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Unformatted text preview: Econometrics for Finance 1 1 Quantitative Methods in Finance Lecture 2 Inference in Classical Linear Regression Model 2 An Introduction to Statistical Inference • We want to make inferences about the likely population values from the regression parameters. Example: Suppose we have the following regression results: • is a single (point) estimate of the unknown population parameter, β . How “reliable” is this estimate? • The reliability of the point estimate is measured by the coefficient’s standard error. & . β = 0 5091 ) 2561 . ( 5091 . ) 38 . 14 ( 3 . 20 ˆ t t x y + = Econometrics for Finance 2 3 Hypothesis Testing: Some Concepts • We can use the information in the sample to make inferences about the population. • We will always have two hypotheses that go together, the null hypothesis (denoted H ) and the alternative hypothesis (denoted H 1 ). • The null hypothesis is the statement or the statistical hypothesis that is actually being tested. The alternative hypothesis represents the remaining outcomes of interest. • For example, suppose given the regression results above, we are interested in the hypothesis that the true value of β is in fact 0.5. We would use the notation H 0 : β = 0.5 H 1 : β ≠ 0.5 This would be known as a two sided test. 4 OneSided Hypothesis Tests • Sometimes we may have some prior information that, for example, we would expect β > 0.5 rather than β < 0.5. In this case, we would do a onesided test: H 0 : β = 0.5 H 1 : β > 0.5 or we could have had H 0 : β = 0.5 H 1 : β < 0.5 • There are two ways to conduct a hypothesis test: via the test of significance approach or via the confidence interval approach. Econometrics for Finance 3 5 The Probability Distribution of the Least Squares Estimators • We assume that u t ∼ N(0, σ 2 ) • Since the least squares estimators are linear combinations of the random variables i.e. • The weighted sum of normal random variables is also normally distributed, so ∼ N( α , Var( α )) ∼ N( β , Var( β )) • What if the errors are not normally distributed? Will the parameter estimates still be normally distributed? • Yes, if the other assumptions of the CLRM hold, and the sample size is sufficiently large. & β = & w y t t & α & β 6 The Probability Distribution of the Least Squares Estimators (cont’d) • Standard normal variates can be constructed from and : and • But var( α ) and var( β ) are unknown, so and & α & β ( ) ( ) 1 , ~ var ˆ N α α α ( ) ( ) 1 , ~ var ˆ N β β β 2 ~ ) ˆ ( ˆ T t SE α α α 2 ~ ) ˆ ( ˆ T t SE β β β Econometrics for Finance 4 7 Testing Hypotheses: The Test of Significance Approach • Assume the regression equation is given by , for t =1,2,..., T • The steps involved in doing a test of significance are: 1. Estimate , and , in the usual way 2. Calculate the test statistic. This is given by the formula where is the value of β under the null hypothesis.under the null hypothesis....
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 Spring '10
 C.Favero
 Finance

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