NONLINEAR DEPENDENCE IN INSURANCE STOCK …保险股的非线性关系….docVIP

NONLINEARITY AND LOW DETERMINISTIC CHAOTIC BEHAVIOR IN INSURANCE PORTFOLIO STOCK RETURNS by JORGE L. URRUTIA LOYOLA UNIVERSITY CHICAGO JOSEPH VU DEPAUL UNIVERSITY CHICAGO PAUL GRONEWOLLER COLORADO STATE UNIVERSITY - FORT COLLINS MONZURUL HOQUE SAINT XAVIER UNIVERSITY CHICAGO 1 NONLINEARITY AND LOW DETERMINISTIC CHAOTIC BEHAVIOR IN INSURANCE PORTFOLIO STOCK RETURNS Abstract This paper investigates the presence of nonlinearity and low deterministic chaos in the time series returns of life-health and property-casualty insurance companies. The motivation of the paper is twofold: First, a primary reason for the weak findings of nonlinearities reported in previous research is the use of aggregate data that can hide nonlinearities at the micro level. We correct this problem by using more desegregate data sets. Second, we choose insurance data because of some unique characteristics of the insurance industry, which can make the insurance sector to be segmented from the capital market as a whole. Tests based on the correlation dimension (CD), and the Brock, Dechert, and Scheinkman (BDS) statistic strongly suggest the presence of nonlinearities in the insurance data. The CD and the BDS statistic applied to the standardized residuals of an EGARCH model indicate that conditional heteroskedasticity is not responsible for the presence of nonlinear structures in the data. On the other hand, tests of chaos based on the locally weighted regression (LWR), indicate that stock insurance returns exhibit low complexity chaotic behavior. This is an important result since most of previous research has failed to report evidence of chaos in the time series of stock returns. Important contributions of this paper are the application of tests of nonlinearities and chaos to more desegregated data sets, and the findings of statistically significant evidence of nonlinearities and low deterministic chaotic behavior in insurance company returns. Our findings could have some implications for insurance price forecasting modeling. I. Introduction This paper investigates the possibility that equity price changes in the insurance industry exhibit nonlinearity and deterministic chaos. The finding of a nonlinear structure would open the possibility for nonlinear price forecasting models that could better explain certain aspects of insurance equity returns than do linear price forecasting models, such as the capital asset pricing model (CAPM) and the arbitrage pricing theory (APT). Chaotic behavior is interesting because it can potentially explain fluctuations in the economy and financial markets, which appear to be random processes. The empirical evidence in favor of chaotic dynamic in equity returns has been generally weak (Hsieh, 1991). An obvious question is if the evidence of the presence of deterministic chaos is weak for equity in general, why should one expect insurance stock returns to behave differently? The answer to this question is twofold. First, Brock and Sayers (1988) point out that one of the main reasons preventing for finding evidence of chaotic behavior in economic data is the use of aggregate data that can hide nonlinearities at the micro level. In this paper we avoid this problem by employing returns for only one 2 economic sector. Second, we have chosen the insurance industry because of its unique characteristics. To begin with it, the pricing of an insurance contract is different from that of other products. In effect, the insurance premium must be determined before losses have occurred. Therefore, the forward nature of most insurance contracts demands that losses be accurately estimated before prices are set. However, Venezian (1985) and Cummins and Outreville (1987) point out that nave loss forecasting techniques omit much current and relevant information and are subject to significant error. In addition, the payout structure of an insurance contract is not linear. Insurance company operations are also heavily regulated and there is no agreement concerning whether the regulation facilitates a competitive product market (Joskow (1973), Cummins and Harrington (1985), Cummins and Outreville (1987), and Witt and Urrutia (1983). Price regulation can create delays that prevent insurance prices from quickly adjusting to changing loss exposures and economic conditions (Cummins and Outreville (1987). We postulate that all these unique characteristics of the insurance industry contribute to the segmentation of the insurance sector from the capital market as a whole. A segmented market could behave differently from the rest of the market. The finding of a nonlinear structure and deterministic chaos in insurance stock returns will strengthen the argument that the insurance sector and the stock market are segmented. II. Background Chaos theory provides a set of diagnostic tools to distinguish between underlying structures that appear random or unpredictable under traditional analysis, but are nonlinear deterministic processes, and underlying structures that contain stochastic components. Thus, low-level chaotic behavior is differentiated from random behavior by the presence of identifiable nonlinear structure. A. Literature Chaos theory has been widely applied to economic data. Brock and Sayers (1988) test for nonlinearities in quarterly data on U.S. real gross national product (GNP) and U.S. real gross private domestic investment, GPDI. These authors calculate the Grassberger-Procaccia (1983), correlation dimension and conclude that more data are needed to establish that U.S. real GNP and real GPDI are generated by a low dimension chaotic deterministic dynamical system. However, they also point out that their results should not discourage new attempts to find evidence of significant nonlinearities and chaos in economic data, mainly because they use aggregate data that can hide nonlinearities at the micro