Econometrics-I-8 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Business
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Applied Econometrics 8. Hypothesis Testing in the Linear Regression Model
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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.
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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. 
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Testing a Hypothesis About a Parameter: Confidence Interval b k  = the point estimate Std.Dev[b k ] = sqr{[ σ 2 ( X’X ) -1 ] kk } = v k Assume normality of  ε  for now:  b k  ~ N[ β k ,v k 2 ] for the true  β k . (b k - β k )/v ~ N[0,1] Consider a range of plausible values of  β k  given the point  estimate b k .  b k  +/- sampling error. Measured in standard error units,  |(b k  –  β k )/   v k | <  z* Larger z*   greater probability (“confidence”) Given normality, e.g., z* = 1.96   95%, z*=1.645 90% Plausible range for  β k  then is b k  ± z* v k
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Estimating the Confidence Interval Assume normality of  ε  for now:  b k  ~ N[ β k ,v k 2 ] for the true  β k . (b k - β k )/v ~ N[0,1] v k  = [ σ 2 ( X’X ) -1 ] kk  is not known because  σ must be  estimated. Using s 2  instead of  σ 2 , (b k - β k )/est.(v k ) ~ 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.  
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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 confidence  interval?   Is the hypothesized value within the range of  plausible values
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Wald Distance Measure Testing more generally about a single parameter.  Sample estimate is b k Hypothesized value is  β k How far is  β k  from b k ?  If too far, the hypothesis is  inconsistent with the sample evidence.  Measure distance in standard error units
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Econometrics-I-8 - Applied Econometrics William Greene...

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