Stock Market Volatility in US Bull and Bear Markets

Journal of Money, Investment and Banking ISSN 1450-288X Issue 1 (2008) © EuroJournals Publishing, Inc. 2008 http://www.eurojournals.com/JMIB.htm Stock Market Volatility in US Bull and Bear Markets J. Cuñado Universidad de Navarra, Pamplona, Spain L. A. Gil-Alana Universidad de Navarra, Pamplona, Spain F. Perez de Gracia Universidad de Navarra, Pamplona, Spain Abstract In this paper we test whether the stock market volatility presents a different behavior in bull and bear phases. Using long range dependence techniques we estimate the order of integration in the squared returns in the US stock market (S&P 500) over the sample period August, 1928 to December, 2006. The results suggest that squared returns present long memory behavior. In general, the estimates of d are above 0 and below 0.5 implying long memory stationarity for the volatility processes. The results also show that in many cases the volatility is more persistent in the bear market than in the bull market. Keywords: Volatility; Bull market; Bear market; Long range dependence; Squared returns. Jel Classification Codes: G12; E40; C32. 1. Introduction A well known stylized fact in stock markets is that volatility tends to vary over time (see, for example, Shiller, 1981; Schwert, 1989 and Campbell et al., 2001 among others). Many empirical papers have related the behavior of stock market volatility with the business cycle (see, Hamilton and Gang, 1996 and, recently, Casarin and Trecroci, 2007), while others papers relate stock volatility with upward and downward trends in stock market (i.e., bull and bear markets). Several authors have found that stock market volatility is higher during bear markets than in bull markets (Maheu and McCurdy, 2000; Edwards et al., 2003; Gomez Biscarri and Perez de Gracia, 2004; Jones et al., 2004; Gonzalez et al., 2005; Guidolin and Timmerman, 2005; Nishina et al., 2006; Tu, 2006; Cunado et al., 2007; etc.). Jones et al. (2004) provide two possible explanations for the higher volatility during bear markets. First, the increased uncertainty and risk observed in the bear market may generate a decline in equity value. Also, in the context of increased uncertainty investors react to bad news more quickly, adding then more volatility to the market. Further, Chordia et al., (2001) also suggest that the different behavior observed in the stock market liquidity in bull and bear markets may be related with volatility. Thus, bear markets tend to attract more investors while bear markets, on the other hand, could be subject to falling liquidity. In line with the above line of research, we also analyze volatility in the US stock market. First, we re-examine the volatility behavior taking into account cycles in the stock market over the period 1928 to 2006. Stock markets typically present periods of stock price increases and decreases about its Journal of Money, Investment and Banking - Issue 1 (2008) 25 long term trend. In order to define bull and bear stock market phases, we use the turning point algorithm developed by Bry and Boschan (1971). The Bry and Boschan algorithm let us identify troughs and peaks in stock market indices and, thus, it permits us to locate the starting and finishing points of the bull and bear markets. More specifically, the novel feature of our work is to analyse whether stock market volatility behaves similarly in the US bull and bear periods. Thus, instead of studying the time varying volatility behavior for the whole sample period we analyze the same issue over different sub-samples previously defined by peaks and troughs’ in the stock market. Second, instead of using classic approaches based on simple standard deviations or variances, we proxy volatility using squared returns, and estimate the models using methodologies based on long range dependence. Long range dependence has been extensively examined in stock markets (see, for example, Lo, 1991; Granger and Ding, 1995a,b; Ding et al., 1993; Baillie et al., 1996; Lobato and Savin, 1998; Sibbertsen, 2004 and recently, Elder and Jin, 2007). The empirical evidence using these models is inconclusive. Thus, some authors find little or no evidence of long memory in stock markets (see, for example, Hiemstra and Jones, 1997, and the references therein), while others such as Lobato and Savin (1998), Panas (2001) and Tolvi (2003) among others found evidence of long memory in absolute and squared returns. The empirical work will be carried out using daily data from the Standard and Poors 500 (S&P). The remainder of the paper is organized as follows. Section 2 provides the testing procedure employed in the article. In Section 3, we apply the methodology described in Section 2 to the data and report the empirical results. Finally, Section 4 contains some concluding comments. 