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ch03Hypothesistesting

# ch03Hypothesistesting - ECON 6002 Econometrics Memorial...

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Adapted from Vera Tabakova’s notes ECON 6002 Econometrics Memorial University of Newfoundland

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3.1 Interval Estimation 3.2 Hypothesis Tests 3.3 Rejection Regions for Specific Alternatives 3.4 Examples of Hypothesis Tests 3.5 The p -value
Slide 3-3 Principles of Econometrics, 3rd Edition y x e = + + β β 1 2 ( ) 0 E e = 1 2 ( ) E y x = β +β 2 var( ) var( ) e y = σ = cov( , ) cov( , ) e e y y i j i j = = 0 e N ~ ( , ) 0 2 σ

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The normal distribution of , the least squares estimator of β, is A standardized normal random variable is obtained from by subtracting its mean and dividing by its standard deviation: The standardized random variable Z is normally distributed with mean 0 and variance 1. ( 29 ( 29 2 2 2 2 ~ 0,1 i b Z N x x = σ - ( 29 2 2 2 2 ~ , i b N x x σ β - 2 b 2 b
This defines an interval that has probability .95 of containing the parameter β 2 . ( 29 1.96 1.96 .95 P Z - = ( 29 2 2 2 2 1.96 1.96 .95 i b P x x - = σ - ( 29 ( 29 ( 29 2 2 2 2 2 2 2 1.96 1.96 .95 i i P b x x b x x - σ - ≤ β ≤ + σ - =

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The two endpoints provide an interval estimator. In repeated sampling 95% of the intervals constructed this way will contain the true value of the parameter β 2 . This easy derivation of an interval estimator is based on the assumption SR6 and that we know the variance of the error term σ 2 . ( 29 ( 29 2 2 2 1.96 i b x x ± σ -
Replacing σ 2 with creates a random variable t: The ratio has a t -distribution with ( N – 2) degrees of freedom, which we denote as . 2 ˆ σ ( 29 ( 29 · ( 29 2 2 2 2 2 2 ( 2) 2 2 2 2 ~ se ˆ var N i b b b t t b x x b - = = = σ - ( 29 ( 29 2 2 2 se t b b = ( 2) ~ N t t -

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In general we can say, if assumptions SR1-SR6 hold in the simple linear regression model, then The t -distribution is a bell shaped curve centered at zero. It looks like the standard normal distribution, except it is more spread out, with a larger variance and thicker tails. The shape of the t -distribution is controlled by a single parameter called the degrees of freedom , often abbreviated as df . ( 29 ( 29 2 ~ for 1,2 se k k N k b t t k b - = =
We can find a “critical value” from a t- distribution such that where α is a probability often taken to be α = .01 or α = .05. The critical value t c for degrees of freedom m is the percentile value . ( 29 ( 29 2 c c P t t P t t = ≤ - = α ( 29 1 2, m t

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Figure 3.1 Critical Values from a t -distribution Slide 3-10 Principles of Econometrics, 3rd Edition
Each shaded “tail” area contains α /2 of the probability, so that 1–α of the probability is contained in the center portion. Consequently, we can make the probability statement Slide 3-11 Principles of Econometrics, 3rd Edition ( ) 1 c c P t t t - = - α [ ] 1 se( ) k k c c k b P t t b - = - α [ se( ) se( )] 1 k c k k k c k P b t b b t b - ≤ β ≤ + = - α

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For the food expenditure data The critical value t c = 2.024, which is appropriate for α = .05 and 38 degrees of freedom.
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ch03Hypothesistesting - ECON 6002 Econometrics Memorial...

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