Hypothesis Testing in the Linear Regression Model参考.ppt

* * * * * * * * * * * * * * * * * * Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 8. Hypothesis Testing in the Linear Regression Model Classical Hypothesis Testing We are interested in using the linear regression to establish 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. Inference in the Linear Model Hypothesis testing: Formulating hypotheses: linear restrictions as a general framework Substantive restrictions: What is a testable hypothesis? Nested vs. nonnested models Methodological issues Classical (likelihood based approach): Are the data consistent with the hypothesis? Bayesian approach: How do the data affect our prior odds? ? The posterior odds ratio. Testing a Hypothesis About a Parameter: Confidence Interval bk = the point estimate Std.Dev[bk] = sqr{[σ2(X’X)-1]kk} = vk Assume normality of ε for now: bk ~ N[βk,vk2] for the true βk. (bk-βk)/vk ~ N[0,1] Consider a range of plausible values of βk given the point estimate bk. bk +/- sampling error. Measured in standard error units, |(bk – βk)/ vk| z* Larger z* ? greater probability (“confidence”) Given normality, e.g., z* = 1.96 ? 95%, z*=1.645?90% Plausible range for βk then is bk ± z* vk Estimating the Confidence Interval Assume normality of ε for now: bk ~ N[βk,vk2] for the true βk. (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. Testing a Hypothesis Using a Confidence Interval Given the range of plausible values The confidence interval approach. Testing the hypothesis that a coefficient equals zero or some other particular value: Is the hypothesized value in the c