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Statistical model used in time series analysis

In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA). The general ARMA model was described in the 1951 thesis of Peter Whittle, Hypothesis testing in time series analysis, and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins.

Given a time series of data

$$



X

t




{\displaystyle X_{t}}

, the ARMA model is a tool for understanding and, perhaps, predicting future values in this series. The AR part involves regressing the variable on its own lagged (i.e., past) values. The MA part involves modeling the error term as a linear combination of error terms occurring contemporaneously and at various times in the past. The model is usually referred to as the ARMA(p,q) model where p is the order of the AR part and q is the order of the MA part (as defined below).

ARMA models can be estimated by using the Box–Jenkins method.

Autoregressive model[edit]

The notation AR(p) refers to the autoregressive model of order p. The AR(p) model is written as

$$



X

t


=

∑

i
=
1


p



φ

i



X

t
−
i


+

ε

t




{\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}}

where

$$



φ

1


,
…
,

φ

p




{\displaystyle \varphi _{1},\ldots ,\varphi _{p}}

are parameters and the random variable

$$



ε

t




{\displaystyle \varepsilon _{t}}

is white noise, usually independent and identically distributed (i.i.d.) normal random variables.^[1][2]

In order for the model to remain stationary, the roots of its characteristic polynomial must lie outside of the unit circle. For example, processes in the AR(1) model with

$$



|


φ

1



|

≥
1


\displaystyle

are not stationary because the root of

$$


1
−

φ

1


B
=
0


{\displaystyle 1-\varphi _{1}B=0}

lies within the unit circle.^[3]

Moving-average model[edit]

The notation MA(q) refers to the moving average model of order q:

$$



X

t


=
μ
+

ε

t


+

∑

i
=
1


q



θ

i



ε

t
−
i





{\displaystyle X_{t}=\mu +\varepsilon _{t}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}\,}

where the

$$



θ

1


,
.
.
.
,

θ

q




{\displaystyle \theta _{1},…,\theta _{q}}

are the parameters of the model,

$$


μ


{\displaystyle \mu }

is the expectation of

$$



X

t




{\displaystyle X_{t}}

(often assumed to equal 0), and the

$$



ε

t




{\displaystyle \varepsilon _{t}}

,

$$



ε

t
−
1




{\displaystyle \varepsilon _{t-1}}

,… are again i.i.d. white noise error terms that are commonly normal random variables.^[4]

ARMA model[edit]

The notation ARMA(p, q) refers to the model with p autoregressive terms and q moving-average terms. This model contains the AR(p) and MA(q) models,^[5]

$$



X

t


=

ε

t


+

∑

i
=
1


p



φ

i



X

t
−
i


+

∑

i
=
1


q



θ

i



ε

t
−
i


.



{\displaystyle X_{t}=\varepsilon _{t}+\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}.\,}

The general ARMA model was described in the 1951 thesis of Peter Whittle, who used mathematical analysis (Laurent series and Fourier analysis) and statistical inference.^[6][7] ARMA models were popularized by a 1970 book by George E. P. Box and Jenkins, who expounded an iterative (Box–Jenkins) method for choosing and estimating them. This method was useful for low-order polynomials (of degree three or less).^[8]

The ARMA model is essentially an infinite impulse response filter applied to white noise, with some additional interpretation placed on it.

Specification in terms of lag operator[edit]

In some texts the models will be specified in terms of the lag operator L.
In these terms then the AR(p) model is given by

$$



ε

t


=

(

1
−

∑

i
=
1


p



φ

i



L

i



)


X

t


=
φ
(
L
)

X

t





{\displaystyle \varepsilon _{t}=\left(1-\sum _{i=1}^{p}\varphi _{i}L^{i}\right)X_{t}=\varphi (L)X_{t}\,}

where

$$


φ


{\displaystyle \varphi }

represents the polynomial

$$


φ
(
L
)
=
1
−

∑

i
=
1


p



φ

i



L

i


.



{\displaystyle \varphi (L)=1-\sum _{i=1}^{p}\varphi _{i}L^{i}.\,}

The MA(q) model is given by

$$



X

t


=

(

1
+

∑

i
=
1


q



θ

i



L

i



)


ε

t


=
θ
(
L
)

ε

t


,



{\displaystyle X_{t}=\left(1+\sum _{i=1}^{q}\theta _{i}L^{i}\right)\varepsilon _{t}=\theta (L)\varepsilon _{t},\,}

where

$$


θ


{\displaystyle \theta }

represents the polynomial

$$


θ
(
L
)
=
1
+

∑

i
=
1


q



θ

i



L

i


.



