Homewor1SolF08 - Copy

# Homewor1SolF08 - Copy - Homework 1 STAT 331 Fall 2008 1(5...

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Homework 1 STAT 331 Fall 2008 1. (5 points) Prove that covariance between residuals and predictor variable is 0. Proof: ) , cov( ) , cov( * 1 * 0 x y x x β ε - - = ) , cov( ) , cov( ) , cov( * 1 * 0 x x x y x - - = ) , cov( ) , cov( ) , cov( * 1 * 1 x x x y x y x - - - = ) , cov( ) , cov( ) , cov( 0 ) , cov( x x x x y x y x - - = = 0 2. (5 points) Show that for SLR squared t-statistic for a slope is equal to F-statistic. For the SLR t t t X Y + + = 1 0 , the t-statistic for the slope is 2 * 1 * 1 ~ ) .( . - = n t e s T under the null hypothesis 0 1 = , where * 1 denotes the least square estimator of 1 , ) .( . * 1 e s denotes the standard error of * 1 , and n represents the total number of observations. [ ] 2 * 1 2 * 1 * 0 * 1 * 0 1 2 1 2 ) ( ) ( ) 2 /( ) ˆ ( ) 1 2 /( ) ˆ ( ) /( ) 1 /( σ = = = + - + = - - - - = - - = n t n t n t n t n t n t X X n Y Y Y Y k n SSE k SSR F [ ] [ ] ) 2 , 1 ( 2 2 1 * 1 * 1 2 2 * 2 1 * 2 * 1 2 2 1 * ~ ) .( . ) ( / ) ( - = = = = - = - = n n t n t n t n t F T e s X X X X Therefore, the squared t-statistic for a slope is equal to F-statistic. Note that this implies that ) ( ) ( |) | | (| ) 2 , 1 ( 2 2 2 F F P F t P T t P n n n = = - - - where F T = 2 . 3. (40 points) DESCRIPTIVE ABSTRACT: The datafile contains 11 years of quarterly sales for four kinds of retail establishments, along with non-agricultural employment and wage and salary disbursements. The current task is to develop a model for 1

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predicting general merchandise dealer sales (GMER) based on national income wage and salary disbursements (WASA). These data are published monthly in the statistical section of the Survey of Current Business. VARIABLE DESCRIPTIONS: 1. TIME: Quarter, from 1st quarter 1979 to 4th quarter 1989 2. WASA: National income wage and salary disbursements (\$ billions) 3. EMPL: Employees on payrolls of non-agricultural establishments (thousands) 4. BLDG: Building material dealer sales (\$ millions) 5. AUTO: Automotive dealer sales (\$ millions) 6. FURN: Furniture and home furnishings dealer sales (\$ millions) 7. GMER: General merchandise dealer sales (\$ millions) Values are aligned and delimited by blanks. YOUR TASK: To forecast general merchandise dealer sales for 1-4 quarters of 1989 using national income wage and salary disbursements. The training window is from the 1 st quarter of 1979 to the 4 th quarter of 1988. You should a) construct a simple linear regression of general merchandise dealer sales and national income wage and salary disbursements, include the summary output of the linear regression, the scatterplot, correlation, covariance and R-squared of GMER vs. WASA;
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Homewor1SolF08 - Copy - Homework 1 STAT 331 Fall 2008 1(5...

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