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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
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 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 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 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.
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 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 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 T1/2[Cm (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
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 nonlinearity due to the existence of deterministic chaos.
D. TESTING FOR CONDITIONAL HETEROSKEDASTICITY
There is growing evidence that stock return volatility is not only time-varying (French, Schwert, and Stambaugh (1987)) but it is also predictable (Schwert and Seguin (1990)). The objective here is to investigate whether the conditional heteroskedasticity captured by ARCH-type models is responsible for the nonlinearity in the data.
Following Hsieh (1991), we proceed to consider whether stock returns are nonlinear-in-variance: Xt = g (xt-1, . . .) (t. This is a general model of conditional heteroskedasticity, which includes ARCH-type models as special cases. Let us take the absolute values of the above equation:
( xt ( = ( g ( xt-1, . . . ) ( ( (t ( .
If g ( ) is differentiable, a Taylor series expansion would yield the result that ( xt ( depends on ( xt-1 (. Thus, if we compute the autocorrelation of the absolute valued data, the finding of correlation of ( xt ( with ( xt-1 ( is evidence of conditional heteroskedasticity.
However, our main interest is to determine whether ARCH-type models capture the nonlinear dependence of insurance stock returns. In order to investigate this issue we fit an EGARCH model to the data (Hsieh (1991)):
ARCH and GARCH impose restrictions on the signs of the parameters to guarantee that estimated variances are positive, which creates numerical problems associated with constrained optimization. The EGARCH model does not impose these restrictions. In addition, EGARCH can accommodate conditional skewness.
If the EGARCH model is correctly specified, then the standardized residuals
Zt = xt / (t, should be IID in large samples. In the above formula, (t is the fitted value of the standard deviation from the variance equation.
The CD and the BDS statistic can be applied to the standardized residuals to test if EGARCH captures all nonlinearities present in the data. If the CD explodes and the BDS statistic does not reject the null of IID, then we can conclude that the presence of nonlinearity in the data is due to heteroskedasticity.
E. TESTING FOR LOW DETERMINISTIC CHAOTIC BEHAVIOR.
The test of low deterministic chaos employed in this paper is one type of nonparametric regression known as the locally weighted regression, LWR (Diebold and Nason (1990), LeBaron (1988)). The LWR can be briefly described as follows: suppose the time series data are generated according to:
Xt+1 = f (xt).
We have observed xt, xt-1, ., and would like to forecast xt+1. Locally weighted regression uses the k nearest neighbors of xt, and a scheme, which gives more weight to closer observations and less weight to farther observations. Several parameters must be selected: (a) the number of nearest neighbors to use: we try 10% of all observations, up to 90%, increasing in steps of 10%; (b) the number of lags of xt to include as arguments of the unknown function f (x ): we use lags 1 through 5; (c) the weighting scheme: we use the tricubic scheme proposed by Cleveland and Devlin (1988). If stock returns are governed by low complexity chaos, we should be able to use LWR to forecast returns better than with simple methods, such as the random walk model (Hsieh (1991)).
IV. Data
The data set consists of daily returns, including dividends, of portfolios of life and health insurance companies and property and casualty insurance companies. The data have been extracted from the daily tapes of the CRSP files of the University of Chicago, and cover the period from January 1984 to December 1998. The insurance stocks trade on the New Stock Exchange, American Stock Exchange, and NASDAQ. Equally weighted portfolios of both types of insurance firms are constructed. Variation in the number of companies in the portfolios is due to entry, exit, and mergers within the insurance industry, and exchange listing and deslisting. A summary of the descriptive statistic of the data is provided in Table 1.
