Unformatted text preview: jT j > tβ
i.e. the sum of the areas to the right of
ˆ
j
tβ and to the left of tβ below the pdf of the relevant t . Suppose we
ˆ
ˆ
j
j
specify H1 : βj > 0. Notice that: Melissa Tartari (Yale) Econometrics 30 / 41 Summary for Simple Hypothesis
Once the p value has been computed, a classical test can be carried
out at any α; computing p values for onesided H1 is also simple:
just divide the twosided p value by 2.
EXAMPLE: Let tβ be the computed value of our statistics. STATA
ˆ
j gives us p = P jT j > tβ
i.e. the sum of the areas to the right of
ˆ
j
tβ and to the left of tβ below the pdf of the relevant t . Suppose we
ˆ
ˆ
j
j
specify H1 : βj > 0. Notice that:
ˆ
if βj < 0 then p value 0.50 since the tβ is symmetric around 0 )
ˆ
j we fail to reject Ho (and do not even bother computing the exact value
of p ) Melissa Tartari (Yale) Econometrics 30 / 41 Summary for Simple Hypothesis
Once the p value has been computed, a classical test can be carried
out at any α; computing p values for onesided H1 is also simple:
just divide the twosided p value by 2.
EXAMPLE: Let tβ be the computed value of our statistics. STATA
ˆ
j gives us p = P jT j > tβ
i.e. the sum of the areas to the right of
ˆ
j
tβ and to the left of tβ below the pdf of the relevant t . Suppose we
ˆ
ˆ
j
j
specify H1 : βj > 0. Notice that:
ˆ
if βj < 0 then p value 0.50 since the tβ is symmetric around 0 )
ˆ
j we fail to reject Ho (and do not even bother computing the exact value
of p )
ˆ
if βj > 0 ) tβ > 0 and we have p value = p i.e. the area under the
ˆ
2
j pdf to the right of tβ
ˆ Melissa Tartari (Yale) j Econometrics 30 / 41 A Single Linear Combination of the Parameters Let Ho : αo + αk βk + αj βj = γ for some real number (αo , αk , αj , γ)
and some j , k . Do we know host to test this hypothesis about a linear
combination of the parameters βk , βj ? Melissa Tartari (Yale) Econometrics 31 / 41 A Single Linear Combination of the Parameters Let Ho : αo + αk βk + αj βj = γ for some real number (αo , αk , αj , γ)
and some j , k . Do we know host to test this hypothesis about a linear
combination of the parameters βk , βj ?
Yes, we do. This is left for a homework. Melissa Tartari (Yale) Econometrics 31 / 41 Con…dence Interval: De…nition
Con…dence intervals (CI) provide a range of likely values for the
population parameters (and not just a point estimate). Melissa Tartari (Yale) Econometrics 32 / 41 Con…dence Interval: De…nition
Con…dence intervals (CI) provide a range of likely values for the
population parameters (and not just a point estimate).
Using the fact that ˆ
βj βj
jx
ˆ
se [ βj jx] tn K 1, simple manipulation leads to a CI for the unknown βj . Melissa Tartari (Yale) Econometrics 32 / 41 Con…dence Interval: De…nition
Con…dence intervals (CI) provide a range of likely values for the
population parameters (and not just a point estimate).
Using the fact that ˆ
βj βj
jx
ˆ
se [ βj jx] tn K 1, simple manipulation leads to a CI for the unknown βj . For instance, given a con…dence level of α%, we obtain the CI
h
i
ˆ
ˆ
ˆ
ˆ
βj c se βj jx , βj + c se βj jx
(4.16) α
where c is the 1 2 percentile in a tn K 1 distribution (e.g.
(1 α) = 0.95 and c is the 97.5th percentile). Melissa Tartari (Yale) Econometrics 32 / 41 Con…dence Interval: De…nition
Con…dence intervals (CI) provide a range of likely values for the
population parameters (and not just a point estimate).
Using the fact that ˆ
βj βj
jx
ˆ
se [ βj jx] tn K 1, simple manipulation leads to a CI for the unknown βj . For instance, given a con…dence level of α%, we obtain the CI
h
i
ˆ
ˆ
ˆ
ˆ
βj c se βj jx , βj + c se βj jx
(4.16) α
where c is the 1 2 percentile in a tn K 1 distribution (e.g.
(1 α) = 0.95 and c is the 97.5th percentile). MEANING: if random samples were obtained over and over again,
with the lower and upper bound of the CI computed each time, then
the unknown population parameter βj would be in the intervals for
95% of the samples.
Melissa Tartari (Yale) Econometrics 32 / 41 Con…dence Interval: Properties Once a CI is constructed, it is easy to carry out twotailed hypothesis
test. If Ho : βj = aj , then Ho is rejected against H1 : βj 6= aj at the
α% signi…cance level if and only if
aj 2 CI
/ Melissa Tartari (Yale) Econometrics 33 / 41 Con…dence Interval: Properties Once a CI is constructed, it is easy to carry out twotailed hypothesis
test. If Ho : βj = aj , then Ho is rejected against H1 : βj 6= aj at the
α% signi…cance level if and only if
aj 2 CI
/
STATA’ command regress reports, by default, a 95% CI along with
s
each coe¢ cient and its standard error. However, any con…dence level
of α% can be obtained by adding level(1 α). Melissa Tartari (Yale) Econometrics 33 / 41 Testing General Hypothesis: Consider a case in which K = 3 i.e.
y = βo + β1 x1 + β2 x2 + β3 x3 + u Melissa Tartari (Yale) Econometrics 34 / 41 Testing General Hypothesis: Consider a case in which K = 3 i.e.
y = βo + β1 x1 + β2 x2 + β3 x3 + u
We now know how to test hypothesis abou...
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 Fall '10
 DonaldBrown
 Econometrics, Normal Distribution, Yale, Jx, Melissa Tartari

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