# AbrahamLedolterRegressionChapter10 - COPYRIGHT Abraham B...

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COPYRIGHT Abraham, B. and Ledolter, J. Introduction to Regression Modeling Belmont, CA: Duxbury Press, 2006
Abraham Abraham˙C10 November 8, 2004 12:5 10 Regression Models for Time Series Situations The standard regression assumptions specify that the errors ε t ( t = 1 , 2 ,..., n ) in the regression model y t = β 0 + β 1 x t 1 + β 2 x t 2 + ··· + β p x tp + ε t are independent, or at least uncorrelated. Independence may be an unreasonable assumption if we estimate the regression model on time series data. In this chapter, we replace the standard case index “ i ” by “ t ” in order to emphasize the fact that we deal with time series and not cross-sectional data. That is, we assume that ( y t , x t 1 , x t 2 ,..., x tp ) represent the measurements on the response y and the p explanatory variables x 1 , x 2 ,..., x p at time t —usually months, quarters, or years. For example, ( y t , x t 1 , x t 2 ) , for t = 1 , 2 ,..., n , may represent the sales, the price, and the amount spent on advertisement in month t . 10.1 A BRIEF INTRODUCTION TO TIME SERIES MODELS In this chapter, we assume that observations become available at equally spaced time periods, such as months, quarters, or years. The situation becomes more complicated if time periods are unequally spaced, and we do not address this issue here. In this section, we introduce several simple models that are very useful for characterizing the correlations among time series observations. Correlations among observations k periods apart are referred to as autocorrelations or serial correlations (see Section 6.2). 10.1.1 FIRST-ORDER AUTOREGRESSIVE MODEL In our previous discussion on generalized least squares (Section 4.6), we men- tionedasituationinwhicherrorswereserially(auto)correlated.Weparameterized 307
Abraham Abraham ˙ C10 November 8, 2004 12:5 308 Regression Models for Time Series Situations the covariance matrix of the vector of errors ε = 1 ,...,ε t ,...,ε n ) as V ( ε ) = σ 2 V = σ 2 1 φ φ 2 · · φ n 1 φ 1 φ φ 2 · φ n 2 φ 2 φ 1 · · · · · · · · · · · · · 1 φ φ n 1 φ n 2 · · φ 1 (10.1) and referred to the associated model as a first-order autoregressive represen- tation. Note that the diagonal elements of the matrix V are all one; hence, the errors have equal variance, V t ) = σ 2 . The matrix V in Eq. (10.1) is in fact a correlation matrix. All correlations among observations one step apart are the same. The lag 1 autocorrelations are Corr 1 2 ) = Corr 2 3 ) =··· Corr t 1 t ) =··· Corr n 1 n ) = φ Correlations must be between 1 and + 1. Hence, we must restrict the autoregres- sive parameter to | φ | < 1. Note that we also exclude values of φ at the boundary. The case φ = 1 will be covered in the next model when we discuss the random walk.