Economics 105: Statistics

1. Economics 105: Statistics Any questions? GH 21 due Wednesday

2. Standard Error of Estimated Slope • If we knew pop var of errors, could use z-test • Need an unbiased estimator of pop variance of  • Now t distribution

3. For this test • the test statistic is • If n > 30, can compare this test statistic to critical values in the z-table (hence, “the rule of 2” or technically, 1.96, for 2-sided test at 5% level) • However, you can obtain exact critical values on t-scale from Excel, under OLS assumptions (1) – (6)!! Hypothesis Testing

4. If and standard assumptions hold, then a 100(1-)% confidence interval for the population slope parameter is given by Confidence Interval

5. Simple Linear Regression Example • A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet) • A random sample of 10 houses is selected • Dependent variable (Y) = house price in $1000s • Independent variable (X) = square feet

6. Sample Data for House Price Model

7. Excel Output The regression equation is:

8. Graphical Presentation • Interpret the coefficients … Slope = 0.10977 Intercept = 98.248

9. t-test for Significance of β1 • t-statistic:

10. Excel Output H0: β1 = 0 H1: β1 ≠ 0  = .05 df= n-2 = 8

11. F-test for Significance of β1 • F-test statistic: where where F follows an F distribution with K numerator and (n – K - 1) denominator degrees of freedom (K = the number of independent variables in the regression model)

12. Excel Output With 1 and 8 degrees of freedom p-value for the F Test

13. H0: β1 = 0 H1: β1 ≠ 0  = .05 df1= 1 df2 = 8 F-test for Significance of β1 (continued) Test Statistic: Decision: Conclusion: Critical Value: F = 5.32 Reject H0 at  = 0.05  = .05 There is sufficient evidence that house size affects selling price 0 F Do not reject H0 Reject H0 F.05 = 5.32

14. Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: d.f. = n - 2 Excel Printout for House Prices: At 95% level of confidence, the confidence interval for the slope is (0.0337, 0.1858)

15. Confidence Interval Estimate for the Slope (continued) Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.74 and $185.80 per square foot of house size This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the .05 level of significance

16. Unbiased estimator • Efficiency of an estimator • Intuition for when var is smaller • We won’t know , so we’ll need to estimate it Properties of OLS Estimator • Gauss-Markov Theorem • Under assumptions (1) - (5) [don’t need normality of errors], is B.L.U.E. of

17. Test for a Correlation Coefficient • Hypotheses H0: ρ = 0 (no correlation between X and Y) H1: ρ≠ 0 (correlation exists) • Or one-sided alternative hypotheses • Test statistic

18. Example: Suicide vs. Unemployment (GH19.xls, data tab) Is there evidence of a positive linear relationship between suicide rate and unemployment rate at the 5% level of significance? H0: ρ= 0 (No correlation) H1: ρ> 0 (positive correlation) =.05 , df=11 - 2 = 9

19. If we want to forecast the average value of Y for a given value of X, call it Xn+1, the point estimate is • How accurate is this prediction or forecast? • A100(1-)% confidence interval for is given by Prediction

20. What affects the width of this confidence interval? • Four factors Prediction

21. What if we want to predict an individual response Y? This is called a prediction interval. • Only difference is the “1” • Which is wider, for the same xn+1 value? Intuition? Prediction