Enter the Black Box: How to Obtain Beta (Part 2

We will keep using the data from this worksheet

Before we were trying to under how to obtain beta from this model:

Income_i=\beta_0+\beta_1Batten_i+\epsilon_i

Now, we want to understand how $\beta_1$ changes once we add a covariate. In this case the full model will be:

Income_i=\beta_0+\beta_1Batten_i+\beta_2 ParentsIncome_i+\epsilon_i

Let’s go through several methods. Each of this inspire different ways of understanding what a covariate is really doing. What I recommend is going through this and trying to understand from your own perspective the intuition of what “controlling for a variable” is doing.

Regression

First run the regression using the data and report what the value of $\beta_1$ is.

‣

Answer

* The value of beta 1 is 14.84 
reg income batten parents

      Source |       SS           df       MS      Number of obs   =        18
-------------+----------------------------------   F(2, 15)        =    246.69
       Model |    15477.94         2     7738.97   Prob > F        =    0.0000
    Residual |      470.56        15  31.3706667   R-squared       =    0.9705
-------------+----------------------------------   Adj R-squared   =    0.9666
       Total |     15948.5        17  938.147059   Root MSE        =     5.601

-------------------------------------------------------------------------------
       income |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
       batten |      14.84   2.743895     5.41   0.000     8.991526    20.68847
parentsincome |     1.2944   .0672114    19.26   0.000     1.151142    1.437658
        _cons |   65.33333   5.027009    13.00   0.000     54.61852    76.04815
-------------------------------------------------------------------------------

Mean Comparison

How would we obtain the value if we were to use averages?

‣

Answer

🗒️

Before we would compare the income for people who went to Batten and then subtract it from the income for people who did not go to Batten. We would then do the same process for all levels of ParentsIncome. This concept then really presents the concept of “holding things constant” make the comparison.

Parents Income	Batten=1	Batten=0	Difference	N
50	142.5	129.25	13.25	6
75	180	165	15	6
100	208.75	192.5	16.25	6

The difference in each bin of parents income is 13.25, 15 and 16.25. In each of these comparisons, there are 6 observations, so when we average out all the difference we obtain: 13.25 (6/18)+15(6/18)+16.25(6/18)=14.83 Using this method we obtain a beta of 14.83

. tab parentsincome

parentsinco |
         me |      Freq.     Percent        Cum.
------------+-----------------------------------
         50 |          6       33.33       33.33
         75 |          6       33.33       66.67
        100 |          6       33.33      100.00
------------+-----------------------------------
      Total |         18      100.00

* There are 3 tiers of parents income, so we will make ///
* the mean comparison under each of these tiers
sum income if batten==1 & parentsincome==50

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
      income |          2       142.5    3.535534        140        145

sum income if batten==0 & parentsincome==50

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
      income |          4      129.25    4.349329        125        135

display  142.5 - 129.25
13.25

sum income if batten==1 & parentsincome==75

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
      income |          3         180           5        175        185

sum income if batten==0 & parentsincome==75

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
      income |          3         165           5        160        170

display 180 - 165
15
sum income if batten==1 & parentsincome==100

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
      income |          4      208.75    8.539126        200        220

sum income if batten==0 & parentsincome==100

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
      income |          2       192.5    3.535534        190        195

display 208.75 - 192.5
16.25

display 13.25*(6/18)+15*(6/18)+16.25*(6/18)
14.833333

Using the formula

Now obtain the value of beta 1 using the following Formula $\beta_1=\frac{Cov(Batten,Income)}{Var(Batten)}$

‣

Answer

🗒️

When using this method we have to adapt some things. First we have to obtain

\widetilde{Batten}

\widetilde{Batten}

is the difference between the predicted value of Batten relative to all their covariates, and the actual value of Batten. First step,is to regress Batten on all the covariates of the model. In this case, parentsincome.

