tut5_09

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

This preview shows pages 1–2. Sign up to view the full content.

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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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?...
View Full Document

## This note was uploaded on 11/25/2010 for the course AS 3859 taught by Professor Braun during the Fall '10 term at UWO.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online