1. MF-852 Financial Econometrics
Lecture 7
Hypothesis Testing in Bivariate Regression
Roy J. Epstein
Fall 2003

2. Topics Two-Sided vs. One-Sided Hypothesis Tests
Confidence Intervals and P-Values
R2 and F in Linear Model
Regression Example: Beta Coefficients
Modeling Strategy: Cocaine and Sentencing

3. Two-Sided Confidence Interval
The 95% confidence interval (“C.I.”) for (normally distributed) xbar is
This is a two-sided test:
H0:  = 0 vs.
H1:   0 (i.e.,  > 0 or  < 0)

4. One-Sided Confidence Interval
One-sided confidence interval: used to find upper or lower limit for .
95% upper limit:
95% C.I. is

5. Example
Children with lead poisoning have lower blood hemoglobin than normal children. Want to find 95% upper limit for  for lead poisoned children.
25 hemoglobin samples yield xbar = 10.6 with standard deviation 2.
95% C.I. is (–, /5)

6. Other Confidence Intervals
Customary to use a 95% C.I.
What is 90% C.I.? 99% C.I.?

7. P-Value
Assuming H0, what is the probability that the sample value would be as extreme as the value actually observed?
Alternative to pre-determined confidence interval.
Lets the data tell you the confidence level.

8. P-Value Example
Sample yields xbar = 7 with standard error of 4. Assume normality.
H0:  = 0
(xbar–0)/4 has standard normal dist.
Critical value is (7–0)/4 = 1.75
P(z  1.75) = 0.04

9. P-Value Example
If H0 was true, then 4% chance of observing z as large as 1.75.
Two-tailed test:
“Significant at 8% level”
C.I. would be
One-tailed test: significant at 4% level.

10. Linear Model: OLS Estimation
Regression model:
Yi =  + Xi + ei
Estimated coefficients are
Predicted Yi =
Predicted ei =
Note:

11. R2
It can be shown that
Total variance of Y equals “predicted variance” + “error variance”
R2 =
fraction of variance explained by model.

12. F Used for hypothesis tests with variances.
Test of significance of R2 (“goodness of fit”)

13. Regression From Last Time

14. OLS Regression Coefficients
The estimated coefficients are random variables.
In this example,
 = – 0.173, standard error = 1.32
 = 0.144, standard error =
R2 = 0.90
F(1,26) =

15. Statistical Significance
Suppose H0:  = 0
Is the estimated  statistically significant?
Suppose H0:  = 0
Is the estimated  statistically significant?
Suppose H0:  = 0 AND  = 0
Is the joint hypothesis accepted or rejected?

16. More Hypothesis Tests Suppose H0:  = 0.16 Suppose H0:  = 2
Do you accept or reject H0?
Suppose H0:  = 2

17. Regression Intuition
Suppose you run a regression of Y just on an intercept (no X variables).
What will be the value of alphahat?
What is the R2 in this regression?
Suppose the model is Y = a + bX.
What is yhat when X=xbar?

18. Example: Beta Coefficient
We will estimate the CAPM.

19. Example: Cocaine Sentencing
You will propose a model and hypotheses!