1974

1975

1976

1977

1978

1979

1980

1981

Figure 13.2. Weekly average total transactions deposits component of M, for May 15, 1974-January 6, 1982. Source: D. A. Pierce, M. R. Grupe, and W. P. Cleveland, "Seasonal Adjustment of the Weekly Monetary Aggregates," Journal of Business and Economics Statistics, Vol. 2 (1984), p. 264.

Nonstationarity

In time-series analysis we do not confine ourselves to the analysis of stationary time series. In fact, most of the time series we encounter are nonstationary. A simple nonstationary time-series model is A, = x, -I- e„ where the mean x, is a function of time and e, is a weakly stationary series, yi,, for example, could be a linear function of t (a linear trend) or a quadratic function t (a parabolic trend). In Figure 13.2 we show the graph of a nonstationary time series. It is clear that apart from a linear trend, holiday season peaks dominate the movements in the series.

13 Some Useful Models for Time Series

In this section we discuss several different types of stochastic processes that are useful in modeling time series: (1) a purely random process, (2) a random walk, (3) a moving-average (MA) process, (4) an autoregressive (AR) process, (5) an autoregressive moving-average (ARMA) process, and (6) an autoregressive integrated moving-average (ARIMA) process.

P-n forAO A purely random process is also called white noise.

Random Walk

This is a process often used to describe the behavior of stock prices (although there are some dissidents who disagree with this random walk theory). Suppose that {e,} is a purely random series with mean x and variance cr. Then a process {X,} is said to be a random walk if

X, = X,, + e,

Let us assume that Xq is equal to zero. Then the process evolves as follows:

X, = e,

X2 = X, + E2 = Ei + e2 and so on We have by successive substitution

,= 1 .

Hence E(X,) = tp. and var(Z,) = ta. Since the mean and variance change with t, the process is nonstationary, but its first difference is stationary. Referring to share prices, this says that the changes in a share price will be a purely random process.

Moving-Average Processes

Suppose that {e,} is a purely random process with mean zero and variance u. Then a process {X,} defined by

X, = poe, + p,e, , + • • • + „¨,„„

is called a moving-average process of order m and is denoted by MA(w). Since the es are unobserved variables, we scale them so that Po = 1- Since £(e,) =

0 for all t. we have ( ,) = 0. Also, var(X,) = P?) smce the e, are

Purely Random Process

This is a discrete process {X,} consisting of a sequence of mutually independent identically distributed random variables. It has a constant mean and a constant variance and the acvf is

y{k) = cov(„ ,+ ) = 0 for A: 5 0

The acf is given by

for A = 0

< 1

This gives the result that p, and P2 must satisfy:

p, + p2 > -1

2- , >-l (13.1)

IP2l< 1

An alternative statement often found in books on time series is that the roots of the equation 1 + P,Z + PjZ + . . . + Z" = 0 all lie outside the unit circle.

independent with a common variance cr. Further, writing out the expressions for X, and X,k in terms of the es and picking up the common terms (since the es are independent), we get

yik) = cov(Z„ X,)

S for* = 0, 1,2, . . . ,m

1 = 0

0 for A: > m

Also considering cov(Z„ X,+i), we get the same expressions as for y{k).

Hence y{-k) = { ). The acf can be obtained by dividing y{k) by var(Z,). For the MA process, p{k) ~ Ofor k> m, that is, they are zero for lags greater than the order of the process. Since ( ) is independent of /, the MA{m) process is weakly stationary. Note that no restrictions on the (5, are needed to prove the stationarity of the MA process.

To facilitate our notation we shall use the lag operator L. It is defined by , = X, j for all j. Thus LX, = Z, „ DX, = X,, LX, = Z,+ „ and so on.

With this notation the { ) process can be written as (since Po = 1)

J, = (1 + p,L + + •• + p„L")e, = p(I.)e, (say)

The polynomial in L has m roots and we can write

, = (1 - iT,L)(l - 1T2L) • • • (1 - iT„L)e,

where ,, ttj, . . . , „ are the roots of the equation

yn, + PiK-"- • • p„ = 0

After estimating the model we can calculate the residuals from e, = [p(L)]~x, provided that [p(L)]" converges. This condition is called the invertibility condition. The condition for invertibility is that it, < 1 for all i. This impUes that an MA(w) process can be written as an AR(oo) process. For instance, for the MA(2) process

X, = {\ + p,L + jz,

, and are roots of the quadratic equation 1 -I- , + = 0-The condition IttJ < 1 gives

-P, ± VP? -4p2