tut5_09

tut5_09 - Coefficients: Estimate Std. Error t value...

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1 The University of Western Ontario Department of Statistical and Actuarial Sciences SS 3859a Regression Analysis Tutorial 5 1. Consider the multiple linear regression model y ˜ = X β ˜ + ± ˜ , where E[ ± ˜ ] = 0 and E[ ± ˜ ± ˜ T ] = σ 2 I , and the noise is assumed to be normally distributed. X is assumed to be an n × p matrix having full rank. Consider the null hypothesis where β i = 0 for some i p . (a) What is the expected value of b β i under the null hypothesis? (b) Show that the standard error of b β i is q ( X T X ) - 1 ii σ . (c) Show that SSE = ± ˜ T ( I - H ) ± ˜ for an appropriately defined matrix H . (d) Show that the SSE is independent of each component of b β ˜ , including the i th component. (e) Identify the distribution of SSE 2 . (f) Identify the distribution of b β i q SSE( X T X ) - 1 ii / ( n - p ) when the null hypothesis is true. (g) How would you calculate a p-value for testing the given hypothesis? 2. Fill in the blanks in the R output below, obtained for 30 observations: lm(formula = y ~ x1 + x2 + x3 + x4)
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Unformatted text preview: Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.6695 4.1254 0.889 0.38222 x1 ______ 0.4380 1.951 0.06232 x2 0.9477 ______ 1.651 0.11122 x3 0.7528 0.5739 _____ 0.20159 x4-1.1851 0.3666-3.232 0.00343 F-statistic: 7.995 on ___ and ___ DF, p-value: 0.0002702 Which coefficients do you think are actually different from 0? 2 3. inverse of X^T X matrix: (Intercept) x1 x2 (Intercept) 0.24515 -0.02352 -0.02280 x1-0.02352 _______ 0.00036 x2-0.02280 0.00036 0.00484 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) _______ ______-1.566 0.1289 x1 0.9200 ______ 2.241 0.0335 x2 2.9071 ______ _____ 2.12e-07 Residual standard error: 6.067 on _____ degrees of freedom F-statistic: 25.19 on ___ and ____ DF, p-value: 6.718e-07 Which coefficients do you think are really different from 0?...
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This note was uploaded on 11/25/2010 for the course AS 3859 taught by Professor Braun during the Fall '10 term at UWO.

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tut5_09 - Coefficients: Estimate Std. Error t value...

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