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Chapter 5 Slides - Chapter 5 Inference in the Simple...

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Chapter 5 Inference in the Simple Regression Model: Interval Estimation, Hypothesis Testing, and Prediction Assumptions of the Simple Linear Regression Model SR1. 1 2 t t t y x e = β + β + SR2. ( ) 0 t E e = 1 2 ( ) t t E y x = β +β SR3. 2 var( ) var( ) t t e y = σ = SR4. cov( , ) cov( , ) 0 i j i j e e y y = = SR5. t x is not random and takes at least two different values SR6. 2 ~ (0, ) t e N σ 2 1 2 ~ [( ), ] t t y N x β + β σ ( optional ) Slide 5.1 Undergraduate Econometrics, 2 nd Edition –Chapter 5
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From Chapter 4 2 2 1 1 2 2 2 2 2 ~ , ( ) ~ , ( ) t t t x b N T x x b N x x σ β - σ β - 2 2 ˆ ˆ 2 t e T σ = - This Chapter introduces additional tools of statistical inference: Interval estimation, prediction, prediction intervals, hypothesis testing . Slide 5.2 Undergraduate Econometrics, 2 nd Edition –Chapter 5
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5.1 Interval Estimation 5.1.1 The Theory A standardized normal random variable is obtained from b 2 by subtracting its mean and dividing by its standard deviation: 2 2 2 ~ (0,1) var( ) b Z N b = (5.1.1) The standardized random variable Z is normally distributed with mean 0 and variance 1. Slide 5.3 Undergraduate Econometrics, 2 nd Edition –Chapter 5
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5.5.1a The Chi-Square Distribution Chi-square random variables arise when standard normal, N (0,1), random variables are squared. If Z 1 , Z 2 , ..., Z m denote m independent N (0,1) random variables, then 2 2 2 2 1 2 ( ) ~ m m V Z Z Z = + + + χ K (5.1.2) The notation 2 ( ) ~ m V χ is read as: the random variable V has a chi-square distribution with m degrees of freedom . Slide 5.4 Undergraduate Econometrics, 2 nd Edition –Chapter 5
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2 ( ) 2 ( ) [ ] var[ ] var 2 m m E V E m V m = χ = = χ = (5.1.3) V must be nonnegative, v 0 the distribution has a long tail, or is skewed , to the right. As the degrees of freedom m gets larger the distribution becomes more symmetric and “bell-shaped.” As m gets large the chi-square distribution converges to, and essentially becomes, a normal distribution. Slide 5.5 Undergraduate Econometrics, 2 nd Edition –Chapter 5
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5.5.1b The Probability Distribution of 2 ˆ σ The random error term e t has a normal distribution, 2 ~ (0, ) t e N σ . Standardize the random variable by dividing by its standard deviation so that / ~ (0,1) t e N σ . 2 2 (1) ( / ) ~ t e σ χ . If all the random errors are independent then 2 2 2 2 2 1 2 ( ) ~ t T T t e e e e = + + + χ σ σ σ σ L (5.1.4) Slide 5.6 Undergraduate Econometrics, 2 nd Edition –Chapter 5
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2 2 2 2 ˆ ˆ ( 2) t t e T V - σ = = σ σ (5.1.5) V does not have a 2 ( ) T χ distribution because the least squares residuals are not independent random variables. All T residuals 1 2 ˆ t t t e y b b x = - - depend on the least squares estimators b 1 and b 2 . It can be shown that only T - 2 of the least squares residuals are independent in the simple linear regression model.
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