NONLINEAR DYNAMICS BEHAVIOR IN THE REIT INDUSTRY: A PRE- AND POST-1993 COMPARISON

Transcription

1 NONLINEAR DYNAMICS BEHAVIOR IN THE REIT INDUSTRY: A PRE- AND POST-1993 COMPARISON By Benjamas Jirasakuldech * Associate Professor of Finance Slippery Rock University of Pennsylvania School of Business 1 Morrow Way Slippery Rock, PA b.jirasakuldech@sru.edu Riza Emekter Assistant Professor of Finance Robert Morris University School of Business 6001 University Boulevard Moon Township, PA emekter@rmu.edu December 1, 2009 JEL Classification: Keywords: C14, C22, C32, F31, G12, and G15 Real Estate Investment Trusts, Nonlinearity, and Asymmetry * Corresponding Author 1

2 NONLINEAR DYNAMICS BEHAVIOR IN THE REIT INDUSTRY: A PRE-AND POST-1993 COMPARISONS Abstract We examine the nonlinear dynamic behavior of EREIT, MREIT and HREIT returns during the pre- and post-1993 REIT organizational and structural changes. There is substantial empirical evidence in favor of nonlinearity in REITs industry and small stock markets for the whole sample periods and across two sub-periods. The nonlinear dependence in all types of REITs is caused by heteroskedasticity. During the most recent time periods, real estate returns exhibit time irreversible pattern with some predictable components. The asymmetric price change patterns in all three types of REITs in general follow a linear model with non-guassian innovations. Similar patterns are reported for EREITs before and after 1993, suggesting that return behavior of EREIT hasn t changed as the sizes and structures of REITs have changed over time. Applying the same tests on Russell 2000, a proxy for small stocks, we find that EREITs and small stocks behave differently during the most recent time periods. The return behavior of small stocks resembles a nonlinear model with Gaussian innovations. 2

3 NONLINEAR DYNAMICS BEHAVIOR IN THE REIT INDUSTRY: A PRE- AND POST-1993 COMPARISONS 1. Introduction Real estate investment trusts (REIT), particularly equity REITs, have evolved as an innovative financial instrument that warrants greater attention from the institutional and individual investors. The market capitalization of REIT has almost reached 200 billion dollars 1. Such significant growth in size of real estate market suggests that REITs have become a popular investment vehicle to be included in a well-diversified portfolio. REITs offer an opportunity to invest in real estate without committing large amount of money with an added advantage of liquidity similar to a stock or bond. It has a relatively low correlation with general stocks and bonds, therefore, it offers some diversification for investors. This study examines the nonlinear dynamic behavior of equity REITs (EREITs), mortgage REITs (MREITs) and hybrid REITs (HREITs) returns during 1972 to During the most recent years, Mortgage and Hybrid REITs have grown significantly. The market capitalizations for MREITs and HREITs increase by 34.31% and 18.46% from 2003 to The significant growth in these markets and its relative importance to EREITs justifies the inclusion of Mortgage REITs and Hybrid REITs in this study. Unlike stock markets, the unique market microstructures of real estate markets such as illiquidity, thin trading, high transaction costs, and short-selling restrictions make REIT markets susceptible to exhibit clustering of price change and nonlinearity. Any 1 Source: NAREIT s web site: 2 The market capitalizations in millions of dollars at year end of 2003 for EREITs, MREITs, and HREITs are 204,800.4, 14,186.51, and 5, At year end of 2007, they are 288,694.6, 19,054.1, and 4,260.3 millions of dollars for EREITs, MREITs, and HREITs respectively. 3

4 information regarding the non-linear structure of REIT helps both researchers and investors to predict and price these securities more precisely and at the end it would lead to better asset allocations. REITs market went through a major transformation at the beginning of 1990s is usually recognized as the reference year for this significant transformation (Mueller and Anikeff (2001), Clayton MacKinnon (2000), Ott, Riddiough and Yi (2005)). Since 1993, the market has grown rapidly, liquidity has increased, and the market has become more integrated with stock market. This significant changes suggest that the information was conveyed to the market more efficiently and quickly, resulting in a more smooth price-generating process. Because of these significant structural changes, we also examine whether the change in REIT industry cause any change in the dynamic behavior of REIT returns by dividing our sample into two sub-periods: pre-and post We also compare the return behavior of REITs with the Russell 2000, an index of small stocks. The Russell 2000 index is used because prior research has shown that REITs stocks behave similarly to small stocks (i.e., Liu and Mei 1992; and Nelling and Gyourko 1998). The (dis)similarity in return behavior will provide additional information as the segmentation of the REIT and small stock markets. In analyzing the non-linear dynamics in real estate markets, we used various techniques, some of which have not been previously implemented on REITs. The econometric techniques used are Brock-Dechert-Scheinkman (BDS) test (1996), Markov chain test developed by McQueen and Thorley (1991), and time reversibility (TR) test developed by Ramsey and Rothman (1996). 4

