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4 but rst why would we care about such an hypothesis

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Unformatted text preview: s We are now ready to learn how regression output can be used to test simple hypothesis of interest involving the βj , for instance Ho : β j = 0 Melissa Tartari (Yale) Econometrics (4.4) 19 / 41 A Simple Hypothesis We are now ready to learn how regression output can be used to test simple hypothesis of interest involving the βj , for instance Ho : β j = 0 (4.4) But …rst, why would we care about such an hypothesis and what is it that we are testing? Melissa Tartari (Yale) Econometrics 19 / 41 A Simple Hypothesis We are now ready to learn how regression output can be used to test simple hypothesis of interest involving the βj , for instance Ho : β j = 0 (4.4) But …rst, why would we care about such an hypothesis and what is it that we are testing? Because βj measures the partial e¤ect of xj on the expected value of y after controlling for all other indep. vars, (4.4) means that, once x1 , ...xj 1 , xj +1 , ..., xK have been accounted for, xj has no e¤ect on E [y jx] i.e. (4.4) is equivalent to Ho : Melissa Tartari (Yale) ϑ E [y jx] =0 ϑxj Econometrics 19 / 41 A Simple Hypothesis - Example Let log wage = βo + β1 educ + β2 exp + β3 exp2 + β4 tenure + u Melissa Tartari (Yale) Econometrics 20 / 41 A Simple Hypothesis - Example Let log wage = βo + β1 educ + β2 exp + β3 exp2 + β4 tenure + u A simple hypothesis of interest could be: Ho : β 4 = 0 Melissa Tartari (Yale) Econometrics 20 / 41 The t-Test We will see that the statistics that we use to test (4.4) is tβ ˆ tβ = ˆ j ˆ β hj i ˆ se β jx j (4.5) j Melissa Tartari (Yale) Econometrics 21 / 41 The t-Test We will see that the statistics that we use to test (4.4) is tβ ˆ tβ = ˆ j ˆ β hj i ˆ se β jx j (4.5) j QUESTION: why is tβ “reasonable” as a test to detect βj 6= 0? ˆ j Melissa Tartari (Yale) Econometrics 21 / 41 The t-Test i h ˆ ˆ Recall: tβ = βj /se βj jx ˆ j Melissa Tartari (Yale) Econometrics 22 / 41 The t-Test i h ˆ ˆ Recall: tβ = βj /se βj jx ˆ j Observe: Melissa Tartari (Yale) Econometrics 22 / 41 The t-Test i h ˆ ˆ Recall: tβ = βj /se βj jx ˆ j Observe: ˆ βj is an unbiased estimator of βj so it is only natural that we look at if for guidance Melissa Tartari (Yale) Econometrics 22 / 41 The t-Test i h ˆ ˆ Recall: tβ = βj /se βj jx ˆ j Observe: ˆ βj is an unbiased estimator of βj so it is only natural that we look at if for guidance i h hi hi ˆ ˆ since se βj jx > 0 always, ) sign tβ = sign βj ˆ j Melissa Tartari (Yale) Econometrics 22 / 41 The t-Test i h ˆ ˆ Recall: tβ = βj /se βj jx ˆ j Observe: ˆ βj is an unbiased estimator of βj so it is only natural that we look at if for guidance i h hi hi ˆ ˆ since se βj jx > 0 always, ) sign tβ = sign βj ˆ j h i ˆ ˆ for a given value of se βj jx , a larger βj ) larger tβ ˆ j Melissa Tartari (Yale) Econometrics 22 / 41 The t-Test i h ˆ ˆ Recall: tβ = βj /se βj jx ˆ j Observe: ˆ βj is an unbiased estimator of βj so it is only natural that we look at if for guidance i h hi hi ˆ ˆ since se βj jx > 0 always, ) sign tβ = sign βj ˆ j h i ˆ ˆ for a given value of se βj jx , a larger βj ) larger tβ ˆ j ˆ Hence, a sample value of βj very di¤erent from 0 provides evidence against Ho : βj = 0 ; however we must recognize that there is ˆ ˆ sampling error in our estimate βj , so the deviation of βj from 0 must h i ˆ ˆ be weighted against the sampling error of βj . Because se βj jx is an h i ˆ estimate of sd βj jx , tβ measures how many estimated standard ˆ j ˆ is away from 0. deviations β j Melissa Tartari (Yale) Econometrics 22 / 41 The t-Test: Digression The formal rendition of this intuitive answer is that the test is unbiased. Let us recap on terminology: a test is a statistics, a statistics is a function (solely) of the sample data (prior to the sampling process), hence it is a random variable and as such it has a distribution (which does not depend on unknown parameters). An unbiased test is a random variable whose distribution under Ho is centered at zero. For instance, E [tβ jHo : βj = 0] = 0, and similarly we will see that ˆ j n o E [FQ jHo : βj = 0jj 2 Q ] = 0. Melissa Tartari (Yale) Econometrics 23 / 41 2- Sided Alternative Consider the case in which Ho : βj = 0 is tested against H1 : βj 6= 0. Melissa Tartari (Yale) Econometrics 24 / 41 2- Sided Alternative Consider the case in which Ho : βj = 0 is tested against H1 : βj 6= 0. Let tβ be the value of the statistics computed using sample ˆ j information. Melissa Tartari (Yale) Econometrics 24 / 41 2- Sided Alternative Consider the case in which Ho : βj = 0 is tested against H1 : βj 6= 0. Let tβ be the value of the statistics computed using sample ˆ j information. First, we choose a signi…cance level α (probability of rejecting Ho when it is true): e.g. α = 5%. Melissa Tartari (Yale) Econometrics 24 / 41 2- Sided Alternative Consider the case in which Ho : βj = 0 is tested against H1 : βj 6= 0. Let tβ be the value of the statistics computed using sample ˆ j information. First, we choose a signi…cance level α (probability of rejecting Ho when it is true): e.g. α = 5%. Then, you can proceed in one of two ways: Melissa Tartari (Yale) Econometrics 24 / 41 2- Sided Alternative Consider the case in which Ho : βj = 0 is tested against H1 : βj 6= 0. Let tβ be the value of the statistics computed using sample ˆ j information. First, we choose a signi…cance level α (probability of rejectin...
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