In 1984, there are only 33 life-heath insurance companies in the sample. They have average sales of $1.16 billion with a standard deviation of $1.46 billion. The average total assets are $3.78 billion and the corresponding standard deviation is $4.68 billion. In 1984, there are 46 property-casualty insurance companies in the sample. They have average sales of $1.46 billion with a standard deviation of $2.40 billion. Their average total assets are $3.52 billion and the corresponding standard deviation is $6.37 billion. In general, from 1984 to 1998, there are more insurance companies in the sample and these companies are larger, on average. In 1998, there are 93 life-health insurance companies in the sample. They have average sales of $2.27 billion with a standard deviation of $4.19 billion. The average total assets increase to $14.12 billion with a standard deviation of $35.31 billion. The large magnitude of two standard deviations indicates that there is considerable variability in sales and total assets of life-health companies in 1998. Property-casualty insurance companies have similar results in 1998 with the number of companies increasing to 102. They have average sales of $2.44 billion with a standard deviation of $5.37 billion. The average total assets are $13.59 billion and the corresponding standard deviation is $32.97 billion. The large standard deviations indicate that there was considerable variability in sales and total assets of the property-casualty insurance companies in 1998.
V. ANALYSIS OF EMPIRICAL RESULTS
Table 2 presents the results of the correlation dimension of Grassberger and Procaccia (1983) in Panels A1 and A2, and the BDS statistic of Brock, Dechert and Scheinkman (1987) in Panels B1 and B2, for the raw data. The correlation dimension increases with the embedding dimension but does not explode. Thus, the correlation dimension suggests the presence of nonlinearity in the insurance data. Unlike the correlation dimension, the BDS is a statistical test that detects departures from the null hypothesis that the returns are independently and identically distributed, IID. The large absolute values BDS statistics, reported in Table 2, strongly reject the null of IID for both portfolios of insurance companies.
The rejection of null of IID is a departure from serial independence in stock returns. Several papers have reported empirical evidence, which suggests that stock returns contain predictable components (Keim and Stambaugh (1986), Fama and French (1987)). However the rejection of the random walk does not necessarily mean that the market is inefficient, or that stock prices are not rational assessments of fundamental values (Lo and MacKinlay (1988)). In effect, Leroy (1973) and Lucas (1978) have shown that rational expectations equilibrium prices need not even form a martingale sequence. Therefore, a rejection of the random walk hypothesis has few implications for the efficiency of market price formation. One explanation for the serial dependence of stock returns is thin trading; that is, some stocks in the insurance portfolio trade infrequently. It is known that small capitalization stocks trade less frequently than larger stocks, and new information is impounded into smaller-stock prices with a lag. Another reason could be noise trading, or trading by investors whose demand for shares is determined by factors other than their expected return (Poterba and Summers (1988)).
The BDS test has good power to detect at least four types of non-IID behavior (Hsieh (1991)): linear dependence, nonstationarity, nonlinear stochastic process, and chaos. Since our data set covers a relatively long period of time, it would be difficult to argue that the behavior of insurance stock prices has remained unchanged during this extended time period. Changes in economic fundamentals, changes in volatility of financial markets, changes in the operating procedures of the insurance industry, changes in insurance regulation laws, etc., can cause nonstationary in the insurance price time series. However, nonstationary in prices does not necessarily imply nonstationarity in returns. In effect, the Augmented Dickey and Fuller (1979) tests (not reported here to save space) indicate that the insurance daily return time series are stationary. Therefore, we can rule out nonstationarity as the cause of rejection of IID returns. We therefore concentrate on the remaining three causes.
We proceed to remove any linear dependence in the data by means of the Brocks (1986) residual test. In order to apply this test we first filter the data by fitting an autoregressive model to the transformed series, and then compute the correlation dimension and the BDS statistic on the residuals. The results of the Brock's residuals test are reported in Table 3. It can be observed that the correlation dimensions of the residuals of the filtered returns, reported in Panels A1 and A2 of Table 3, increase with the embedding dimension but do not explode, which is an indication of nonlinearity. In addition, the large values of BDS statistics, reported in Panels B1 and B2 of Table 3, reject the null of IID for the filtered data.
We have detected nonlinearity in the insurance returns. However, nonlinearity can be stochastic or chaotic. That is, our finding of nonlinearity in the insurance data can be due to the presence of heteroskedasticity or chaos. In this respect, auto-correlation coefficients and the Q-statistics (not reported here to save space) indicate the presence of heteroskedasticity in the insurance return time series. Therefore, we must perform additional tests in the data in order to discriminate between stochastic nonlinearities and chaotic nonlinearities.
