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# Midterm - Copy - Midterm STAT 331 Fall 2008 1. Derive a...

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Midterm STAT 331 Fall 2008 1. Derive a covariance between OLS intercept and slope estimates. COV ( ) = E ( ) – E( )E( ) Since Therefore, COV ( ) = E ( ) – E( )E( ) = E ( ) – (E( E( ) = = = = = - 2. DESCRIPTIVE ABSTRACT: The observations contain five series, consisting of annual hog supply (AnHogSp), hog prices (HogPr), corn supply (CornSp), corn prices (CornPr) and farm wages (Wages) for the period 1868-1948. Values are aligned and delimited by blanks. YOUR TASK: To forecast hog prices in 1947 and 1948, i.e. you use the data from 1868 to 1946 as a training set for your model. You should construct a simple linear regression model with a potential predictor annual hog supply (AnHogSp). You should a) obtain a scatter plot of dependent variable vs. regressor and discuss your findings; 1

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Scatterplot > plot(data\$AnHogSp,data\$HogPr, xlab="Annual hog supply", ylab="Hog prices",main="Hog Prices Against Annual Hog Supply") > abline(lm(data\$HogPr~data\$AnHogSp)) 1. Outliers 2. Residuals values are very high 3. Slope not zero important predictor 4. ANhogSUP are positivly related and important predictor b) construct a linear regression of hog prices vs. annual hog supply; Summary > l<-lm(data\$HogPr~data\$AnHogSp) > summary(l) Call: lm(formula = data\$HogPr ~ data\$AnHogSp) Residuals: Min 1Q Median 3Q Max -0.49812 -0.14989 0.01380 0.13189 0.45736 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.4326 1.2391 -1.156 0.251 data\$AnHogSp 1.2504 0.1894 6.600 4.7e-09 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2048 on 77 degrees of freedom Multiple R-squared: 0.3613, Adjusted R-squared: 0.353 F-statistic: 43.56 on 1 and 77 DF, p-value: 4.7e-09 2
Therefore the linear regression model is Y i = -1.4326+ 1.2504X i + i c) construct 90% confidence interval for the intercept and slope; T-Value > qt(.95,77) [1] 1.664885 Intercept β 0 = -1.4326 ± 1.664885 * 1.2391 = (-3.495559, 0.630359 Slope β 1 = 1.2504 ± 1.664885 * 0.1894 = (0.9350708, 1.565729) d) test whether your slope is significant at 99% confidence level using t-statistic; T-Value > qt(.995,77) [1] 2.641198 > anova(l) Analysis of Variance Table Response: data\$HogPr Df Sum Sq Mean Sq F value Pr(>F) data\$AnHogSp 1 1.8279 1.8279 43.564 4.7e-09 *** Residuals 77 3.2309 0.0420 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 T-statistics = (SSR/(SSE/n-2))^1/2 = (1.8279/0.0420)^1/2 = 6.597077275 Since 6.597077275 is bigger than 2.641198 therefore we can reject slope equals to zero with 99% confidence interval. AnHOGSP is a very useful predictor e) provide residual diagnostics, i.e. verify that residuals are homoscedastic (residual plot), uncorrelated (acf plot, the Durbin-Watson test, the runs test) and normally distributed 3

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(Shapiro-Wilk test, QQ plot, histogram). HOMOSCEDASTIC
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## This note was uploaded on 06/10/2010 for the course STAT 331 taught by Professor Yuliagel during the Spring '08 term at Waterloo.

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Midterm - Copy - Midterm STAT 331 Fall 2008 1. Derive a...

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