{\displaystyle \theta (L)=1+\sum _{i=1}^{q}\theta _{i}L^{i}.\,}

Finally, the combined ARMA(p, q) model is given by

$$



(

1
−

∑

i
=
1


p



φ

i



L

i



)


X

t


=

(

1
+

∑

i
=
1


q



θ

i



L

i



)


ε

t



,


{\displaystyle \left(1-\sum _{i=1}^{p}\varphi _{i}L^{i}\right)X_{t}=\left(1+\sum _{i=1}^{q}\theta _{i}L^{i}\right)\varepsilon _{t}\,,}

or more concisely,

$$


φ
(
L
)

X

t


=
θ
(
L
)

ε

t





{\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}\,}

or

$$





φ
(
L
)


θ
(
L
)




X

t


=

ε

t



.


{\displaystyle {\frac {\varphi (L)}{\theta (L)}}X_{t}=\varepsilon _{t}\,.}

Alternative notation[edit]

Some authors, including Box, Jenkins & Reinsel use a different convention for the autoregression coefficients.^[9] This allows all the polynomials involving the lag operator to appear in a similar form throughout. Thus the ARMA model would be written as

$$



(

1
−

∑

i
=
1


p



ϕ

i



L

i



)


X

t


=

(

1
+

∑

i
=
1


q



θ

i



L

i



)


ε

t



.


{\displaystyle \left(1-\sum _{i=1}^{p}\phi _{i}L^{i}\right)X_{t}=\left(1+\sum _{i=1}^{q}\theta _{i}L^{i}\right)\varepsilon _{t}\,.}

Moreover, starting summations from

$$


i
=
0


{\displaystyle i=0}

and setting

$$



ϕ

0


=
−
1


{\displaystyle \phi _{0}=-1}

and

$$



θ

0


=
1


{\displaystyle \theta _{0}=1}

, then we get an even more elegant formulation:

$$


−

∑

i
=
0


p



ϕ

i



L

i




X

t


=

∑

i
=
0


q



θ

i



L

i




ε

t



.


{\displaystyle -\sum _{i=0}^{p}\phi _{i}L^{i}\;X_{t}=\sum _{i=0}^{q}\theta _{i}L^{i}\;\varepsilon _{t}\,.}

Alternative interpretation[edit]

In digital signal processing, the ARMA model is represented as a digital filter with white noise at the input and the ARMA process at the output.

Fitting models[edit]

Choosing p and q[edit]

Finding appropriate values of p and q in the ARMA(p,q) model can be facilitated by plotting the partial autocorrelation functions for an estimate of p, and likewise using the autocorrelation functions for an estimate of q. Extended autocorrelation functions (EACF) can be used to simultaneously determine p and q.^[10] Further information can be gleaned by considering the same functions for the residuals of a model fitted with an initial selection of p and q.

Brockwell & Davis recommend using Akaike information criterion (AIC) for finding p and q.^[11] Another possible choice for order determining is the BIC criterion.

Estimating coefficients[edit]

ARMA models in general can be, after choosing p and q, fitted by least squares regression to find the values of the parameters which minimize the error term. It is generally considered good practice to find the smallest values of p and q which provide an acceptable fit to the data. For a pure AR model the Yule-Walker equations may be used to provide a fit.

Unlike other methods of regression (i.e. OLS, 2SLS, etc.) often employed in econometric analysis, ARMA model outputs are used primarily for the cases of forecasting time-series data. Their coefficients are then as such only utilized for prediction. Other areas of econometrics look at the causal inference, time-series forecasting using ARMA is not. The coefficients should then only be seen as useful for predictive modelling.