5 The literature on nonlinearity has extensively employed the BDS test to uncover nonlinear dependence in many financial time series, including stock and REIT markets, commodity futures markets, and exchange rates markets. The main problem with the BDS test is that the BDS rejection could mean the possible deviations from independence in the linear, nonlinear, or chaos form. To rule out the linear and ARCH effect, we first remove potential linear dependence by fitting an autoregressive model for each security returns. We further examine whether the nonlinear dependence in the linearly filtered return series could be caused by an ARCH-type conditional heteroskedasticity by applying the BDS test on the residuals from GARCH model. 3 While the BDS may detect the presence of nonlinearity, it does nothing to indicate the source of the rejection of the null hypothesis or to characterize the nature of the nonlinearity, an essential step toward understanding the underpinning of the nonlinearity. Markov chain test can detect non-linear dependence and at the same time provide further information about the nature of the dependence. The Markov chain captures the nonlinear nature by allowing the transitional probability from one state to another to vary depending on the sequences of prior states rather than one single parameter. If the returns have unequal transitional probabilities, then it is likely that the return series exhibits a nonlinear dependency, a departure from a purely random process. The Markov chain test overcomes the major limitation of the BDS tests which do not reveal the form of nonlinearity by providing additional information as to the predictability pattern of REIT returns. 3 Once the IID is rejected for both linear-filtered and ARCH-filtered return series, one can then proceed to investigate whether the cause of departure from IID can possibly be explained by deterministic chaos. 5

6 Time reversibility test reveals whether the dependence is due to non-linear structure of the model or due to Gaussian innovations. The nonlinear TR test is used to complement the BDS and Markov chain tests to further characterize the return-generating process for REITs. The TR test examines the symmetry of time series behavior with respect to time reversal. If the REITs returns series are time reversible or symmetric, the statistical properties should be invariant to reversal of time direction. Therefore, a random walk property in the return implies that the returns are time reversible. Ramsey and Rothman (1996) pointed out that the TR test is superior to other nonlinear techniques in detecting the random walk hypothesis in several aspects. First, the TR test allows for the intertemporal dependence to be detected at higher frequencies than what is suggested by previous literature documenting long-horizon dependence assets returns. 4 Second, the intertemporal dependence based on the TR test is not sensitive to the underlying ARCH effects as found in the BDS test. Third, the TR test can identify the source of asymmetric behavior. Ramsey and Rothman (1996) argued that time irreversibility could be the result of the nonlinearity in the underlying model or non-gaussian innovations. Fourth, while much of the literature testing for nonlinearity provides no guidance on the appropriate specification of the time series model, the TR test provides additional information on the nature of nonlinearity, which can assist in model development. Lastly, Rothman (1992) showed that the TR test is more powerful than the Brock, Dechert and Scheinkman (1988) BDS and Hinich s (1982) bispectrum tests in detecting for nonlinearity. This study applies a battery of tests to several REIT series for the first time and intends to help both researchers and practitioners to come up with better fitting models 4 See for example, Lo and MacKinlay (1988) and Poterba and Summers (1988) 6

7 for forecasting the real estate behavior. These tests reveal that, both real estate and small stock markets inherent some non-linear structures during 1971 to 2008 and a period before and after 1993 major structural change in real estate markets. The nonlinear dependence in all types of REITs can be explained by ARCH-type conditional heteroskedasticity. During the most recent time period, real estate returns exhibit time irreversible pattern with some predictable components. The asymmetric price change patterns in all three types of REITs in general follow a linear model with non-guassian innovations. Similar patterns are reported for EREITs before and after 1993, suggesting that return behavior of EREIT hasn t changed as the sizes and structures of REITs have changed over time. Applying the same tests on Russell 2000, we find that return behavior of small stocks exhibit nonlinear model, suggesting that EREITs and small stocks behave differently during the most recent time period. The remainder of the paper is organized as follows. In the next section, we review the literature for REIT returns behavior. Section 3 describes our data and summarizes the descriptive statistics of EREITs, MREITs, HREITs, and the Russell In section 4, we present the descriptions of methodologies including the BDS, Markov chain and time reversibility tests. Their empirical results are presented in section 5. Section 6 offers some concluding remarks. 2. Literature Review Return distribution of securities have historically been assumed to follow the linear stochastic process which is also known as the random walk model. There have been many recent studies, including Brock, Dechert, and Scheinkman (1996), Brock, 7

8 Hsieh, and LeBaron (1991), Hsieh (1991), Willey (1992), Duett, Hershbarger, and Pandey (1994), and Peters (1991; 1992) to name a few, show that assets return distributions are characterized by non-linear dynamics, suggesting that random walk model may not be an appropriate model in modeling the return behavior. Savit (1989) s study shows that asset return distributions may be governed by non-linear deterministic chaos such that the forecasting errors grow exponentially, which generate the price behavior that appears to be random. Scheinkman and LeBaron (1989) report significant evidence of nonlinear dependence for weekly but not daily valueweighted CRSP U.S. stock returns, indicating that nonlinearities play an important role in explaining the variation of U.S. equity returns. Applying the BDS test to U.S. stock returns, Hsieh (1991) find some support that stock returns follows a nonlinear dynamic system and such nonlinearity can be attributable to the conditional heteroskedasticity as opposed to the chaotic behavior. Employing the Hurst statistic and the BDS test, Opong et al. (1999) show that price behavior of UK stock markets exhibit more frequent cycles and patterns than what a true random series would generate. Other studies examining price behavior of foreign equities indices including Wiley (1992), Antoniou et al. (1997), Chu (2003), and Appaih-Kusi and Menyah (2003) find that the return dynamics exhibit a much more complex nature and cannot be captured by a simple linear stochastic or random walk model. Literature on non-linearity has also expanded to examine the nature of stochastic process of assets in futures markets. Frank and Stengos (1986) examine the nonlinear behavior in futures markets and find evidence of nonlinear structure for gold and silver commodities. Blank (1990) further shows that the nonlinear dynamics of futures prices 8