We apply the CD and BDS procedures to the standardized residuals to test if EGARCH captures all nonlinearities present in insurance stock returns. Table 4 presents the standardized residuals from the EGARCH model for the life-health and property-casualty insurance portfolios. Again, the correlation dimensions increase with the embedding dimension but do not explode, and the BDS statistic rejects the null of IID. These results allow us to conclude that conditional heteroskedasticity is not responsible for the presence of nonlinearities in the time series returns of the insurance portfolios.
Next, we proceed to test for the presence of low deterministic chaos. The test employed in this paper is the locally weighted regression, LWR (Diebold and Nason (1990), LeBaron (1988)). If stock returns are governed by low complexity chaos, we should be able to use LWR to forecast returns better than the random walk model (Hsieh (1991)). We measured forecastability by the mean squared errors of the LWR. The results reported in Table 5 indicate that the mean squared errors obtained from the LWR are all smaller than those of the random walk. Thus, the findings from the locally weighted regression support our hypothesis that the existence of nonlinearities in the time series of insurance stock returns is due to the presence of low deterministic chaos. This is an important result since most of previous research has failed to report empirical evidence that stock returns exhibit low complexity chaotic behavior.
VI. SUMMARY AND CONCLUSIONS
The purpose of this paper is to empirically investigate the possibility that the time series of insurance stock returns exhibit nonlinear structure and low deterministic chaos. The data correspond to daily returns of life-health and property-liability insurance portfolios, covering the January 1984 to December 1998 time period.
The motivation of our research is twofold. First, Brock and Sayers (1988) argue that the main reason for the weak findings of nonlinearities in macroeconomic data and general equity market data is the use of aggregate data that can hide nonlinearities at the micro level. In this paper we avoid this problem by using a more desegregated data set: stock returns for only one economic sector: the insurance industry. Second, we choose the insurance sector because certain unique characteristics of the insurance industry imply that the insurance sector is segmented from the capital market as a whole. A segmented sector could have time series properties different from those of the market as a whole. A series of diagnostic tests based on the correlation dimension and the BDS statistic are employed to detect nonlinear dependence: the Grassberger and Procaccia diagnostic test (1983), the Brock and Sayers residual diagnostic test (1988), and the Brock, Dechert and Scheinkman BDS statistic (1987).
The correlation dimension and the BDS statistic confirm the existence of non-linearity in the raw and filtered insurance returns. Because the existence of nonlinearity in the data can be due to the presence of either stochastic or chaotic behavior, we also run tests to discriminate between these two processes. In order to investigate whether conditional heteroskedasticity is responsible for the non-linearity in insurance stock returns, we apply the correlation dimension and the BDS statistic to the standardized residuals of the EGARCH model. These tests indicate that conditional heteroskedasticity is not responsible for the presence of nonlinearities in the time series returns of the insurance portfolios.
In testing for chaotic behavior we employ the nonparametric regression known as locally weighted regression, LWR. Our results indicate that the mean squared errors obtained from the LWR are smaller than those of the random walk. Thus, the locally weighted regression supports our hypothesis that the existence of nonlinearities in the insurance data is due to the presence of low deterministic chaos. This is an important result since most of previous research has failed to report empirical evidence that the time series of stock returns exhibit low deterministic chaos.
Contributions of the paper to the empirical analysis of returns time series are: the application of the tests of nonlinearity and chaos to more desegregated data sets, such as the insurance sector, and the findings of nonlinearity and low deterministic chaotic behavior in the insurance returns. The discovery of chaos in the insurance returns time series could not help to predict insurance prices, but it would help to understand the insurance pricing mechanism better. Moreover, our findings of nonlinear structure in the insurance returns might have some implications for insurance forecasting modeling. They could open an avenue for the development of nonlinear models, such as models based on neural networks and nonlinear regressions, that could explain the behavior of insurance prices better than the linear models currently used by the insurance industry, such as the capital asset pricing model. Indeed, there is little theoretical justification for assuming linearity in insurance price forecasting models. In this respect, our paper not only provides insight into insurance market dynamics, but the reported evidence of nonlinear dependence opens the possibility that nonlinear prediction techniques might be applied to insurance price data.
