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109

should be distinguished from higher-order difference operators such as

& , = ,- ,-2-

Higher-order differences are useful for time series with seasonal components. For example, to eliminate seasonal effects in monthly data one can use the 12th difference operator given by

b\2yt=yt-yt-n-

The lag operator is defined as

Ly,=y,-i.

By defining L2yt = L(Lyt) = Lyt x = y, 2 and more generally powers of L as Lky, - y, k, polynomials, rational functions and power series in the lag operator can be constructed. For example,

, + 1-1 + a2y, 2 = (1 + a,L + a2L2)yt.

The ordinary operators of algebra also apply to time series, element by element. For our purposes the values of a time series will be real numbers, so they obey the usual rules of algebra relating to the operations of scalar multiplication and addition:

+ = {y. + Vx,}. 11.1.2 Stationary Processes and Mean-Reversion

Time series may have both stochastic and deterministic components; for example, a series with a deterministic trend and a stochastic white noise component is

, = a + $t + e„ (11.1)

where e, ~ i.i.d.( , 2). Most time series models of financial markets will have a stochastic component, and so the unconditional expectation and variance of the rth observation on the time series can be calculated (§1.2). For example, in the model (11.1)

E(yt) = a + $t and V(yt) = a2 for all t.

Also the sth-order autocovariance of {y,}, that is, the unconditional covariance of y. with v, j, is

cov(>„ y, s) = E[(yt - E(yt))(y, s - E(yt s))].

So. for example, the time series (11.1) has cov(>>„ , = [ , , 5] - 0 for all t and = 0.

A time series {y,} is covariance-stationary if the expectation, variance and jutoco\ariance are the same at every date t, that is,

Volatility and correlation are concepts that only apply to stationary processes. It makes no sense to try to estimate volatility or correlation on price data



> E(y,) is a finite constant; »- V(y,) is a finite constant; »- cov(yt, y, s) depends only on the lag s.

This is a weak form of stationarity that is usually what is meant when a time series is said to be simply stationary. A stronger form of stationarity, where not just the autocovariances but the whole joint distribution is independent of the date at which it is measured and depends only on the lag, is referred to as strict stationarity.

When prices appear to be trending, this is normally due to a stochastic rather than a deterministic trend

The time series (11.1) is not stationary. Even though V(yt) is a finite constant and cov(y„ y, s) depends only on the lag s (in fact it is zero for all t and s 0), the unconditional mean of y, is not independent of time. Any series with a trend in the mean will not be stationary and this is why financial asset prices, or their logarithms, tend not to be stationary. But the trends in financial markets do not normally follow the model (11.1). When prices appear to be trending, this is normally due to a stochastic rather than a deterministic trend. The basic distinction between these two models of trends is discussed in §11.1.4.

Figure 11.1 illustrates the very different behaviour of asset prices and asset returns. Whereas prices or log prices in most markets are represented by non-stationary time series models, a stationary process is used for the first difference in prices - or, more usually, the first difference in log prices since these are approximately the returns.2 Note that although returns often exhibit autoregressive conditional heteroscedasticity, this does not normally preclude them from being stationary (§4.1).

A simple example of a stationary time series is the process generated by an autoregressive model of order 1, the AR(1) model. Autoregressive models for time series are discussed in detail in §11.2.1; here we use the simplest of these models to illustrate the stationarity concept. Consider a simple version of the AR(1) model that does not have a constant term,

, = ay, x + zt,

(11.2)

where , ~ i.i.d.(0, ct2). The general AR(1) model is stable only if ct < 1, and in this case it defines a stationary process.3 To verify that this is true, suppose E(yt) = k\ and V(yt) = k2 for all t, where kx and k2 are finite constants. Then taking expectations and variances of (11.2) gives

kx = akt and k2 - a2k2 + ct2,

so if I a I < 1 then, for all t,

2When x is small 1 (1+ )» . The return , =( ,- -1)/ - = ( / -i) - 1, so 1 + r, = ( / -i) and taking logs gives the return as the first difference of the log prices.

3This is shown in §11.2, using the moving average representation of an autoregressive model.



(a) Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96

(b) Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Figure 11.1 HSBC: (a) price in Hong Kong dollars; (b) daily return.

= 0 and V(yt) = o7(l - a2). (11.3)

Since E(yt) - 0, the autocovariance at lag s is E(y,y, s). Now

E(y,y,-i) = ,-1 + £,)y,-i) = aE(y2 x) + E(e,yt x) = aV(yt x) - aa2/(l - a2)

( , ,-2) = ,-1 + z,)yt-2) = uE(yt xyt 2) = 2 2/(1 - 2), and in general

E(y,yt) = asa2/(l-a2), (11.4)

which depends only on the lag, s. So (11.2) defines a stationary time series if I ot I < 1.

A stationary process can never drift too far from its mean because of the finite variance

The mean-reversion property of stationary series is well known. A stationary process can never drift too far from its mean because of the finite variance. The speed of mean-reversion is determined by the autocovariance: mean-reversion



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