Lecture Module 6. Agenda. Professor Spearot. 1 P-values. 2 Comparing Parameters. 3 Predictions. 4 F-Tests

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1 Lecture Module 6 Professor Spearot Agenda 1 P-values 2 Comparing Parameters 3 Predictions 4 F-Tests

2 Confidence intervals Housing prices and cancer risk Housing prices and cancer risk (Davis, 2004) Housing Data - two similar Nevada Counties 1 Churchill county 2 Lyon county - 31 new cases of Pediatric Leukemia (PL), Similarly sized counties should expect 1 new case. - A similar county, located directly to the west. Specification log(price) = β 1 Risk + Other + u 1 log(price): log value of housing prices 2 Risk : Index of perceived PL risk 3 Other: Other controls (lot size, floor space, etc.) Housing data from

3 Confidence intervals Housing prices and cancer risk Estimation (other effects suppressed for brevity) log(price) = 0.156Risk (0.017) n = 10204, R 2 = 0.63 What is the 99% confidence interval β 1? t crit = Use the formula: β 1 t crit se( β 1 ) < β 1 < β 1 + t crit se( β 1 ) < β 1 < < β 1 < At the 99% level, cancer risk negatively affects housing prices.

4 P-values A different approach How likely is it that I falsely reject the null? P-value: The probability that the null hypothesis is falsely rejected In the crime and enrollment example H 0 : β Enroll = 1, H A : β Enroll 1 T-statistic tstat = = The p-value is the value of the following pvalue = Pr( T > 2.45) The p-value is the probability that I randomly draw a value from the t-distribution that is larger (in absolute terms) than the estimated T-statistic

5 P-value is the area of the shaded regions put together x% of the distribution in this region x% of the distribution in this region t stat t stat

6 P-values A different approach To find the p-value 1 Calculate the t-statistic 2 Find the closest match to your t-statistic on the normal table 3 The p-value is the significance level that generates this t-statistic. In the crime and enrollment example 1 t stat = Pr( T > 2.45)? Pr( T > 2.45) = Pr(T > 2.45 T < 2.45) = Pr(T > 2.45 T < 2.45) = Pr(T > 2.45) + Pr(T < 2.45) = 2 Pr(T > 2.45) = 2 ( ) = Thus, we will falsely reject the null less than 1.42% of the time.

7 Linear combinations of parameters Returns to schooling Suppose that we estimated log(wage) = (0.021) (0.0068) obs = 6763, R 2 = JC Univ exper (0.0023) (0.0002) 1 JC is years at a junior college 2 Univ is years at a university 3 exper is experience. Clearly, β 1 < β 2. What does this mean? Is this significant? How do we test H 0 : β 1 = β 2 β 1 = β 2 is the same as β 1 β 2 = 0

8 Linear combinations of parameters Returns to schooling Define θ = β 1 β 2. H 0 : θ = 0. H A : θ 0. How do we integrate θ into our regression? Solve for β 1 = θ + β 2 Substitute for β 1 log(wage) = β 0 + β 1 JC + β 2 Univ + β 3 exper + u log(wage) = β 0 + θ + β 2 JC + β2 Univ + β 3 exper + u Collect parameters log(wage) = β 0 + θjc + β 2 (Univ + JC) + β 3 exper + u Univ + JC is total schooling θ the effect of a JC year, controlling for total schooling.

9 Linear combinations of parameters Returns to schooling Original Model: log(wage) = (0.021) Modified Model: (0.0068) obs = 6763, R 2 = log(wage) = (0.021) (0.0069) obs = 6763, R 2 = JC Univ exper (0.0023) (0.0002) JC (Univ + JC) exper (0.0023) Can we reject H 0 : θ = 0 in favor of H A : θ 0? 95% Confidence interval? Use the formula (0.0002) < θ < < θ <

10 Comparing Parameters Earnings Example Regression equation: wage = β 0 + β 1 educ + β 2 exper + β 3 tenure + u I claim that experience and tenure have identical effects on the wage. Derive an equation that allows me to test this hypothesis. Use the Wage dataset to test this hypothesis

11 Predictions Introduction Suppose you start with the equation: y = β 0 + β 1 x β k x k + u You may wish to predict y for various individuals which are not in the sample. Solving for predictions is easy Plug x 1 = c 1,..., x k = c k into equation Produce prediction, θ Also might want Standard Errors. Why? The prediction may be precise or imprecise - need standard errors to figure this out.

12 Predictions Getting standard errors Prediction: θ = β 0 + β 1 c β k c k Solve for β 0 : β 0 = θ β 1 c 1 β k c k Plug into estimating equation y = θ β 1 c 1 β k c k + β1 x β k x k + u Simplify: y = θ + β 1 x1 c βk xk c k + u Estimate gives us prediction θ and standard error.

13 Predictions Earnings Example Predicted earnings of a person with - 10 years of education - 2 years of experience - 1 year of tenure Want prediction and standard error. Estimate: wage = θ + β 1 (educ 10) + β 2 (exper-2) + β 3 (tenure 1) + u Confidence interval for θ?

14 Multiple restrictions The F-Test Baseball player salaries: log(salary) = (0.29) ( ) (0.0121) years gamesyr (0.0026) avg hrun rbi (0.0161) obs = 353, SSR = , R 2 = Call this model the "unrestricted model". How do we test H 0 : β avg = β hrun = β rbi = 0? What is the alternative hypothesis? H A : Some combination is significant. Restricted Baseball model log(salary) = (0.11) (0.0125) (0.0072) years gamesyr (0.0013) obs = 353, SSR = , R 2 =

15 Multiple restrictions The F-Test We compare models by computing an F-statistic: Intuition (SSR R SSR UR ) q F = SSR UR n k 1 - SSR R is the SSR for the restricted model. - SSR UR is the SSR for the unrestricted model. - q is the number of restrictions - n k 1 is DOF for the unrestricted model F = Average loss in explanatory power under H 0 Average unexplained variation under H A If F is high we lose a ton by our restrictions. For our model: F stat = (SSR R SSR UR ) q SSR UR n k 1 = ( ) = 9.55

16 Multiple restrictions The F-Test Under the null, F stat is distributed according to an "F-distribution" Even if restrictions are "good", large F stat is randomly possible, but unlikely Compare Fstat to the "F Distribution" 1 Choose a significance level (say 5%) 2 Using the 5% F-table, find the critical value, F crit. 3 If F > F crit, reject the null!! Looking at the table, F crit = 2.60 Reject the null! Some combination of those variables is significant.

17 The F-Test General Restrictions Unrestricted model log(wage) = β 0 + β 1 log(educ) + β 2 log(exper) + β 3 log(tenure) + u Test H 0 : β 1 = 0, β 2 = 0 Restricted model Create a new dependent variable log(wage) = β 0 + β 3 log(tenure) + u Run regressions, conduct F Test.

18 The F-Test General Regression Significance Unrestricted model log(wage) = β 0 + β 1 educ + β 2 log(exper) + β 3 log(tenure) + u Test H 0 : β 1 = 0, β 2 = 0, β 3 = 0 Restricted model log(wage) = β 0 + u Create a new dependent variable Run regressions, conduct F Test.

19 The F-Test General Restrictions Unrestricted model log(wage) = β 0 + β 1 educ + β 2 log(exper) + β 3 log(tenure) + u Test H 0 : β 1 = 1, β 2 = 0.5 New restricted model log(wage) log(educ) 0.5 log(exper) = β 0 + β 3 log(tenure) + u Create a new dependent variable Run regressions, conduct F Test.