Modern Stochastics: Theory and Applications

Stochastic processes are widely used in modeling and forecasting numerous real life phenomena such as natural disasters, activity of the sun, sales of a company and market movements, to mention a few. When stationary processes are considered, modeling is traditionally based on fitting an autoregressive moving average (ARMA) process. However, in this paper, we challenge this conventional approach. Instead of fitting an ARMA model, we apply the AR$(1)$ characterization in modeling any strictly stationary processes. Moreover, we derive consistent and asymptotically normal estimators of the corresponding model parameter.

One of the reasons why ARMA processes have been in a central role in modeling of time-series data is that for every autocovariance function $\gamma (\cdot )$ vanishing at infinity and for every $n\in \mathbb{N}$ there exists an ARMA process X such that $\gamma (k)=\gamma _{X}(k)$ for every $k=0,1,..,n$. For a general overview of the theory of stationary ARMA processes and their estimation, the reader may consult for example [1] or [5].

ARMA processes, and their extensions, have been studied extensively in the literature. A direct proof of consistency and asymptotic normality of Gaussian maximum likelihood estimators for causal and invertible ARMA processes was given in [18]. The result was originally obtained, using asymptotic properties of the Whittle estimator, in [7]. The estimation of the parameters of strictly stationary ARMA processes with infinite variances was studied in [16], again, by using Whittle estimators. Portmanteau tests for ARMA models with stable Paretian errors with infinite variance were introduced in [12]. An efficient method for evaluating the maximum likelihood function of stationary vector ARMA models was presented in [14]. Fractionally integrated ARMA models with a GARCH noise process, where the variance of the error terms is also of ARMA form, was studied in [13]. Consistency and asymptotic normality of the quasi-maximum likelihood estimators of ARMA models with the noise process driven by a GARCH model was shown in [3]. A least squares approach for ARMA parameter estimation has been studied at least in [9] by contrasting its efficiency with the maximum likelihood estimation. Also estimators of autocovariance and their limiting behavior have been addressed in numerous papers. See for example [2, 8, 11] and [15].

Modeling an observed time-series with an ARMA process starts by fixing the orders of the model. This is often done by an educated guess, but there also exists methods for estimating the orders, see e.g. [6]. After the orders are fixed, the related parameters can be estimated, for example, by using the maximum likelihood or least squares estimators. These estimators are expressed in terms of optimization problems and do not generally admit closed form representations. The final step is to conduct various diagnostic tests to determine whether the estimated model is sufficiently good or not. These tests are often designed to recognize whether the residuals of the model support the underlying assumptions about the error terms. Depending on whether one considers strict or weak stationarity, the error process is usually assumed to be an IID process or white noise, respectively. If the tests do not support the assumptions about the noise process, then one has to start all over again. Tests for the goodness of fit of ARMA models have been suggested e.g. in [4].

The approach taken in this paper is based on the discrete version of the main theorem of [17] leading to an AR$(1)$ characterization for (any) strictly stationary processes. Note that this approach covers, but is not limited to, strictly stationary ARMA processes. It was stated in [17] that a process is strictly stationary if and only if for every fixed $0<H<1$ it can be represented in the AR$(1)$ form with $\phi ={e}^{-H}$ and a unique, possibly correlated, noise term. Although the representation is unique only after H is fixed, we show that in most of the cases, given just one value of the autocovariance function of the noise, one is able to determine the AR$(1)$ parameter and, consequently, the entire autocovariance function of the noise process. It is worth emphasizing that since the parameter–noise pair in the AR$(1)$ characterization is not unique, it is natural that some information about the noise has to be assumed. Note that conventionally, when applying ARMA models, we have assumptions about the noise process much stronger than being IID or white noise. That is, the autocovariance function of the noise is assumed to be identically zero except at the origin. When founding estimation on the AR$(1)$ characterization, one does not have to select between different complicated models. In addition, there is only one parameter left to be estimated. Yet another advantage over classical ARMA estimation is that we obtain closed form expressions for the estimators.

The paper is organized as follows. We begin Section 2 by introducing some terminology and notation. After that, we give a characterization of discrete time strictly stationary processes as AR$(1)$ processes with a possibly correlated noise term together with some illustrative examples. The AR$(1)$ characterization leads to Yule–Walker type equations for the AR$(1)$ parameter ϕ. In this case, due to the correlated noise process, the equations are of quadratic form in ϕ. For the rest of the section, we study the quadratic equations and determine ϕ with as little information about the noise process as possible. The approach taken in Section 2 leads to an estimator of the AR$(1)$ parameter. We consider estimation in detail in Section 3. The end of Section 3 is dedicated to testing the assumptions we make when constructing the estimators. A simulation study to assess finite sample properties of the estimators is presented in Section 4. Finally, we end the paper with three appendices containing a technical proof, detailed discussion on some special cases and tabulated simulation results.