9 are consistent with deterministic chaos. On contrary, study by Yang and Brorsen (1993) shows that daily future returns of twelve commodities are generated by non-linear conditional heteroskedasticity rather than deterministic chaos. Nonlinear dynamics in exchange rates have also been studied extensively. Hsieh (1989) employs GARCH model and applies the BDS test on daily British pound, Canadian dollar, Deutsche mark, Japanese yen, and Swiss franc and reports evidence of nonlinearities in these exchange rates. Cecen and Erkal (1996) apply the BDS test on the hourly data of the British pound, Deutsche Mark, Swiss franc and Japanese and show strong evidence of nonlinear stochastic dependence but little evidence of lowdimensional chaos. Using daily currency value, Gilmore (2001) reports similar finding of nonlinear dependence but no chaos. Employing time-delay embedding techniques and local linear predictor, Cao and Soofi (1999) conclude that exchange rates are not generated by purely random process but most likely generated by nonlinear deterministic systems with dynamic noise. Nonlinear behavior of real estate investment trust (REITs) return is first investigated by Ambrose et al. (1992). Using the rescaled range analysis, they fail to find evidence of fractal structure, a subset of nonlinear dynamics in U.S. REITs and other equity markets and conclude that REITs return behaviors were best characterized by a random process. Newell et al. (1996) apply a vast variety of nonlinear tests including the BDS test, Lo s test, correlation dimension, Lyapunov exponents and three other moment test on the Australian property trust return during They find that the Australian real estate return dynamics are governed by nonlinear structure; therefore, the nonlinear stochastic models seem to be the appropriate model in capturing the underlying 9

10 nonlinear dynamics process. Employing the same nonlinear tests, Newell and Matysiak (1997) report that the UK real estate property is not driven by the chaotic behavior. Rather the underlying dynamic behavior of UK property is explained by non-linear return-generating process. Liow and Webb (2008) apply the BDS test and a nonlinear logistic model to examine the nonlinear return dependence in six major real estate markets including the US, UK, Japan, Australia, Hong Kong, and Singapore. The results indicate that majority of real estate markets with the exception of UK exhibit some nonlinear structure, but provide no information as to whether the nonlinearity is deterministic or stochastic. Lastly, Lee and Chiu (2008) show that the dynamic process of REIT returns exhibits an exponential smooth transition, suggesting that the nonlinear model is better suited than linear model in describing the behavior of REITs. Numerous research in real estate markets in contrast show that the return dynamics of REIT is consistent with linear stochastic or random walk model. Seck (1996) applied autocorrelation and variance ratio tests to equity REITs (EREITs) and the S&P 500 during He found evidence consistent with a random walk in both markets, suggesting some sort of similarity and substitutability between REITs and the general stock market. Ambrose, Ancel, and Griffiths (1992) reported similar results for Mortgage REITs, Equity REITs, and the S&P 500, concluding that real estate and stock markets are not segmented. Applying the unit root test, variance ratio test, and runs test on international real estate markets in Europe, Asia, and North America, Kleiman, Payne, and Sahu (2002) found evidence of a random walk and weak-form efficiency in these markets. 10

11 3. Data The data used for this study are monthly price indices including dividends for three categories of REITs: equity REITs (EREITs), mortgage REITs (MREITs), and hybrid REITs (HREITs). 5 The REIT data are obtained from the National Association of Real Estate Investment Trusts, Inc. (NAREIT), and cover January 1972 to December 2008 for a total of 444 observations. For comparison purposes, we also examine the return dynamics of small-cap stock market as proxied by the Russell 2000 index. The use of Russell 2000 as a proxy of small stock is consistent with a number of studies that shows REITs seem to behave more like small stocks [See for example: Liu and Mei (1992) and Nelling and Gyourko (1998)]. The monthly Russell 2000 index data with dividends are obtained from Frank Russell company. The Russell 2000 index data from January 1979 to December 2008 are used because the index was not introduced until Total data for Russell 2000 is 360 observations. From the price index data, P t, we first calculate continuously compounded monthly returns, r t 100 *ln( Pt / Pt 1), where P t is the end-of-month closing value for the REITs index and P t-1 is the previous end-of-month closing value. The continuously compounded real return is then calculated by taking the difference between the continuously compounded monthly return and the continuously compounded U.S. inflation rate. The continuously compounded U.S. inflation rate is calculated as the first 5 EREITs are real estate firms that own and operate income-producing real estate. MREITs are real estate firms that lend money either directly to real estate owners or indirectly through the acquisition of loans and mortgage-back securities. HREITs are real estate companies that own real estate properties and also make loans to owners and operators. 11

