Econometrics-I-8 计量经济分析(第六版英文)课件

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.
Applied Econometrics
William Greene Department of Economics Stern School of Business
Inference in the Linear Model
Hypothesis testing: Formulating hypotheses: linear restrictions as a
times
(random vector - hypothesized value)
= Normalized distance measure
= (q - q0 )'[Var(q - q0 )]-1 (q - q0 )
Distributed as chi-squared(J) if (1) the distance is
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.
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*=1.64590% Plausible range for βk then is bk ± z* vk
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:
zero or some other particular value: Is the hypothesized value in the confidence interval? Is the hypothesized value within the range of plausible values
normally distributed and (2) the variance matrix is
the true one, not the estimate.
Robust Tests
The Wald test generally will (when properly constructed) be more robust to failures of the narrow model assumptions than the t or F
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 posteri.
The Wald Statistic
Most test statistics are Wald distance measures
W = (random vector - hypothesized value)' times
[Variance of difference]-1
inconsistent with the sample evidence. Measure distance in standard error units
t = (bk - βk)/Estimated vk. If t is “large” (larger than critical value), reject the
Testing a Hypothesis Using a Confidence Interval
Given the range of plausible values The confidence interval approach. Testing the hypothesis that a coefficient equals
Wald Distance Measure
Testing more generally about a single parameter. Sample estimate is bk Hypothesized value is βk How far is βk from bk? If too far, the hypothesis is