Have Individual Stocks Become More Volatile? 11 Panel A.Market volatility 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.00 WA 8 Panel B.Market volatility,MA(12) 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 器器只品高品留品高 Figure 2.Annualized market volatility MKT.The top panel shows the annualized variance within each month of daily market returns,calculated using equation(17),for the period July 1962 to December 1997.The bottom panel shows a backwards 12-month moving average of MKT.NBER-dated recessions are shaded in gray to illustrate cyclical movements in volatility. fair amount of high-frequency noise.Market volatility was particularly high around 1970,in the mid-1970s,around 1980,and at the very end of the sample.The stock market crash in October 1987 caused an enormous spike in market volatility which is cut off in the plot.The value of MKT in October 1987 is 0.672,about six times as high as the second highest value.The plot also shows NBER-dated recessions shaded in gray.A casual look at the plot suggests that market volatility increases in recessions.We will study the cyclical behavior of MKT and the other volatility measures below. Next,consider the behavior of industry volatility IND in Figure 3.Com- pared with market volatility,industry volatility is slightly lower on average. As for MKT,there is a slow-moving component and some high-frequency
fair amount of high-frequency noise. Market volatility was particularly high around 1970, in the mid-1970s, around 1980, and at the very end of the sample. The stock market crash in October 1987 caused an enormous spike in market volatility which is cut off in the plot. The value of MKT in October 1987 is 0.672, about six times as high as the second highest value. The plot also shows NBER-dated recessions shaded in gray. A casual look at the plot suggests that market volatility increases in recessions. We will study the cyclical behavior of MKT and the other volatility measures below. Next, consider the behavior of industry volatility IND in Figure 3. Compared with market volatility, industry volatility is slightly lower on average. As for MKT, there is a slow-moving component and some high-frequency Figure 2. Annualized market volatility MKT. The top panel shows the annualized variance within each month of daily market returns, calculated using equation ~17!, for the period July 1962 to December 1997. The bottom panel shows a backwards 12-month moving average of MKT. NBER-dated recessions are shaded in gray to illustrate cyclical movements in volatility. Have Individual Stocks Become More Volatile? 11
12 The Journal of Finance Panel A.Industry volatility 0.007 0.008 0.005 0.004 0.003 0.002 0.001 0 Panel B.Industry volatility,MA(12) 0.0025 0.002 0.0015 0.001 0.0005 0 铝器只科寸炽R品器高品88别高8 Figure 3.Annualized industry-level volatility IND.The top panel shows the annualized variance within each month of daily industry returns relative to the market,calculated using equations (18)and(19),for the period from July 1962 to December 1997.The bottom panel shows a backwards 12-month moving average of IND.NBER-dated recessions are shaded in gray to illustrate cyclical movements in volatility. noise.IND was particularly high in the mid-1970s and around 1980.The effect of the crash in October 1987 is quite significant for IND,although not as much as for MKT.More generally,industry volatility seems to increase during macroeconomic downturns. Figure 4 plots firm-level volatility FIRM.The first striking feature is that FIRM is on average much higher than MKT and IND.This implies that firm-specific volatility is the largest component of the total volatility of an average firm.The second important characteristic of FIRM is that it trends up over the sample.The plots of MKT and IND do not exhibit any visible upward slope whereas for FIRM it is clearly visible.This indicates that the
noise. IND was particularly high in the mid-1970s and around 1980. The effect of the crash in October 1987 is quite significant for IND, although not as much as for MKT. More generally, industry volatility seems to increase during macroeconomic downturns. Figure 4 plots firm-level volatility FIRM. The first striking feature is that FIRM is on average much higher than MKT and IND. This implies that firm-specific volatility is the largest component of the total volatility of an average firm. The second important characteristic of FIRM is that it trends up over the sample. The plots of MKT and IND do not exhibit any visible upward slope whereas for FIRM it is clearly visible. This indicates that the Figure 3. Annualized industry-level volatility IND. The top panel shows the annualized variance within each month of daily industry returns relative to the market, calculated using equations ~18! and ~19!, for the period from July 1962 to December 1997. The bottom panel shows a backwards 12-month moving average of IND. NBER-dated recessions are shaded in gray to illustrate cyclical movements in volatility. 12 The Journal of Finance
