"Frontmatter". In: Analysis of Financial Time Series

SIMPLE AUTOREGRESSIVE MODELS 33where B is called the back-shift operator such that Bρ l = ρ l−1 . This differenceequation determines the properties of the ACF of a stationary AR(2) time series. Italso determines the behavior of the forecasts of r t . In the time series literature, somepeople use the notation L instead of B for the back-shift operator. Here L stands forlag operator. For instance, Lr t = r t−1 and Lψ k = ψ k−1 .Corresponding to the prior difference equation, there is a second order polynomialequationx 2 − φ 1 x − φ 2 = 0.Solutions of this equation are the characteristic roots of the AR(2) model, and theyare√x = φ 1 ± φ1 2 + 4φ 2.2Denote the two characteristic roots by ω 1 and ω 2 . If both ω i are real valued, then thesecond order difference equation of the model can be factored as (1−w 1 B)(1−w 2 B)and the AR(2) model can be regarded as an AR(1) model operates on top of anotherAR(1) model. The ACF of r t is then a mixture of two exponential decays. Yet if φ 2 1 +4φ 2 < 0, then ω 1 and ω 2 are complex numbers (called a complex conjugate pair),and the plot of ACF of r t would show a picture of damping sine and cosine waves.In business and economic applications, complex characteristic roots are important.They give rise to the behavior of business cycles. It is then common for economictime series models to have complex-valued characteristic roots. For an AR(2) modelin Eq. (2.10) with a pair of complex characteristic roots, the average length of thestochastic cycles isk =360 ◦cos −1 [φ 1 /(2 √ −φ 2 )] ,where the cosine inverse is stated in degrees.Figure 2.4 shows the ACF of four stationary AR(2) models. Part (b) is the ACF ofthe AR(2) model (1−0.6B+0.4B 2 )r t = a t . Because φ 2 1 +4φ 2 = 0.36+4×(−0.4) =−1.24 < 0, this particular AR(2) model contains two complex characteristic roots,and hence its ACF exhibits damping sine and cosine waves. The other three AR(2)models have real-valued characteristic roots. Their ACFs decay exponentially.Example 2.1. As an illustration, consider the quarterly growth rate of U.S.real gross national product (GNP), seasonally adjusted, from the second quarter of1947 to the first quarter of 1991. This series is used in Chapter 4 as an example ofnonlinear economic time series. Here we simply employ an AR(3) model for thedata. Denoting the growth rate by r t , we can use the model building procedure of thenext subsection to estimate the model. The fitted model is