LONG MEMORY, VOLATILITY, RISK AND DISTRIBUTION

Transcription

1 LONG MEMORY, VOLATILITY, RISK AND DISTRIBUTION Clive W.J. Granger Department of Economics University of California, San Diego La Jolla, CA USA Tel: ( Fax: ( This Version 28 October 2002

2 1. Long Memory Models A time series process with an unbounded spectrum as frequency goes to zero is called a long memory process. A class of processes with this property are the fractionally integrated series generated by (1B d X t Y t (1 with B the lag operator and where Y t is a stationary series and X t is the fractionally integrated series of order d, denoted X t ~I(d. These were introduced independently by Granger and Joyeux (1980 and by Hoskins (1981, and extend a continuous time model due to Mandelbrot to discrete time. In the following, only the case 0 d 1 will be considered. It can be shown that so that the process can be nonstationary. Variance X t $ const as t &" if 0d½ $ c log t if d ½ $ ct 2d1 if d >½ For ease, consideration will aimed at the simple model (1B d X t ε t (2 where ε t is white noise, corr (ε t, ε s 0, t U s, E[ε t ] m. The properties of the process X t are discussed in the Special Issue of the Journal of Econometrics, volume 73 (1996, edited by Richard Baillie and Max King. One finds that E[X t ] _ mt d (3 so that the process is non-stationary if m U

3 There is also a useful moving average representation X t t j0 c j ε tj where c j _ cj d1. (4 The auto-correlations are ρ k corr (X t, X tk _ ck 2d1 and the spectrum is f x (w c.w 2d $" as w $ 0 Equations (5 and (6 are known as the long memory properties. (5 (6 Stock returns have been found not to have the long memory properties, but most volatility measures do have these properties. Several papers with Ding, including Ding, Granger, and Engle (1993 found that r t d or r n m d to be long memory. The longest memory was obtained for d = 1, so that absolute returns had longer memory effects than variances, for example. It is easy to fall into false logic at this point. Noting that I(d processes are long memory, that volatility has the long memory properties, it is simple to conclude that volatility is I(d, but this does not necessarily follow. Although I(d is the only linear process with long memory, there are other generating mechanisms which also produce these properties, as will be seen later. In fact, it is possible to prove that volatility cannot be I(d. As volatility is necessarily positive, it follow from the moving average representation shown in equation (4, with c j coefficients which are all positive, then the white noise ε t terms -2-

4 must all be positive, as otherwise a negative volatility could be generated. If the C s are positive, they must have a positive mean m, and so from (3 the volatility series will contain a trend. As d estimates are near 0.5 and long series are used, these trends in volatility should be observable. However, virtually all of the volatility series based on stock returns do not have a square root trend, strongly suggesting that volatility, although having the long memory properties, will not be described by an I(d model. I first presented this result to a time series conference at Yale University in October 2000 but work still continues to appear assuming that volatility, particularly variance, is I(d. There are a variety of non-linear models that generate long memory properties. For example, in a recent paper Dittman and Granger (2002 showed that if X t ~I(d and Y t g(x t where and 1 J < k < ", g J U 0, H j (x k g(x g g j H j (x (7 jj are Hermitic polynomials, and J is called the Hermitic rank, then Y t has the long memory properties similar to those of I(d, with d max (0, (d0.5 J0.5. Some examples are g(x d = 0.2 d = 0.4 X 2 (J 2 X 3 (J 1 X 3 3X(J Thus, a simple polynomial of an I(d process, 0 < d < ½ will sometimes produce a process with long memory properties, but it is clearly not I(d. If 0.5 < d < 1, simulations suggest that g(x t has long memory properties similar to I(d -3-

