The Simple Autoregressive Model

Martingale Difference Processes

The adapted sequence \(\{s_{t},\mathcal{F}_{t}\}\) is called a martingale if for every \(t\) the following conditions hold:

  • \[E|s_{t}|<\infty\]
  • \(E(s_{t}\mid\mathcal{F}_{t-1})=s_{t-1}\) a.s.

The adapted sequence \(\{x_{t},\mathcal{F}_{t}\}\) is called a martingale difference (m.d.) if for every \(t\) the following conditions hold:

  • \[E|x_{t}|<\infty\]
  • \(E(s_{t}\mid\mathcal{F}_{t-1})=0\) a.s. Here is a fundamental property of m.d. processes.

theorem Let \(\{x_{t}\}\) be an m.d. sequence and let \(g_{t-1}=g(x_{t-1},x_{t-2},\cdots,)\) be any measurable, integrable function of the lagged values of the sequence. Then \(x_{t}g_{t-1}\) is also an m.d., and \(x_{t}\) and \(g_{t-1}\) are uncorrelated.

proof By law of iterated expectations, we have

\[\begin{eqnarray*} E(x_{t}g_{t-1})&=&E(E(x_{t}g_{t-1}\mid\mathcal{F}_{t-1}))\\ &=&E(g_{t-1}E(x_{t}\mid\mathcal{F}_{t-1}))\\ &=&E(g_{t-1}\cdot 0)\\ &=&0 \end{eqnarray*}\]

It means that \(x_{t}g_{t-1}\) is also an m.d. For uncorrelation, we show that

\[\begin{eqnarray*} Cov(x_{t},g_{t-1})&=&E(x_{t}g_{t-1})-E(x_{t})E(g_{t-1})\\ &=&E(x_{t}g_{t-1})-E(E(x_{t}\mid\mathcal{F}_{t-1})E(g_{t-1})\\ &=&0-0\\ &=&0 \end{eqnarray*}\]

In particular, putting \(g_{t-1}=x_{t-j}\) for any \(j>0\), the theorem implies

\[\begin{eqnarray*} Cov(x_{t},x_{t-j})&=&0 \end{eqnarray*}\]

The m.d. property implies uncorrelatedness of the sequence, although it is a weaker property than independence.

Given an arbitrary sequence \(\{y_{t}\}\) satisfying $$E y_{t} <\infty\(and\)\sigma\left(y_{t},y_{t-1},\cdots\right)\subset \mathcal{F}{t}\(, an m.d. sequence can always be generated as the centred sequence\){x{t}}$$ where
\[\begin{eqnarray*} x_{t}&=&y_{t}-E(y_{t}\mid\mathcal{F}_{t-1}) \end{eqnarray*}\]

We consider the law of large number and central limit theorem for m.d. processes. theorem

Let \(\{x_{t}\}\) be an m.d. sequence. Then \(\mathop{plim} \bar{x}_{n}=0\) if either of the following condition hold.

  • The sequence is strictly stationary and $$E x_{t} <\infty$$
  • $$E x_{t} ^{1+\delta}<\infty\(,\)\forall t$$

The following is central limit theorem for m.d.

theorem Let \(\{x_{t},\mathcal{F}_{t}\}\) be a m.d. sequence with \(E(x_{t}^{2})=\sigma_{t}^{2}\) and let \(\bar{\sigma}_{n}^{2}=n^{-1}\sum_{t=1}^{n}\sigma_{t}^{2}\). If

  • \[\mathop{plim} n^{-1}\sum_{t=1}^{n}\left(x_{t}^{2}-\sigma_{t}^{2}\right)=0\]
  • either
    • the sequence is strictly stationary