In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation (or recurrence relation) which should not be confused with a differential equation. Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable. Unlike the moving-average (MA) model, the autoregressive model is not always stationary, because it may contain a unit root. Large language models are called autoregressive, but they are not a classical autoregressive model in this sense because they are not linear.

Definition

The notation AR(p) indicates an autoregressive model of order p. The AR(p) model is defined as where are the parameters of the model, and is white noise. This can be equivalently written using the backshift operator B as so that, moving the summation term to the left side and using polynomial notation, we have An autoregressive model can thus be viewed as the output of an all-pole infinite impulse response filter whose input is white noise. Some parameter constraints are necessary for the model to remain weak-sense stationary. For example, processes in the AR(1) model with are not stationary. More generally, for an AR(p) model to be weak-sense stationary, the roots of the polynomial must lie outside the unit circle, i.e., each (complex) root z_i must satisfy |z_i |>1 (see pages 89,92 ).

Intertemporal effect of shocks

In an AR process, a one-time shock affects values of the evolving variable infinitely far into the future. For example, consider the AR(1) model. A non-zero value for at say time t=1 affects X_1 by the amount. Then by the AR equation for X_2 in terms of X_1, this affects X_2 by the amount. Then by the AR equation for X_3 in terms of X_2, this affects X_3 by the amount. Continuing this process shows that the effect of never ends, although if the process is stationary then the effect diminishes toward zero in the limit. Because each shock affects X values infinitely far into the future from when they occur, any given value Xt is affected by shocks occurring infinitely far into the past. This can also be seen by rewriting the autoregression (where the constant term has been suppressed by assuming that the variable has been measured as deviations from its mean) as When the polynomial division on the right side is carried out, the polynomial in the backshift operator applied to has an infinite order—that is, an infinite number of lagged values of appear on the right side of the equation.

Characteristic polynomial

The autocorrelation function of an AR(p) process can be expressed as where y_k are the roots of the polynomial where B is the backshift operator, where \phi(\cdot) is the function defining the autoregression, and where \varphi_k are the coefficients in the autoregression. The formula is valid only if all the roots have multiplicity 1. The autocorrelation function of an AR(p) process is a sum of decaying exponentials.

Graphs of AR(p) processes

The simplest AR process is AR(0), which has no dependence between the terms. Only the error/innovation/noise term contributes to the output of the process, so in the figure, AR(0) corresponds to white noise. For an AR(1) process with a positive \varphi, only the previous term in the process and the noise term contribute to the output. If \varphi is close to 0, then the process still looks like white noise, but as \varphi approaches 1, the output gets a larger contribution from the previous term relative to the noise. This results in a "smoothing" or integration of the output, similar to a low pass filter. For an AR(2) process, the previous two terms and the noise term contribute to the output. If both \varphi_1 and \varphi_2 are positive, the output will resemble a low pass filter, with the high frequency part of the noise decreased. If \varphi_1 is positive while \varphi_2 is negative, then the process favors changes in sign between terms of the process. The output oscillates. This can be likened to edge detection or detection of change in direction.

Example: An AR(1) process

An AR(1) process is given by:where is a white noise process with zero mean and constant variance. (Note: The subscript on \varphi_1 has been dropped.) The process is weak-sense stationary if |\varphi|<1 since it is obtained as the output of a stable filter whose input is white noise. (If \varphi=1 then the variance of X_t depends on time lag t, so that the variance of the series diverges to infinity as t goes to infinity, and is therefore not weak sense stationary.) Assuming |\varphi|<1, the mean is identical for all values of t by the very definition of weak sense stationarity. If the mean is denoted by \mu, it follows fromthatand hence The variance is where is the standard deviation of. This can be shown by noting that and then by noticing that the quantity above is a stable fixed point of this relation. The autocovariance is given by It can be seen that the autocovariance function decays with a decay time (also called time constant) of. The spectral density function is the Fourier transform of the autocovariance function. In discrete terms this will be the discrete-time Fourier transform: 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: which yields a Lorentzian profile for the spectral density: where is the angular frequency associated with the decay time \tau. An alternative expression for X_t can be derived by first substituting for X_{t-1} in the defining equation. Continuing this process N times yields For N approaching infinity, \varphi^N will approach zero and: It is seen that X_t is white noise convolved with the \varphi^k kernel plus the constant mean. If the white noise 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. For, the process will be a geometric progression (exponential growth or decay). In this case, the solution can be found analytically: whereby a is an unknown constant (initial condition).

