This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 = + + SR2. ( ) t E e = 1 2 ( ) t t E y x = + SR3. 2 var( ) var( ) t t e y = = SR4. cov( , ) cov( , ) 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 2 1 2 ~ [( ), ] t t y N x + ( optional ) Slide 5.1 Undergraduate Econometrics, 2 nd Edition Chapter 5 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   2 2 2 t e T = 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 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 5.5.1a The ChiSquare Distribution Chisquare 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 = + + + K (5.1.2) The notation 2 ( ) ~ m V is read as: the random variable V has a chisquare distribution with m degrees of freedom . Slide 5.4 Undergraduate Econometrics, 2 nd Edition Chapter 5 2 ( ) 2 ( ) [ ] var[ ] var 2 m m E V E m V m = = = = (5.1.3) V must be nonnegative, v 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 bellshaped. As m gets large the chisquare distribution converges to, and essentially becomes, a normal distribution. Slide 5.5 Undergraduate Econometrics, 2 nd Edition Chapter 5 5.5.1b The Probability Distribution of 2 The random error term e t has a normal distribution, 2 ~ (0, ) t e N . Standardize the random variable by dividing by its standard deviation so that / ~ (0,1) t e N . 2 2 (1) ( / ) ~ t e . If all the random errors are independent then 2 2 2 2 2 1 2 ( ) ~ t T T t e e e e = + + + L (5.1.4) Slide 5.6 Undergraduate Econometrics, 2 nd Edition Chapter 5 2 2 2 2 ( 2) t t e T V = = (5.1.5) V does not have a 2 ( ) T distribution because the least squares residuals are not independent random variables....
View Full
Document
 Summer '09

Click to edit the document details