T1. Studentized residuals The residuals e; = Yi - ĝi have a natural correlation structure when the linear model assumpti

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T1 Studentized Residuals The Residuals E Yi Gi Have A Natural Correlation Structure When The Linear Model Assumpti 1 (280.79 KiB) Viewed 47 times

T1. Studentized residuals The residuals e; = Yi - ĝi have a natural correlation structure when the linear model assumptions are met. In particular, they have a variance that is proportional to 1 - hii, where hii is the ith diagonal of the hat matrix H = x(x'x)-?x'. So sometimes internally studentized residuals are used instead for diagnostics: ei ri= Vô2(1 – hii) Here you'll explore this idea. Assume throughout that Y = xB+e with e ~ N(0, oʻI). Let e denote the residual vector Y - Ý. Note that û = xß = x(x'x)-'x'Y = HY , so e = (I – HY. i. Show that (I – H) is symmetric: (I – H)' = (I – H). ii. Show that (I – H) is idempotent: (I – H)(I – H) = (I – H). iii. Show that vare = 02(I - H). iv. Do the residuals have equal variances? Explain in one sentence based on your answer in (iii). v. Write the vector of studentized residuals r as a linear function of e: find a matrix v such that r = ve. vi. What is varr? Find an expression in terms of 1, H, v, and o2. vii. Do the studentized residuals have equal variances? Explain in one sentence based on your answer in (iii). (Hint: what's the ith diagonal of the variance-covariance matrix varr?) viii. Are the studentized residuals independent for general x? Answer in 1-2 sentences; you do not need to provide a rigorous proof, but please explain your reasoning.