220222 b Var b � b Var c 1 b 1 c 2 b 2 c 2 1 b Var b 1 c 2 2 b Var b 2 2 c 1 c

220222 b var b ? b var c 1 b 1 c 2 b 2 c 2 1 b var b

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2.20–2.22 b Var b λ = b Var ( c 1 b 1 + c 2 b 2 ) = c 2 1 b Var ( b 1 ) + c 2 2 b Var ( b 2 ) + 2 c 1 c 2 b Cov ( b 1 , b 2 ) , (3.9) David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 5
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The standard error of b λ is the square root of the estimated variance se b λ = se ( c 1 b 1 + c 2 b 2 ) = q b Var ( c 1 b 1 + c 2 b 2 ) (3.10) David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 6
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If in addition SR6 holds, or if the sample is large, the least squares estimators b 1 and b 2 have normal distributions. It is also true that linear combinations of normally distributed variables are normally distributed, so that b λ = c 1 b 1 + c 2 b 2 N λ , Var b λ . David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 7
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We can estimate the average (or expected) expenditure on food as: [ Food Exp = 83.42 + 10.21 · Income If the household income is $ 2, 000, which is 20 since income is measured in $ 100 units in this example, then the average expenditure is: b E ( Food Exp | Income = 20 ) = b 1 + b 2 20 = 83.42 + 10.21 · 20 = 287.61 We estimate that the expected food expenditure by a household with $ 2, 000 income is $ 287.61 per week David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 8
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The t -statistic for the linear combination is: t = b λ - λ r b Var b λ = b λ - λ se b λ = ( c 1 b 1 + c 2 b 2 ) - ( c 1 β 1 + c 2 β 2 ) se ( c 1 b 1 + c 2 b 2 ) t ( N - 2 ) David Atkin (UCLA) Lecture Note 4C: Hypotheses Testing of Linear Combination of Parameters Spring, 2015 9
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Substituting the t value into Pr ( - t c t t c ) = 1 - α , we get: Pr (( c 1 b 1 + c 2 b 2 ) - t c · se ( c 1 b 1 + c 2 b 2 ) ( c 1 β 1 + c 2 β 2 ) ( c
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