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Econ 488

Econ 488. Lecture 5 – Hypothesis Testing Cameron Kaplan. Classical Assumptions. Regression is linear, correctly specified, and has additive error term E( ε i )=0 Correlation between X ki and ε i is 0 for all k. ε t is uncorrelated with ε t +1 for all t.

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Econ 488

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  1. Econ 488 Lecture 5 – Hypothesis Testing Cameron Kaplan

  2. Classical Assumptions • Regression is linear, correctly specified, and has additive error term • E(εi)=0 • Correlation between Xki and εi is 0 for all k. • εtis uncorrelated with εt+1 for all t. • Var(εi)=σ2 [No Heteroskedasticity] • No perfect multicollinearity and sometimes: • εi~N(0,σ2)

  3. Sampling Distribution of • is assumed to be normally distributed because the stochastic error is assumed to be normally distributed (assumption 7) • Usually, we take a sample of size N from a population to produce a single estimator of β, which we call . • But what if we took a different sample? • We should get a different result for

  4. Sampling Distribution of β

  5. Sampling Distribution of • In OLS, is unbiased, so E( )=β • OLS estimators also have the smallest variance possible at any sample size (efficiency) • Finally, OLS estimators are consistent. As N increases, variance shrinks. • As N->∞, β->

  6. Consistency

  7. Hypothesis Testing • Most times, we only take one sample, so we only get one estimate of • How do we know if is meaningful I we can only observe one value in the distribution?

  8. Example • Suppose we are interested in whether school size has an effect on student performance. • Specifically, do students at small schools do better? • We estimate the following equation: • math10i = β0+β1enrolli+β2staffi+β3totcompi+εi

  9. Example • math10i = β0+β1enrolli+β2staffi+β3totcompi+εi • Where: • math10 = % of students passing the 10th grade math portion of the Michigan Educational Assessment Program (MEAP) test • enroll = school size • staff = number of staff/1000 students (to control for how much attention students get) • totcomp = average annual teaching compensation (to control for teacher quality)

  10. Hypothesis Testing • We need to develop a null and alternative hypothesis before running the regression. • Null Hypothesis (H0) • Usually, you want to reject the null hypothesis • Most common null hypothesis: “there is no effect of X on Y” or “β1=0” • Alternative Hypothesis (HA or H1) • Usually, what you are trying to prove

  11. Hypothesis Testing • In our example, we would pick • H0:β1≥0 “there is no negative effect of school size on student performance” • HA:β1<0 “There is a negative effect of school size on student performance” • Test this using meap93.gdt

  12. Example 2 • Consider the wage equation • log(wagei)=β0+β1educi+β2exeri+β3tenurei+εi • The null hypothesis H0: β2=0 says: • once education and tenure have been accounted for, the number of years in the workforce has no effect on hourly wage • If β2>0, prior work experience contributes to productivity, and to wage.

  13. Alternative Hypothesis • Usually, we want to reject the null hypothesis. • We form an alternative hypothesis – values we don’t expect. • One-sided Alternatives • We expect there to be a sign on a particular variable based on our economic model • e.g. HA: βK>0.

  14. Hypothesis Testing • log(wagei)=β0+β1educi+β2exeri+β3tenurei+εi • In our example, we might set our hypotheses as • H0:β2≤0 • HA:β2>0 • We believe that the effect of experience on wages is positive, holding education and tenure fixed.

  15. Hypothesis Testing • log(wagei)=β0+β1educi+β2exeri+β3tenurei+εi • What should the null and alternative hypotheses for the other coefficients be? • H0:β1≤0 • HA:β1>0 • H0:β3≤0 • HA:β3>0

  16. Two sided alternatives • Yi=β0+β1X1i+…+βkXki+εi • H0:β1=0 • HA:β1≠0 • Under the alternative, X1i has a significant effect on the dependent variable without specifying if it’s positive or negative • You should use this if you don’t know what sign βk has (not well defined by theory) • Or…sometimes it is better to use because it prevents us from forming our hypothesis after looking at the results

  17. Other Hypotheses • Although H0:βk=0 is the most common null hypothesis, sometimes, we want to test whether or not βk is equal to some other constant – usually 1 or -1. • Example: Suppose we want to look at the effect of college enrollment on crime. • log(crimei)=β0+β1log(enrolli)+εi • This is a constant elasticity model, where β1 is the elasticity of crime with respect to enrollment.