5 Another example of a simple nonlinear process that generates long memory is X t sign (X t1 ε t where sign X 1ifX>0 if X0 1ifX<0 This process becomes of a regular switching form if ε t has a fairly small variance, but (8 if X t 1 and ε t takes a large enough negative value so that X t1 1, there will be a sequence of values near -1 until another switch to +1 occurs. The autocorrelations can be long memory as shown by Teräsvirta and Granger (1999, Economic Letters. 2. Breaks There have been several papers pointing out that a stationary process with occasional level shifts will have the long memory properties, for example Granger and Hyung (2001 (based on Hyung s Ph.D. thesis and Diebold and Inoue (2001. The breaks need to be not too frequent and stochastic in magnitude. A break process considered by Hyung and Franses (2002 takes the form y t m t ε t m t m t1 q t η t (9 with ε t, η t being zero mean, white noise and where q t follows an iid binomial distribution, so that q t 1 with probability p, q t 0 with a probability 1-p. The expected number of breaks is effected by p and the magnitude by. The break process σ 2 2 does not suffer from the trending problem when used with volatilities and is found to fit as well, if not better in other respects, than an I(d model, by Granger and Hyung (2001. Starica and Granger (2001 compare the forecasting abilities of the break and I(d models on variance data and finds the break forecasts to be superior. The interpretation -4-

6 of the break model in terms of stock market fundamentals is also probably easier. Neither model has a firm theoretical foundation. A recent attempt to build a theoretical model that produces long memory is by Kirman and Teyssiere (2002. The model involves two types of agents, roughly equal in number, with different and changing beliefs. Bubble-like phenomena occur in prices, and thus regime switching is observed. Simulated returns from the model have long memory volatilities but no trends in volatility. The theory is seen to be closer to a break process than to an I(d process. Although quite sophisticated, it is still too simple to capture the complexities of the actual market place, but this theory is still useful to have. 3. Where Next? The justification of the study of volatility measures such as variance or absolute returns is that they are related to risk. However the economists who study uncertainty have been firmly stating for years that variance is an unsatisfactory measure of risk. I have discussed this topic recently in Granger (2002. For most investors, who are not going short, risks only occur for low returns; that is, in the lower half of the return distribution. A large positive return may increase uncertainty but this is not risk; yet variance gives equal weight to the two side of the mean in the distribution. This is also true for most other, purely statistical, measures of volatility. As we move further into a new century, with considerable increases in financial data, computer power, and investor sophistication, we can expect these investors to make decisions based on the complete predictive, or conditional, distribution for future returns, rather than forecasts of just a few summary statistics, such as the mean and -5-

7 variance. 4. Some Definitions for Distributions a. Persistence. A basic idea in time series analysis is that of persistence. For example, I(0 series are considered to be non-persistent but an I(1 series is persistent. For simple time series, involving conditional expectations, a series X t is persistent if corr (X t, X th 1 as h & but is not persistent if this correlation declines in magnitude in h increases. For distributions, denote the joint density of X nh and X n by f (X nh, X n. Then the process is said to be persistent if f (X nh, X n ;/ f (X nh f (X n (10 h & so that asymptotically and are not independent. Otherwise the process is not persistent. b. Measures of Dependence X nh X n There are many possible measures of dependence between a pair of variables X nh, X n, as reviewed in Joe (1997, including rank correlation coefficients. A potentially useful measure has been investigated by Maasoumi, Racine, and Granger (2002. S(h ½ [f ½ (x nh, x n f ½ (x nh f ½ (x n ] 2 dx nh dx n (11 If S(h $ 0 as h &, then we have persistence. If S(h $ c(h θ, θ > 0 and c > 0, we then get long memory in distribution which is similar to α- or φ-mixing. If S(h $ exp (ch,c>0, we get short memory in distribution. -6-

8 c. Copula It is important to note that one-to-one (monotonic transforms of the variables X n $ g(x n, X nh $ g(x nh, will have the properties of persistence or non-persistence unaltered and the value of S(h will be unchanged, I believe. Defining U n F(X n where F(x is the distribution function corresponding to f(x n, the predictive distribution of X n. Then the joint distribution function can be written as F(X nh, X n c(u nh (12 where c( is the copula distribution function. Taking partial derivatives with respect to X nh giving f(x nh, X n f(x nh f (X n c(u nh (13 where f(x nh,f(x n are marginal densities and c(u nh is the copula density function defined on the unit box (0, t (0, 1. If X nh, X n are independent, c(u nh 1. Using this concept, the measure of dependence introduced in (11 becomes S(h ½ [1c ½ (U nh ] 2 f (X nh f(x n dx nh dx n. Thus, for persistence, one needs c(u nh 1 as h &. To consider long memory in distribution, a sensible strategy would be to form the marginal distribution f (X n, to then form U n and U nh and finally to consider the joint density between U n and U nh, which is the copula c(u nh. The corresponding measure of dependence will be S n (h ½ [1c ½ (U nh ] 2 du n du nh -7-

