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