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Unformatted text preview: University of California, Davis ARE 106 - Winter 2011 Homework 4 A quiz based on these homework problems will be held Thursday, February 3, 2011 at the start of class. Conceptual Questions 1. Explain what R 2 means in words. There are two ways to think of R 2 . First, it is often referred to as the goodness of fit, because it is the square of the correlation between the data, y , and the predicted ˆ y . The sum of squares decomposition also shows that it can be thought of as the amount of variation in the data that can be explained by the x terms included in the model. 2. Which is biggest: SSR, SSE, or SST? Why? SST is always bigger than SSR or SSE. It is the total amount of variation in the sample, while SSR is just the explainable portion of the variation and SSE is the random portion. Remember SSR, SST and SSE are ALWAYS POSITIVE. 3. Suppose you are estimating a simple linear regression model. Answer the following questions. (Hint: use the algebraic transformations presented in class.) (a) If you multiply all the x values by 100, but not the y values, what happens to the parameter values, β 1 and β 2 and the parameter estimates b 1 and b 2 ? What happens to the least squares residuals? What happens to ˆ σ 2 , the variance of the error term? Recall that the scaling x by 100 effects the econometric model like this: y = β 1 + β 2 100 100 x + e To multiply x by 100 we need to divide that term by 100. The coefficient in the transformed model becomes β * 2 = β 2 / 100. The estimated b 2 in the transformed mode must be similarly scaled. For the standard deviation, recall that the variance of aX is a 2 var ( X ). The variance of (1 / 100) β 2 is (1 / 100) 2 var ( β 2 ). The standard deviation is the square root of the variance, so the standard deviation of the slope parameter will be scaled by 1 / 100. The residuals in the transformed equation remain the same so when we estimate the model we will have identical residuals with the same variance, ˆ σ 2 . (b) If you multiply all the y values by 100, but not the x values, what happens to the parameter values, β 1 and β 2 and the parameter estimates b 1 and b 2 ? What happens to the least squares residuals? What happens to ˆ σ 2 , the variance of the error term? 1 University of California, Davis ARE 106 - Winter 2011 Recall that the scaling y by 100 effects the econometric model like this: 100 y = 100 β 1 + 100 β 2 x + 100 e All of the coefficients will be increased by 100. The residuals will be scaled in a similar way. By our rule about the variance of a random variable multiplied by a constant, this means that the variance of the error term will be 100 2 multiplied by the original ˆ σ 2 . 4. Explain why the estimated R 2 remains the same no matter how the data are scaled....
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This note was uploaded on 03/18/2011 for the course ARE 106 taught by Professor Havenner during the Spring '09 term at UC Davis.
- Spring '09