2. Testing procedure The procedures implemented in this paper are based on the concept of long memory. Let us consider a zero-mean covariance stationary process {xt, t = 0, ±1, …} with autocovariance function, u = E(xt, xt+u). The time domain definition of long memory states that: u u . Now, defining the spectral density function as 1 f( ) 2 0 u cos ( u ) , 2 u 1 the frequency domain definition of long memory states that the spectral density function is unbounded at some frequency in the interval [0, ). Most of the empirical literature on long memory has concentrated on the case where the singularity or pole in the spectrum takes place at the 0-frequency. This is the case of the standard I(d) models examined in this paper, which adopt the form: L) d x t ut , t (1) (1 0 , 1, ... , 1 where L is the lag-operator (Lxt = xt-1) and ut is I(0). Note that, for any real d, the polynomial in the left-hand-side of the above expression can be expanded as: d d (d 1) 2 (1 L) d ( 1) j L j 1 dL L ... . 2 j 0 j Thus, if d is an integer value, xt in (1) will be a function of a finite number of past observations, while if d is real, xt depends strongly upon values of the time series far away in the past, and higher the d is, higher is the level of association between the observations. There exist several sources that might produce I(d) processes: Aggregation is the usual argument: Robinson (1978) and Granger (1980) showed that fractionally integrated data could arise as a result of aggregation when data are aggregated across heterogeneous autoregressive (AR) processes, 1 For the purpose of the present work, we define I(0) as a covariance stationary process with spectral density that is positive and finite at the zero frequency. 26 Journal of Money, Investment and Banking - Issue 1(2008) involving heterogeneous dynamic relationships at the individual level, which are then aggregated to form the time series. Cioczek-Georges and Mandelbrot (1995), Taqqu et al. (1997), and Chambers (1998) also use aggregation to motivate long memory processes, while Parke (1999) uses a closely related discrete time error duration model. More recently, Diebold and Inoue (2001) propose another source of long memory based on regime-switching models.2 There exist many procedures for estimating and testing the fractional differencing parameter d. In this paper we focus on parametric approaches, and model the d-differenced processes as white noise but also in terms of weakly autocorrelated processes. We implement a testing procedure suggested by Robinson (1994). His method does not require preliminary differencing and thus, it allows us to test any real value d encompassing thus stationary and nonstationary hypotheses. We use the following formulation: yt xt , (2) ' zt with xt given by (1), testing the null hypothesis: Ho : d do , (3) for any real value do. Here, yt is the observed time series, and zt is a (kx1) vector of deterministic regressors that may include, for example, an intercept, or an intercept with a linear trend. Thus, it includes tests for I(0) stationarity (do = 0); for unit roots (do = 1) as well as other fractionally integrated possibilities. The test is based on the Lagrange Multiplier (LM) principle using the Whittle function which is an approximation to the likelihood function. Thus, the value of d producing the lowest statistic should be an approximation to the maximum likelihood estimate. The functional form of the test statistic can be found in any of the numerous empirical applications of the tests. (See, Gil-Alana and Robinson, 1997; Gil-Alana, 2000; etc.). 