12 difference of the natural log of the U.S. monthly consumer price index (CPI). The CPI data is obtained from the Federal Reserve Bank of Saint Louis. 6 Panel A of Table 1 reports a statistical description of the monthly real returns of EREITs, MREITs, HREITs, and the Russell 2000 Index. During the study period, the Russell 2000 index produces a higher average monthly real return (0.54%) than do EREITs (0.52%). Both HREITs and MREITs yield negative real returns of 0.085% and 0.036%, respectively. The small stock index also shows a higher volatility of 6.46% when compared with the volatility of EREITs of 4.70%. Among all the equities, the market for HREITs exhibits the most volatility of 6.57%. The statistics also show that the distribution of returns of all types of REITs and small stock equities are negatively skewed and significant, suggesting that investing in these markets has a high probability of earning negative returns. All the kurtosis values are much larger than 3, indicating the presence of fat tails compared with the normal distribution. The Jarque-Bera also shows evidence of departure from normal distribution for all securities under investigation. Panel B of Table 1 reports the autocorrelation coefficients at lags 1 to 6 and lag 12 for all four types of securities. The autocorrelation coefficients are relatively small for all three securities, except for the HREITs indicating a slow decaying process. The Ljung- Box (1978) Q statistics reject the null hypothesis of no autocorrelation at lags 6 and 12 for EREITs and lags 6 and 12 for EREITs, HREITs, and MREITs at the 1 percent significance level. 7 This provides evidence of serial dependence in the higher moments Under the null hypothesis of no serial correlation, the Ljung-Box (1978) Q-statistics are distributed as chisquare with m degree of freedom. 12

13 of the return distributions of all REITs securities under investigation. 8 However, the null hypothesis of no autocorrelation cannot be rejected for Russell 2000 at both lags 6 and 12 at the traditional significance level. 4. Methodology (4.1) The BDS Test Prior studies by Hsieh (1989), Frank and Stengos (1989), Poshakwale (2002), Newell et al. (1996), Newell and Matysiak (1997), and Liow and Webb (2008), to name a few, follow the Brock et. al. (1996), hereafter referred to as BDS test, as a means of detecting nonlinear dependence in many financial time series. The BDS test is powerful in distinguishing a pure random process from nonlinear stochastic dynamics or from deterministic chaos. 9 The BDS test is based on a null hypothesis that a univariate time series { Z t : t 1,... T} is a random sample of independent and identically distributed (IID) observations. The non-iid alternatives can take the form of non-stationarity, linear, or nonlinear dependence. However, the BDS test cannot reveal the type of non-linear present. In other words, this test cannot distinguish a nonlinear deterministic system from a nonlinear stochastic system. The BDS statistic is calculated as: T [ C ( l) ( C1( l)) m m ] Wm ( l) (1) ( l) m where C m (l) is the correlation integrals that correspond to various embedding dimensions m and l is distance. m (l) is an estimate of standard deviation and the test statistics; W m (l) has a standard normal distribution. For a small sample size of The unit root tests are performed on the real returns of all types of REITs and Russell The null hypothesis of a unit root is rejected for all the securities at the 1% significance level. 9 According to Hsieh (1991), chaos is defined as a nonlinear deterministic system that appears to be random. 13

14 observations in our case, a bootstrapping is used to generate the p-value for the BDS test statistics. The power of the BDS test is sensitive to the choice of l and m. Following Hsieh (1989) s study, we calculate the BDS test statistics when the values of l range from 0.5 to 1.5 times standard deviation of the real return series and m ranges from 2 to 6. Time series literature has shown that the rejection of IID in any financial time series may be due to the nonstationarity of the underlying economics factors. We, therefore, employed the augmented Dickey-Fuller (1979) and Philips and Perron (1989) unit root tests to test for the stationarity in these four securities. If nonstationarity is present, the nonstationarity must be removed. Since the BDS test can detect linear dependence, we first filter the real return of each security with an autoregressive moving (ARMA) model to remove potential linear dependence. 10 The residuals from linearly filtered real returns are then subjected to the BDS test of general nonlinear dependence. We further examine whether the nonlinear dependence in the linearly filtered real return series could be caused by an ARCH-type conditional heteroskedasticity. We employed a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model to remove conditional heteroskedasticity. After filtering with GARCH model, the BDS test statistics are calculated for the residuals. 11 If the null hypothesis of IID is rejected for residuals from linearly filtered returns but not GARCH filtered returns, it indicates the presence of nonlinear dependence in the form of GARCH-type conditional 10 The autoregressive models for each currency excess returns are not reported here but are available from the authors upon request. The autoregressive (AR) and the moving average (MA) terms are chosen using a number of criteria: the Akaike Information Criterion (AIC), the Schwartz Information Criterion (SBC), the absence of serial correlation in the residuals, and the significance of the coefficients on the AR or MA models. 11 The diagnostics on the standardized residuals, the Ljung-Box statistics Q(12), Q 2 (12), indicate the absence of autocorrelation in the standardized residuals and the squared standardized residuals, and the LM test indicates that remaining ARCH effects are also insignificant, supporting the validity of our EGARCH specification for real returns of all securities. 14

15 heteroskedasticity. If the linear-filtered and GARCH-filtered series are proved to be non IID, it indicates possible presence of non-linear dependence. We further test whether non-linear dependence is caused by the deterministic chaos. (4.2) Second-order Markov Chain Test The Markov chain test developed by McQueen and Thorely (1991) is used to complement the BDS and Chaos tests to investigate the random walk hypothesis vs. nonlinear return dependence as well as the predictability components of real returns of REITs and Russell The advantages of Markov chain test over the BDS and variance ratio tests are (1) it does not require a pre-filtering of any linear dependency in the series under investigation, (2) it accommodates the issue of nonlinearity by modeling the return pattern as a two-state Markov process and allowing the current state to depend on the prior two sequences of the returns, rather than one single parameter, (3) it does not require the series to be normally distributed, and (4) it also reveals the predictability pattern, a significant implications for profitable trading strategy. According to the Markov chain test, if the returns series follows a random walk process, then the probability of obtaining either a negative or a positive return in the current period should be the same, regardless of prior returns. Therefore, a rejection of the null hypothesis of random walk indicates the presence of nonlinear dependence in the return dynamics behavior. A two-state second-order Markov process, I t, can be defined as follows: 1 if Rt 0 I t (2) 0 if R 0 t 15

