📝

Enter the Black Box: How to Obtain Beta (Part 1 - No Covariates)

Using the worksheet on “calculating beta by hand” we learned how to obtain beta in OLS, and in short one uses the following equation

β^=∑i=1N(Xi−Xˉ)(Yi−Yˉ)∑i=1N(Xi−Xˉ)2\hat{\beta}=\frac{\sum^{N}_{i=1} ({X_i-\bar{X}})({Y_i-\bar{Y}})}{\sum^{N}_{i=1}({X_i-\bar{X}})^2}

Now, notice what the expression on the top and the bottom represent.

❓
What do those expression represent?
‣
Answer
✅
The top expression represents the covariance of X and Y, and the bottom expression represents the variance of X. Therefore, the other way to express beta hat is β^=Cov(X,Y)var(X)\hat{\beta}=\frac{Cov(X,Y)}{var(X)}

Where, X is the variable “attached” to beta.

Getting to work

Knowing these concepts, and the interpretation of regression where the main variable is discrete, let’s go over the following example. For the following data:

You can use this link to download it

https://www.dropbox.com/scl/fo/vzm55313fdzl1ned2s56i/h?rlkey=774jljwee9r6kg3wsx49ifsw8&dl=0

The data is called:

battenincome.dta2.5KB

Mean Comparison

Imagine that we are trying to find the effect of batten on income. We can run the following regression:

Incomei=β0+β1Batteni+ϵiIncome_i=\beta_0+\beta_1Batten_i+\epsilon_i

Let’s say for the sake of the example that Income is an average of earnings for the 4 years after Batten.

image
  1. The value of β1=29.22\beta_1=29.22. Obtain this value using averages.
    2. ‣
    Answer
    ‣
    First we obtain the mean income for people that went to Batten, then we obtain the income for people that did not go to Batten, and then we take the difference to obtain β1\beta_1.
    . sum income if batten==1

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
      income |          9    184.4444    27.88867        140        220

. sum income if batten==0

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
      income |          9    155.2222    27.11908        125        195

. display  184.4444 - 155.2222
29.2222

  2. The value of β1=29.22\beta_1=29.22. Obtain this value using the following Formula β1=Cov(Batten,Income)Var(Batten)\beta_1=\frac{Cov(Batten,Income)}{Var(Batten)}
    3. ‣
    Answer
    . * First we obtain the covariance between Batten and Income

. 
. 
. correlate batten income, covariance
(obs=18)

             |   batten   income
-------------+------------------
      batten |  .264706
      income |  7.73529  938.147


. * Then we obtain the variance of Batten

. sum batten, det

                           batten
-------------------------------------------------------------
      Percentiles      Smallest
 1%            0              0
 5%            0              0
10%            0              0       Obs                  18
25%            0              0       Sum of Wgt.          18

50%           .5                      Mean                 .5
                        Largest       Std. Dev.      .5144958
75%            1              1
90%            1              1       Variance       .2647059
95%            1              1       Skewness              0
99%            1              1       Kurtosis              1

. * Now we divide:

. display 7.73529/ .2647059
29.222205

  3. The value of β1=29.22\beta_1=29.22. Obtain this value using a regression
    4. ‣
    Answer
    . reg income batten

      Source |       SS           df       MS      Number of obs   =        18
-------------+----------------------------------   F(1, 16)        =      5.08
       Model |  3842.72222         1  3842.72222   Prob > F        =    0.0386
    Residual |  12105.7778        16  756.611111   R-squared       =    0.2409
-------------+----------------------------------   Adj R-squared   =    0.1935
       Total |     15948.5        17  938.147059   Root MSE        =    27.507

------------------------------------------------------------------------------
      income |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      batten |   29.22222   12.96672     2.25   0.039     1.734006    56.71044
       _cons |   155.2222   9.168855    16.93   0.000     135.7851    174.6593
------------------------------------------------------------------------------

What these three exercise should show you, are ways of obtaining the same number. They all carry the same interpretation but seeing how to obtain them is important. You can do this for a continuous variable and it should work as well. Notice that for a continuous variable using average could get really complex because you would have to do it for each value change, therefore method (2) seems the most sensible to use for the continuous variable

InstagramX