level. Brock and Sayers (1988) apply the Brock, Dechert and Scheinkman (1987), BDS, statistic to the residuals of the best-fitted trend stationary and difference stationary linear models. The test provides evidence of strong nonlinear dependence in residuals of employment, unemployment, industrial production, and pig iron production. Empirical nonlinear science has enjoyed not only a boom in economics, but also in finance, with numerous applications to stock returns. However, the evidence 3 supporting the existence of low level deterministic chaos in equity returns is generally weak. Scheinkman and LeBaron (1989) estimate the correlation dimension for 1226 weekly observations on the CRSP value-weighted U.S. stock returns index starting in the early 1960s. They arrive at a correlation dimension of 6. In general, they find evidence of nonlinear dependence for the weekly value weighted CRSP index but not for the daily value weighted CRSP index. Their results suggest that a substantial part of the variation of weekly returns comes from nonlinearities as opposed to randomness. This result is not compatible with the random walk theory, which predicts that the returns are generated by independent and identically distributed (IID), random variables (Granger and Morgenster (1963), and Fama (1970). They conclude that nonlinearities may play an important role in explaining financial asset returns. Brock and Baek (1991) follow up on the study of Scheinkman and LeBaron (1989). They use embedding dimensions from 1 to 14 and obtain correlation dimension estimates between 7 and 9. The null hypothesis of IID is rejected in favor of a lower dimensional alternative; this finding is consistent with the results of Scheinkman and LeBaron. However, Brock and Baek warn that the rejection of the null of IID for stock returns in favor of some lower dimensional alternative does not necessarily mean chaos is present. Schwert (1989) uses the BDS method to test the adequacy of a broad class of models to predict stock market volatility as a function of past information, such as macroeconomic variables and past returns. Hsieh (1991) applies the BDS test to investigate the goodness-of-fit of a broad class of parametric models of conditional volatility. He finds strong evidence to reject the hypothesis that stock returns are IID. However, the cause of nonlinearity does not appear to be the presence of chaos. In effect, Hsieh attributes the nonlinear dependence in the weekly CRSP index returns to conditional heteroskedasticity (e.g., predictable variance changes). Brock, Lakonishok, and LeBaron (1992) combine nonlinear methods with moving averages and other technical analysis methods to make predictions of stock prices. The consensus of recent work in finance is that the temporal behavior of stock returns is not a totally random process. However, the predictable component does not appear to be sufficiently correlated with identifiable variables to provide successful forecasting of stock price changes with linear models. In fact, most of these empirical studies have reported the presence of nonlinearities on stock returns. Nevertheless, the evidence in favor of chaotic dynamic in equity returns is generally weak. B. Chaos Chaos is a nonlinear deterministic process, which looks random (Hsieh 1991, 1993). The attractiveness of chaotic dynamics is precisely its ability to generate large movements, which appear to be random, with greater frequency than linear models. In the economic literature there are two ways to generate output fluctuations. In the Box- Jenkins times-series models, the economy is in equilibrium, but it is constantly being perturbed by external shocks (wars, weather, etc). The dynamic behavior of the economy 4 comes about as a result of these external shocks. In the chaotic growth models, on the other hand, the economy follows nonlinear dynamic processes, which are self-generating and never die down. That is, under chaotic behavior, the economic fluctuations are internally generated. This has some intuitive appeal. In addition, chaotic dynamics is nonlinear. This is also appealing because nonlinear models can generate rich types of behavior. For example, the system can have sudden bursts of volatility and occasional large movements. This characteristic is important for stock markets because asset prices exhibit large movements (crashes), which cannot be explained by linear models. The tent map is the simplest chaotic process. It can be generated by picking a number between 0 and 1 and then generate the sequence of number xt using the following rule: Xt = 2xt- 1, if xt -1 0.5, and Xt = 2(1 xt - 1), if xt 1 0.5 Intuitively, the tent map takes the interval 0, 1, stretches it to twice the length, and folds it in half. Repeated stretching and folding pulls separates points close to each other. This type of stretching and folding is characteristic of chaotic maps. It makes prediction difficult, thus creating the illusion of randomness. Other chaotic systems are the pseudo random number generators, the logistic map, the Henon map (Henon 1976), the Lorenz map (Lorenz 1963), and the Mackey- Glass equation (Mackey and Glass 1977). In general chaotic maps are obtained by a deterministic rule: xt = f(xt 1, xt 2 ). Here, xt can be either a scalar or a vector. In order to generate chaotic behavior, f(x) must be a nonlinear function. However, nonlinearity alone is not sufficient to generate chaotic behavior. III. METHODOLOGY The methodology used in this paper follows that of Brock (1986), Brock, Dechert and Scheinkman (1987), Brock, Lebaron, and Hsieh (1991) and Hsieh (1989, 1991, and 1993). Several diagnostic tests are employed to detect nonlinear dependence: Grassberger and Procaccia diagnostic test (1983), Brock and Sayers residual diagnostic test (1988), and the Brock, Dechert and Scheinkman, BDS, statistic (1987). We also test for heteroskedasticity and low deterministic