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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 tTest 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 tTest 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 tTest
i
h
ˆ
ˆ
Recall: tβ = βj /se βj jx
ˆ
j Melissa Tartari (Yale) Econometrics 22 / 41 The tTest
i
h
ˆ
ˆ
Recall: tβ = βj /se βj jx
ˆ
j Observe: Melissa Tartari (Yale) Econometrics 22 / 41 The tTest
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 tTest
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 tTest
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 tTest
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 tTest: 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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 Fall '10
 DonaldBrown
 Econometrics, Normal Distribution, Yale, Jx, Melissa Tartari

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