Autoregressive Model - Example: An AR(1)-process

Example: An AR(1)-process

An AR(1)-process is given by:

where is a white noise process with zero mean and variance . (Note: The subscript on has been dropped.) The process is wide-sense stationary if since it is obtained as the output of a stable filter whose input is white noise. (If then has infinite variance, and is therefore not wide sense stationary.) Consequently, assuming, the mean is identical for all values of t. If the mean is denoted by, it follows from

\operatorname{E} (X_t)=\operatorname{E} (c)+\varphi\operatorname{E} (X_{t-1})+\operatorname{E}(\varepsilon_t),

that

and hence

In particular, if, then the mean is 0.

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:

\Phi(\omega)= \frac{1}{\sqrt{2\pi}}\,\sum_{n=-\infty}^\infty B_n e^{-i\omega n} =\frac{1}{\sqrt{2\pi}}\,\left(\frac{\sigma_\varepsilon^2}{1+\varphi^2-2\varphi\cos(\omega)}\right).

This expression is periodic due to the discrete nature of the, which is manifested as the cosine term in the denominator. If we assume that the sampling time is much smaller than the decay time, then we can use a continuum approximation to :

which yields a Lorentzian profile for the spectral density:

\Phi(\omega)= \frac{1}{\sqrt{2\pi}}\,\frac{\sigma_\varepsilon^2}{1-\varphi^2}\,\frac{\gamma}{\pi(\gamma^2+\omega^2)}

where is the angular frequency associated with the decay time .

An alternative expression for can be derived by first substituting for in the defining equation. Continuing this process N times yields

For N approaching infinity, will approach zero and:

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

Read more about this topic:  Autoregressive Model