3. Empirical results In this section, we report the empirical results of applying the methodology described in Section 2 to the data. First, we define the bull and bear phases in the US case using long time series data. In particular, we provide the peaks and troughs of the US stock market using the Bry and Boschan`s algorithm. Finally, we compare the volatility behavior across different bull and bear phases. We use data for the Standard and Poors 500 (S&P), daily, from August 1st, 1928 to December th 29 , 2006. All data are obtained from Global Financial Data web page (http://www.globalfindata.com/) and they are all in natural logs. Bull and bear phases Many papers have analyzed the behavior of stock prices across bull and bear markets. In this paper we follow the algorithm suggested by Bry and Boschan (1971) already used in stock markets by Edwards et al. (2003), Kaminsky and Schmuckler (2003), Pagan and Sossounov (2003), Gomez Biscarri and Perez de Gracia (2004), Gonzalez et al. (2005) and Candelon et al. (2007) among others.3 An alternative approach followed by Turner et al. (1989), Maheu and McCurdy (2000), Ang and Bekaert (2002), Guidolin and Timmermann (2005) and Tu (2006) includes two regimes in stock returns (i.e., bull and bear markets) using switching models. However the regime switching models do not date the phase of the cycles. The turning point algorithm developed by Bry and Boschan (1971) identifies troughs and peaks in stock market indices and thus indicates the starting and finishing points of the bull and bear markets. The turning points identified using the Bry and Boschan’s algorithm are presented in Table 1. The dates of the bull and bear phases for the period 1928 to 1946 are already obtained by Gonzalez et al. (2005). For the period 1946 to 2000, the dating phases of the bull and bear markets are 2 3 Complete surveys of fractional integration can be found in Beran (1994), Baillie (1996) and Robinson (2003). This algorithm was first applied to the location of business cycles. Some examples of this dating procedure are Watson (1994) and Harding and Pagan (2000, 2002). Journal of Money, Investment and Banking - Issue 1 (2008) 27 taken from Pagan and Sossounov (2003). The dating of the last two cycles (i.e., bull and bear periods) is obtained from our own calculations using the Bry and Boschan’s algorithm. Table 1: Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5 Cycle 6 Cycle 7 Cycle 8 Cycle 9 Cycle 10 Cycle 11 Cycle 12 Cycle 13 Cycle 14 Cycle 15 Cycle 16 Cycle 17 Cycle 18 Cycle 19 Cycle 20 Cycle 21 Cycle 22 Cycle 23 Cycle 24 Cycle 25 Bull and bear periods for each cycle Bull period ----1932m7 – 1932m8 1933m3 – 1934m1 1935m4 – 1937m2 1938m4 – 1938m12 1940m6 – 1941m7 1942m5 – 1943m6 1943m12 – 1946m5 1948m3 – 1948m6 1949m7 – 1952m12 1953m9 – 1956m7 1958m1 – 1959m7 1960m11 – 1961m12 1962m7 – 1966m1 1966m10 – 1968m11 1970m7 – 1971m4 1971m12 – 1972m12 1974m10 – 1976m12 1978m3 – 1980m11 1982m8 – 1983m6 1984m6 – 1987m8 1987m12 – 1990m5 1990m11 – 1994m1 1994m7 – 2000m9 2003m3 – 2006m12 Bear period 1929m8 - 1932m6 1932m9 – 1933m2 1934m2 – 1935m3 1937m3 – 1938m3 1939m1 -1940m5 1941m8 – 1942m4 1943m7 – 1943m11 1946m6 – 1948m2 1948m7 – 1949m6 1953m1 – 1953m8 1956m8 – 1957m12 1959m8 – 1960m10 1962m1 – 1962m6 1966m2 – 1966m9 1968m12 – 1970m6 1971m5 – 1971m11 1973m1 – 1974m9 1977m1 – 1978m2 1980m12 – 1982m7 1983m7 – 1984m5 1987m9 - 1987m11 1990m6 – 1990m10 1994m2 – 1994m6 2000m10 – 2003m2 ----- Stock market volatility In order to measure stock market volatility we use squared returns which have been already used in the financial literature (see, Lobato and Savin, 1998; Gil-Alana, 2003; Cavalcante and Assaf, 2004; Cotter, 2005 and Elder and Jin, 2007 among others). An alternative measure is the absolute return employed by Ding et al. (1993), Granger and Ding (1996), Bollerslev and Wright (2000), Gil-Alana (2005), Cavalcante and Assaf (2004), Sibbertsen (2004), Cotter (2005) and Elder and Jin (2007). Using this alternative measure of volatility the results were completely in line with those reported in the present paper.4 4 The results for the absolute returns are avalaible upon request. 