Basic Data Analysis for Time Series with R

13 ADJUSTING FOR AR(1) CORRELATION IN COMPLEX MODELS

13.1 INTRODUCTION

Using two cases (a t-test and a regression) from the Ramsey and Schafer (“The Statistical Sleuth,” 2002, Chapter 15), the Semmelweis intervention data (a t-test with a different structure than the first case from Ramsey and Schafer), the NYC temperatures (adjusted) data (a simple periodic model), and the Boise river flow data (a complex periodic model), the basic adjustment for AR(1) noise will be presented—it will later be verified that all of these models have residuals indicative of AR(1) models. An approach will be derived for adjusting sample means for AR(1) serial correlation, then a more general approach, often called filtering, is presented, that can be generalized to all AR(m) models in all regression settings. In the first few examples, simulated data will be introduced before the final analysis of the real data. The simulated data can be used to demonstrate how things work out in the best-case scenario—when the true signal is known and the noise is known to be AR(1). In statistics, things can only work out so well, and it is always good to know how much noise there really is in the best-case scenario.

13.2 THE TWO-SAMPLE T-TEST—UNCUT AND PATCH-CUT FOREST

13.2.1 The Sleuth Data and the Question of Interest

The first two examples to be discussed come directly from The Statistical Sleuth (Ramsey and Schafer, 2002) which has an excellent chapter on AR(1) models, Chapter 15, although this is the ...