chaos in the insurance data. We employ the EGARCH model to test for heteroskedasticity, and the test of chaos is based on the locally weighted regression (LWR), method. A description of the several tests employed in this paper follows. A.THE GRASSBERGER AND PROCACCIA TEST Grassberger and Procaccia (1983) developed the notion of correlation dimension (CD). According to Brock and Sayers (1988) dimension is an indication of the number of nonlinear factors that describe the data. For example, a single point has zero-dimension. A line has one dimension. A solid has three dimensions. A white noise process and a 5 purely random process have infinite dimension. A chaotic system has a positive but finite dimension. The key in detecting nonlinearities with the Grassberger and Procaccia test is to observe the changes in the correlation dimension as we increase the embedding dimension. If the CD does not explode with the embedding dimension, then there is evidence of nonlinearity in the data. The correlation dimension is developed in several steps: Step 1:Construct the m-histories of the data in order to obtain the embedding dimension. An m-history is a point in m-dimensional space; m is called the embedding dimension. The m-histories are denoted as follows: 1-history: xt1 = xt 2-history: xt2 = (xt 1, xt) . . . m-history: xtm = (xt m + 1, , xt) Step 2:Grassberger and Procaccia (1983) and Swinney (1985) use a form of correlation integral to define the correlation dimension: Step 3:Calculate the slope of the graph of log Cm (e) versus loge for small values of e. More precisely, we want to calculate the following quantity: The correlation dimension is estimated for increasingly larger values of the embedding dimension. If nonlinearity is present, the CD estimates will stabilize at some value. If this stabilization does not occur, that is, if the CD explodes as the embedding dimension increases, the system is considered high- dimensional, or stochastic. T is actually determined by the number of observations available for the analysis. This places limits on possible values for e, the distance between a pair of observations ,/T e |:|,0),(lime 2 m x m tTm xxTststC e|max 1,.,0 itismi xx elog/eloglim 0mm CV : thatsense in the other,each toclose are which ,x pairs offraction theis integraln correlatio The norm.max or sup theis | | where s m t m x 6 and their chosen neighbors. For a given m, e cannot be too small because Cm (e) will contain too few observations. Also, e cannot be too large because Cm (e), will contain too many observations. Barnett and Choi (1989) suggest selecting a small value for e, without allowing it to reach zero, in order to eliminate noise in the data. Hsieh (1989) defines e in terms of multiples of the series standard deviation. These multiples are 1.50, 1.25, 1.00, 0.75, and 0.50. In this paper we use the program “Chaos Data Analyzer,“ developed by Sprott and Rowlands. The program automatically chooses the optimal e and leave two parameters under user control: the embedding dimension, and the parameter n, which is the number of sample intervals over which each pair of points is followed before a new pair is chosen. Essentially, if n is too large, the trajectories get too far apart, and if n is too small, the calculation becomes too slow. We report results for embedding dimensions 2 through 10 and n of 1, 2, 4, and 8. It is important to keep in mind that the correlation dimension is a graphical procedure and not a statistical test. B. BROCKS RESIDUAL TEST The Brock (1986)s residual test consists of filtering the raw data in order to remove any linear structure. The usual procedure is fitting an autoregressive model to the transformed series. Then, the CD and the BDS statistic are computed on the residuals to test for nonlinearity. C. THE BROCK, DECHERT AND SCHEINKMAN, BDS, STATISTIC. Brock, Dechert, and Scheinkman (1987) developed a statistic test of nonlinearity, known as the BDS statistic. The null hypothesis is that the data are independently and identically distributed, IID. Rejection of the null of IID implies the presence of nonlinearity in the time series. BDS have shown that tests based on their statistic have a higher power for tests of stochastic or chaotic independence than other statistical techniques. BDS demonstrate that under the null hypothesis (xt) is IID with a nondegenerate density F (Hsieh 1989), Cm (e, T) C1 (e)m with probability equal to unity, as T , For any fixed m and e. In addition, they show that T1/2Cm (e, T) C1 (e, T)m has a normal limiting distribution with zero mean and variance equal to Where: C1 (e,T) is a consistent estimate of C(e), and ,124e 1 1 2222 2 22 m j mmjjmm m KCmCmCKK 7 is a consistent estimate of K(e). Therefore, m (e) can be estimated by m (e, T), which uses C1 (e, T) and K (e, T) instead of C (e) and K (e). The BDS statistic has a standard normal limiting distribution and is calculated by BDS show that, under the null of IID, Wm N (0, 1), as T . If the residuals from the estimated linear (nonlinear) model are actually IID, the BDS statistic should be asymptotically N (0,1). Large values of the BDS statistic would indicate strong evidence of nonlinearity in the data. However, the rejection of the null of IID by the BDS statistic does not necessarily mean the time series exhibits a low complexity chaotic behavior. In effect, rejection of IID can be consistent with any of the following four types of non-IID behavior: linear dependence, nonstationarity, nonlinear stochastic processes (ARCH-types models), and chaos (nonlinear deterministic process). The linear dependence can be easily ruled out, by running the BDS test in the filtered data. Nonstationarity can also be ruled out by computing the BDS statistic in the first difference of the data (i.e., returns). However, additional tests are needed to discriminate between nonlinearity due to stochastic behavior (heteroskedasticity) and nonlin