Throughout the paper we consider strictly stationary processes.

The following class of stochastic processes was originally introduced in [17].

Next, we consider the AR$(1)$ characterization of strictly stationary processes. The continuous time analogy was proved in [17] together with a sketch of a proof for the discrete case. For the reader’s convenience, a detailed proof of the discrete case is presented in Appendix A.

It is worth to note that the noise Z in Corollary 1 is unique only after the parameter H is fixed. The message of this result is that every strictly stationary process is an AR$(1)$ process with a strictly stationary noise that may have a non-zero autocovariance function. The following examples show how some conventional ARMA processes can be represented as an AR$(1)$ process.

Next we show how to determine the AR$(1)$ parameter ${\phi }^{(H)}$ in (3) provided that the observed process X is known. In what follows, we omit the superindices in (3). We assume that the second moments of the considered processes are finite and that the processes are centered. That is, $\mathbb{E}(X_{t})=\mathbb{E}(Z_{t})=0$ for every $t\in \mathbb{Z}$. Throughout the rest of the paper, we use the notation cov$(X_{t},X_{t+n})=\gamma (n)$ and cov$(Z_{t},Z_{t+n})=r(n)$ for every $t,n\in \mathbb{Z}$. Note that if $\gamma (N)=r(N)=0$, then Lemma 1 yields only $\gamma (N+1)+\gamma (N-1)=0$ providing no information about the parameter ϕ. As such, in order to determine the parameter ϕ, we require that either $\gamma (N)\ne 0$ or $r(N)\ne 0$. At first glimpse it seems that Corollary 2 is not directly applicable. Indeed, in principle it seems like there could be complex-valued solutions although representation (3) together with (4) implies that there exists a solution $\phi \in (0,1)$. Furthermore, it is not clear whether the true value is given by (5) or (6). We next address these issues. We start by proving that the solutions to (4) cannot be complex. At the same time we are able to determine which one of the solutions one should choose.

Note that if $r(N)=0$, as $\phi <1$, the discriminant is always positive. Let $a_{N}=\frac{r(N)}{\gamma (N)}$. The proof above now gives us the following identity This enables us to consider the choice between (5) and (6). Assume that $\gamma (N)>0$. If ${\phi }^{2}-1+a_{N}>0$, then ϕ is given by (5) (as $\phi \in (0,1)$). Similarly, if ${\phi }^{2}-1+a_{N}<0$, then ϕ is determined by (6). Finally, contrary conclusions hold in the case $\gamma (N)<0$. In particular, we can always choose between (5) and (6) provided that either $a_{N}\le 0$ or $a_{N}\ge 1$. Moreover, from (4) it follows that if and only if provided that the denominators differ from zero. Since (5) and (6) can be written as we observe that one can always rule out one of the solutions (5) and (6) provided that $a_{N}\ne a_{N+k}$. Therefore, it always suffices to know two values of the autocovariance r such that $a_{N}\ne a_{N+k}$, except the worst case scenario where $a_{j}=a\in (0,1)$ for every $j\in \mathbb{Z}$. A detailed analysis of this particular case is given in Appendix B.

We end this section by observing that in the case of vanishing autocovariance function of the noise, we get the following simplified form for the AR$(1)$ parameter.

Recall that the representation (2) is unique only after H is fixed. As a simple corollary for Theorem 2 we obtain the following result giving some new information about the uniqueness of the representation (2).

Corollary 2 gives natural estimators for ϕ provided that we have been able to choose between (5) and (6), and that a value of $r(n)$ is known. We emphasize that in our model it is sufficient to know only one (or in some cases two) of the values $r(n)$, whereas in conventional ARMA modeling much stronger assumptions are required. (In fact, in conventional ARMA modeling the noise process is assumed to be white noise.) It is also worth to mention that, generally, estimators of the parameters of stationary processes are not expressible in a closed form. For example, this is the case with the maximum likelihood and least squares estimators of conventionally modeled ARMA processes, see [1]. Within our method, the model fitting is simpler. Finally, it is worth to note that assumption of one known value of $r(n)$ is a natural one and cannot be avoided. Indeed, this is a direct consequence of the fact that the pair $(\phi ,Z)$ in representation (3) is not unique. In fact, for practitioner, it is not absolutely necessary to know any values of $r(n)$. The practitioner may make an educated guess and proceed in estimation. If the obtained estimate then turns out to be feasible, the practitioner can stop there. If the obtained estimate turns out to be unreasonable (not on the interval $(0,1)$), then the practitioner have to make another educated guess. The process is similar to selecting p and q in traditional ARMA$(p,q)$ modeling.

