Marietta College

Marietta College Spring 2011 Econ 420: Applied Regression Analysis Dr. Jacqueline Khorassani Week 8

Tuesday, March 1 Exam 2: Tuesday, March 22 Exam 3: Monday, April 25, 12- 2:30PM

Return and discuss Asst 11 • # 4 on page 151 • # 5 on Page 151

Note • Null hypothesis is what you expect to reject • Alternative hypothesis is what you expect to be true. • This is the reason for your research

Question 4, Page 151 • When is it important to minimize type I error? • Type I error: Reject a true null hypothesis • Seth’s example • Sally’s example

When is it important to minimize type II error? • Type II error: Fail to reject a false null hypothesis • Tina’s example • Yuan’s example • Charlotte's example

# 5 on Page 151 • For βN: Reject H0: βN ≥0 if |-4.42| > tcand betahat is negative. • For βP: Reject H0: βP≤0 if |4.88| > tcand betahat is positive • For βI: Reject H0: βI≤0 if |2.37| > tcand betahat is positive • (a) tC = 1.943; reject the null hypothesis for all three coefficients. • (b) tC= 1.311; reject H0 for all three coefficients. • (c) tC = 6.965; cannot reject the null hypothesis for any of the three coefficients.

Collect Asst 12 • 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

Return and discuss Asst 13 • 95% of the time βH = (8.27 ± 2.131 * 3.26) • 95% of the time βH = (8.27 ± 6.95) • 95% of the time (1.32 ≤βH ≤ 15.22) • 95% of the time βG = (-3.06 ± 2.131 * 28) • 95% of the time βG = (-3.06 ± 59.67) • 95% of the time (-62.73 ≤βG ≤ 56.61)

Chapter 6: Effects of Omitted Variables • Example • True equation is Yi = f (X1i,X2i) Where Yi = GPA of the ith individual X1i= hours of study of the ith individual X2i = IQ score of the ith individual • True equation will be Yi= β0 + β1X1i + β2X2i + єi

We fail to include X2iin our model • How would the model look now? • Will it have the same error term? Yi= β0 + β1X1i + ui • What is the error equal to? ui= β2X2i+ єi • Does this violate any assumptions? • Go back and study the assumptions to answer this question (Page 94) • Violates assumption 1. Why? • May violate assumption 3. Why? • If X1 is correlated with X2 • Since X2 is correlated with u • X1 will be correlated with u too • Other assumptions?

Effects of Omitted Variables • OLS may not be BLUE • The estimated coefficient of X1 (that is, B^1) may be biased • Graph • That is, E(β^1) ≠ β1 • Instead E(β^1) = β1 + Bias • Where Bias = β2 . f(rX1,X2) • Bias depends on the f(rX1,X2)(a function of (multiple of) the correlation coefficient between X1 & X2) and the coefficient of X2 in true regression line.

The Sign (Direction) of Bias • Under what condition is the bias zero? • Bias is zero • if X2 does not affect Y (β2is zero), or • if X2 is not correlated with X1 • How do you expect IQ (X2) to affect GPA (Y)? • positively • How are IQ (X2) and Hours of study (X1) correlated? • ? • What is the direction of bias in our example? • Will β^1 be bigger or smaller than it actually should be?

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

Return and discuss Asst 12 • 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 steps • Show your work

Overall results • Great! • Some lost a few points! • Any volunteers to say why they lost points?

Problem Areas • A couple of you did not use the right sample • What is wrong with this formulation? Yi= β0 + β1X1i + β2X2i Remember that if you don’t include an error term in your formulation, you need to make sure your βs wear hats. 3. Alternative hypothesis is what you expect to be true. • It is formulated before your estimate the equation based on your theory • How do you expect ADVICE & EDUC to affect drinking? • We expect the coefficients on ADVICE & EDUC to be negative. • We expect the coefficients on DIVSEP & UNEMP to be positive • You only formulate a two-sided test IFF there is no reason (theory) that supports a one-sided test.

Problem Areas 4. About the critical t • You are doing one-sided test in all cases • If the table does not provide the exact critical t for your degrees of freedom, choose the closest critical t. • 1.658 in this case

Let’s fast forward to Wednesday October 26, 201111 am • Nate comes to my office for his personal meeting about his Econ 421 project. • Nate is not happy! • He doesn’t understand his estimation results!

His project? • Determinants of crime rate • Sample: 50 US states in 2010 • Model Yi= β0 + β1 X1i + β2 X2i + єi Yi is crime rate in state i X1i is per capita income in state i X2i is the number of police officers per 10,000 population in state i

Problem? • He expected the coefficient on state per capita income (X1i) to be negative; but his estimated coefficient is positive • What might have caused it? • An omitted variable • Is the bias in the estimated coefficient of income X1ipositive or negative? • Let’s see the graph of distribution of ^1 • Bias in ^1 is positive • Let’s call the omitted variable X3i • What is the approximate size of the bias in ^1? • Bias in ^1 #β3 * rX1,X3> 0

Bias in ^1#β3 * rX1,X3>0 • Asst 14: Can you help Nate to figure out which of the following omitted variables is more likely to have caused the bias in his estimated coefficient on income? Explain your answer. • Number of college educated individuals per 1000 population • Percentage of population living in poverty • State’s unemployment rate • Percentage of population who lives in urban areas.

Correction for Omitted Variables • Study the theoretical literature again • Include the omitted variable based on the Expected bias analysis

Asst 15: Due Tuesday, March 15 in class • Recall our height-weight regression • Estimate the regression that has gender and height as its independent variables • Is the coefficient of gender likely to be biased? Why or why not? • Suppose that we suspect the coefficient of gender to be biased downward. Suggest an omitted variable that is likely to be the cause of this bias. Discuss your reasoning.

Irrelevant Variable Problem • Suppose the true regression model GPA = f (Hours of study) + error • Our model GPA = f (Hours of Study, Weight ) + error Yi= β0 + β1X1i + β2X2i + єi • Does our model violate assumption 1? • Yes, incorrect specification • Any other assumptions are violated? • Not if weight is truly an irrelevant variable • If weight is truly an irrelevant variable, the estimated coefficient on hours of study is unbiased.

Does the estimated coefficient on X1i have the minimum variance? No, • Because we are wasting our degrees of freedom to estimate an extra coefficient unnecessarily • Graph

So the estimated coefficient onX1iis not efficient How does this affect t-test? • variance of the estimated coefficient on hours of study goes up • standard error of the estimated coefficient of hours of study goes up • What does this do to t-statistic? • Goes down • What does this do to probability of not rejecting a false null hypotheses? • Goes up • t-test may suggest an insignificant coefficient of hours of study while it shouldn’t. • What type of error is this? • Type II

Should we include X in the set of our independent variable? • Yes, if (order matters) • Theory calls for its inclusion (the most important criterion) • t- test: the estimated coefficient of X is significant in the right direction • Note: this does not mean that if the estimated coefficient is insignificant you have to drop the variable from your model. • As you include X, the adjusted R squared goes up. • As you include X, the other variables’ coefficients change significantly. • As you include X, AIC and SC go down

Akaike’s Information Criterion (AIC) Schwarz Criterion (SC) • AIC = log (RSS/N) + 2 (K+1)/N • SC = log (RSS/N) + log (N)(K+1)/N • Note: As you include X K goes up and RSS goes down or stays the same • EViews reports both

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