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Welcome to Econ 420 Applied Regression Analysis

Learn about hypothesis testing in applied regression analysis. Understand the steps involved, two-sided and one-sided hypotheses, type I and type II errors, and their implications.

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Welcome to Econ 420 Applied Regression Analysis

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  1. Welcome to Econ 420 Applied Regression Analysis Study Guide Week Four Ending Wednesday, September 19 (Assignment 4 which is included in this study guide is due before Wednesday, September 19 at 10:00 PM)

  2. Notes • Exam 1 • Thursday, September 27 at 8:00 A.M. in Thomas 223 • Covers Chapters 1, 2, and 3 • Assignment 3 • Will be grades and emailed to you soon

  3. Answer Key to Assignment 2:Here is how your EViews output should look. • Dependent Variable: W • Method: Least Squares • Date: XX/XX/07 • Time: XX:XX • Sample: 1 8 • Included observations: 8 • Variable Coefficient Std. Error t-Statistic Prob. C - 231.1391 92.54581 -2.497565 0.0467 H 5.879205 1.314659 4.472039 0.0042 R-squared 0.769223 Mean dependent var 181.8750 Adjusted R-squared 0.730760 S.D. dependent var 32.39681 S.E. of regression 16.81016 Akaike info criterion 8.694162 Sum squared resid 1695.489 Schwarz criterion 8.714023 Log likelihood -32.77665 F-statistic 19.99913 Durbin-Watson stat 1.128086 Prob(F-statistic) 0.004228

  4. What you should have recognized. • Coefficient value -231.1391 represents the W intercept (when H = 0) of the regression equation. • Coefficient value 5.879205 represents the slope of the estimated regression line (the value of W increases by 5.879205 when the Value of H increases by 1 inch, ignoring the effect of other variables on W. • The mean of the dependant variable value weight is 181.8750= the average weigh value of the 8 observations • The standard deviation of the dependent variable value 32.39681 is the average weight distance of the observations away from the mean. • The sum of the squared residuals = 1695.489 = is the sum of the squared differences between the predicted Ws and the actual (See Equation 1-8 on Page 12)

  5. What you could not recognize • Everything else on the output.

  6. Hypothesis Testing (Chapter 3, up to page 53) • Remember we don’t see the true line • So we don’t know the true intercept or the slope coefficients. • We collect a sample. • We use OLS to estimate the coefficients. • Hypothesis testing refers to using sample information to draw a conclusion about the true population coefficient .

  7. Four Steps of Hypothesis Testing Step One • Set the null and alternative hypotheses about the true coefficients. • Alternative hypothesis is consistent with our common sense or theory. It is what we expect to find. It is what we expect to fail to reject. • Null hypothesis is what we expect to not find. It is what we expect to reject.

  8. Two Sided Hypotheses • Suppose all you want to show is that something affects something else. But you don’t want to show the direction of the relationship • In this case, you will set a two sided hypothesis • Example • All you want to show is that calorie intake does affect the weight. • In this case you would set up a two sided test.

  9. Suppose B3 is the true coefficient of calorie intake, then the two sided hypothesis looks like this

  10. You expect to • Reject the null hypothesis H0 in favor of alternative hypothesis, HA • And if you do, • You have found empirical evidence that calorie intake does matter. • However, you are not making any statements with regards to the nature of the relationship between calorie in take and weight.

  11. One sided hypotheses • Suppose you want to show that something has a positive (or negative) effect on something else. In this case, you will set a one sided hypothesis • Example • you want to show that calorie intake affect the weight in a positive way • In this case you would set up a one sided test

  12. Suppose B3 is the true coefficient of calorie intake, then the one sided hypothesis looks like this • H0: B3 ≤0 • HA: B3 >0

  13. You expect to • Reject the null hypothesis H0 in favor of alternative hypothesis, HA • And if you do, • You have found empirical evidence that calorie intakes have a positive effect on weight. • Notes • A one sided test is stronger than a two sided test. • You must test a stronger hypothesis when possible.

  14. Type I Errors • Refers to rejecting a true null hypothesis • Example • Weight = ……..+ B3 calorie in take + error • H0: B3 ≤0 • HA: B3 >0 • If you reject a true H0, you will conclude that the higher the calorie intake the higher the weight; while in reality, calorie intake does not matter.

  15. Type II Error • Refers to failing to reject a false null hypothesis • Example • Weight = ……..+ B3 calorie in take + error • H0: B3 ≤0 • HA: B3 >0 • If you fail to reject a false H0, you will conclude that calorie intake does not matter; while in reality, it does.

  16. Type I/Type II Errors

  17. Which type of error is more serious? • –   Type I error: We conclude that the higher calorie in take, the height the weight. (While in reality there is no correlation between the two.) So we put a lot of effort into watching our calorie intake while we should not. • –   Type II error: We conclude that calorie intake does not affect the weight, while it actually does. So, we do not watch our diet. (no effort) • –   When testing hypothesis we try to minimize type I error

  18. Four Steps of 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 312) • Degrees of freedom = n-k-1 • Where n=sample size, k= number of independent variables • Formulate the decision rule

  19. The decision rule Otherwise, fail to reject the null hypothesis; where t = t-statistics And t- stat is B^ is the estimated B Bnullis the value of B under null hypothesis (usually zero) SE (B^) is the standard error of B^

  20. Four Steps of Hypothesis Testing • Step Three • Estimate the regression equation and find the t- statistic • Formula (Page 48) • T-stat. for the null hypothesis equal to zero is reported by EViews.

  21. Four Steps of Hypothesis Testing • Stop Four • Apply the decision rule to either • Reject the null hypothesis • Or fail to reject the null hypothesis

  22. Assignment 4 (Due before 10 PM on Wednesday, September 19, 30 points) • Set up the appropriate null and alternative hypotheses for our height- weight equation that we estimated before. Test your hypothesis at alpha = 10 percent. Don’t skip any steps. Evaluate your results. {Note: EViews output includes both the stand errors and the t-stats [for null hypotheses that have zero in them (= 0. ≥0 or ≤0 ]} • #3, Page 61 • # 8, Page 62

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