2. NOTATION & ASSUMPTIONS 2 Y i =  1 +  2 X 2i +  3 X 3i + U i Zero mean value of U i No serial correlation Homoscedasticity Zero covariance between U i and X i No specification bias No exact collinearity between the X variables

3. ESTIMATION 3

4. BETA COEFFICIENTS 4 Occasional you’ll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaning Idea is to replace y and each x variable with a standardized version – i.e. subtract mean and divide by standard deviation Coefficient reflects standard deviation of y for a one standard deviation change in x

5. VARIANCE AND STANDARD ERROR 5

6. THE T TEST 6

7. ONE-SIDED ALTERNATIVES 7 y i =  0 +  1 x i1 + … +  k x ik + u i H 0 :  j = 0 H 1 :  j > 0 c 0   Fail to reject reject

8. TWO-SIDED ALTERNATIVES 8 y i =  0 +  1 X i1 + … +  k X ik + u i H 0 :  j = 0 H 1 :  j > 0 c 0   -c  reject fail to reject

9. THE COEFFICIENT OF DETERMINATION A MEASURE OF “GOODNESS OF FIT” 9 Verbally, R-square measure the proportion or percentage of the total variation in Y explained by the regression model. Verbally, R-square measure the proportion or percentage of the total variation in Y explained by the regression model.

10. ADJUSTED R-SQUARED 10 Recall that the R2 will always increase as more variables are added to the model. The adjusted R2 takes into account the number of variables in a model, and may decrease.

11. CONT.. 11 It’s easy to see that the adjusted R2 is just (1 – R2)(n – 1) / (n – k – 1), but most packages will give you both R2 and adj-R2. You can compare the fit of 2 models (with the same y) by comparing the adj-R2. You cannot use the adj-R2 to compare models with different y’s (e.g. y vs. ln(y)).

12. THE F STATISTIC 12 0 c   f( F ) F reject fail to reject Reject H 0 at  significance level if F > c

13. 13