金融时间序列预测与决策及matlab软件（中山大学 夏南新）--chapter 4 univariate non-linear stochastic models

Chapter 4,Univariate non-linear stochastic models,4.1 Martingales, random walks and non-linearity,A martingale is a stochastic process that is a mathem-atical model of a fair game. The term martingale re-fers a usage that may be felt to be rather apposite wh-en considering the behaviour of financial data! A martingale may be formally defined as a stocha-stic process having the following properties:(a) for each ;(b) , whenever , where is the algebra comprising events determined by observ-ations over the interval , so that when,. This is known as the martingale property.While the history can, in general, include observations on any number of variables, it is often restricted to be just the past history of itself, i.e., . Written as , the martingale property implies that the MMSE forecast of a future increment of a martingale is zero. This property can be generalised to situations, quite common in finance, where,In which case we have a submartingale, and to the case where the above inequality is reversed, giving us a supermartingale. The martingale given by can be written equivalently asWhere is the martingale increment or martingale difference. When written in this form, the sequence looks superficially identical to the random walk, a model that was first introduced formally in chapter 2.,There could be defined to be strict white noise, so that it is both a stationary and independent sequence. Moreover, it is possible for to be uncorrelated but not necessarily stationary, such behaviour is allowed for in martingale differences; this implies that there could be dependent between higher conditional mom-ents, most notably conditional variances. The possibility of this form of dependence in finan-cial time series, which often go through protracted quiet periods interspersed with bursts of turbulence, leads naturally to the consideration of non-linear sto-,chastic processes capable of modelling such volatility. Non-linearity can, however, be introduced in many other ways, some of which may violate the martingale model. As an illustration, suppose that is generated by the process , with being defined aswhere is strict white noise. It follows immediately that has zero mean, constant variance, and ACF given by,For all , each of the terms in the ACF has zero expectation, so that, as far as its second-order properties are concerned, behaves just like an ind-ependent process. However, the MMSE forecast of a future observation, , is not zero (the unconditio-nal expectation), but is the conditional expectation,It then follows that is not a martingale, becauseand the non-linear structure of the process could be used to improve the forecasts of over the simple no-change forecast associated with the martingale model.,4.2 Testing the random walk hypothesis,4.2.1 Autocorrelation tests diagnostic testsAs noted in example 2.1, the portmanteau statistics and may be used in these circumstan-ces. On the random walk null, both statistics are distr-ibuted as , so that the null would be rejected for sufficiently high values.4.2.2 Calendar effectsA turn-of-the-month effect, in which the four-day ret-urn around the turn of a month is greater than the av-erage total monthly return; an intramonth effect, in,which the return over the first half of a month is sign-ificantly larger than the return over the second half; and a variety of intraday effects. 4.2.3 Consequences of non-linearities for random walk tests What happens if this innovation is just white noise (no strict white noise), so that the sequence is merely uncorrelated rather than independent? Suppose that is constructed from the strict white-noise process by, for , forIgnoring the sample mean , the autocorrelations are given by Keeping fixed at a positive number and letting , we have,which are the autocorrelations of the sequence . These have variances of approximately , so that , which can be arbitrarily high, and certainly bigger than the value of unity that occurs when is strict white noise rather than just uncorre-lated, as it is here by construction.,4.3 Stochastic volatility,A non-linear process is to allow the variance (or con-ditional variance) of the process to change either at certain discrete points in time or continuously. For a non-linear stationary process , the variance , is a constant for all , but the conditional variance depends on the observations and thus can change from period to period.4.3.1 Stochastic volatility models Suppose that the sequence is generated by the product process,Where is a standardised process, so that and for all , and is a sequence of posit-ive random variables usually such that : is thus the conditional standard deviation of . Typically is assumed to be normal and independent of : we will further assume that it is strict white noise. Equation can then be shown to be obtained as the discrete time approximation to the stochastic differential equation,Where and is standard Brownian motion. This is the usual diffusion process used to pr-ice financial assets in theoretical models of finance. The above assumption together imply that has mean , varianceAnd autocovariances,i.e., it is white noise. However, note that both the squared and absolute deviations, and , can be autocorrelated. What models are plausible for the conditional stan-dard deviation ? Since it is a sequence of positive random variables, a normal distribution is inappropr-iate, but as it is likely that will be skewed to the right, a log-normal distribution would seem to be a plausible choice. Let us define,Where and is independent of . A common interpretation of is that it represents the random and uneven flow of new information into financial markets. We then haveSince is always stationary, will be (weakly) stationary if and only if is, which will be the case if . Assuming this, then using the properties,of the log-normal distribution shows that all even moments of and will exist, being given byWhere and . All odd moments are zero. The moment measure of kurtosis then given by,so that the process has fatter tails than a normal distri-bution. The autocorrelation function of follows from the fact that,HenceandTaking logarithms of (4.2) yields,which shows that , but with non-normal innovations: if is normal then has mean 1.27 and variance 4.93 and a very long left-hand tail, caused by taking logarithms of very small numbers. The autocorrelation function of isNote that it is possible that some values of may be zero, in which case their logarithms cannot be taken.,One way of overcoming this difficulty is to employ the transformation used by Koopman et al.