9 And to see how this measure changes with h. So far, I have not seen this procedure attempted with actual data. d. Breaks A predictive distribution involving parameters can be called a model, for example the density f(x, θ I t, when θ may be a vector. If θ changes value in a step form, such as e. Quantiles θ θ 0, t<t 0 θ 1 tt 0 then one has a break in distribution. The break can be observed in various moments, not just the mean or variance, for example. An upward break can also occur in a tail, but being found in the upper quantiles, say, but not other quantiles. Note that the distribution need not be symmetric. As an example, Sin and Granger (2001 consider the distribution of returns for the Hong Kong Stock Exchange Index by looking at quantiles. After the 1977/8 Asian financial crisis, it was found that the break produced distributions that were longer tailed. What is not clear is if breaks in distribution will produce the long memory property for the measure of dependence discussed above. This possibility has to be investigated. A useful and relatively simple way to model a conditional distribution is by considering quantiles. For a random variable X with distribution function F(x, the α-probability quantile q α is defined by F(X<q α α. If one takes a monotonic transformation -8-

10 so that then thus X h(z, Z h 1 (X F(h(z<q α α F(Z<h 1 (q α α So the α-quantile for Z is h 1 (q α, which will be denoted q α (Z. As the predictive distribution changes over time, so each quantile q α, n will form a time series. For some pre-specified finite set of probabilities, spanning the line (0, 1, the corresponding set of quantiles will be a ordered set of time series. For a given α, quantiles corresponding to a probability β, such that β > α, will be said to be above below q α., and those with β < α, will be said to be q α Quantiles are not bounded and are not necessarily positive, for example, but if q α, n evolves it will have an impact on other quantiles. For example, if q α, n contains an upward trend in n, then so will all quantiles above it. A simple case to consider is a break. If q α, n is a stationary sequence which contains an upward break in mean at parameter time N, then necessarily all quantiles above it must contain the same break, but quantiles below it need not be affected. It is worth remembering that if a break occurs in a tail quantile, say α = 0.85, so that all of the tail of the distribution is changed, the median will not be altered although the mean will be. A downward break in a quantile will effect all quantiles below it. The relationship between volatility measures and quantiles is an -9-

11 important, and interesting, one. Suppose one observes a time-varying variance which is changing around a constant value, but at some point in time, this mean level of variance jumps to a higher value, so that there is a break in variance. This break can be reflected by various changes in the distribution. For example, either: i. The lower quantiles move further out and all other quantiles stay the same so that the distribution becomes longer lower-tailed; or ii. The upper quantiles move further out, and all other quantiles stay the same with the distribution becoming longer upper-tailed; or iii. Some mixture of these two effects, the mixture need not be symmetric. All of these three explanations will lead to a break in variance, or any other standard volatility measure, and the one cannot say which is occurring from seeing the break in volatility. It is worth noting that all the explanations will leave the median unaltered, but not the mean. The only case where the mean is guaranteed unchanged is when the movements in the quantiles in the two tails are balanced. This is a very special case of a mean-preserving spread. There are many other examples. If one outer quantile moves outwards but other quantiles near the mean moves towards it, the distribution can become longer-tailed on one side, but the mean not be altered. This is a basic concept in the risk literature; see for example Rothschild and Stiglitz (1971. A further distinction between properties of volatilities and quantiles -10-

