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Mathematics and Statistics for Financial Risk Management by Michael B. Miller

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CHAPTER 9

Time Series Models

In this chapter, we provide an introduction to time series analysis. Time series describe how random variables evolve over time and form the basis of many financial models.

RANDOM WALKS

A time series is an equation or set of equations describing how a random variable or variables evolves over time. Probably the most basic time series is the random walk. For a random variable X, with a realization xt at time t, the following conditions describe a random walk:

(9.1) Numbered Display Equation

In other words, X is equal to its value from the previous period, plus a random disturbance, t; t is mean zero, with a constant variance. The last assumption, combined with the fact that t is mean zero, tells us that the 's from different periods will be uncorrelated with each other. In time series analysis, we typically refer to xt–1 as the first lagged value of xt, or just the first lag of xt. By this convention, xt–2 would be the second lag, xt–3 the third, and so on.

We can also think in terms ...

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