slides_Ch4_W[1]

Example let t be the computed value of our statistics

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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 one-sided H1 is also simple: just divide the two-sided 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 one-sided H1 is also simple: just divide the two-sided 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 two-tailed 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 two-tailed 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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