Part 8 hypothesis testing methodology p bayesian n

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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 ™    11/50
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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. ™    12/50
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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 = 0 , Substantive restrictions: What is a "testable hypothesis?"  Substantive restriction on parameters  Reduces dimension of parameter space  Imposition of restriction degrades estimation criterion ™    13/50
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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 ™    14/50
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Part 8: Hypothesis Testing The General Linear Hypothesis: H0: R - q = 0 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. ™    15/50
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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?). Ru2 = unrestricted model, Rr2 = restricted model fit. F = { (Ru2 – Rr2)/J } / [(1 – Ru2)/(n-K)] = F [J,n-K]. (2) Is Rb - q close to 0 ? Basing the test on the discrepancy vector: m = Rb - q. Using the Wald criterion : m (Var[ m ])-1 m has a chi-squared distribution with J degrees of freedom But, Var[ m ] = R [2( X’X )-1] R . If we use our estimate of 2, we get an F[J,n-K], instead. (Note, this is based on using ee /( n - K ) to estimate 2.) These are the same for the linear model ™    16/50
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Part 8: Hypothesis Testing Testing Fundamentals - I p SIZE of a test = Probability it will incorrectly reject a “true” null hypothesis.
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