  18. Other hypotheses • log(crimei)=β0+β1log(enrolli)+εi • We could test, H0:β1=0 & HA:β1≠0 • But more interesting would be to test if β1=1 • If β1>1, then a 1% increase in enrollment leads to a greater than 1% increase in crime, so crime is a bigger problem at large campuses • Set up our hypotheses as follows • H0:β1=1 • HA:β1≠1

  19. t-test • Yi=β0+β1X1i+…+βkXki+εi • t-statistic: • = estimated regression coefficient of the kth variable • = The border value (usually zero) implied by the null hypothesis • = The estimated standard error of the coefficient on the kthvariable

  20. t-test • For example, suppose our hypotheses were: • H0:β1=0 • HA:β1>0 • Then, suppose that we estimate that =6, and that =2 • We would calculate t as

  21. How does the t-test work? Distribution of if null is true β1 Suppose we found a value of way out here It’s not very likely that the null hypothesis is true…

  22. t-test • How does this look for our example? • =6 and =2 0 -2 2 6

  23. t-test • We want to know, if H0 really is true (i.e. β1 really is 0), how likely is it that we could have observed a value of 6? • Not very. • We can probably say that H0 is not true. • But we need a rule to decide.

  24. Hypothesis Testing • How do we decide when to reject the null? • Choose a level of significance • Rule of thumb: 5% level of significance • This means that we will rule out H0if we would have expected a value of at least as extreme as 6 less than 5% of the time. • Instead of trying to figure out this probability using the sampling distribution, we transform the distribution to the t-distribution • The t-distribution is almost the same as the standard normal distribution.

  25. t-test • In our example, t=6-0/2 = 3 • Suppose our sample size was 23 • We need to compare our t-statistic to the critical t-value, which distinguishes the acceptance region from the rejection region. • Look at inside cover of book • We want the t-value for 23-2-1= 20 degrees of freedom. • For a one sided test with 5% significance, this is tc=1.725 • Decision Rule: Reject H0 if |tk|>tc, and has the sign implied by HA, otherwise do not reject. • Here, we reject the null in favor of the alternative, suggesting that X1 is significant

  26. Choosing a Level of Significance • Rule of thumb – Significance level = 5% • If significance level is too low, we risk what is called a type II error, where we reject the null hypothesis when it is actually true. • If we reject H0 at the 5% level, we say that the coefficient is “statistically significant at the 5% level” • Sometimes researchers use asterisks • * means significant at 10% • ** means significant at 5% • *** means significant at 1%

  27. Confidence Intervals • Confidence Interval - The range that contains the population value a specified percent of the time. • The two-sided t-critical value at a specific significance level gives the (1-sig level) confidence interval. • So, the 5% significance level is equivalent to the 95% CI.

  28. Confidence Intervals • For our example, the t-critical value was 2.086 • So the 95% CI= 6 ± 2*2.086 = 6±4.172 • Or 1.828 to 10.172 • We could say that with 95% confidence, the true value of β is between 1.828 and 10.172 • Notice that 0 is not in this range. • We can reject H0

  29. P-value • Alternative to t-test • If the true population value was really 0, what is the probability we would have observed a value as extreme as 6? • If p is small, reject the null. • This is calculated automatically by most econometrics software • Reject the null if p is less than the significance level. 0 -2 2 6

  30. Example • Student performance and school size using data.

  31. F-test (Appendix Ch. 5) • What if you want to test a hypothesis that involves multiple coefficients? • For example: Suppose we run this regression (data7-2.gdt): • wagei= β0+β1educi+β2experi+β3clericali+β4mainti+β5craftsi+εi • clerical, maint, and crafts are job type “dummies” • We want to test whether job type matters • We would need to test whether β3, β4, and β5 are “jointly significant. • H0:β3=β4=β5=0 • HA: The null hypothesis is not true.

  32. F-test • Steps • 1. Run full regression, get RSS • 2. Run constrained regression (without job type variables), get RSSM • RSS = RSS from step 1 • RSSM = RSS from step 2 • M = # of excluded coeffs • N = # observations • K = # of coefficients in overall equation

  33. F-stat • Calculate F-stat, and compare it to the critical value of F (from F-table) • Degrees of freedom numerator = M • Degrees of freedom denominator = N-K-1 • If F>Fcrit reject null hypothesis • The variables are jointly significant if you can reject the null.

  34. F-test • In Gretl • Run the model • Select test>omit variables • Gives F-stat and related p-value

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