Explicit mean/difference form of AR(1) process

The AR(1) model is the discrete-time analogy of the continuous Ornstein-Uhlenbeck process. It is therefore sometimes useful to understand the properties of the AR(1) model cast in an equivalent form. In this form, the AR(1) model, with process parameter, is given by By rewriting this as and then deriving (by induction), one can show that

Choosing the maximum lag

The partial autocorrelation of an AR(p) process equals zero at lags larger than p, so the appropriate maximum lag p is the one after which the partial autocorrelations are all zero.

Calculation of the AR parameters

There are many ways to estimate the coefficients, such as the ordinary least squares procedure or method of moments (through Yule–Walker equations). The AR(p) model is given by the equation 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.

Yule–Walker equations

The Yule–Walker equations, named for Udny Yule and Gilbert Walker, are the following set of equations. where m = 0, …, p, yielding p + 1 equations. Here \gamma_m is the autocovariance function of Xt, is the standard deviation of the input noise process, and is the Kronecker delta function. Because the last part of an individual equation is non-zero only if m = 0, the set of equations can be solved by representing the equations for m > 0 in matrix form, thus getting the equation which can be solved for all The remaining equation for m = 0 is which, once are known, can be solved for An alternative formulation is in terms of the autocorrelation function. The AR parameters are determined by the first p+1 elements \rho(\tau) of the autocorrelation function. The full autocorrelation function can then be derived by recursively calculating Examples for some Low-order AR(p) processes

Estimation of AR parameters

The above equations (the Yule–Walker equations) provide several routes to estimating the parameters of an AR(p) model, by replacing the theoretical covariances with estimated values. Some of these variants can be described as follows: Other possible approaches to estimation include maximum likelihood estimation. Two distinct variants of maximum likelihood are available: in one (broadly equivalent to the forward prediction least squares scheme) the likelihood function considered is that corresponding to the conditional distribution of later values in the series given the initial p values in the series; in the second, the likelihood function considered is that corresponding to the unconditional joint distribution of all the values in the observed series. Substantial differences in the results of these approaches can occur if the observed series is short, or if the process is close to non-stationarity.

Spectrum

The power spectral density (PSD) of an AR(p) process with noise variance is

AR(0)

For white noise (AR(0))

AR(1)

For AR(1)

AR(2)

The behavior of an AR(2) process is determined entirely by the roots of it characteristic equation, which is expressed in terms of the lag operator as: or equivalently by the poles of its transfer function, which is defined in the Z domain by: It follows that the poles are values of z satisfying: which yields: z_1 and z_2 are the reciprocals of the characteristic roots, as well as the eigenvalues of the temporal update matrix: AR(2) processes can be split into three groups depending on the characteristics of their roots/poles: with bandwidth about the peak inversely proportional to the moduli of the poles: The terms involving square roots are all real in the case of complex poles since they exist only when \varphi_2<0. Otherwise the process has real roots, and: The process is non-stationary when the poles are on or outside the unit circle, or equivalently when the characteristic roots are on or inside the unit circle. The process is stable when the poles are strictly within the unit circle (roots strictly outside the unit circle), or equivalently when the coefficients are in the triangle. The full PSD function can be expressed in real form as:

Implementations in statistics packages

Impulse response

The impulse response of a system is the change in an evolving variable in response to a change in the value of a shock term k periods earlier, as a function of k. Since the AR model is a special case of the vector autoregressive model, the computation of the impulse response in vector autoregression#impulse response applies here.

n-step-ahead forecasting

Once the parameters of the autoregression have been estimated, the autoregression can be used to forecast an arbitrary number of periods into the future. First use t to refer to the first period for which data is not yet available; substitute the known preceding values Xt-i for i=1, ..., p into the autoregressive equation while setting the error term equal to zero (because we forecast Xt to equal its expected value, and the expected value of the unobserved error term is zero). The output of the autoregressive equation is the forecast for the first unobserved period. Next, use t to refer to the next period for which data is not yet available; again the autoregressive equation is used to make the forecast, with one difference: the value of X one period prior to the one now being forecast is not known, so its expected value—the predicted value arising from the previous forecasting step—is used instead. Then for future periods the same procedure is used, each time using one more forecast value on the right side of the predictive equation until, after p predictions, all p right-side values are predicted values from preceding steps. There are four sources of uncertainty regarding predictions obtained in this manner: (1) uncertainty as to whether the autoregressive model is the correct model; (2) uncertainty about the accuracy of the forecasted values that are used as lagged values in the right side of the autoregressive equation; (3) uncertainty about the true values of the autoregressive coefficients; and (4) uncertainty about the value of the error term for the period being predicted. Each of the last three can be quantified and combined to give a confidence interval for the n-step-ahead predictions; the confidence interval will become wider as n increases because of the use of an increasing number of estimated values for the right-side variables.