(1995)Where is the sample variance of and is a small number, set by Koopman et al. to be 0.02.,4.3.2 Estimation of stochastic volatility models The main difficulty with using stochastic volatility (SV) models is that they are rather difficult to estim-ate. Quasi-maximum likelihood(QML) technique of Koopman et al. (1995) available in STAMP 5.0 soft-ware package and the Kalman filter can all estimate the volatility .,Example 4.1 A stochastic volatility model for the dollar/sterling exchange rate In this example we fit the SV modelto the daily series of dollar/sterling first differences initially examined in example 2.5, where it was found to be close to zero mean white noise. To use the QML, the model is rewritten as,Where , or as Whereand,QML estimation using STAMP 5.0 yields the follow-ing estimates: , , and , and a plot of the exchange rate vo-latility, given by the smoothed estimates (of the sq-uare root) of , are shown in figure 4.1. The conditional variance-equation is close to a rand-om walk and the time-varying nature of the volatility can clearly be seen.,4.4 ARCH process,4.4.1 Development of generalised ARCH processes In the previous section the process determining the conditional standard deviations of was assumed not to be a function of . For example, for the log-normal model of equation (4.3), was depend-ent upon the information set . When the conditional standard deviations are a function of , i.e.,A simple example iswhere and are both positive. With and independent of , is then white noise and conditionally normal, i.e. so thatIf the kurtosis exceeds 3, the unconditional distribution is fatter tailed than the normal.,This model was first introduced by Engle (1982) and is known as the first-order autoregressive conditi-onal heteroskedastic process. A more convenient notation is to define , so that the model can be written asDefine , the model can also be written as,Since , the model corresponds directly to an model for the squared innovati-ons . However, as , the errors are obviously heteroskedastic. Suppose the parameters in the equation are defined as and , where and are independent. Thus which is consistent with being generated by a ran-dom coefficient process,where and has mean zero. A natural extension is the process, where (4.4) is replaced byWhere and , . The proc-ess will be weakly stationary if all the roots of the ch-aracteristic equation associated with the para-meters, , lie outside the unit circle, i.e., if, in which case the unconditional variance is . In terms of and , the conditional variance function isOr, equivalentlyTo obtain more flexibility about lag structure and po-sitive variance, a further extension, to the generalised process, was proposed (Bollerslev,1986, 1988): the process has the con-ditional variance functionWhere and , . All the coeffic-ients in the corresponding model must be positive. The roots of lie outside the unit circle, this positivity constraint is satisfied if and only if all the coefficients in are non-negative.,The equivalent form of the process is So that , where . This process will be weakly stationary if and only if the roots of lie outside the unit circle, i.e., if . This also ensure that is wea-kly stationary, but it is only a sufficient, rather than a necessary, condition for strict stationarity. Because processes are thick tailed, the conditions for,weak stationarity are often more stringent than those for strict stationarity. Nelson (1990a) shows that and will be strictly stationary in the model if and only ifand this will be satisfied if, for example, , and , although the conditions for weak stationarity are clearly violated. These complications with stationarity conditions carry over to the concept of volatility persistence in,models. If in (4.5) then contains a unit root and we say that the model is integrated , or (see Engle and Bollerslev, 1986). It is often the case that is very close to unit for financial ti-me series. Unfortunately, whether or not shocks persist can depend on which definition is adopted. For example, consider the model,from which we have thatIn the model with the conditi-onal expectation will tend to infinity as increases, i.e.Yet models are strictly stationary and, in this case, converges to a finite li-mit whenever . The implication of this is,that any apparent persistence of shocks may be a con-sequence of thick-tailed distributions rather than of inherent non-stationarity. Persistence may also be characterised by the impu-lse response coefficients. The process can be written, with , asor asThe impulse response coefficients are found from the,coefficients in the lag polynomial , , , The cumulative impulse response is zero beca-use contains a unit root or, equivalently, beca-use , which exponentially tends to zero in the limit as long as . However, when , so that we have an process., and hence shocks persist indefinitely.4.4.1 Modifications of GARCH processes To simplify the exposition, w shall concentrate on variants of the processAn early alternative was to model conditional standa-rd deviations rather than variances (Taylor, 1986, Sc-hwert, 1989),This makes the conditional variance the square of a weighted average of absolute shocks, rather than the weighted average of squared shocks. Consequently, large shocks have a smaller effect on the conditional variance than in the standard model. A non-symmetric response to shocks is made expl-icit in Nelsons (1991) exponential modelwhere,The news impact curve, , relates revisions in conditional volatility, here given by , to ne-ws, . It embodies a non-symmetric response sin-ce when and when . (Note that volatility will be at a minimum when there is no news, ). This asymmetry is potentially useful as it allows vola-tility to respond more rapidly to falls in a market than,to corresponding rises, which is an important stylized fact for many financial assets and is known as the le-verage effect. It is easy to show that is strictwhite noise with zero mean and constant variance, so that is an process and will be stationary if . A model which nests (4.6), (4.7) and (4.8) is the non-linear model (Higgins and Bera, 1992), a general form of which is,while an alternative is the threshold processwhere being the indicator function. If , we have the threshold model of Zakoian (1994), while for we have the GJR model of Glosten, Jagannathan and Runkle (1993), which allo-ws a quadratic response of volatility to news but with different coefficients for good and bad news, althou-,gh it maintains the assertion that the minimum volati-lity will result when there is no news. Ding, Granger and Engle (1993) and Hentschel (1995) define very general classes of models which nest all the above models; the details of how to do so may be found in the two papers. Hentschels model, for example, can be written using the Box and Cox (1964) transformation as,where Several variants cannot be nested within (4.9). En-gles (1990) asymmetric and Se-ntanas (1995) quadratic are two such models. These can be written in the simple case being considered here aswhere a negative value of means that good news increases volatility less than bad news. It is the prese-,nce of a quadratic from in that precludes them from being included as special cases of (4.9). An alternative way of formalising themodel (4.6) is to define , where is the unconditional variance, or long-run volatility, to which the process reverts to EVIEWS extends this formalisation to allow reversion to a varying level defined by,Here is long-run volatility, which converges to through powers of , while is the transitory component, converging to zero via powers of . This component model can also be com-bined with the model to allow asymmetries in both the permanent and transi