Econometrics-I-8

# Econometrics-I-8 - Econometrics I Professor William Greene...

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Part 8: Hypothesis Testing Econometrics I Professor William Greene Stern School of Business Department of Economics

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Part 8: Hypothesis Testing Econometrics I Part 8 – Interval Estimation  and Hypothesis Testing ™    1/50
Part 8: Hypothesis Testing Interval Estimation p b = point estimator of p We acknowledge the sampling variability. n Estimated sampling variance n b = + sampling variability induced by  ™    2/50

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Part 8: Hypothesis Testing p Point estimate is only the best single guess p Form an interval, or range of plausible values p Plausible  likely values with acceptable degree of probability. p To assign probabilities, we require a distribution for the variation of the estimator. p The role of the normality assumption for  ™    3/50
Part 8: Hypothesis Testing Confidence Interval bk = the point estimate Std.Err[bk] = sqr{[σ2( XX )-1]kk} = vk Assume normality of ε for now: n bk ~ N[βk,vk2] for the true βk. n (bk-βk)/vk ~ N[0,1] Consider a range of plausible values of βk given the point estimate bk. bk sampling error. n Measured in standard error units, n |(bk – βk)/ vk| < z* n Larger z*  greater probability (“confidence”) n Given normality, e.g., z* = 1.96  95%, z*=1.64590% n Plausible range for βk then is bk ± z* vk ™    4/50

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Part 8: Hypothesis Testing Computing the Confidence Interval Assume normality of ε for now: n bk ~ N[βk,vk2] for the true βk. n (bk-βk)/vk ~ N[0,1] vk = [σ2( X’X )-1]kk is not known because σ2 must be estimated. Using s2 instead of σ2, (bk-βk)/Est.(vk) ~ t[n-K]. (Proof: ratio of normal to sqr(chi-squared)/df is pursued in your text.) Use critical values from t distribution instead of standard normal. Will be the same as normal if n > 100. ™    5/50
Part 8: Hypothesis Testing Confidence Interval Critical t[.975,29]  =  2.045 Confidence interval based on t:           1.27365 –  2.045 * .1501 Confidence interval based on normal: 1.27365 –  1.960 * .1501 ™    6/50

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Part 8: Hypothesis Testing Testing a Hypothesis Using a Confidence Interval Given the range of plausible values Testing the hypothesis that a coefficient equals zero or some other particular value: Is the hypothesized value in the confidence interval? Is the hypothesized value within the range of plausible values? If not, reject the hypothesis. ™    7/50
Part 8: Hypothesis Testing Test a Hypothesis About a Coefficient ™    8/50

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Part 8: Hypothesis Testing Classical Hypothesis Testing We are interested in using the linear regression to support or cast doubt on the validity of a theory about the real world counterpart to our statistical model. The model is used to test hypotheses about the underlying data generating process. ™    9/50
Part 8: Hypothesis Testing Types of Tests p Nested Models: Restriction on the parameters of a particular model y = 1 + 2x + 3z + , 3 = 0 p Nonnested models: E.g., different RHS variables yt = 1 + 2xt + 3xt-1 + t yt = 1 + 2xt + 3yt-1 + wt p Specification tests:  ~ N[0,2] vs. some other distribution ™    10/50

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