Figure 1.5 The importance of the conditional mean.

time period throughout the process, but for the purposes of estimating and forecasting conditional volatility it is often assumed that the conditional mean is a constant.

A linear regression model such as the capital asset pricing model introduced in §8.1.1 could be used to estimate and forecast the conditional mean. Not necessarily very well though, since returns are extremely difficult to predict. However, there are some notable exceptions where the exclusion of a proper time-varying conditional mean could produce misleading results. The returns series depicted in Figure 1.5 shows the same sort of characteristics as, for example, internet stocks during 1999 and 2000. In the 1999 technology boom, returns on many internet-related companies were high, but in 2000 the boom turned into a slump and below normal returns were experienced in many such companies.

Figure 1.5 shows that the conditional volatility would be very high if a constant conditional mean were assumed throughout the entire period. However, if the model had a conditional mean that varied over time (being high during the first part of the data period and low during the second part of the data period) then the conditional volatility would be much lower at every point in time.

The conditional volatility has no place in the standard framework for linear regression, because standard linear regression assumes that returns are homoscedastic - that is, their conditional variance is the same throughout the process (this assmption is depicted in Figure 1.4a). The term conditional heteroscedasticity means that the conditional variance changes over time (this is depicted in Figure 1.4b). The most popular models for time-varying volatility are the GARCH models described in Chapter 4.

1.4 Constant and Time-Varying Correlation Models

Suppose that two stationary return processes rx and r2 are jointly covariance-stationary (§11.4). That is, their joint distribution has certain stability

properties over time. One of these properties is that the contemporaneous covariance cov{ru, rlt) is a constant, irrespective of the time at which it is measured; at every point in time t, cov(ru, r2t) = an-

It is only under this assumption of joint stationarity that one can define the unconditional correlation. It is the constant correlation given by

corr(r1(, r2t) = cov(r1(, 2,)1 /{ ( ) { )) (1.6)

Thus the correlation is independent of the time at which it is measured; in alternative notation it is denoted p12 = a/oo.

A scatter plot of two return series, such as those in Figures 1.2 and 1.3, gives a graphical representation of unconditional correlation. Scatter plots may be regarded as observations on the joint density function: the denser the points, the higher the frequency of joint values in that range. Where there is insignificant correlation scatter plots correspond to flat joint densities, like the Table Top mountain. Concentrated scatter plots come from joint densities with peaks or ridges, and here the unconditional correlation may be significantly different from zero.

If it exists, the unconditional correlation is one parameter, p, that has the same value throughout the process, just like unconditional volatility. But the estimates of unconditional volatility and correlation will be different at different times because of differences in the sample data. The smaller the sample the bigger the differences, because sampling errors are inversely proportional to the square root of sample size. It is important to note that there is nothing in an unconditional model to explain the variation in volatility or correlation estimates over time, except for sampling error.

It is important to note that there is nothing in an unconditional model to explain the variation in volatility or correlation estimates over time, except for sampling error

Unfortunately, the correlation one speaks of in financial markets does not always exist. Although it is generally satisfactory to assume that individual return processes are stationary it is by no means always the case that two return processes will be jointly stationary. For example, two arbitrary returns series such as a Latin American Brady bond and a stock in the Nikkei 225 could be totally unrelated. In that case they are not likely to be jointly stationary, so unconditional correlations between these returns do not exist. Of course, it is always possible to calculate a number, according to some statistical formula, and to suppose that this number represents correlation. But often these numbers change considerably from day to day, and this is a sure sign that the two returns processes are not jointly stationary - that unconditional correlation does not exist.

Of course, it is always possible to calculate a number, according to some statistical formula, and to suppose that this number represents correlation

It is not only in obviously unrelated markets that joint stationarity might fail to hold. In currency markets, commodity markets and equity markets it is not uncommon for time-varying correlation estimates to jump around considerably from day to day. Commonly, cross-market correlation estimates are even

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Figure 1.6 Bivariate vech GARCH correlation estimates of US dollar-sterling and Japanese yen-sterling exchange rates.

Some standard correlation estimation methods induce an apparent stability that is purely an artefact of the method

more unstable. Unfortunately, some standard correlation estimation methods induce an apparent stability that is purely an artefact of the method, and the unstable nature of underlying conditional correlations may be obscured (§3.1).

If correlation estimates are very unstable, what can be done to hedge correlation risk? In the absence of any specialized derivatives contract to hedge correlation it will be necessary to adjust mark-to-model values for uncertainty in correlation estimates (§5.3.1). Alternatively, one might consider using other measures of co-movement between assets, as described in Chapter 12.

The joint distribution between two returns series can also be viewed in a conditional framework, where the parameters that govern the joint distribution of returns are assumed to vary over time. Conditional correlation models allow the correlation in the conditional joint distribution to be different at different points in time. The notation is a natural extension of the notation for conditional volatility introduced above: the conditional covariance cov,(r,„ r2t), also denoted a12>„ is divided by the product of the conditional standard deviations of each return. Thus (1.6) is extended to the parameters of the conditional joint distribution, to give the conditional correlation, denoted corr((r„, r2t), or p12 ,.

A time-varying correlation model, such as the bivariate GARCH model described in §4.5.2, can be used to obtain time series estimates (and forecasts) of the conditional correlations between two returns series. However, these estimates are often quite unstable over time. Figure 1.6 illustrates a daily series