In statistics, autoregressive moving average (ARMA) models, sometimes called Box-Jenkins models after the iterative Box-Jenkins methodology usually used to estimate them, are typically applied to time series data.

Given a time series of data X_t, the ARMA model is a tool for understanding and, perhaps, predicting future values in this series. The model consists of two parts, an autoregressive (AR) part and a moving average (MA) part. The model is usually then referred to as the ARMA(p,q) model where p is the order of the autoregressive part and q is the order of the moving average part (as defined below).

Autoregressive model

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

$X\_t\; =\; c\; +\; sum\_\{i=1\}^p\; varphi\_i\; X\_\{t-i\}+\; varepsilon\_t\; .,$

where $varphi\_1,\; ldots,\; varphi\_p$ are the parameters of the model, $c$ is a constant and $varepsilon\_t$ is white noise. The constant term is omitted by many authors for simplicity.

An autoregressive model is essentially an all-pole infinite impulse response filter with some additional interpretation placed on it.

Some constraints are necessary on the values of the parameters of this model in order that the model remains stationary. For example, processes in the AR(1) model with |φ₁| ≥ 1 are not stationary.

Example: An AR(1)-process

An AR(1)-process is given by:

$X\_t\; =\; c\; +\; varphi\; X\_\{t-1\}+varepsilon\_t,,$

where $varepsilon\_t$ is a white noise process with zero mean and variance $sigma^2$. (Note: The subscript on $varphi\_1$ has been dropped.) The process is wide sense stationary if . (If $varphi=1$ then $X\_t$ exhibits a unit root and can also be considered as a random walk, which is not wide sense stationary.) Assuming and denoting the mean by $mu$, we get

$mbox\{E\}(X\_t)=mbox\{E\}(c)+varphimbox\{E\}(X\_\{t-1\})+mbox\{E\}(varepsilon\_t)Rightarrow\; mu=c+varphimu+0.$

Thus

$mu=frac\{c\}\{1-varphi\}.$

In particular, if $c\; =\; 0$, then the mean is 0.

The variance can be shown to equal

$textrm\{var\}(X\_t)=E(X\_t^2)-mu^2=frac\{sigma^2\}\{1-varphi^2\}.$

The autocovariance is given by

$B\_n=E(X\_\{t+n\}X\_t)-mu^2=frac\{sigma^2\}\{1-varphi^2\},,varphi^$.>

It can be seen that the autocovariance function decays with a decay time (also called time constant) of $tau=-1/ln(varphi)$ [to see this, write $B\_n=Kphi^$

> where $K$ is independent of $n$. Then note that $phi^$=e^{|n|lnphi}> and match this to the exponential decay law $e^\{-n/tau\}$].

The spectral density function is the Fourier transform of the autocovariance function. In discrete terms this will be the discrete-time Fourier transform:

$Phi(omega)=$

frac{1}{sqrt{2pi}},sum_{n=-infty}^infty B_n e^{-iomega n} =frac{1}{sqrt{2pi}},left(frac{sigma^2}{1+varphi^2-2varphicos(omega)}right).

This expression is periodic due to the discrete nature of the $X\_j$, which is manifested as the cosine term in the denominator. If we assume that the sampling time ($Delta\; t=1$) is much smaller than the decay time ($tau$), then we can use a continuum approximation to $B\_n$:

$B(t)approx\; frac\{sigma^2\}\{1-varphi^2\},,varphi^$>

which yields a Lorentzian profile for the spectral density:

$Phi(omega)=$

frac{1}{sqrt{2pi}},frac{sigma^2}{1-varphi^2},frac{gamma}{pi(gamma^2+omega^2)}

where $gamma=1/tau$ is the angular frequency associated with the decay time $tau$.

An alternative expression for $X\_t$ can be derived by first substituting $c+varphi\; X\_\{t-2\}+varepsilon\_\{t-1\}$ for $X\_\{t-1\}$ in the defining equation. Continuing this process N times yields

$X\_t=csum\_\{k=0\}^\{N-1\}varphi^k+varphi^NX\_\{t-N\}+sum\_\{k=0\}^\{N-1\}varphi^kvarepsilon\_\{t-k\}.$

For N approaching infinity, $varphi^N$ will approach zero and:

$X\_t=frac\{c\}\{1-varphi\}+sum\_\{k=0\}^inftyvarphi^kvarepsilon\_\{t-k\}.$

It is seen that $X\_t$ is white noise convolved with the $varphi^k$ kernel plus the constant mean. If the white noise $varepsilon\_t$ is a Gaussian process then $X\_t$ is also a Gaussian process. In other cases, the central limit theorem indicates that $X\_t$ will be approximately normally distributed when $varphi$ is close to one.