12 comes from considering what models can generate them. A quantile can be an I(1 series such as a random walk. If one quantile is a random walk, it will follow that they are all random walks, I believe, and so the whole predictive distribution is a random walk. The best forecast of the next distribution is today s distribution. However, a volatility, such as a variance, cannot be a random walk because variances must be positive. Of course, one can have random walks with reflecting barriers, or such-like devices, but they have rather different properties than an unconstrained I(1. A similar argument applies to the I(d, d a fraction, process. A quantile can be fractionally integrated but, as has been noted earlier in the paper, a volatility cannot be. Both can share certain second-moment properties but not first moments. The conclusion from this argument is that if one truly believes in fractionally integrated models, then the place to look is the quantiles of a distribution rather than the volatilities. If it is just the long memory properties that is of interest, then both should display them, but the breaking process will probably give an explanation for them both. 5. Models for Quantiles For a predictive distribution F t (x, define the α quantile q t (α to be F t (q t (αα. For a set of increasing probabilities covering most of (0, 1, α 1, α 2,, α m there will be corresponding quantiles q(α j such that. q(α i <q(α j if α i <α j Each quantile can be time varying and can thus be modeled as a time series using the -11-

13 tick cost function discussed by Koenker and Bassett (1978, with some generalizations being applied by Sin and Granger (2001. However, providing a model for a vector of ordered series is not simple. A better approach is to consider the quantile increments or spacings positive. If one knows all of the δq(α i q(α i q(α i1. Each of these will be δq(α i, i = 1,..., m-1 and any quantile, then all the quantiles can be deduced. The task is now to model a vector of m-1 positive series, the quantile spacings. However, it is convenient to consider a further transformation by taking logarithms of these quantiles, and so removing the positive constraint. Thus the final set of time series to be analyzed are Z(α i log δq t (α i, i = 1,..., m-1. Linear models are an obvious starting class to consider, of the form Z t (α µ(αβ 1 (αx t β 2 (αz t1 (α β 2 (αz t1 (α1ε t (α for each α α i. Here X t is the basic time series whose conditional distribution is under study, µ is a constant and the lags of the log quantile increments will help with stability across time and across α s. There are a number of interesting features of this model: i. Potentially, a point forecast can be given for each quantile that is of q m1 (α i made at time n, but it cannot be easily evaluated as no realization of a quantile occurs. ii. Long memory, even unit roots, could occur in the Z s but will not necessarily influence all quantiles. For example, if Z t (α i is I (d, with d 1, then quantile q(α i1 must be I (d even though q(α i need not -12-

14 be. Further, if Z t (α i1 is also I (d, with the same d, then quantile q(d i2 need not be I (d because there could be cointegration and the two effects cancel in the sum. This can occur because negative coefficients are possible in the Z s. Thus, different parts of the distribution can have different time series properties; one tail could be stationary and the other tail be unit root, for example. By examining the time series behavior of the quantiles, conceptually this could be detected. -13-

15 References Diebold, F.X. and A. Inoue (2001: Long memory and regime switching. Journal of Econometrics 105, Ding, Z., C.W.J. Granger, and R.F. Engle (1993: A long memory property of stock returns and a new model. Journal of Empirical Finance 1, Dittman, I. And C.W.J. Granger (2002: Properties of nonlinear transformations of fractionally integrated processes. To appear, Journal of Econometrics. Granger, C.W.J. (2002: Some comments on risk. To appear, Journal of Applied Econometrics. Granger, C.W.J. and R. Joyeux (1980: An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis 1, Hosking, J.R.M. (1981: Fractional differencing. Biometrika 68, Joe, H. (1997: Multivariate Models and Dependence Concepts. Chapman and Hall: London. Kirman, A. and G. Teyssière (2002: Bubbles and long range dependence in asset price volatilities. GREQAM, Marseilles, Working paper. Koenker, R. and G. Bassett (1978: Regression Quantiles. Econometrica 46, Maasoumi, E., J. Racine, and C.W.J. Granger (2002: A dependence metric for nonlinear time series. Working paper, Southern Methodist University. Rothschild, M. and J. Stiglitz (1971: Increasing risk I. Journal of Economic Theory 2, Sin, C.-Y. And C.W.J. Granger (2001: Modeling the absolute returns of different stock indices exploring the forecastability of alternative measures of risk. Forthcoming, Journal of Forecasting. Starica, C. and C.W.J. Granger (2001: Non-stationarities in stock returns. Working paper, Chalmers University of Technology. Teräsvirta, T. and C.W.J. Granger (1999: A simple nonlinear model with misleading linear properties. Economic Letters 62,