arma (Series)

Apply ARMA model to input series

Syntax

Y = arma(X, E, AR, MA, Range)

Input arguments

X [ tseries ]

Input time series from which initial condition will be constructed.

E [ tseries ]¨

Input time series with innovations; NaN values in E on Range will be replaced with 0.

AR [ numeric | empty ]

Row vector of AR polynominal coefficients; if empty, AR = 1; see Description.

MA [ numeric | empty ]

Row vector of MA polynominal coefficients; if empty, MA = 1; see Description.

Range [ numeric | char ]

Range on which the output series observations will be constructed.

Output arguments

X [ tseries ]

Output time series constructed by running an ARMA model on the input series X and E; the output time series also includes p initial conditions where p is the order of the AR polynomial.

Options

zzz=default [ zzz | ___ ]

Description

Description

The output series is constructed as follows:

\[ A(L) X_t = M(L) E_t \]

where \(A(L) = A_0 + A_1 L + \cdots\) and \(M(L)=M_0 + M_1 L + \cdots\) are polynomials in lag operator \(L\) defined by the vectors AR and MA:

$$ X_t = \frac{1}{A_1} \left( -A_2 X_{t-1} - A_3 X_{t-2} - \cdots + M_0 E_t + M_1 E_{t-1} + \cdots \right) $$ .

Note that the coefficient \(A_0\) is AR(1), \(A_1\) is AR(2), and so on.

Examples

Construct an AR(1) process with autoregression coefficient 0.8, built from normally distributed innovations:

X = Series(0:20, 0);
E = Series(1:20, @randn);
X = arma(X, E, [1, -0.8], [ ], 1:20);
plot(X);