Some Examples

Some Examples[1]#

First-Order-Markov:#

Consider the first-order autoregressive process \((1-\lambda L)x_t = \epsilon_t\), \(\epsilon_t\) white noise, \(|\lambda| < 1\), \(\epsilon_t = x_t - P[x_t|x_{t-1}, x_{t-2},\ldots]\); we can write \(x_t = (1/(1-\lambda L))\epsilon_t\). We have

\[\begin{split} \begin{aligned} P_{t-1}x_t &= \left[L^{-1}(1 + \lambda L + \lambda^2 L^2 + \ldots)\right]_{+}(1-\lambda L)x_{t-1}\\ &= ( \lambda + \lambda^2 L + \ldots)(1 - \lambda L)x_{t-1}\\ &= \left(\frac{\lambda}{1 - \lambda L}\right)(1-\lambda L)x_{t-1} = \lambda x_{t-1} \end{aligned} \end{split}\]

More generally,

\[ P_{t-k}x_t = \left[L^{-k}(1 + \lambda L + \ldots)\right]_{+}(1-\lambda L)x_{t-k} = \lambda^k x_{t-k}. \]

Thus we have

\[ P_t x_{t+k} = \lambda^k x_t. \]

First-Order Moving Average:#

Suppose \(x_t = (1 + \beta L)\epsilon_t\), \(\epsilon_t\) white, \(|\beta| < 1\), \(\epsilon_t = x_t - P[x_t|x_{t-1}, x_{t-2},\ldots]\). Then we have

\[ P_{t-1}x_t = \left[L^{-1}(1 + \beta L)\right]_{+}\left(\frac{1}{1 + \beta L}\right)x_{t-1}, \qquad P_{t-1}x_t = \frac{\beta}{1 + \beta L}x_{t-1}. \]

We also have that for \(k \geq 2\),

\[ P_{t-k}x_t = \left[L^{-k}(1 + \beta L)\right]_{+}\left(\frac{1}{1 + \beta L}\right)x_{t-k} = 0, \]

which can also be seen directly by projecting on \(\{x_{t-k}, x_{t-k-1},\ldots\}\) both sides of \(x_t = (1 + \beta L)\epsilon_t\).

First-Order Moving Average, Autoregressive:#

Suppose we have

\[\begin{split} x_t = \left(\frac{1 + a L}{1 - \beta L}\right)\epsilon_t, \qquad \epsilon_t \text{ white},\quad |a|<1, |\beta| < 1,\\ \epsilon_t = x_t - P[x_t|x_{t-1}, x_{t-2},\ldots] \end{split}\]

We then have

\[\begin{split} \begin{aligned} P_{t-1}x_t &= \left(\frac{L^{-1}(1 + a L)}{(1 - \beta L)}\right)_{+}\left(\frac{1 - \beta L}{1 + a L}\right)x_{t-1}\\ &= \left(\frac{L^{-1}}{1 - \beta L} + \frac{a}{1 - \beta L}\right)_{+}\left(\frac{1-\beta L}{1 + a L}\right)x_{t-1}\\ P_{t-1}x_t &= \left(\frac{\beta + a}{1 - \beta L}\right)\left(\frac{1 - \beta L}{1 + a L}\right)x_{t-1}, \qquad P_{t-1}x_t = \left\{\frac{a + \beta}{1 + a L}\right\}x_{t-1}, \end{aligned} \end{split}\]

which expresses the forecast of \(x_t\) as a geometric distributed lag of past \(x\)’s. The first-order mixed moving average, autoregressive model for \(x_t\) thus provides a rationalization for the familiar “adaptive expectations” model. As we let \(\beta \to 1\) (from below, in order to assure that the roots condition \(|\beta| < 1\) is met), \(P_{t-1}x_t\) approaches

\[ P_{t-1}x_t = \{(1 + a)/(1 + a L)\}x_{t-1}, \]

which with \(a < 0\) is equivalent with Cagan’s (1956) adaptive expectations scheme

\[ P_{t-1}x_t = \{(1 - \lambda )/(1 - \lambda L)\}x_{t-1}, \]

with \(a = - \lambda\). Notice that as \(\beta \to 1\) (from below), we approach the situation in which \((1-L)x_t = (1 + a L)\epsilon_t\), so that the first difference of \(x_t\) follows a first-order moving average. The parameter \(a\) must be negative in order that \(\lambda > 0\).

For the general case in which \(k \geq 1\), we have

(249)#\[\begin{split} \begin{aligned} P_{t-k}x_t &= \left(\frac{L^{-k}(1+aL)}{(1-\beta L)}\right)_{+}\left(\frac{1-\beta L}{1+aL}\right)x_{t-k}\\ &=\left(\frac{\beta^k}{1-\beta L} + \frac{a \beta^{k-1}}{1-\beta L}\right)_{+}\left(\frac{1-\beta L}{1+aL}\right)x_{t-k}\\ &=\left(\frac{\beta^k}{1-\beta L} + \frac{a \beta^{k-1}}{1-\beta L}\right)\left(\frac{1-\beta L}{1+aL}\right)x_{t-k} = \frac{\beta^{k-1}(\beta + a)}{(1+aL)} x_{t-k}. \end{aligned} \end{split}\]

We can write this alternatively as

\[ P_t x_{t+k} = (\beta^{k-1}(\beta + a)/(1 + a L))x_t \]

Notice that as \(\beta \to 1\) (from below) we approach the situation in which

\[ P_t x_{t+k} = ((1 + a)/(1 + a L))x_t, \]

so that the same forecast is made for all horizons \(k \geq 1\). In this sense there is a well-defined concept of “permanent \(x\).” This was first pointed out in the economics literature by Muth (1960), who showed that the hypothesis of rational expectations in conjunction with the model for income \((1 - L)x_t = (1+aL)\epsilon_t\), provides a rationalization both for the concept of permanent income and the geometric distributed lag formula that Friedman had earlier used to estimate permanent income in empirical work.