Calculation of the AR parameters

The AR(p) model is given by the equation

$X\_t\; =\; sum\_\{i=1\}^p\; varphi\_i\; X\_\{t-i\}+\; varepsilon\_t.,$

It is based on parameters $varphi\_i$ where i = 1, ..., p. There is a direct correspondence between these parameters and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function (which is itself obtained from the covariances). This is done using the Yule-Walker equations:

$$

gamma_m = sum_{k=1}^p varphi_k gamma_{m-k} + sigma_varepsilon^2delta_m

where m = 0, ... , p, yielding p + 1 equations. $gamma\_m$ is the autocorrelation function of X, $sigma\_varepsilon$ is the standard deviation of the input noise process, and $delta\_m$ is the Kronecker delta function.

Because the last part of the equation is non-zero only if m = 0, the equation is usually solved by representing it as a matrix for m > 0, thus getting equation

$begin\{bmatrix\}$

gamma_1 gamma_2 gamma_3 vdots end{bmatrix}

=

begin{bmatrix} gamma_0 & gamma_{-1} & gamma_{-2} & dots gamma_1 & gamma_0 & gamma_{-1} & dots gamma_2 & gamma_{1} & gamma_{0} & dots vdots & vdots & vdots & ddots end{bmatrix}

begin{bmatrix} varphi_{1} varphi_{2} varphi_{3}

vdots 

end{bmatrix}

solving all $varphi$. For m = 0 have

$$

gamma_0 = sum_{k=1}^p varphi_k gamma_{-k} + sigma_varepsilon^2

which allows us to solve $sigma\_varepsilon^2$.

The above equations (the Yule-Walker equations) provide one route to estimating the parameters of an AR(p) model, by replacing the theoretical covariances with estimated values. One way of specificying the estimated covariances is equivalent to a calculation using least squares regression of values X_t on the 'p previous values of the same series.

Derivation

The equation defining the AR process is

$X\_t\; =\; sum\_\{i=1\}^p\; varphi\_i,X\_\{t-i\}+\; varepsilon\_t.,$

Multiplying both sides by X_t − m and taking expected value yields

$E[X\_t\; X\_\{t-m\}]\; =\; Eleft[sum\_\{i=1\}^p\; varphi\_i,X\_\{t-i\}\; X\_\{t-m\}right]+\; E[varepsilon\_t\; X\_\{t-m\}].$

Now, E[X_tX_t − m] = γ_m by definition of the autocorrelation function. The values of the noise function are independent of each other, and X_t − m is independent of ε_t where m is greater than zero. For m > 0, E[ε_tX_{t − m] = 0. For m = 0,}

$E[varepsilon\_t\; X\_\{t\}]$

Eleft[varepsilon_t left(sum_{i

1}^p varphi_i,X_{t-i}+ varepsilon_tright)right]

sum_{i

1}^p varphi_i, E[varepsilon_t,X_{t-i}] + E[varepsilon_t^2] = 0 + sigma_varepsilon^2,

Now we have, for m ≥ 0,

$gamma\_m\; =\; Eleft[sum\_\{i=1\}^p\; varphi\_i,X\_\{t-i\}\; X\_\{t-m\}right]\; +\; sigma\_varepsilon^2\; delta\_m.$

Furthermore,

$Eleft[sum\_\{i=1\}^p\; varphi\_i,X\_\{t-i\}\; X\_\{t-m\}right]$

sum_{i

1}^p varphi_i,E[X_{t} X_{t-m+i}]

sum_{i

1}^p varphi_i,gamma_{m-i},

which yields the Yule-Walker equations:

$gamma\_m\; =\; sum\_\{i=1\}^p\; varphi\_i\; gamma\_\{m-i\}\; +\; sigma\_varepsilon^2\; delta\_m.$

for m ≥ 0. For m < 0,

$gamma\_m\; =\; gamma\_\{-m\}\; =\; sum\_\{i=1\}^p\; varphi\_i\; gamma\_\{|m|-i\}\; +\; sigma\_varepsilon^2\; delta\_m.$

Moving average model

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

$X\_t\; =\; varepsilon\_t\; +\; sum\_\{i=1\}^q\; theta\_i\; varepsilon\_\{t-i\},$

where the θ₁, ..., θ_q are the parameters of the model and the ε_t, ε_t-1,... are again, the error terms. The moving average model is essentially a finite impulse response filter with some additional interpretation placed on it.

