statsmodels.tsa.arima_process.ArmaProcess

class statsmodels.tsa.arima_process.ArmaProcess(ar=None, ma=None, nobs=100)

Theoretical properties of an ARMA process for specified lag-polynomials

Parameters:

ararray_like, optional

Coefficient for autoregressive lag polynomial, including zero lag. Must be entered using the signs from the lag polynomial representation. See the notes for more information about the sign. If not provided, defaults to [1.0].

maarray_like, optional

Coefficient for moving-average lag polynomial, including zero lag. If not provided, defaults to [1.0].

nobsint, optional

Length of simulated time series. Used, for example, if a sample is generated. See example.

Attributes:

arroots

Roots of autoregressive lag-polynomial

isinvertible

Arma process is invertible if MA roots are outside unit circle

isstationary

Arma process is stationary if AR roots are outside unit circle

maroots

Roots of moving average lag-polynomial

Methods

acf([lags])	Theoretical autocorrelation function of an ARMA process
acovf([nobs])	Theoretical autocovariances of stationary ARMA processes
arma2ar([lags])	A finite-lag AR approximation of an ARMA process
arma2ma([lags])	A finite-lag approximate MA representation of an ARMA process
from_coeffs([arcoefs, macoefs, nobs])	Create ArmaProcess from an ARMA representation
from_estimation(model_results[, nobs])	Create an ArmaProcess from the results of an ARIMA estimation
from_roots([maroots, arroots, nobs])	Create ArmaProcess from AR and MA polynomial roots
generate_sample([nsample, scale, distrvs, ...])	Simulate data from an ARMA
impulse_response([leads])	Compute the impulse response function (MA representation) for ARMA process
invertroots([retnew])	Make MA polynomial invertible by inverting roots inside unit circle
pacf([lags])	Theoretical partial autocorrelation function of an ARMA process
periodogram([nobs])	Periodogram for ARMA process given by lag-polynomials ar and ma

Notes

Both the AR and MA components must include the coefficient on the zero-lag. In almost all cases these values should be 1. Further, due to using the lag-polynomial representation, the AR parameters should have the opposite sign of what one would write in the ARMA representation. See the examples below.

The ARMA(p,q) process is described by

\[y_{t}=\phi_{1}y_{t-1}+\ldots+\phi_{p}y_{t-p}+\theta_{1}\epsilon_{t-1} +\ldots+\theta_{q}\epsilon_{t-q}+\epsilon_{t}\]

and the parameterization used in this function uses the lag-polynomial representation,

\[\left(1-\phi_{1}L-\ldots-\phi_{p}L^{p}\right)y_{t} = \left(1+\theta_{1}L+\ldots+\theta_{q}L^{q}\right)\epsilon_{t}\]

Examples

ARMA(2,2) with AR coefficients 0.75 and -0.25, and MA coefficients 0.65 and 0.35

>>> import statsmodels.api as sm
>>> import numpy as np
>>> np.random.seed(12345)
>>> arparams = np.array([.75, -.25])
>>> maparams = np.array([.65, .35])
>>> ar = np.r_[1, -arparams] # add zero-lag and negate
>>> ma = np.r_[1, maparams] # add zero-lag
>>> arma_process = sm.tsa.ArmaProcess(ar, ma)
>>> arma_process.isstationary
True
>>> arma_process.isinvertible
True
>>> arma_process.arroots
array([1.5-1.32287566j, 1.5+1.32287566j])
>>> y = arma_process.generate_sample(250)
>>> model = sm.tsa.ARIMA(y, order=(2, 0, 2), trend='n').fit()
>>> model.params
array([ 0.790456  , -0.2314185 ,  0.70071724,  0.40606905,  0.98005244])

The same ARMA(2,2) Using the from_coeffs class method

>>> arma_process = sm.tsa.ArmaProcess.from_coeffs(arparams, maparams)
>>> arma_process.arroots
array([1.5-1.32287566j, 1.5+1.32287566j])

Methods

acf([lags])	Theoretical autocorrelation function of an ARMA process
acovf([nobs])	Theoretical autocovariances of stationary ARMA processes
arma2ar([lags])	A finite-lag AR approximation of an ARMA process
arma2ma([lags])	A finite-lag approximate MA representation of an ARMA process
from_coeffs([arcoefs, macoefs, nobs])	Create ArmaProcess from an ARMA representation
from_estimation(model_results[, nobs])	Create an ArmaProcess from the results of an ARIMA estimation
from_roots([maroots, arroots, nobs])	Create ArmaProcess from AR and MA polynomial roots
generate_sample([nsample, scale, distrvs, ...])	Simulate data from an ARMA
impulse_response([leads])	Compute the impulse response function (MA representation) for ARMA process
invertroots([retnew])	Make MA polynomial invertible by inverting roots inside unit circle
pacf([lags])	Theoretical partial autocorrelation function of an ARMA process
periodogram([nobs])	Periodogram for ARMA process given by lag-polynomials ar and ma

Properties

arroots	Roots of autoregressive lag-polynomial
isinvertible	Arma process is invertible if MA roots are outside unit circle
isstationary	Arma process is stationary if AR roots are outside unit circle
maroots	Roots of moving average lag-polynomial