Have Individual Stocks Become More Volatile? 13 Panel A.Firm volatility 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 2ww 0.004 0.002 0 8 Panel B.Firm volatility,MA(12) 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 8留只g炽R品剑高品品品别高号 Figure 4.Annualized firm-level volatility FIRM.The top panel shows the annualized vari. ance within each month of daily firm returns relative to the firm's industry,calculated using equations (20)-(22),for the period from July 1962 to December 1997.The bottom panel shows a backwards 12-month moving average of FIRM.NBER-dated recessions are shaded in gray to illustrate cyclical movements in volatility. stock market has become more volatile over the sample but on a firm level instead of a market or industry level.Apart from the trend,the plot of FIRM looks similar to MKT and IND.Firm-level volatility seems to be higher in NBER-dated recessions and the crash also has a significant effect. Looking at the three volatility plots together,it is clear that the different volatility measures tend to move together,particularly at lower frequencies. For example,all three volatility measures increase during the oil price shocks in the early to mid-1970s.However,there are also some periods in which the volatility measures move differently.For example,IND is very high com- pared to its long-term mean during the early 1980s while MKT and FIRM
stock market has become more volatile over the sample but on a firm level instead of a market or industry level. Apart from the trend, the plot of FIRM looks similar to MKT and IND. Firm-level volatility seems to be higher in NBER-dated recessions and the crash also has a significant effect. Looking at the three volatility plots together, it is clear that the different volatility measures tend to move together, particularly at lower frequencies. For example, all three volatility measures increase during the oil price shocks in the early to mid-1970s. However, there are also some periods in which the volatility measures move differently. For example, IND is very high compared to its long-term mean during the early 1980s while MKT and FIRM Figure 4. Annualized firm-level volatility FIRM. The top panel shows the annualized variance within each month of daily firm returns relative to the firm’s industry, calculated using equations ~20!–~22!, for the period from July 1962 to December 1997. The bottom panel shows a backwards 12-month moving average of FIRM. NBER-dated recessions are shaded in gray to illustrate cyclical movements in volatility. Have Individual Stocks Become More Volatile? 13
4 The Journal of Finance Table I Autocorrelation Structure Raw Data Downweighted Crash Autocorrelation MKT IND FIRM MKT IND FIRM P1 0.149 0.529 0.591 0.494 0.591 0.776 P2 0.115 0.419 0.560 0.383 0.463 0.727 P3 0.113 0.393 0.514 0.313 0.438 0.686 Pa 0.020 0.364 0.418 0.160 0.415 0.584 Pe 0.069 0.339 0.414 0.183 0.384 0.572 P12 0.004 0.275 0.340 0.087 0.316 0.471 Note:This table reports the autocorrelation structure of monthly volatility measures con- structed from daily data.MKT is market volatility constructed from equation (17),IND is industry-level volatility constructed from equations(18)and(19),and FIRM is firm-level vol- atility constructed from equations(20)-(22).All measures are value-weighted variances.The columns denoted"downweighted crash"replace the observation in October 1987 with the second- largest observation in the respective series.P:denotes the ith monthly autocorrelation. remain fairly low during this period.Another interesting episode is the last year of our sample.Market volatility increased significantly in 1997 while IND and FIRM did not. It is evident from the plots that the stock market crash in October 1987 had a significant effect on all three volatility series.This raises the issue whether this one-time event might overshadow the rest of the sample and distort some of the results.To avoid this we report many results for both the raw data set and a modified version where we replace the October 1987 observation with the second largest observation in the data set.This admit- tedly ad hoc procedure decreases the influence of the crash but leaves it as an important event in the sample. B.Stochastic versus Deterministic Trends Figures 2 to 4 suggest the strong possibility of an upward trend in idiosyn- cratic firm-level volatility.A first important question is whether such a trend is stochastic or deterministic in nature.The possibility of a stochastic trend is suggested by the persistent