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Marietta College Spring 2011 Econ 420: Applied Regression Analysis Dr. Jacqueline Khorassani Week 7

Tuesday, February 22 Exam 2: Tuesday, March 22 Exam 3: Monday, April 25, 12- 2:30PM

Return and discuss ASST 8 • # 5 , Page 111

General Notes • Serial correlation Is not about the independent variables being correlated; it is about the error terms being correlated • A dummy variable can only take two values (0 and 1) • If I define a variable to take a value of -1 under certain conditions, it doesn’t mean that I expect the coefficient to be negative

(a) Classical Assumption 2 (b) Classical Assumption 6. (c) R: A one-unit increase in yesterday’s R will result in a 0.1% increase in today’s Dow Jones average, holding constant the other independent variables in the equation. M: The Dow Jones will fall by 0.017% on Mondays, holding constant the other independent variables in the equation. (d) Technically, C is not a dummy variable because it can’t take on three different values. Saunders assumed (at least implicitly) that all levels of cloud cover between 0% and 20% have the same impact on the Dow and also that all levels of cloud cover between 21% and 99% have the same impact on the Dow. (e) Disagree; Low R2 bar; bad specification; regression does prove causation

Collect Asst 10 • # 9, Page 114 • #11, Page 116

Asst 11 • # 4 on page 151 • # 5 on Page 151

Three Steps in Hypothesis Testing • Step Two • Choose the level of significance (alpha) • Alpha measures the probability of rejecting a true null hypothesis (type I error) • The smaller alpha the smaller the probability of type I error • Find the critical tc (page 585) • Degrees of freedom = n-k-1 • Where n=sample size, k= number of independent variables • Formulate the decision rule

The decision rule And t has the sign implied by HA Otherwise, fail to reject the null hypothesis (where t = t-statistics)

Three Steps in Hypothesis Testing B^ is the estimated B Bnullis the value of B under null hypothesis (usually zero) SE (B^) is the standard error of B^ Step Three Estimate the regression equation and calculate the t- statistic as follows • t-stat. for the null hypothesis equal to zero is reported by EViews. • Apply the decision rule to either • Reject the null hypothesis in favor of alternative hypothesis • Or fail to reject the null hypothesis

Let’s practice Set up the appropriate null and alternative hypotheses for our height- weight equation that has gender in it too Test your hypothesis at alpha = 10 percent. Don’t skip any steps. Evaluate your results. Note: EViews output includes both the standard errors and the t-stats [for null hypotheses that have zero in them (= 0, ≥0 or ≤0 )]

Step 1 • Null and alternative hypotheses for coefficient on height H0: βh ≤0 HA: βh >0 • Null and alternative hypotheses for coefficient on gender H0: βg ≤ 0 HA: βg >0

Step 2 for beta height • α =10% • df = 17-2-1= 14 tc for both coefficients is 1.345 Decision rule If |th or g |>tc, and th or g>0 then reject H0 in favor of HA • 90% confident that height (gender) has a positive effect on weight in the population • If not, fail to reject H0 that height (gender)does not have a significant positive effect on weight

Step 3: Estimate the equation and complete the tests th= 8.27/3.26 = 2.54 tg = -3.06/28 = -0.11 Dependent Variable: W Method: Least Squares Date: 02/22/11 Time: 12:15 Sample: 1 17 Included observations: 17 Variable Coefficient Std. Error t-Statistic Prob. C -417.7228 210.0895 -1.988309 0.0667 H 8.275483 3.261114 2.537625 0.0237 G -3.061951 28.01293 -0.109305 0.9145 |th|>tC , Reject H0 at 10% level . |tg|<tC , Can’t reject H0 at 10% level .

Thursday, February 24 • Exam 2: Tuesday, March 22 • Exam 3: Monday, April 25, 12- 2:30PM

Collect Asst 11 • # 4 on page 151 • # 5 on Page 151

Return and discuss Asst 10 • # 9, Page 114 • #11, Page 116

# 9, Page 114 (part a) • Not obvious problems • May violate Assumption 3 (maybe the higher T, the higher error) • May violate Assumption 4 (maybe in one period there is a report that frozen yogurt adds five years to your life– this assumption has to do with an EXTERNAL SHOCK) • May violate Assumption 5 (maybe the higher T, the higher the variance of error) Note • Assumption 6 has to do with PERFECT LINEAR correlation (you don’t necessarily place an ad in every period that the school is in session)

(b) Holding constant the other independent variables, the store will sell 134.4 more frozen yogurts per two week it places an ad. • (c) School is not in session during the prime yogurt-eating summer months, so the variable might be picking up the summer time increased demand for frozen yogurt from nonstudents. • (d) Price of ice cream

#11, Page 116 • The coefficient of DIVSEP implies that a divorced or separated individual will drink 2.85 more drinks than otherwise, holding constant the other independent variables in the equation. The coefficient of UNEMP implies that an unemployed individual will drink 14.20 more drinks than otherwise, holding constant the other independent variables in the equation. The signs of the estimated coefficients make sense The size of the coefficients may not make sense (14.2>2.85)

The coefficient of ADVICE implies that an individual will drink 11.36 more drinks, holding constant the other independent variables in the equation, if a physician advises them to cut back on drinking alcohol. This coefficient certainly has an unexpected sign! We know that error picks up the effect of missing variables We have a missing variable such as the degree of alcohol addiction. Since alcohol addiction is correlated with error And alcohol addiction is correlated with ADVICE Then ADVICE is correlated with error

(c) We’d expect each sample to produce different estimates of ADVICE. This entire group is called a sampling distribution of -hats. • The estimated ADVICE for this subsample is 8.62, which is a little lower than the coefficient for the entire sample. The other coefficients for this sub-sample differ even more from the coefficients for the entire sample, and the estimated coefficient of EDUC actually has an unexpected sign. These results are clear evidence of the advantages of large samples.

Asst 12: Due on Tuesday in class • Conduct a 5% t-test of significance on all of the coefficients included in the equation you estimated in part d of question 11 on page 116. • Don’t skip any the 3 steps • Show your work • Note: if the exact value of critical t is unavailable, use the closest value that is available to you

P- value The lowest level of significance (alpha) at which you can reject the null hypothesis in a two sided test. EViews reports it

Theses are the p values Dependent Variable: W Method: Least Squares Date: 02/22/11 Time: 12:15 Sample: 1 17 Included observations: 17 Variable Coefficient Std. Error t-Statistic Prob. C -417.7228 210.0895 -1.988309 0.0667 H 8.275483 3.261114 2.537625 0.0237 G -3.061951 28.01293 -0.109305 0.9145 You can reject the null hypothesis that height does not affect weight at 3% level of significance (with 97% level of confidence)

Limitation of Statistical Significance • It does not provide a theoretical validity • It does not test the importance • The importance is determined based on the absolute value of coefficients • Example

Confidence Interval • The range within which β is likely to fall a specified percentage of the time • 95% of the time β = (β hat ± critical t for two sided test at 5% level of significance * SE of β hat) • 90% of the time β = (β hat ± critical t for two sided test at 10% level of significance * SE of β hat)

Asst 13: Build a 95% confidence interval for the coefficients of H and G Dependent Variable: W Method: Least Squares Date: 02/22/11 Time: 12:15 Sample: 1 17 Included observations: 17 Variable Coefficient Std. Error t-Statistic Prob. C -417.7228 210.0895 -1.988309 0.0667 H 8.275483 3.261114 2.537625 0.0237 G -3.061951 28.01293 -0.109305 0.9145

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