Data Generation Processes

Let \(x_{t}\) be an \(m\times 1\) vector of economic variables generated at time \(t\). Such variables are typically inter0related both contemporaneously and across time. The collection \(\{x_{t},-\infty< t < \infty\}\) is called a vector-valued random sequence.

The data generation process (DGP) is represented by the conditional density

\begin{equation} D_{t}\left(x_{t}\mid\mathcal{X}_{t-1}\right) \end{equation} where \(\mathcal{X}_{t-1}=\sigma\left(x_{t-1},x_{t-2},x_{t-3},\cdots\right)\). This is a shorthand for the \(\sigma\)-field representing knowledge of the past history of the system. Notice that

::: remark \(D_{t}\) is allowed to depend on time, because the data are not assumed to be stationary. :::

DGPs and Models

A dynamic econometric model is a family of functions of the data which are intended to mimic aspects of the DGP, either \(D_{t}\) itself or functions derived from \(D_{t}\) such as moments. Formally, a model is a family of functions

\begin{equation} {M\left(x_{t},x_{t-1},x_{t-2},\cdots,d_{t};\psi\right),\psi\in\Psi},\Psi\subset\mathbb{R}^{p} \end{equation}

In particular, let \(M_{D}\) be a model of complete DGP.

• Parameters \(\psi\): \(p\) in number; parameters are constants that are common to every \(t\).
• Parameter space: \(\Psi\) denotes the set of admissible parameter values.
• The vector \(d_{t}\) represents variables, treated as non-stochastic, which are intended to capture the changes in the DGP over time.

The relationship between the DGP and the model is a difficult issue. The axiom of correct specification is the assumption that there exists a model element that is identical to the corresponding function of the DGP. \(M_{D}\) is correctly specified if there exists \(\psi_{0}\in\Psi\) such that \begin{equation} M_{D}\left(x_{t},x_{t-1},x_{t-2},\cdots,d_{t};\psi_{0}\right) = D_{t}\left(x_{t}\mid\mathcal{X}_{t-1}\right) \end{equation} In general, correct specification in practical modelling exercises is an implausible assumption.

Sequence Properties

Stationarity

A random sequence \(\{x_{t}\}\) is said to be stationary in the wide sense/covariance-stationary if the mean, the variance and the sequence of \(j\)th-order autocovariances for \(j>0\) are all independent of \(t\). It is said to be stationary in the strict sense if for every \(k>0\), the joint distributions of all collections \(\left(x_{t},x_{t+1},x_{t+2},\cdots,x_{t+k}\right)\) do not depend in any way on \(t\).

Mixing

In a mixing sequence the realization of the sequence at time \(t\) contains no information about the realization at either \(t-j\) or \(t+j\), when \(j\) is sufficiently large. The mixing property ensures that points in the sequence appear randomly sampled when they are far enough apart.

::: remark Stationarity and mixing are quite distinct properties. :::

Ergodicity

A stationary sequence having the property that a random event involving every member of the sequence always has probability either 0 or 1 is called ergodic.

::: theorem If \(\{x_{t}\}\) is a stationary ergodic sequence, and \(E(x_1)\) exists, \(\bar{x}_{n}\to E(x_{1})\) with probability 1. :::