Standardized regression coefficients

[lal mohan kumar]

Sorry if I am posting a mundane question. I did enough google search in vain hence thought of posting this in the Stata forum.
In one article, I saw that the beta coefficient of the independent variable on the dependent variable (.795) is multiplied with the standard deviation of the independent variable(.082). However, they also divided the above product by the mean of the dependent variable (.590). They interpreted this as “One standard deviation increase in the independent variable raises dependent variable by 11%” (=0.795* 0.082/0.590).
I have read about standardized regression coefficient from Prof Richard William’s teaching note https://www3.nd.edu/~rwilliam/stats1/x92b.pdf, but nowhere in it I could find the product of beta and s.dev of an independent variable gets divided by the mean of the dependent variable.
My question is that

1. Is there such practice to divide by mean of the dependent variable for standardization? Also, can we interpret as the way paper interpreted?
2. How to get such coefficients in Stata. I guess, the option listcoef, help won't help

Any help in this regard will he highly helpful

[Joro Kolev]

A beta weights regression in Stata is simply a regression where all variables (dependent and regressors) have been standardised to have a standard deviation of 1. Here:

. clear

. sysuse auto
(1978 Automobile Data)

. foreach var of varlist price mpg headroom {
  2. summ `var'
  3. gen `var'std = `var'/r(sd)
  4. }

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
       price |         74    6165.257    2949.496       3291      15906

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
         mpg |         74     21.2973    5.785503         12         41

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
    headroom |         74    2.993243    .8459948        1.5          5

. regress pricestd mpgstd headroomstd

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(2, 71)        =     10.44
       Model |    16.58487         2  8.29243499   Prob > F        =    0.0001
    Residual |   56.415129        71  .794579282   R-squared       =    0.2272
-------------+----------------------------------   Adj R-squared   =    0.2054
       Total |   72.999999        73  .999999987   Root MSE        =    .89139

------------------------------------------------------------------------------
    pricestd |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mpgstd |  -.5082417   .1146017    -4.43   0.000    -.7367509   -.2797324
 headroomstd |  -.0958064   .1146017    -0.84   0.406    -.3243156    .1327029
       _cons |   4.300164   .7033396     6.11   0.000     2.897744    5.702583
------------------------------------------------------------------------------

. regress price mpg headroom, beta

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(2, 71)        =     10.44
       Model |   144280501         2  72140250.4   Prob > F        =    0.0001
    Residual |   490784895        71  6912463.32   R-squared       =    0.2272
-------------+----------------------------------   Adj R-squared   =    0.2054
       Total |   635065396        73  8699525.97   Root MSE        =    2629.2

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|                     Beta
-------------+----------------------------------------------------------------
         mpg |  -259.1057   58.42485    -4.43   0.000                -.5082416
    headroom |  -334.0215   399.5499    -0.84   0.406                -.0958063
       _cons |   12683.31   2074.497     6.11   0.000                        .
------------------------------------------------------------------------------

[Joro Kolev]

The interpretation of any regression slope is that One Unit Increase in X leads to The Slope Number of Units Increase in Y.

Therefore the interpretation of the beta weights is that, e.g., One Standard Deviation increase in MPG leads to -.5082416 Standard Deviations Increase in PRICE. (Note the minus sign, so another way to say it is 0.5 standard deviations decrease in Price).

[lal mohan kumar]

Thanks Joro for the prompt reply continuing with your example,

. clear

 . sysuse auto
(1978 Automobile Data)

. regress price mpg headroom

      Source |       SS       df       MS              Number of obs =      74
-------------+------------------------------           F(  2,    71) =   10.44
       Model |   144280501     2  72140250.4           Prob > F      =  0.0001
    Residual |   490784895    71  6912463.32           R-squared     =  0.2272
-------------+------------------------------           Adj R-squared =  0.2054
       Total |   635065396    73  8699525.97           Root MSE      =  2629.2

------------------------------------------------------------------------------
       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         mpg |  -259.1057   58.42485    -4.43   0.000    -375.6015   -142.6098
    headroom |  -334.0215   399.5499    -0.84   0.406    -1130.701    462.6585
       _cons |   12683.31   2074.497     6.11   0.000     8546.885    16819.74
------------------------------------------------------------------------------

. listcoef, help

regress (N=74): Unstandardized and standardized estimates

  Observed SD:  2.9e+03
  SD of error:  2.6e+03

--------------------------------------------------------------------------------
             |          b        t    P>|t|    bStdX    bStdY   bStdXY     SDofX
-------------+------------------------------------------------------------------
         mpg |  -259.1057   -4.435    0.000  -1.5e+03   -0.088   -0.508     5.786
    headroom |  -334.0215   -0.836    0.406  -282.580   -0.113   -0.096     0.846
    constant |   1.27e+04    6.114    0.000        .        .        .         .
--------------------------------------------------------------------------------
       b = raw coefficient
       t = t-score for test of b=0
   P>|t| = p-value for t-test
   bStdX = x-standardized coefficient
   bStdY = y-standardized coefficient
  bStdXY = fully standardized coefficient
   SDofX = standard deviation of X

bStdXY is the coefficient which implies both the X and the Y variables are standardized to have a mean of 0 and a standard deviation of 1. Whereas bStdX= -259.1057 * 5.79=-1.5e+03; where -259.1057 is coefficient of mpg on price and 5.79 is the standard deviation of mpg, where only mpg is standardized but the price is not. Am I right? Also, as my question posted no where one will use the mean of dependent variable for scaling/dividing right?

Similar to this

 
 regress price mpg headroom, beta

Can we standardize beta for panel data? Is there a way as Stata says, "option beta not allowed"

[lal mohan kumar]

For the last part,

Can we standardize beta for panel data? Is there a way as Stata says, "option beta not allowed"

. I got an answer from the Forum, https://www.statalist.org/forums/for...d-effect-model.