16 where R t is the real returns of each security at time t. To test the randomness of each security returns, a two-order transition probability ( ij ) is constructed based on the behavior of process {I t }: ij = Prob[I t = 0 I t-2 = i, I t-1 = j ], (3) where i and j can take the value of either 1 or 0. For example, the transition probability ( 00 ), 00 = Prob[I t = 0 I t-2 = 0, I t-1 = 0], (4) is the probability that a negative return will continue to persist in the current period given two preceding negative returns. Likewise, (1-00 ) is the probability that a sequence of two negative returns will revert to a positive return in the current period. The random walk hypothesis posits that the probability of either the state 0 (i.e., I t = 0) or state 1 (I t = 1) should be invariant to any prior two-state sequence. Accordingly, the following seven null hypotheses should be tested: H 1,0 : 00 01against H 1,A : H 2,0 : against H 2,A : H 3,0 : 00 11against H 3,A : H 4,0 : against H 4,A : H 5,0 : against H 5,A : H 6,0 : against H 6,A : H 7,0 : against H 7,A : Therefore, the Markov chain test is built on the assumption of a nonlinear equilibrium model by focusing on the period of the established trends. The Markov 16

17 Chain tests can be performed by first obtaining the Maximum Likelihood Estimates (MLE) of four transition probabilities, i.e., [ ], by maximizing the following log likelihood function: 11 0 ) log 0 [ Nij log ij M ij log(1 ij )], ij 00 L ( S,, (5) T with respect to four transition probabilities. N ij and M ij are numbers of transition counts. The estimated maximum likelihood estimator of ij is 12 ˆ ij ( N ij N ij M ij. (6) ) The Likelihood Ratio Test (LRT) procedure is applied to test the null hypothesis: LRT = 2[L UR - L R ], (7) where L UR and L R are the log likelihood functions, which are obtained using the MLE of the unconstrained and constrained parameters, respectively. The LRT statistic is asymptotically distributed as restrictions. 2 n with n degrees of freedom, where n is the number of (4.3) Time Reversibility Test The nonlinear TR test developed by Ramsey and Rothman (1996) is further applied to characterize the dynamic behavior of EREITs, MREITs, HREITs, and small stocks. The purpose of the TR test is to test for a symmetric pattern of the time series along the time axis. In a time series that exhibits symmetric or time reversible patterns, the statistical properties should be the same when viewing forward or backward in time. In other words, reversing the time axis will not alter the dynamic behaviors of a time 12 The corresponding asymptotic variance, 2 ( ) ˆ (1 ˆ ij )/( N M ). ij 17 ij ij ij

18 reversible series. Therefore, if the equity returns follow a random walk, then they are time reversible or symmetric. The TR test differs from the BDS and Markov chain approaches in that it reveals the nature of stochastic process generating the stock return series, which is useful for modeling the return behavior. The TR test has been used to examine the asymmetric price change or nonlinear behavior in wide range of economics and financial data including agricultural land prices, inflation, unemployment rates, bond yields, and CRSP stock returns (i.e, Lavin and Zorn; 2001 and Ramsey and Rothman; 1996). A stationary time series { X t } is time reversible if for every positive integer n, every t 1, t 2,..., tn R, and the vectors X, X,... X ) and X, X,..., X ) have t t t t t t ( 1 2 n ( 1 2 n identical joint probability distributions. The TR test detects the departure from IID hypothesis by testing for the equality between certain pairs of moments from joint probability distributions of a time reversible time series. Time series { X t } is time reversible if i j j i E [ X X ] E[ X X ] or (8) t t k t t k i, j i j j i E [ X X ] E[ X X ] 0, for all i, j, k N. (9) t t k t t k The TR test statistic is the difference between a sample estimate of the symmetric bicovariance function of a mean zero stationary time series { X t } with T observations, T ˆ k T k 1 X 2 2,1( ) ( ) t X and (10) t k t k 1 T ˆ k T k 1 2 1,2 ( ) ( ) X t X, (11) t k t k 1 ( k) 2,1( k) 1, 2( ) for various integer values of k. (12) 2,1 k 18