Throughout this section, we assume that $(X_{1},\dots ,X_{T})$ is an observed series from a centered strictly stationary process that is modeled using the representation (3). We use $\hat{\gamma }_{T}(n)$ to denote an estimator of the corresponding autocovariance $\gamma (n)$. For example, $\hat{\gamma }_{T}(n)$ can be given by or more generally where $\bar{X}$ is the sample mean of the observations. For this estimator the corresponding sample covariance (function) matrix is positive semidefinite. On the other hand, the estimator is biased while it is asymptotically unbiased. Another option is to use $T-n-1$ as a denominator. In this case one has an unbiased estimator, but the sample covariance (function) matrix is no longer positive definite. Obviously, both estimators have the same asymptotic properties. Furthermore, for our purposes it is irrelevant how the estimators $\hat{\gamma }_{T}(n)$ are defined, as long as they are consistent, and the asymptotic distribution is known.

We next consider estimators of the parameter ϕ arising from characterization (3). In this context, we pose some assumptions related to the autocovariance function of the observed process X. The justification and testing of these assumptions are discussed in Section 3.1. From a priori knowledge that $\phi \in (0,1)$ we enforce also the estimators to the corresponding closed interval. However, if one prefers to use unbounded versions of the estimators, one may very well do that. The asymptotic properties are the same in both cases. We begin by defining an estimator corresponding to the second part (2) of Corollary 2.

We continue by defining an estimator corresponding to the first part (1) of the Corollary 2. For this we assume that, for reasons discussed in Section 2, we have chosen the solution (5) (cf. Remark 5 and Section 3.1). As above, we show that consistency and asymptotic normality follow from the same properties of the autocovariance estimators. In the sequel we use a short notation In addition, we denote and

Before proving the asymptotic normality, we present some short notation. We set and where

Note that the asymptotics given in Remarks 6 and 7 hold also if one forces the corresponding estimators to the interval $[0,1]$ as we did in Definitions 4 and 5.

We present a simulation study to assess the finite sample performance of the estimators. In the simulations, we apply the estimator corresponding to the first part (1) of Corollary 2. We simulate data from AR$(1)$ processes and ARMA$(1,2)$ processes with $\theta _{1}=0.8$ and $\theta _{2}=0.3$ as the MA parameters. (Note that these processes correspond to Examples 1 and 2.) We assess the effects of the sample size T, AR$(1)$ parameter φ, and the chosen lag N. We consider the sample sizes $T=50,500,5000,50000$, lags $N=1,2,3,\dots ,10$, and the true parameter values $\varphi =0.1,0.2,0.3,\dots ,0.9$. For each combination, we simulate 1000 draws. The sample means of the obtained estimates are tabulated in Appendix C.

Histograms given in Figures 1, 2 and 3 reflect the effects of the sample size T, AR$(1)$ parameter φ, and the chosen lag N, respectively. In Figure 1, the parameter $\varphi =0.5$ and the lag $N=3$. In Figure 2, the sample size $T=5000$ and the lag $N=3$. In Figure 3, the parameter $\varphi =0.5$ and the sample size $T=5000$. The summary statistics corresponding to the data displayed in the histograms are given in Appendix C.

Figure 1 exemplifies the rate of convergence of the estimator as the number of observations grows. One can see that with the smallest sample size, the lower bound is hit numerous times due to the large variance of the estimator. In the upper series of the histograms, the standard deviation reduces from 0.326 to 0.019, whereas in the lower series it reduces from 0.250 to 0.008. The faster convergence in the case of ARMA$(1,2)$ can be explained with the larger value of $\gamma (3)$ reducing the variance in comparison to the AR$(1)$ case. The same phenomenon recurs also in the other two figures.

Figure 2 reflects the effect of the AR$(1)$ parameter on the value of $\gamma (3)$ and consequently on the variance of the estimator. The standard deviation reduces from 0.322 to 0.020 in the case of AR$(1)$ and from 0.067 to 0.009 in the case of ARMA$(1,2)$.

In Figure 3 one can see how an increase in the lag increases the variance of the estimator. In the topmost sequence, the standard deviation increases from 0.014 to 0.326 and in the bottom sequence from 0.015 to 0.282.

We wish to emphasize that in general smaller lag does not imply smaller variance, since the autocovariance function of the observed process is not necessarily decreasing. In addition, although the autocovariance $\gamma (N)$ appears to be the dominant factor when it comes to the speed of convergence, there are also other possibly significant terms involved in the limit distribution of Theorem 6.