Econ 399 Chapter4c - -Using our CLM assumptions we can...

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4.3 Confidence Intervals -Using our CLM assumptions, we can construct CONFIDENCE INTERVALS or CONFIDENCE INTERVAL ESTIMATES of the form: ) ˆ ( * ˆ j j se t CI -Given a significance level α (which is used to determine t*), we construct 100(1- α)% confidence intervals -Given random samples, 100(1- α)% of our confidence intervals contain the true value B j -we don’t know whether an individual confidence interval contains the true value
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4.3 Confidence Intervals -Confidence intervals are similar to 2-tailed tests in that α/2 is in each tail when finding t* -if our hypothesis test and confidence interval use the same α: 1) we can not reject the null hypothesis (at the given significance level) that B j =a j if a j is within the confidence interval 2) we can reject the null hypothesis (at the given significance level) that B j =a j if a j is not within the confidence interval
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4.3 Confidence Example -Going back to our Pepsi example, we now look at geekiness: 43 N 62 . 0 5 . 0 3 . 0 3 . 4 ˆ 2 21 . 0 25 . 0 1 . 2 R Pepsi Geek ol o C -From before our 2-sided t* with α=0.01 was t*=2.704, therefore our 99% CI is: ] 976 . 0 , 376 . 0 [ ) 25 . 0 ( 704 . 2 3 . 0 ) ˆ ( * ˆ CI CI se t CI j j
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4.3 Confidence Intervals -Remember that a CI is only as good as the 6 CLM assumptions: 1) Omitted variables cause the estimates (B j hats) to be unreliable -CI is not valid 2) If heteroskedasticity is present, standard error is not a valid estimate of standard deviation -CI is not valid 3) If normality fails, CI MAY not be valid if our sample size is too small
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4.4 Complicated Single Tests -In this section we will see how to test a single hypothesis involving more than one B j -Take again our coolness regression: 43 N 62 . 0 5 . 0 3 . 0 3 . 4 ˆ 2 21 . 0 25 . 0 1 . 2 R Pepsi Geek ol o C -If we wonder if geekiness has more impact on coolness than Pepsi consumption: 2 1 2 1 0 : : a H H
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4.4 Complicated Single Tests -This test is similar to our one coefficient tests, but our standard error will be different -We can rewrite our hypotheses for clarity: 0 : 0 : 2 1 2 1 0 a H H -We can reject the null hypothesis if the estimated difference between B 1 hat and B 2 hat is positive enough
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4.4 Complicated Single Tests -Our new t statistic becomes: ) ˆ ˆ ( ˆ ˆ 2 1 2 1 se t -And our test continues as before: 1) Calculate t 2) Pick α and calculate t* 3) Reject if t<t*
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4.4 Complicated Standard Errors -The standard error in this test is more complicated than before -If we simply subtract standard errors, we may end up with a negative value -this is theoretically impossible -se must always be positive since it estimates standard deviations
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4.4 Complicated Standard Errors -Using the properties of variances, we know that: ) ˆ , ˆ ( 2 ) ˆ ( ) ˆ ( ) ˆ ˆ ( 2 1 2 1 2 1 Cov Var Var Var -Where the variances are always added and the covariance always subtracted -transferring to standard deviation, this becomes: 12 2 2 2 1 2 1 2 )} ˆ ( { )} ˆ ( { ) ˆ ˆ ( s se se se
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