19 Under the null hypothesis that { X t } is time reversible, the expected value of ˆ 2, 1( k) should be zero for all lags k. To test for symmetric behavior, the TR test is applied directly on the real returns of each currency, and the TR test statistics are standardized by var[ ˆ 2, 1( k ) ] 1/2, which is obtained through Monte Carlo simulation. The precondition for the TR test is the stationarity of the underlying series, and the results of the unit root test indicate that this precondition is met. Since there is no exact small sample expression for var[ ˆ 2, 1( k ) ] 1/2, the sample distribution of the TR test statistic is generated through Monte Carlo simulations. An ARMA model is first identified and estimated for each excess return series. Based on the fitted ARMA model, a Monte Carlo simulation is performed 1,000 times to obtain the estimates of the standard deviation of ˆ 2, 1( k ). The significance of the standardized TR test statistics for each lag is ascertained based on the empirical sample distribution from Monte Carlo simulations. In addition to testing for time reversibility on the individual lag k, the joint test for a set of ˆ 2, 1( k ) = 0 is also performed based on the time reversibility portmanteau statistic, n [ ˆ2, 1 k m P m, n ( k) / var[ ˆ 2, 1( k ) ] 1/2 ] 2, (13) where P, is distributed as 2 with n m 1degrees of freedom. m n If the dynamic behavior of REITs and stock returns is time irreversible or asymmetric, we can further identify whether the asymmetry is due to the functional form or the asymmetric innovations of the data-generating process. Ramsey and Rothman (1996) provided that time irreversibility can come from two sources: (1) the nonlinearity of the model but Gaussian innovations (Type I time irreversibility) or (2) the linearity of the model but non-gaussian innovations (Type II time irreversibility). This information 19

20 is useful particularly for the analysts in developing the appropriate forecasting model that accounts for the nonlinearity. To differentiate between Type I and Type II time irreversibility, the standardized TR test statistics are calculated on the ARMA residuals, where var[ ˆ 2, 1( k ) ] 1/2 is calculated via the following theorem stated by Ramsey and Rothman (1996). Theorem 1: Let { X t } be a stationary sequence of independently and identically distributed random variables (IID standard errors) for which E[ X t ] 0 t and assume E [ 4 X t ] 1/ 2 var[ ˆ2,1( k )] =. Then, 2( T k, (14) ) /( k) 2 2 ( T 2k) /( T ) where E [ X ], E[ X ], and E[ X ]. 2 t 3 t 4 t If the time series process is Type I time irreversibility, the approximation via an ARMA (p, q) will reduce the power of the test and, hence, reject the null hypothesis of the time reversibility on the residuals most of the time. Therefore, if the null hypothesis of time reversibility is rejected under both the raw data and ARMA residuals, the asymmetric behavior is due to nonlinearity in the functional form. On contrary, if it is rejected on the raw data but not on the ARMA residuals, asymmetric behavior is due to non-gaussian innovations. 5. Empirical Results 5.1 The BDS Test Results According to Hsieh (1991), the BDS test can detect four different types of departures from IID: non-stationarity, linear dependence, non-linear stochastic process, and chaos. Changes in the underlying economic factors can cause the non-stationarity in the security returns. To rule out nonstationarity as a cause of rejection of IID, we 20

21 performed the Augmented Dickey-Fuller (1979) and Perron (1989) unit root tests on the security returns. The null hypothesis of a unit root or non-stationarity in the real return of all securities is rejected at the 1% significance level for all types of REITs and Russell 2000 for the full period and two sub-periods. The stationarity in the real returns of EREITs, HREITs, MREITs, and Russell 2000 suggests that the departure from IID is not caused by the nonstationarity of the underlying economic variables. We then proceed to rule out the linear dependence as a possible cause of departure from IID by fitting the REITs and small stock return series with an autoregressive moving average (ARMA) model to remove any linear dependence. To further distinguish the nonlinearity caused by heteroskedasticity and chaotic nonlinearity, we employ EGARCH model to remove the conditional heteroskedasticity. The residuals from linear filtered returns and EGARCH filtered returns are subject to BDS test. Table 2 reports the results of the BDS test for the full period for both linearly filtered and EGARCH filtered returns of each security for values of m ranging from 2 to 6 and l equal to 0.5 and 1.5 he null hypothesis of IID is rejected for all three types of REITs: EREITs, HREITs, and MREITs and the Russell 2000 at the traditional significance level. The significance of the BDS statistics for linear pre-filtered returns points to the possible nonlinear dependence in these series. When the BDS tests are performed on the residuals from the EGARCH model, the results indicate a rejection of IID only in the Russell The findings of IID in only the EGARCH filtered and but not in the linear-filtered EREITs, MREITs, and HREITs returns indicate that the nonlinear dynamic behavior of REITs securities but not the Russell 2000 appear to be 13 The fitted EGARCH models for security are not reported here but are available from the authors upon request. 21

22 explained by EGARCH- type conditional heteroskedasticity. Our findings of nonlinear dependencies in the REITs markets conform to the findings of Liow and Webb (2008). When the BDS analysis is performed on the pre- and post-january 1993 data of REITs and Russell 2000, the null hypothesis of IID is rejected for all three types of REITs in both sub-periods. The results are reported in Table 3. During the pre- and post- 1993, both EREITs and MREITs return behaviors exhibit non-linear dependence which is caused by the conditional heteroskedasticity. For HREITs, the conditional heteroskedasticity appears to explain nonlinear dependence only in the post-1993 period. The results are quite different for small stocks. Russell 2000 exhibits a random walk process before 1993 but not after 1993 period and the source of nonlinear dependence in the post-1993 is not due to the heteroskedasticity. The results here suggest that EGARCH specification seems to be best fitted model for most REITs. The result for small stocks suggest that the EGARCH model as used in various studies may be misspecified and fail to capture all dependencies or the remaining unspecified hidden structures in the data exhibit a much more complicated nature which may be described by a complex nonlinear threshold model. 5.2 Markov Chain Test Results It is evident from the BDS test that some degree of nonlinear dependence still exists in the EGARCH-M filtered returns of some securities. This dependence could be the result of a misspecified EGARCH model, which fails to capture all the dependence in the residuals. It also may indicate the presence of some nonlinear pattern in the data, suggesting that a simple linear random walk model may not be able to capture the dynamic behavior in the security returns series. This finding motivates our use of 22