Implementations in statistics packages[edit]

Spectrum[edit]

The spectral density of an ARMA process is

where

$$



σ

2




{\displaystyle \sigma ^{2}}

is the variance of the white noise,

$$


θ


{\displaystyle \theta }

is the characteristic polynomial of the moving average part of the ARMA model, and

$$


ϕ


{\displaystyle \phi }

is the characteristic polynomial of the autoregressive part of the ARMA model.^[13][14]

Applications[edit]

ARMA is appropriate when a system is a function of a series of unobserved shocks (the MA or moving average part) as well as its own behavior. For example, stock prices may be shocked by fundamental information as well as exhibiting technical trending and mean-reversion effects due to market participants.^{[citation needed]}

Generalizations[edit]

The dependence of

$$



X

t




{\displaystyle X_{t}}

on past values and the error terms ε_t is assumed to be linear unless specified otherwise. If the dependence is nonlinear, the model is specifically called a nonlinear moving average (NMA), nonlinear autoregressive (NAR), or nonlinear autoregressive–moving-average (NARMA) model.

Autoregressive–moving-average models can be generalized in other ways. See also autoregressive conditional heteroskedasticity (ARCH) models and autoregressive integrated moving average (ARIMA) models. If multiple time series are to be fitted then a vector ARIMA (or VARIMA) model may be fitted. If the time-series in question exhibits long memory then fractional ARIMA (FARIMA, sometimes called ARFIMA) modelling may be appropriate: see Autoregressive fractionally integrated moving average. If the data is thought to contain seasonal effects, it may be modeled by a SARIMA (seasonal ARIMA) or a periodic ARMA model.

Another generalization is the multiscale autoregressive (MAR) model. A MAR model is indexed by the nodes of a tree, whereas a standard (discrete time) autoregressive model is indexed by integers.

Note that the ARMA model is a univariate model. Extensions for the multivariate case are the vector autoregression (VAR) and Vector Autoregression Moving-Average (VARMA).

Autoregressive–moving-average model with exogenous inputs model (ARMAX model)[edit]

The notation ARMAX(p, q, b) refers to the model with p autoregressive terms, q moving average terms and b exogenous inputs terms. This model contains the AR(p) and MA(q) models and a linear combination of the last b terms of a known and external time series

$$



d

t




{\displaystyle d_{t}}

. It is given by:

$$



X

t


=

ε

t


+

∑

i
=
1


p



φ

i



X

t
−
i


+

∑

i
=
1


q



θ

i



ε

t
−
i


+

∑

i
=
1


b



η

i



d

t
−
i


.



{\displaystyle X_{t}=\varepsilon _{t}+\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}+\sum _{i=1}^{b}\eta _{i}d_{t-i}.\,}

where

$$



η

1


,
…
,

η

b




{\displaystyle \eta _{1},\ldots ,\eta _{b}}

are the parameters of the exogenous input

$$



d

t




{\displaystyle d_{t}}

.

Some nonlinear variants of models with exogenous variables have been defined: see for example Nonlinear autoregressive exogenous model.

Statistical packages implement the ARMAX model through the use of “exogenous” (that is, independent,) variables. Care must be taken when interpreting the output of those packages, because the estimated parameters usually (for example, in R^[15] and gretl) refer to the regression:

$$



X

t


−

m

t


=

ε

t


+

∑

i
=
1


p



φ

i


(

X

t
−
i


−

m

t
−
i


)
+

∑

i
=
1


q



θ

i



ε

t
−
i


.



{\displaystyle X_{t}-m_{t}=\varepsilon _{t}+\sum _{i=1}^{p}\varphi _{i}(X_{t-i}-m_{t-i})+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}.\,}

where

$$



m

t




{\displaystyle m_{t}}

incorporates all exogenous (or independent) variables:

$$



m

t


=
c
+

∑

i
=
0


b



η

i



d

t
−
i


.



{\displaystyle m_{t}=c+\sum _{i=0}^{b}\eta _{i}d_{t-i}.\,}

See also[edit]

References[edit]

Further reading[edit]

• Mills, Terence C. (1990). Time Series Techniques for Economists. Cambridge University Press. ISBN 0521343399.
• Percival, Donald B.; Walden, Andrew T. (1993). Spectral Analysis for Physical Applications. Cambridge University Press. ISBN 052135532X.
• Francq, C.; Zakoïan, J.-M. (2005), “Recent results for linear time series models with non independent innovations”, in Duchesne, P.; Remillard, B. (eds.), Statistical Modeling and Analysis for Complex Data Problems, Springer, pp. 241–265, CiteSeerX 10.1.1.721.1754.