. reg batten parentsincome

      Source |       SS           df       MS      Number of obs   =        18
-------------+----------------------------------   F(1, 16)        =      1.28
       Model |  .333333333         1  .333333333   Prob > F        =    0.2746
    Residual |  4.16666667        16  .260416667   R-squared       =    0.0741
-------------+----------------------------------   Adj R-squared   =    0.0162
       Total |         4.5        17  .264705882   Root MSE        =    .51031

-------------------------------------------------------------------------------
       batten |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
parentsincome |   .0066667   .0058926     1.13   0.275     -.005825    .0191583
        _cons |   5.55e-17   .4580176     0.00   1.000    -.9709539    .9709539
-------------------------------------------------------------------------------

* Then we obtain the predicted value of batten given this model
predict battenhat, xb

* Now we substract the actual value of Batten minus the predicted Value,
* we name this variable batten_tilda:
gen batten_tilda=batten-battenhat

* Notice that this is also the same process of obtaining the residuals of 
reg baten parents income
predic res_batten, res

🗒️

Once we have obtain

\widetilde{Batten}

, now we can use our formula

\beta_1=\frac{Cov(\widetilde{Batten},Income)}{Var(\widetilde{Batten})}

. We’ll do this in STATA

correlate batten_til income, covariance
(obs=18)

             | batten~a   income
-------------+------------------
batten_tilda |  .245098
      income |  3.63725  938.147

sum batten_til, det

                        batten_tilda
-------------------------------------------------------------
      Percentiles      Smallest
 1%    -.6666667      -.6666667
 5%    -.6666667      -.6666667
10%    -.6666667            -.5       Obs                  18
25%          -.5            -.5       Sum of Wgt.          18

50%    -1.49e-08                      Mean          -1.32e-08
                        Largest       Std. Dev.      .4950738
75%           .5             .5
90%     .6666666             .5       Variance        .245098
95%     .6666666       .6666666       Skewness      -1.65e-08
99%     .6666666       .6666666       Kurtosis         1.3104

display 3.63725/.245098
14.839982

FWL Way

Here is another method (similar to the one before) in which it shows how obtain the same beta and provides similar intuition. It’s called using the Frisch–Waugh–Lovell theorem.

reg batten parentsincome

      Source |       SS           df       MS      Number of obs   =        18
-------------+----------------------------------   F(1, 16)        =      1.28
       Model |  .333333333         1  .333333333   Prob > F        =    0.2746
    Residual |  4.16666667        16  .260416667   R-squared       =    0.0741
-------------+----------------------------------   Adj R-squared   =    0.0162
       Total |         4.5        17  .264705882   Root MSE        =    .51031

-------------------------------------------------------------------------------
       batten |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
parentsincome |   .0066667   .0058926     1.13   0.275     -.005825    .0191583
        _cons |   5.55e-17   .4580176     0.00   1.000    -.9709539    .9709539
-------------------------------------------------------------------------------

predict res_batten, res

reg income parentsincome

      Source |       SS           df       MS      Number of obs   =        18
-------------+----------------------------------   F(1, 16)        =    167.82
       Model |  14560.3333         1  14560.3333   Prob > F        =    0.0000
    Residual |  1388.16667        16  86.7604167   R-squared       =    0.9130
-------------+----------------------------------   Adj R-squared   =    0.9075
       Total |     15948.5        17  938.147059   Root MSE        =    9.3145

-------------------------------------------------------------------------------
       income |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
parentsincome |   1.393333   .1075549    12.95   0.000     1.165327     1.62134
        _cons |   65.33333   8.360044     7.81   0.000     47.61083    83.05583
-------------------------------------------------------------------------------

predict res_y, res

reg res_y res_batten

      Source |       SS           df       MS      Number of obs   =        18
-------------+----------------------------------   F(1, 16)        =     31.20
       Model |  917.606684         1  917.606684   Prob > F        =    0.0000
    Residual |  470.559999        16  29.4099999   R-squared       =    0.6610
-------------+----------------------------------   Adj R-squared   =    0.6398
       Total |  1388.16668        17  81.6568637   Root MSE        =    5.4231

------------------------------------------------------------------------------
       res_y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   res_batte |      14.84   2.656765     5.59   0.000      9.20791    20.47209
       _cons |   8.28e-10   1.278237     0.00   1.000    -2.709741    2.709741
------------------------------------------------------------------------------

* This provides the beta of 14.84

Enter the Black Box: How to Obtain Beta (Part 2 - With Covariates)

Regression

Mean Comparison

Using the formula

FWL Way