23 nonlinear Markov chain and TR tests to further reveal the nature of the nonlinear dependence and predictable components, which provides important implications in deriving appropriate forecasting model for real estate returns. The results of the second-order Markov chain test on the real returns of EREITs, MREITs, HREITs, and the Russell 2000 index for the entire sample and two sub-periods are reported in Tables 4 and 5, respectively. The unconstrained estimates of ˆ 00 and ˆ 11 indicate the presence of positive serial dependence in the returns behavior of the MREITs and HREITs during the pre The transition probabilities of ˆ 11 are always smaller than ( 1 ˆ 11 ), and ( 1 ˆ 00) are smaller than ˆ 00, suggesting that a positive (negative) return is more likely to follow two prior positive (negative) returns. For the EREITs and Russell 2000, a positive return is more likely to follow two prior sequences of either positive or negative returns. For the post-1993, the transition probability shows that EREITs tend to exhibit positive serial dependence. The transition probability values also point to the existence of predictability components in MREITs, HREITs, and Russell 2000, where the positive return is more likely to occur after the two occurrences of either positive or negative returns. Under the null hypothesis of symmetry or random walk, the probability of observing either positive or negative returns should be the same and independent of any prior two sequences. From 1972 to 2008, the probabilities of observing a negative return following two sequences of negative returns ( ˆ 00 ) are 50.50% for EREITs, 48.60% for MREITs, 52.3% for HREITs, and 38.70% for Russell For the pre-1993, they are 43.6% for EREITs, 50.7% for MREITs, 55.70% for HREITs, and 36.4% for Russell For the post-1993, they are 59.5% for EREITs, 44.7% for MREITs, 45.9% for 23

24 HREITs, and 35.3% for Russell The probabilities of observing a negative return following two sequences of positive returns ( ˆ 11) for full period, pre-1993, and post are 34.6%, 36.1%, and 33.8% for EREITs, 37.10%, 44.4%, and 32.1% for MREITs, 36.3%, 44.4%, and 29.2% for HREITs, and 36.0%, 31.7%, and 39.4% for Russell The positive returns are more likely to persist than negative returns for both real estate and equity markets because ˆ 00 < ( 1 ˆ 11 ). The Likelihood Ratio Test for the random walk hypothesis is performed by reestimating ˆ 00 and ˆ 11 with the restriction of imposed according to each of seven null hypotheses. Under the less restrictive null hypothesis (H1 to H6), the probability of observing a negative real return in the current period should be identical and independent of the pattern of returns in the two preceding periods. Under the more restrictive null hypothesis (H7), the transition probability of obtaining a positive or negative return should be the same regardless of the sequence of prior returns. We find significant evidence of non-random behavior for all three types of REITs in the full sample period and both sub-periods. The null hypothesis of random walk or no predictability is rejected in at least six out of seven tests during at the 10% significance level. For the pre-1993, evidence of random walk is much stronger for MREITs and HREITs. Out of seven null hypothesis of random walk, only one is rejected for both securities at the 10% significance level. For the post-1993, the random walk hypothesis is rejected for at least five out of seven for real estate securities. The non-random walk and nonlinear dynamic 14 For the full period, pre- and post-1993 periods, (1-00 ) are 49.5%, 56.4%, and 40.5% for EREITs, 48.6%, 50.7%, and 44.7% for MREITs, 52.3%, 55.7%, 45.9% and for HREITs, and for 38.7%, 36.4%, and 35.3% Russell 2000, respectively. For the full period, pre- and post-1993 periods, (1-11 ) are 65.4%, 63.9%, and 66.2% for EREITs, 62.9%, 55.6%, and 67.9% for MREITs, 63.7%, 55.6%, 70.8% and for HREITs, and for 64.0%, 68.3%, and 60.6% Russell 2000, respectively. 24

25 behavior is consistent with the findings of departure from IID in all three types of REITs based on the BDS approach. Results for the Russell 2000 are similar to those for REITs. The null hypothesis is rejected in six of seven tests in the whole sample period. However, the rejection is less significance in the post-1993 compared to the pre The numbers of null hypothesis rejected are twice larger in the first sub-period. Only three out of seven null hypotheses are rejected during the post Overall, the nonlinear Markov chain model suggests that the real return of all types of REITs exhibit a non-random behavior due to the possible existence of nonlinear dependence during the pre-1993 and the post Evidence of departure from IID is much stronger after the 1993 period for real estate markets but the opposite holds for the Russell MREITs and HREITs exhibit a positive serial dependence, a predictable pattern consistent with three positive or three negative consecutive returns during pre- 1993, while EREITs show similar predictable component in the most recent period. The return behavior of Russell 2000 exhibits a positive return after two negative or positive returns, a pattern similar to EREIT during the pre-1993 and similar to MREITs and HREITs during the most recent period. 5.3 Time Reversibility Test Results Panel A of Tables 6 and 7 report the results of the TR test on the real returns of EREITs and Russell 2000 for lags 1 to 10 and the portmanteau statistic P 1, 10 for the full period and the pre and post-1992 periods. The portmanteau statistics, P1, 10 reject the null hypothesis of time reversibility, ˆ 2, 1( k ) = 0 at the 1% significance level for all three types 25

