1. Interpreting Bi-variate OLS Regression
Stata Regression Output
Regression plots and RSS
R2 -- Coefficient of Determination
Adjusted R2
Sample Covariance/Correlation
Hypothesis Testing
Standard Errors
T-tests and P-values

2. Data Use the “caschool.dat” file Data description:
CaliforniaTestScores.pdf
Build a Stata do-file as you go
Model:
Test score=f(student/teacher ratio)

3. Stata Regression Model:
Regressing Student Teacher Ratio onto Test Score
histogram str, percent normal
histogram testscr, percent normal

4. Regression Output regress testscr str
Source | SS df MS Number of obs =
F( 1, 418) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
testscr | Coef. Std. Err t P>|t| Beta
str |
_cons |

5. Regression Descriptive Statistics
cor testscr str, means
Variable | Mean Std. Dev Min Max
testscr |
str |
| testscr str
testscr |
str |

6. Regression Plot
twoway (scatter testscr str) (lfitci testscr str)

7. Measuring “Goodness of Fit”
Root of Mean Squared Error (“Root MSE”)
Measures spread around the regression line
Coefficient of Determination (R2)
“model” or explained sum of squares
“total” sum of squares

8. Explaining R2 For each observation Yi, variation around the mean can
be decomposed into that which is “explained” by the
regression and that which is not:
Book terminology:
TSS = (all)2
RSS = (unexplained)2
ESS = (explained)2
unexplained deviation
explained deviation
Stata terminology:
Residual = (unexplained)2
Model = (explained)2
Total = (all)2

9. Sample Covariance & Correlation
Sample covariance for a bivariate model is defined as:
Sample correlations (r) “standardize” covariance by dividing by the product of the X and Y standard deviations:
Sample correlations range from
-1 (perfect negative relationship) to
+1 (perfect positive relationship)

10. Standardized Regression Coefficients (aka “Beta Weights” or “Betas”)
Formula:
In our example:
Interpretation: the number of std. deviations change in Y one should expect from a one-std. deviation change in X.

11. Hypothesis Tests for Regression Coefficients
For our model: Yi = *Xi+ei
Another sample of 420 observations would lead to different estimates for b0 and b1. If we drew many such samples, we’d get the sample distribution of the estimates
We need to estimate the sample distribution, (because we usually can’t see it) based on our sample size and variance

12. To do that we calculate SEbs (Bivariate case only)

13. Interpreting Standard Errors
Assuming that we estimated the
sample standard error correctly, we
can identify how many standard
errors our estimate is away from
zero.
For our model:
b0 = , and SEb0 = 9.467
b1 = , and SEb1 = .4798
The T-test reports the number of
standard errors our estimate falls
away from zero. Thus, the “T” for
b1 is 4.75 for our model. (rounding!)
Estimated Sampling Distribution for b1
b1 = -2.28
(which is 4.75 SEb1 “units”
away from b1)
b1 - SEb1=-2.76
b1 + SEb1= -1.8

14. Classical Hypothesis Testing
Assume that b1 is zero. What is the probability that your sample would have
resulted in an estimate for b1 that is 4.75 SEb1’s away from zero?
To find out, determine the cumulative density of the estimated sampling
distribution that falls more than 4.75 SEb1’s away from zero.
See Table 2, page 757, in Stock & Watson. It reports discrete “p-values”, given
the sample size and t-values. Note the distinction between 1 and 2 sided tests
In general, if the t-stat is above 2,
the p-value will be < which is
the acceptable upper limit in a
classical hypothesis test.
Note: in Stata-speak,
a p-value is a “p>|t|”
Assume that b1 = 0.0
(null hypothesis)
Estimated b1 = 2.27
(working hypothesis)

15. Coming up... For Next Week For Next Week:
Use the caschool.dta dataseet
Run a model in Stata using Average Income (avginc) to predict Average Test Scores (testscr)
Examine the univariate distributions of both variables and the residuals
Walk through the entire interpretation
Build a Stata do-file as you go
For Next Week:
Read Chapter 8 of Stock & Watson