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Advanced Statistical Regression

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[20 Points] Suppose that n independent paired data points (x1, y1), …,(xn, yn) satisfy the linear regression model, y = β0 + β1(x − x¯) + , where x is a random regressor, is uncorrelated with x and has mean zero and variance σ 2 , and ¯x is the sample mean of the x’s. Let ˆy denote the fitted value of y from this regression model. Prove that the coefficient of determination R2 is equal to the square of the sample correlation between y and ˆy. 2. [20 Points] In a linear regression model, y = β0 + β1x + , where x is a non-random regressor, and has mean zero and variance σ 2 . Suppose that n independent paired data points (x1, y1), …,(xn, yn) satisfy this model. After fitting this model to a set of n = 25 paired data points, we obtain R2 = 0.82, Syy = 50. Construct a 95% prediction value of y at x = ¯x, where ¯x is the sample mean of the n = 25 x’s. 3. [40 Points] You are given the following sample of (x, y) data points: (8, 10),(8, 10),(7, 9),(6, 10),(3, 6),(4, 8),(4, 8),(5, 9),(3, 7),(6, 9). A simple linear regression model is fitted to these data points. (a) Estimate the y-intercept and slope. (b) Estimate the variance of y. 1 (c) Test statistical significance for slope. (d) Construct a 95% confidence interval of the slope. (e) Construct an ANOVA table for testing slope. (f) State any assumptions used in your analyses of (a) – (e) above. Make sure to match any assumption you state with (some combination of) (a) – (e) specifically. Discuss your answers in detail. 4. [20 Points] Let the least-squares residuals be ei = yi −yˆi for i = 1, …, n, obtained from a simple regression model, y = β0 + β1x + , where ˆyi is the predicted value corresponding to yi from this simple regression analysis. Derive the variance of ei and discuss comparing this variance with the variance of the random error i .


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