26 of REITs and Russell 2000, suggesting strong evidence of irreversibility and significant departure from IID behavior. This asymmetric behavior supports the findings of the BDS and Markov chain tests. To further identify the underlying source of this asymmetry which cannot be revealed by the BDS approach, we performed the TR test on the ARMA residuals derived from a fitted model of each series. 15 The results are reported in Panel B of Tables 6 and 7 for the entire sample and two sub-periods, respectively. The test results are different depending on the time period. During 1972 and 2008, asymmetric return behavior of EREIT is driven by the asymmetric innovations in the data-generating process consistent with Type II time irreversibility. Similar asymmetric behaviors are found across two subperiods. The asymmetry found in MREITs, HREITs, and Russell 2000, on the other hands, is caused by nonlinearity in the functional form, which is consistent with Type I time irreversibility. In the early period, asymmetry exerts stronger influence in the return behavior of all REITs and small stocks. The source of asymmetry for MREITs and HREITs is due to the nonlinear behavior in the functional form. In the more recent time period, asymmetry in all three types of REITs, and small stock, is greatly reduced as shown by fewer rejections across horizons and particularly in the first two moments. The remaining asymmetry in MREITs and HREITs is clearly Type II asymmetry caused by asymmetric innovations, and not Type I. The results for the Russell 2000 index show evidence of Type II asymmetry in the pre-1993 and Type I asymmetry in the post-1993 period, an opposite pattern found in MREITs and HREITs. 15 The appropriate model for the autoregressive (AR) and the moving average (MA) is based on a number of criteria: the significance of the coefficients in the ARMA models, Akaike Information Criterion (AIC), the Schwartz Information Criterion (SBC), and the absence of serial correlation in the residuals. The results are not reported here but are available upon request. 26

27 In sum, the results of the TR test indicate that the EREIT, but not MREITs and HREITs markets are more symmetric than the Russell 2000 returns prior to 1993, but are less symmetric than the Russell 2000 after The underlying source of asymmetry suggests that a simple bilinear model will provide a better representation of the returns dynamics than the random walk or linear ARMA alternatives in modeling and forecasting the return behavior of MREITs and HREITs before For the period after 1993, a linear model is the appropriate model that will provide a more accurate forecast of the real estate markets. The 1993 real estate market structural changes do not cause any changes in the nonlinear behavior of equity REITs. When compared with small stocks lately, it suggests that real estate markets are best characterized by linear but non- Gaussian innovations, while the Russell 2000 index is best forecasted by using non-linear with Gaussian innovations. 6. Conclusions We examine the nonlinear dynamic behaviors as well as the return predictability and forecasting of REITs during periods. We also examine whether the 1993 major structural changes in real estate market is associated with the changes in return behavior of REITs by dividing our sample intervals into the periods before and after 1993 tax reform Act. To further provide information as to whether REITs and stock markets are segmented, the behavior of REITs is also compared with the Russell 2000, a proxy of small stock Unlike most previous studies on nonlinear behavior in equity and real estate markets which only suggest the existence of nonlinear dependence or the lack thereof, we 27

28 uncover the underlying source and nature of the nonlinearity, an important implication for appropriate modeling and forecasting REIT behavior. The results based on the BDS approach indicate significant departure from independently and identically distributed for real estate and stock markets throughout the period of 1972 to The nonlinear dependence in these markets is dominated by the presence of ARCH-type conditional heteroskedasticity. Similar nonlinear dynamic behavior is documented for EREITs and MREITs during the pre- and post-1993 periods. The nonlinear EGARCH model appears to provide best fit for return behavior of HREITs after1993 major structural change in REIT industry, not before. The nonlinear dependency in real estate markets are further supported by the Markov chain test. Besides rejecting the random walk, the Markov chain tests also uncover the predictable components based on the transition probability from one state to another. Before and after 1993, the return dynamic behaviors of EREITs, MREITs, and HREITs tend to follow a non-random walk process. A positive serial dependence, where a positive (negative) return follows two prior positive (negative) returns, best describes the return characteristics of MREITs and HREITs before 1993 and EREITs after During the most recent time period, MREITs and HREITs exhibit a predictable pattern where a positive return is most likely to occur after two occurrences of either positive or negative return. Evidence of time irreversibility or asymmetric price pattern contradicting to the random walk is also found in all types of REITs, suggesting that the pattern of price increases differs from that of price decreases in the real estate market. This asymmetry is much more visible in the early time period. The underlying source of asymmetry suggests that a simple bilinear model will provide a better representation of the returns 28

29 dynamics than the random walk or linear ARMA alternatives in modeling and forecasting the return behavior of MREITs and HREITs before Especially in the second time period, asymmetry is greatly reduced and appears to result from a returns pattern that is linear but non-gaussian, suggesting that the linear model is appropriate in modeling REIT returns behavior. The 1993 real estate market structural changes do not cause any changes in the nonlinear behavior of equity REITs. On the whole, this paper provides substantial empirical evidence in favor of nonlinearity in REITs industry and small stock markets. REITs and small stocks behave differently during the most recent time period, suggesting that REIT stock returns may not be proxied as well by the Russell 2000 as they have been in the past. The real estate markets are best predicted by linear model. This is not the case for small stocks which is indeed nonlinear. 29