It was recently proved that any strictly stationary stochastic process can be viewed as an autoregressive process of order one with coloured noise. Furthermore, it was proved that, using this characterisation, one can define closed form estimators for the model parameter based on autocovariance estimators for several different lags. However, this estimation procedure may fail in some special cases. In this article, a detailed analysis of these special cases is provided. In particular, it is proved that these cases correspond to degenerate processes.

Stationary processes are an important tool in many practical applications of time series analysis, and the topic is extensively studied in the literature. Traditionally, stationary processes are modelled by using autoregressive moving average processes or linear processes (see monographs [

One of the most simple example of an autoregressive moving average process is an autoregressive process of order one. That is, a process

Statistical inference for AR(1)-process or Ornstein–Uhlenbeck process is well-established in the literature. Furthermore, recently a generalised continuous time Langevin equation, where the Brownian motion

When the model becomes more complicated, the number of parameters increases and the estimation may become a challenging task. For example, it may happen that standard maximum likelihood estimators cannot be expressed in closed form [

The authors in [

The rest of the paper is organised as follows. In Section

Let

By the Cauchy–Schwarz inequality and equality

In many applications of stationary processes, it is assumed that the covariance function

In most situations, a natural assumption regarding the covariance of the noise is

Examples of covariance functions corresponding to

Throughout this section, without loss of generality, we assume

Clearly,

Using (

Before proving our main theorems we need several technical lemmas.

We denote with

The modulo condition above means only that either

We write

Let

We are now ready to prove Theorem

Throughout the proof we denote

Since

Denote

Examples of excluded intervals. In part (a) we have set

Consider first the case

Note that in the periodic case the covariance matrix

Suppose

In this section we present a simulation study in order to compare the classical Yule–Walker estimator with our quadratic type estimator in the case of an AR

If

Figure

Classical Yule–Walker estimates of an AR

We have shown (Theorem

The discussed estimation procedure has recently been applied in practice in [

In addition to the above mentioned generalized ARCH model, it would be interesting to study whether our method can be applied in modeling and estimation of different temporal (stationary) models. For example, one could consider GARCH-X models or even integer valued processes such as INAR