Autoregressive moving average model

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,

$X\_t\; =\; varepsilon\_t\; +\; sum\_\{i=1\}^p\; varphi\_i\; X\_\{t-i\}\; +\; sum\_\{i=1\}^q\; theta\_i\; varepsilon\_\{t-i\}.,$

Note about the error terms

The error terms ε_t are generally assumed to be independent identically-distributed random variables (i.i.d.) sampled from a normal distribution with zero mean: ε_t ~ N(0,σ²) where σ² is the variance. These assumptions may be weakened but doing so will change the properties of the model. In particular, a change to the i.i.d. assumption would make a rather fundamental difference.

Specification in terms of lag operator

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

$varepsilon\_t\; =\; left(1\; -\; sum\_\{i=1\}^p\; varphi\_i\; L^iright)\; X\_t\; =\; varphi\; X\_t,$

where φ represents the polynomial

$varphi\; =\; 1\; -\; sum\_\{i=1\}^p\; varphi\_i\; L^i.,$

The MA(q) model is given by

$X\_t\; =\; left(1\; +\; sum\_\{i=1\}^q\; theta\_i\; L^iright)\; varepsilon\_t\; =\; theta\; varepsilon\_t\; ,\; ,$

where θ represents the polynomial

$theta=\; 1\; +\; sum\_\{i=1\}^q\; theta\_i\; L^i\; .,$

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

$left(1\; -\; sum\_\{i=1\}^p\; varphi\_i\; L^iright)\; X\_t\; =\; left(1\; +\; sum\_\{i=1\}^q\; theta\_i\; L^iright)\; varepsilon\_t\; ,\; ,$

or more concisely,

$varphi\; X\_t\; =\; theta\; varepsilon\_t\; ,$

Alternative notation

Some authors, including Box, Jenkins & Reinsel (1994) use a different convention for the autoregression coefficients. This allows all the polynomials involving the lag operator to appear in a similar form throughout. Thus the ARMA model would be written as

$left(1\; +\; sum\_\{i=1\}^p\; phi\_i\; L^iright)\; X\_t\; =\; left(1\; +\; sum\_\{i=1\}^q\; theta\_i\; L^iright)\; varepsilon\_t\; ,\; .$

Fitting models

ARMA models in general can, after choosing p and q, be 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.

Implementations in statistics packages

• In R, library tseries includes an arma function. The function is documented in "Fit ARMA Models to Time Series"
• MATLAB includes a function ar to estimate AR models, see here for more details
• gretl probably can also estimate ARMA models, see here where it's mentioned

Applications

ARMA is appropriate when a system is a function of a series of unobserved shocks (the MA 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.

Generalizations

The dependence of 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. See multiscale autoregressive model for a list of references.

Note that the ARMA model is an 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)

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 $d\_t$. It is given by:

$X\_t\; =\; varepsilon\_t\; +\; sum\_\{i=1\}^p\; varphi\_i\; X\_\{t-i\}\; +\; sum\_\{i=1\}^q\; theta\_i\; varepsilon\_\{t-i\}\; +\; sum\_\{i=1\}^b\; eta\_i\; d\_\{t-i\}.,$

where $eta\_1,\; ldots,\; eta\_b$ are the parameters of the exogenous input $d\_t$.

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

See also

• ARIMA
• Predictive analytics
• Radial basis function
• Linear predictive coding

References

• George Box, Gwilym M. Jenkins, and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control, third edition. Prentice-Hall, 1994.
• Mills, Terence C. Time Series Techniques for Economists. Cambridge University Press, 1990.
• Percival, Donald B. and Andrew T. Walden. Spectral Analysis for Physical Applications. Cambridge University Press, 1993.
• Pandit, Sudhakar M. and Wu, Shien-Ming. Time Series and System Analysis with Applications. John Wiley & Sons, Inc., 1983.

External links

• Electronic Statistics Textbook