4️⃣

Worksheet #4 FWL

We will keep using the data shown in that example and can be found here: https://github.com/dstellotri/rmda

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+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.

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Answer

Mean Comparison

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

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Answer

Using the formula

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

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Answer

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