Abstract
A time series is a sequence of observations in chronological order, for example, daily log returns on a stock or monthly values of the Consumer Price Index (CPI). A common simplifying assumption is that the data are equally spaced with a discrete-time observation index; however, this may only hold approximately. For example, daily log returns on a stock may only be available for weekdays, with additional gaps on holidays, and monthly values of the CPI are equally spaced by month, but unequally spaced by days. In either case, the consecutive observations are commonly regarded as equally spaced, for simplicity. In this chapter, we study statistical models for time series. These models are widely used in econometrics, business forecasting, and many scientific applications.
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Notes
- 1.
It is the returns, not the stock prices, that have time-invariant behavior. Stock prices themselves tend to increase over time, so this year’s stock prices tend to be higher and more variable than those a decade or two ago.
- 2.
Best linear prediction is discussed in Sect. 11.9.1
- 3.
However, there is a technical issue here. It must be assumed that Y 0 has a finite mean and variance, since otherwise Y t will not have a finite mean and variance for any t > 0.
- 4.
We discuss higher-order AR models in more detail soon.
- 5.
See Chap. 9 for an introduction to multiple regression.
- 6.
Some textbooks and some software write MA models with the signs reversed so that model (12.21) is written as \(Y _{t}-\mu =\epsilon _{t} -\theta \epsilon _{t-1}\). We have adopted the same form of MA models as R’s arima() function. These remarks apply as well to the general MA and ARMA models given by Eqs. (12.24) and (12.25).
- 7.
An analog is, of course, differentiation and integration in calculus, which are inverses of each other.
References
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Ruppert, D., Matteson, D.S. (2015). Time Series Models: Basics. In: Statistics and Data Analysis for Financial Engineering. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2614-5_12
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