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# Elements of Time Series Analysis

The methods of time series analysis are the main tools used for analysis of price dynamics, as well as for formulating and back-testing trading strategies. Here, we present an overview of the concepts used in this book. For more details, readers can consult Alexander (2001), Hamilton (1994), Taylor (2005), and Tsay (2005).

## THE AUTOREGRESSIVE MODEL

First, we consider a univariate time series y(t) that is observed at moments t = 0, 1, …, n. The time series in which the observation at moment t takes place depends linearly on several lagged observations at moments t − 1, t − 2, …, tp

is called the autoregressive process of order p, or AR(p). The white noise ε(t) satisfies the following conditions:

The lag operator Lp = y(tp) is often used in describing time series. Note that L0 = y(t). Equation (B.1) in terms of the lag operator,

has the form

It is easy to show for AR(1) that

Obviously, contributions of old noise converge with time to zero ...

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