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Table 1
Descriptive statistics of life-health and property-casualty insurance portfolios(Jan. 1984 to Dec. 1998)
Life-Health Insurance PortfolioProperty-Casualty Insurance PortfolioDaily Mean Return0.0009820.000809Daily Standard Deviation0.0386470.031479Max Daily Return1.4000001.068966Minimum Daily Return-0.722222-0.857143Skewness2.8215461.455458Kurtosis78.48348865.420977Number of Daily Returns269,867243,333
Table 2
Correlation dimensions and the BDS statistics at embedding dimensions 2 through 10 and n equaling 1, 2, 4, and 8, for the daily raw returns of life -health and property-casualty insurance portfolios.
Panel A1: Correlation Dimension, Life -Health Insurance Portfolio
Embedding Dimension
n234567891012.0733.0373.8614.5135.1335.5615.8876.2436.21422.0603.0493.8684.5405.1485.5015.9876.0546.38942.0743.0603.8914.5815.1755.5245.9656.0286.36482.0713.0613.8854.5795.2025.5465.9886.0346.383
Panel A2: Correlation Dimension, Property-Casualty Insurance Portfolio
Embedding Dimension
n234567891012.0863.0753.9024.6145.2145.5806.0146.2506.36022.0783.0383.8944.5925.2055.4995.9846.2146.39042.0603.0193.8664.5775.1475.5435.9856.1056.43382.0633.0403.8784.5875.0615.5535.9446.1106.428
Panel B1: BDS Statistic, Life-Health Insurance Portfolio
Embedding Dimension
n23456789101-3.163-6.651-8.990-10.089-10.110-9.431-8.374-7.191-6.0332-3.325-6.977-9.567-10.684-13.627-9.852-8.682-7.387-6.1514-3.440-7.171-9.850-10.952-10.852-9.995-8.761-7.436-6.1828-3.735-7.811-10.636-11.803-11.727-10.838-9.545-8.149-6.811
Panel B2: BDS Statistic, Property-Casualty Insurance Portfolio
Embedding Dimension
n23456789101-3.091-6.579-9.274-10.624-10.769-10.099-8.975-7.707-6.4742-3.282-7.098-9.807-11.028-11.032-10.242-9.066-7.769-6.5124-3.459-7.324-9.985-11.201-11.211-10.415-9.186-7.827-6.5328-3.698-7.885-10.813-12.082-12.029-11.118-9.803-8.378-7.020
Table 3
Correlation dimensions and the BDS statistics at embedding dimensions 2 through 10 and n equaling 1, 2, 4, and 8, for the filtered daily returns of life and health and property-casualty insurance portfolio.
Panel A1: Correlation Dimension, Life-Health Insurance Portfolio
Embedding Dimension
n234567891011.9582.5242.9123.5663.8974.2214.8045.3015.55421.9502.5062.9943.6253.9814.5674.9105.1185.47141.9872.4432.9143.6044.1634.3674.6685.1745.61381.9182.5242.9183.3953.9074.3164.8035.1435.413
Panel A2: Correlation Dimension, Property-Casualty Insurance Portfolio
Embedding Dimension
n234567891011.9572.5913.1073.6754.0454.2114.7405.2555.69621.9832.5903.0783.5974.0824.5804.9005.3105.65442.0002.6372.8663.5884.0054.3424.5925.0855.55681.9262.5672.7903.4383.8874.3214.9045.2975.645
Panel B1: BDS Statistic, Life-Health Insurance Portfolio
Embedding Dimension
n23456789101-1.511-3.569-4.471-4.131-3.156-2.328-1.716-1.229-0.8492-2.890-5.557-7.416-8.390-8.129-7.557-6.535-5.502-4.5314-1.455-3.161-4.214-4.116-3.349-2.449-1.742-1.235-0.8618-2.632-4.600-4.635-4.301-3.500-2.626-1.929-1.358-0.934
Panel B2: BDS Statistic, Property-Casualty Insurance Portfolio
Embedding Dimension
n23456789101-2.232-5.170-7.244-7.729-6.694-5.952-5.005-4.063-3.1922-3.210-5.922-7.041-7.822-7.076-6.182-5.080-4.088-3.2174-2.436-5.124-7.158-7.862-7.354-6.248-5.114-4.115-3.2478-3.163-6.432-7.908-8.380-7.810-6.686-5.552-4.440-3.478
Table 4
EGARCH standardized residual correlation dimensions and BDS statistics, at embedding dimensions 2 through 10 and n equaling 1, 2, 4, and 8, for the daily raw returns of life and health and property-casualty insurance portfolios.
