the final selected  is a “round” number like -1,-1/2,0,1/2
Algorithm 1:
1) Power Transform Y :
 Yi   1
0

Yi '     

log Y 
 0
i

over a range of    2, 2
n
 RSS    
n
2) Obtain the maximized Profile log-likelihood for each : L      log 



1
log Yi 




2
n
i 1


where RSS     Residual sum of squares when using Yi '    in the Regression model
^
3) Choose  that maximizes L   


 ^  1
4) Approximate 95%Confidence Interval for :  : L     L       1,2 
  2


Algorithm 2:
1) Power Transform Z :
 Yi   1

   1
  Y 
'
Zi       
 

Y log Y 
i

0
over a range of    2, 2
 0
1/ n
 1  n
  n

where Y  exp    log Yi      Y i 
 n  i 1
  i 1 

aka the geometric mean
 RSS    
1
2) Obtain the maximized Profile log-likelihood for each : Lmax      log 

2
n


'
where RSS     Residual sum of squares when using Z i    in the Regression model
^
3) Choose  that maximizes L    or equivalently minimizes RSS   
2

 t


 /2, 
^

 
4) Approximate 95%Confidence Interval for :  : RSS     RSS   min 1 

 
 



where   degrees of freedom for RSS   