28 Table 2: Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5 Cycle 6 Cycle 7 Cycle 8 Cycle 9 Cycle 10 Cycle 11 Cycle 12 Cycle 13 Cycle 14 Cycle 15 Cycle 16 Cycle 17 Cycle 18 Cycle 19 Cycle 20 Cycle 21 Cycle 22 Cycle 23 Cycle 24 Cycle 25 Journal of Money, Investment and Banking - Issue 1(2008) Estimates of d for the volatility processes using white noise ut Bull period ----------0.01 (-0.10, 0.15) 0.10 (0.02, 0.20) 0.13 (0.06, 0.22) 0.13 (0.06, 0.22) 0.32 (0.24, 0.43) -0.03 (-0.08, 0.05) 0.16 (0.13, 0.20) 0.12 (-0.05, 0.33) 0.11 (0.09, 0.14) 0.09 (0.05, 0.14) 0.08 (0.02, 0.16) 0.07 (-0.02, 0.20) 0.33 (0.28, 0.38) 0.13 (0.09, 0.18) 0.22 (0.14, 0.31) 0.05 (-0.04, 0.16) 0.17 (0.14, 0.20) 0.10 (0.06, 0.14) 0.07 (0.01, 0.14) 0.06 (0.03, 0.10) 0.07 (0.04, 0.12) 0.08 (0.05, 0.11) 0.17 (0.15, 0.20) 0.12 (0.10, 0.14) Bear period 0.29 (0.26, 0.33) 0.14 (0.07, 0.23) 0.05 (0.00, 0.12) 0.10 (0.02, 0.20) 0.14 (0.09, 0.20) 0.11 (0.04, 0.21) 0.25 (0.16, 0.39) 0.14 (0.11, 0.18) 0.12 (0.07, 0.19) 0.00 (-0.08, 0.12) 0.12 (0.07, 0.17) 0.08 (0.03, 0.16) 0.28 (0.17, 0.44) 0.21 (0.14, 0.31) 0.24 (0.20, 0.29) 0.09 (0.01, 0.20) 0.11 (0.08, 0.15) 0.03 (-0.03, 0.11) 0.00 (-0.05, 0.07) -0.02 (-0.10, 0.08) 0.07 (-0.06, 0.27) 0.10 (0.02, 0.23) 0.10 (-0.03, 0.26) 0.14 (0.11, 0.18) ---------- Notes: In bold, cases with non-overlapping intervals. Table 2 displays the estimated values of d using the Whittle function along with the 95% confidence intervals of the non-rejection values of d using Robinson's (1994) parametric approach, for squared returns of the S&P 500 index. Here we use the different sub-samples for the bull and bear markets, and assume first that the disturbances are white noise. Though we compute the results for the three standard cases of no regressors, an intercept and an intercept with a linear trend, we only report here those cases based on an intercept, since the intercept-coefficient was found to be statistically significant in all cases, unlike the trend coefficient that was insignificant in the majority of cases. In most of the cases d is higher than 0 and the confidence bands are constrained in the interval (0, 0.5) implying stationary long memory volatility processes. There are a few exceptions: for example, d takes the value of –0.01 in bull phase number 2 and –0.03 in bull cycle number 7, while it takes the value of –0.02 in bear cycle number 20. The intervals exclude the case of d = 0 in many cases. The exceptions are cycles 2, 7, 9, 13 and 17 (in the bull market) and cycles 10, 18, 19, 20 and 23 (in the bear market). In these cases the null hypothesis of stationarity I(0) cannot be rejected at the 5% level. We do not find systematically higher orders of integration in one period over the other. In fact, in 11 out of the 23 three cases d-bull is higher than d-bear; in two cases the orders are the same, and in another 10 cases dbull is smaller than d- bear. Finally, note that non-overlapping intervals are obtained in the following cases: we find statistical evidence of d-bull smaller than d-bear in cycle number 7, where d-bull is – 0.03 while d-bear is 0.25, and also in cycle number 15 where d-bull is 0.13 while d-bear is 0.24. On the other hand, we also find that in two cases d-bull is higher than d-bear: in cycle number 6 d-bull is 0.32 and d-bear is 0.11, and in cycle number 18 d-bull is 0.17 and d-bear is 0.03. Thus, using this simple specification we do not obtain evidence of significantly higher orders of integration in the volatility of bull (bear) markets. Journal of Money, Investment and Banking - Issue 1 (2008) Table 3: Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5 Cycle 6 Cycle 7 Cycle 8 Cycle 9 Cycle 10 Cycle 11 Cycle 12 Cycle 13 Cycle 14 Cycle 15 Cycle 16 Cycle 17 Cycle 18 Cycle 19 Cycle 20 Cycle 21 Cycle 22 Cycle 23 Cycle 24 Cycle 25 29 Estimates of d for the volatility processes using Bloomfield ut Bull period ---------0.20 (0.00, 0.61) 0.05 (-0.15, 0.25) 0.17 (0.08, 0.26) 0.18 (0.03, 0.35) 0.09 (-0.08, 0.23) -0.02 (-0.10, 0.11) 0.26 (0.18, 0.39) -0.27 (-0.42, 0.14) 0.23 (0.17, 0.28) 0.07 (0.02, 0.15) 0.07 (-0.03, 0.24) -0.14 (-0.20, 0.00) 0.18 (0.12, 0.25) 0.25 (0.14, 0.36) 0.15 (0.03, 0.30) -0.01 (-0.12, 0.16) 0.32 (0.25, 0.41) 0.09 (0.05, 0.16) 0.22 (0.11, 0.35) 0.07 (0.03, 0.12) 0.11 (0.05, 0.18) 0.19 (0.13, 0.26) 0.16 (0.13, 0.20) 0.30 (0.26, 0.36) Bear