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is quick when autocovariances are small and slow when autocovariances are large.

The speed of mean-reversion in the AR(1) model (11.2) depends on the size of a, as shown in Figure 11.2. If a = 0 then {y,} is just white noise and mean-reversion is instantaneous because cov(y(, yt s) - 0. As a increases in absolute value the speed of mean-reversion decreases. In the limit when a - 1, then {>>,} is a random walk (without drift), which is non-stationary, and there is no mean-reversion.

11.1.3 Integrated Processes and Random Walks

There is an enormous literature in financial economics concerning the validity of various forms of the efficient market hypothesis (see Cuthbertson, 1996, The efficient market for a review). The efficient market hypothesis implies that in liquid markets, hypothesis implies that where asset prices will be the result of unconstrained demand and supply the best forecast of the equilibria, the current price should accurately reflect all the information that is price on any future date available to the players in the market. Future changes in prices can only be the is simply the price today resuit 0f news, which by definition is unpredictable, so the best forecast of the price on any future date is simply the price today. Put another way, the price today is just yesterdays price plus a random term.

The efficient market hypothesis is related to basic option pricing models. The fundamental assumption of these models is that the underlying asset price S follows a geometric Brownian motion process,

dS/S = rdt + odZ, (11.5)

where and a are constants representing the drift in asset prices and the volatility of returns respectively, and Z is a Wiener process. That is, increments dZ are independent and normally distributed with mean zero and variance dt.

To see how (11.5) relates to the efficient market hypothesis, apply Itos lemma to (11.5). This gives the corresponding continuous time process that will be followed by log prices:

d\nS = (r-a2/2)dt + adZ. (11.6)

The discrete time version of (11.6) is InP, - \nPf i = + , or

lnP, = c + lniVi + e(, (11-7)

where - r - a2/2 and the error term e, ~ NID(0, ct2), is the returns process.

The model (11.7) is the random walk model that is commonly applied to model log prices in efficient financial markets. In (11.7) e, is normally and independently distributed with mean zero and constant variance ct2. However, market efficiency only implies that the distribution of the return conditional on



Figure 11.2 Realizations from AR(1) model: (a) a = 0.5; (b) a = 0.9; (c) a = 0.99.

Market efficiency only implies that the distribution of the return

all information up to time t is independent and identically distributed. That is, conditional on all e, \I, ~ i.i.d.(0, o2), where /, denotes the information set at time t. So the information up to time t efficient markets hypothesis implies the model (11.7) but with a less restrictive is independent and assumption on the increments. identically distributed



Upon visual inspection there will be no obvious drift in the data, but the term stochastic trend still applies because the data are generated by an integrated process

The random walk model allows for trends in asset prices by including a constant term in (11.7) that corresponds to the expected return. Thus if > 0 log prices are trending upwards and if < 0 they are trending downwards. Even when = 0 we say that (11.7) has a stochastic trend. Upon visual inspection there will be no obvious drift in the data, but the term stochastic trend still applies because the data are generated by an integrated process.

A time series is integrated of order n, written , ~ I(n), if the stochastic part is non-stationary but it becomes stationary after differencing a minimum of n times. Thus a process that is already stationary is denoted ). Note that the deterministic trend model (11.9) is not stationary, but neither is it integrated; it is a trend-stationary process that is also referred to as 7(0) + trend.

A random walk is an example of an integrated process of order 1. In financial markets the general class of 1(1) processes is characterized by the model

lnPf = c + \nP1 l + 8,.

(11.8)

where 8, ~ 1(0). The distinction between (11.8) and (11.7) is that in the random walk model the error process is not just stationary, it is white noise. In less than fully efficient markets it is possible that log asset prices are not pure random walks because returns are autocorrelated, but they may still be 1) processes.

11.1.4 Detrending Financial Time Series Data

It is important to understand that the trend in (11.8) is not a deterministic trend. The 1) process (11.7) is fundamentally different from the model

lnP, = c + p7+e„

( .9)

which has a stationary component and a deterministic trend component. Neither (11.8) nor (11.9) is a stationary series, and the data generated by the two models may seem very similar indeed to the eye. But the transform required to make data generated by (11.8) into a stationary series is a first difference transform and for this reason the integrated process model (11.8) is also commonly referred to as a difference-stationary process. On the other hand, the stationarity transform for data generated by (11.9) is to take deviations from a fitted trend line, so (11.9) is called a trend-stationary process.

It is not unknown for technical analysis to fit a trend line to price or log price data and to take deviations from that trend as a series that can be predicted

It is not unknown for technical analysis to fit a trend line to price or log price data and to take deviations from that trend as a series that can be predicted. But fitting a trend and taking deviations will not make the series predictable if the market is efficient. All it does is to remove the drift in the random walk. The deviations from trend are still a random walk, just without the drift, but they still have a stochastic trend. They will have no mean-reversion and therefore cannot be predicted well using univariate models.



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