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

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

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)

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

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 (–, 10.6 + 2/5)

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

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.

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

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.

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

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

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

Regression From Last Time

OLS Regression Coefficients The estimated coefficients are random variables. In this example,  = – 0.173, standard error = 1.32  = 0.144, standard error = 0.0094 R2 = 0.90 F(1,26) = 234.26

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?

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

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?

Example: Beta Coefficient We will estimate the CAPM.

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