Panel A1: Correlation Dimension, Life-Health Insurance Portfolio
Embedding Dimension
n234567891012.0683.0323.8584.5355.1385.5745.9346.3356.29922.0813.0163.8544.5135.1525.5945.9956.4236.35042.0853.0573.8924.5765.2075.6475.9686.3256.32182.0893.0303.8694.5735.1455.6035.9586.2816.323
Panel A2: Correlation Dimension, Property-Casualty Insurance Portfolio
Embedding Dimension
n234567891012.0763.0433.8784.5995.2335.6636.0456.3826.39622.0693.0393.8984.5725.2205.6405.9956.4046.33742.0823.0363.8774.5685.1205.5545.9086.2616.35082.0713.0403.8674.5315.1295.5775.9756.3126.360
Panel B1: BDS Statistic, Life-Health Insurance Portfolio
Embedding Dimension
n23456789101-3.180-8.627-12.558-14.996-15.896-15.663-14.673-13.316-11.8232-3.744-8.577-12.628-15.036-15.937-15.729-14.754-13.366-11.8464-3.820-8.477-12.541-14.885-15.835-15.629-14.666-13.307-11.8148-3.671-8.411-12.494-15.060-16.161-16.074-15.179-13.854-12.358
Panel B2: BDS Statistic, Property-Casualty Insurance Portfolio
Embedding Dimension
n23456789101-3.694-8.598-12.913-15.729-16.904-16.792-15.827-14.428-12.8742-3.753-8.723-12.926-15.601-16.726-16.597-15.681-14.333-12.8214-3.779-8.643-12.737-15.397-16.560-16.493-15.598-14.252-12.7368-3.717-8.630-12.941-15.737-16.960-16.918-16.047-14.714-13.200
Table 5
Root mean squared forecast errors for the daily raw returns of life -health and property-casualty insurance portfolios. The errors are obtained by running locally weighted regressions with tricubic weights, lags 1 through 5, and the number of nearest neighbors equaling the fractions 0.1 through 0.9 of the data. The root mean squared forecast error of the random walk model is also provided.
Panel A1: Life-Health Insurance Portfolio
FractionLag 1Lag 2Lag 3Lag 4Lag 5.100.00660360.00659980.00659860.00662580.0065970.200.00661190.00661280.0066073 0.0066400.0066043.300.00661660.00661720.00661230.00663830.0066064.400.00662030.00661750.00661590.00663850.0066102.500.00662220.00661840.00662030.00663920.0066112.600.00662380.00661960.0066228 0.0066400.0066125.700.00662420.00662000.00662250.00664140.0066134.800.00662540.00662110.00662230.00664260.0066144.900.00662730.00662170.00662140.00664410.0066161Random Walk0.00913280.00913280.00913280.00913280.0091328
Panel A2: Property-Casualty Insurance Portfolio
FractionLag 1Lag 2Lag 3Lag 4Lag 5.100.00669810.00677200.00678250.00682080.0067707.200.00669910.00678780.0067854 0.00682920.0067831.300.00670170.00679070.00679000.00683130.0067877.400.00670290.00679280.00679000.00683100.0067914.500.00670440.00679570.00679070.00683120.0067932.600.00670620.00679640.0067921 0.00683160.0067969.700.00670750.00679810.00679310.00683210.0067989.800.00670990.00679980.00679380.00683250.0067998.900.00671130.00680050.00679480.00683330.0068017Random Walk0.00866260.00866260.00866260.00866260.0086626
PAGE 16
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