period 0.38 (0.31, 0.45) 0.32 (0.19, 0.49) 0.11 (0.02, 0.24) 0.05 (-0.15, 0.25) 0.16 (0.06, 0.26) 0.04 (-0.04, 0.20) 0.36 (0.14, 0.59) 0.35 (0.27, 0.45) 0.44 (0.25, 0.71) 0.05 (-0.08, 0.32) 0.22 (0.14, 0.34) 0.16 (0.04, 0.29) 0.10 (-0.01, 0.29) 0.29 (0.14, 0.54) 0.36 (0.27, 0.51) 0.23 (0.01, 0.54) 0.23 (0.17, 0.31) -0.01 (-0.07, 0.09) 0.00 (-0.07, 0.11) 0.01 (-0.12, 0.17) 0.07 (-0.17, 0.51) 0.13 (0.00, 0.40) 0.05 (-0.21, 0.41) 0.34 (0.28, 0.44) ---------- Notes: In bold, cases with non-overlapping intervals. We also estimated d assuming that the disturbances follow the exponential spectral model of Bloomfield (1973). This is a non-parametric approach that produces autocorrelations decaying exponentially as in the AR case. In the context of fractional integration, Gil-Alana (2004) showed that Bloomfield’s model approximates fairly well ARMA processes. The results based on this approach are given in Table 3. In general, the estimated values of d are above 0 suggesting once more that the US stock market volatility presents long memory in bull and bear phases. However there are a few exceptions where the value of d is negative: bull cycles number 7, 9, 13 and 17 and bear cycle number 18. Further, in seven cases in the bull market (cycles 2, 3, 6, 7, 9, 12 and 13) and in nine cases in the bear market (cycles 4, 6, 10, 18, 19, 20, 21, 22 and 23) the null hypothesis of d = 0 cannot be rejected. Moreover, non-overlapping intervals are obtained in the following cases. In cycles phase number 7, 9, 17 and 24 we find evidence of d-bull smaller than d-bear, while in cycle number 18 we find that the value of d in the bull market is significantly higher than the estimated d in the bear market. Finally, in a single cycle the estimate value of d in bull and bear markets is the same: cycle number 21. Though the results presented in this paper do not produce conclusive evidence, we found that generally d is greater after the burst of the bubble for the squared returns, especially if the disturbances are autocorrelated (in 14 out of the 23 cycles; see Table 3). This result has been already documented in the financial literature (see, for example, Maheu and McCurdy, 2000; Edwards et al., 2003; Jones et al., 2004; Guidolin and Timmerman, 2005, Tu, 2006 and Cunado et al., 2007 among many others). Jones et al. (2004) using stock market returns from 1885 to 2002 find that there is some persistence in volatility, and that US volatility is higher in bear markets than in bull markets. The higher uncertainty and risk during bear markets together with a decline in equity price and stock market liquidity could be related to the higher persistence volatility observed in the bear markets. 30 Journal of Money, Investment and Banking - Issue 1(2008) 4. Concluding comments In this paper we have examined the US stock market volatility, daily, from 1928 to 2006. Using long memory techniques we have investigated whether the stock market volatility presents a different behavior in bull and bear phases. In order to define bull and bear phases, we employed the standard procedure of Bry and Boschan (1971), already used by many practitioners. The empirical evidence suggests that persistence in stock market volatility takes place in both the bull and bear US markets. In most of the cases, the estimated values of d are above 0 and below 0.5 implying long memory stationarity and mean-reverting behaviour. However, given the large confidence intervals for the fractional differencing parameter, it is difficult to draw clear conclusions about the systematic higher order of integration in the volatility of the bear markets. A follow-up step in this article could be to examine the robustness of our results in other US aggregate stock market composite indices. Another possible extension could be to analyze the mean reversion in stock prices across bull and bear phases. Finally, the possibility of non-linear models in the context of fractional integration is another issue that will be examined in future papers. Acknowledgement Juncal Cunado and Luis A. Gil-Alana gratefully acknowledge financial support from the Spanish Ministry of Science and Technology (SEJ2005-07657/ECON). 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