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Time Series Analysis - ARIMA models - AR(2) process

[Home] [Up] [Basics] [AR(1) process] [AR(p) process] [MA(1) process] [MA(2) process] [MA(q) process] [ARMA(1,1) process] [ARMA(p,q) process] [Wold's decomp.] [Non stationarity] [Differencing] [Behavior] [Inverse Autocorr.] [Unit Root Tests] [AR(2) process]

c. The AR(2) process

The AR(2) process is defined as

Time Series Analysis - ARIMA models - AR(2) process


where Wt is a stationary time series, et is a white noise error term, and Ft is the forecasting function.

The process defined in (V.I.1-94) can be written in the form


and therefore


Now, for (V.I.1-96) to be valid, it easily follows that


and that


and that


and finally that


The model is stationary if the yi weights converge. This is the case when some conditions on f1 and f2 are imposed. These conditions can be found on using the solutions of the polynomial of the AR(2) model. The so-called characteristic equation is used to find these solutions


The solutions of x1 and x2 are


which can be either real or complex. Notice that the roots are complex if

When these solutions, in absolute value, are smaller than 1, the AR(2) model is stationary.

Later, it will be shown that these conditions are satisfied if f1 and f2 lie in a (Stralkowski) triangular region restricted by


The derivation of the theoretical ACF and PACF for an AR(2) model is described below.

On multiplying the AR(2) model by Wt-k, and taking expectations we obtain


From (V.I.1-97) and (V.I.1-98) it follows that


Now it is possible to combine (V.I.1-104) with (V.I.1-105) such that


from which it follows that




Eq. (V.I.1-106) can be rewritten as


such that on using (V.I.1-108) it is obvious that


According to (V.I.1-107) the ACF is a second order stochastic difference equation of the form


where (due to (V.I.1-108))


are starting values of the difference equation.

In general, the solution to the difference equation is, according to Box and Jenkins (1976), given by


In particular, three different cases can be worked out for the solutions of the difference equation


of (V.I.1-102). The general solution of eq. (V.I.1-113) can be written in the form



Remark that for the case the following stationarity conditions



has two solutions

due to (V.I.1-114) and

due to


Hence we find the general solution to the difference equation


In order to impose convergence the following must hold


Hence two conditions have to be satisfied


which describes a part of a parabola consisting of acceptable parameter values for

Remark that this parabola is the frontier between acceptable real-valued and acceptable complex roots (cfr. Triangle of Stralkowski).


in goniometric notation.

The general solution for the second-order difference equation can be found by


On defining


the ACF can be shown to be real-valued since


On using the property


eq. (V.I.1-126) becomes




In eq. (V.I.1-128) it is shown that the ACF is oscillating with period = 2p/q and a variable amplitude of


as a function of k.

A useful equation can be found to compute the period of the pseudo-periodic behavior of the time series as


which must satisfy the convergence condition (c.q. the amplitude is exponentially decreasing)


The pattern of the theoretical PACF can be deduced from relations (V.I.1-25) - (V.I.1-28).

The theoretical ACF and PACF are illustrated below. Figure (V.I.1-2) contains two possible ACF and PACF patterns for real roots while figure (V.I.1-3) shows the ACF and PACF patterns when the roots are complex.

(figure V.I.1-2)

(figure V.I.1-3)

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AR(1) process
AR(p) process
MA(1) process
MA(2) process
MA(q) process
ARMA(1,1) process
ARMA(p,q) process
Wold's decomp.
Non stationarity
Inverse Autocorr.
Unit Root Tests
AR(2) process
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