Percentage explained by the standard deviation of linear combination of independent variables

[Jason Su]

Hello, everyone, I have a question about using standard deviation to explain changes in the dependent variable when I review relevant empirical microeconomic literature.

Suppose the estimated regression model is as follows:
z=beta_1*x+beta_2*y+epsilon

The literature which I refer comes out with the conclusion that a standard deviation of (beta_1*x+beta_2*y) explains 15% of the standard deviation of z.

How can I obtain the value 15% here if I know the standard deviations of x, y, and z?

Your comments are appreciated and I'm grateful for your suggestions.

[Nick Cox]

R-square is the ratio variance(fitted) / variance(observed) for the outcome. So, I imagine someone is reporting its square root in a non-standard way.

You will recall that SD is the root of variance. The SDs of x and y in your notation are irrelevant here.

Hmm. As the square root is larger than the original over this range, I wonder if this is some sort of rhetoric to make results seem better than they are.

[Carlo Lazzaro]

Jason:
a different take rests on -esize- postestimation command:

. use "C:\Program Files\Stata17\ado\base\a\auto.dta"
(1978 automobile data)

. regress price mpg i.foreign

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(2, 71)        =     14.07
       Model |   180261702         2  90130850.8   Prob > F        =    0.0000
    Residual |   454803695        71  6405685.84   R-squared       =    0.2838
-------------+----------------------------------   Adj R-squared   =    0.2637
       Total |   635065396        73  8699525.97   Root MSE        =    2530.9

------------------------------------------------------------------------------
       price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         mpg |  -294.1955   55.69172    -5.28   0.000    -405.2417   -183.1494
             |
     foreign |
    Foreign  |   1767.292    700.158     2.52   0.014     371.2169    3163.368
       _cons |   11905.42   1158.634    10.28   0.000     9595.164    14215.67
------------------------------------------------------------------------------

. estat esize

Effect sizes for linear models

-----------------------------------------------------------------
             Source | Eta-squared     df     [95% conf. interval]
--------------------+--------------------------------------------
              Model |   .2838475       2      .109358    .4223785
                    |
                mpg |   .2821436       1     .1178997    .4292908
            foreign |   .0823465       1     .0031748    .2185883
-----------------------------------------------------------------
Note: Eta-squared values for individual model terms are partial.

Shamelessly delving into the expalanation of -estat esize-, Example 14, -regress postestimatin- entry, Stata .pdf manual:

The overall model effect size is 0.284. This means that roughly 28.4% of the variation in -price- is
explained by the model. The partial effect size for –mpg- is 0.282. This means that roughly 28.2% of the variation in -price- is explained by –mpg- after you remove the variation explained by all other terms.

You can replicate (mutatis mutandis) the last sentence for -i.foreign-.