EMET2007 Lecture 4

# We replace var bi with var bi r d note that stata

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Unformatted text preview: Var bi s N (0, 1) dβ we do not know Var bi . We replace Var bi with Var bi . β β r dβ β Note that Stata reports se bi = Var bi so we compute b β βi ri dβ Var bi Lecture 4 (econometrics and simple linear regression) = b βi βi se bi β EMET2007/6007 s tn 2 13 th March 2013 12 / 50 Under assumptions SLR 1 - SLR 6: b s N β , Var b βi βi i b βi βi β se bi s N (0, 1) The estimators are normally distributed around the true parameters with the variance that was derived earlier The standardized estimators follow a standard normal distribution Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 13 / 50 Testing hypotheses about a population parameter Under assumptions SLR 1 - SLR 6: (t-distribution for standardized estimators) b βi βi s tn 2 β se bi If the standardization is done using the estimated standard deviation (se bi = standard error), the normal distribution is replaced by a β t-distribution Note: The t-distribution is close to the standard normal distribution if n 2 is large. Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 14 / 50 Null hypothesis (for more general hypotheses, see below) H0 : β 1 = 0 The population parameter is equal to zero, i.e. after controlling for the other independent variables, there is no e¤ect of x on y t statistic (or t ratio) tb = β 1 b βi se bi β The t-statistic will be used to test the above null hypothesis. The farther the estimated coe¢ cient is away from zero, the less likely it is that the null hypothesis holds true. But what does “far” away from zero mean? Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 15 / 50 Distribution of the t-statistic if the null hypothesis is true tb = β 1 b βi β se bi s tn 2 Goal: De…ne a rejection rule so that, if it is true, H0 is rejected only with a small probability (= signi…cance level, e.g. 5%) Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 16 / 50 Testing against one-sided alternatives (H0 : β1 0, H1 : β1 > 0) Reject the null hypothesis in favour of the alternative hypothesis if...
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