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Econometrics-I-8

# &#152&#152;™™™™™ ™ 9/50 part 8

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Unformatted text preview: &#152;&#152;™™™™™ ™ 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 &#152;&#152;™™™™ ™ 10/50 Part 8: Hypothesis Testing Methodology p Bayesian n Prior odds compares strength of prior beliefs in two states of the world n Posterior odds compares revised beliefs n Symmetrical treatment of competing ideas n Not generally practical to carry out in meaningful situations p Classical n “Null” hypothesis given prominence n Propose to “reject” toward default favor of “alternative” n Asymmetric treatment of null and alternative n Huge gain in practical applicability &#152;&#152;™™™™ ™ 11/50 Part 8: Hypothesis Testing Neyman – Pearson Methodology p Formulate null and alternative hypotheses p Define “Rejection” region = sample evidence that will lead to rejection of the null hypothesis. p Gather evidence p Assess whether evidence falls in rejection region or not. &#152;&#152;™™™™ ™ 12/50 Part 8: Hypothesis Testing Inference in the Linear Model Formulating hypotheses: linear restrictions as a general framework Hypothesis Testing J linear restrictions Analytical framework: y = X + Hypothesis: R- q = , Substantive restrictions: What is a "testable hypothesis?" Substantive restriction on parameters Reduces dimension of parameter space Imposition of restriction degrades estimation criterion &#152;&#152;™™™™ ™ 13/50 Part 8: Hypothesis Testing Testable Implications of a Theory p Investors care about interest rates and expected inflation: I = b1 + b2r + b3dp + e p Investors care about real interest rates I = c1 + c2(r-dp) + c3dp + e No testable restrictions implied. c1 =b1, c2=b2-b3, c3=b3. p Investors care only about real interest rates I = f1 + f2(r-dp) + f3dp + e. f3 = 0 &#152;&#152;™™™™ ™ 14/50 Part 8: Hypothesis Testing The General Linear Hypothesis: H0: R - q = A unifying departure point: Regardless of the hypothesis, least squares is unbiased. E[ b ] = The hypothesis makes a claim about the population R – q = 0. Then, if the hypothesis is true, E[Rb – q] = 0. The sample statistic, Rb – q will not equal zero. Two possibilities: Rb – q is small enough to attribute to sampling variability Rb – q is too large (by some measure) to be plausibly attributed to sampling variability Large Rb – q is the rejection region. &#152;&#152;™™™™ ™ 15/50 Part 8: Hypothesis Testing Approaches to Defining the Rejection Region (1) Imposing the restrictions leads to a loss of fit. R2 must go down. Does it go down “a lot?” (I.e., significantly?)....
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&#152&#152;™™™™™ ™ 9/50 Part 8 Hypothesis...

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