fluctuations in volatility shown in the figures. Table I reports autocorrelation coefficients for the three volatility mea- sures using both the raw data and the data set that downweights the crash. Because the crash had an enormous but short-lived effect on market vola- tility,the autocorrelation of MKT is considerably larger when the crash is downweighted.The effect of the crash is much smaller for IND and FIRM. All these series exhibit fairly high serial correlation,which raises the pos- sibility that they contain unit roots. To check this,in Table II we employ augmented Dickey and Fuller (1979) p-tests and t-tests,based on regressions of time series on their lagged values and lagged difference terms that account for serial correlation.The number
remain fairly low during this period. Another interesting episode is the last year of our sample. Market volatility increased significantly in 1997 while IND and FIRM did not. It is evident from the plots that the stock market crash in October 1987 had a significant effect on all three volatility series. This raises the issue whether this one-time event might overshadow the rest of the sample and distort some of the results. To avoid this we report many results for both the raw data set and a modified version where we replace the October 1987 observation with the second largest observation in the data set. This admittedly ad hoc procedure decreases the influence of the crash but leaves it as an important event in the sample. B. Stochastic versus Deterministic Trends Figures 2 to 4 suggest the strong possibility of an upward trend in idiosyncratic firm-level volatility. A first important question is whether such a trend is stochastic or deterministic in nature. The possibility of a stochastic trend is suggested by the persistent fluctuations in volatility shown in the figures. Table I reports autocorrelation coefficients for the three volatility measures using both the raw data and the data set that downweights the crash. Because the crash had an enormous but short-lived effect on market volatility, the autocorrelation of MKT is considerably larger when the crash is downweighted. The effect of the crash is much smaller for IND and FIRM. All these series exhibit fairly high serial correlation, which raises the possibility that they contain unit roots. To check this, in Table II we employ augmented Dickey and Fuller ~1979! r-tests and t-tests, based on regressions of time series on their lagged values and lagged difference terms that account for serial correlation. The number Table I Autocorrelation Structure Raw Data Downweighted Crash Autocorrelation MKT IND FIRM MKT IND FIRM r1 0.149 0.529 0.591 0.494 0.591 0.776 r2 0.115 0.419 0.560 0.383 0.463 0.727 r3 0.113 0.393 0.514 0.313 0.438 0.686 r4 0.020 0.364 0.418 0.160 0.415 0.584 r6 0.069 0.339 0.414 0.183 0.384 0.572 r12 0.004 0.275 0.340 0.087 0.316 0.471 Note: This table reports the autocorrelation structure of monthly volatility measures constructed from daily data. MKT is market volatility constructed from equation ~17!, IND is industry-level volatility constructed from equations ~18! and ~19!, and FIRM is firm-level volatility constructed from equations ~20!–~22!. All measures are value-weighted variances. The columns denoted “downweighted crash” replace the observation in October 1987 with the secondlargest observation in the respective series. ri denotes the ith monthly autocorrelation. 14 The Journal of Finance
Have Individual Stocks Become More Volatile? 15 TableⅡ Unit Root Tests Raw Data Downweighted Crash MKT IND FIRM MKT IND FIRM Constant p-test -328 -103 -80.3 -175 -88.5 -46.5 t-test -12.17 -4.59 -3.98 -8.55 -4.28 -3.29 Lag order 2 5 5 1 4 5 Constant trend p-test -330 -125 -145 -177 -91.7 -79.1 t-test -12.24 -5.60 -6.35 -8.60 -4.36 -4.34 Lag order 1 2 2 1 4 5 Note:This table reports unit-root tests for monthly volatility series constructed from daily data.MKT is market volatility constructed from equation(17),IND is industry-level volatility constructed from equations(18)and(19),and FIRM is firm-level volatility constructed from equations (20)-(22).All measures are value-weighted variances.The columns denoted "down- weighted crash"replace the observation in October 1987 with the second-largest observation in the respective series.The unit-root tests are based on regressions that include a constant,or a constant and time trend.The 5 percent critical values for the Dickey-Fuller p-test are -8.00 when a constant is included in the regression and-21.5 when a constant and a linear trend are included.The 5 percent critical values for the t-test are-2.87 with a constant and-3.42 with a constant and a trend.The number of lags is determined by the "general to specific"method recommended in Campbell and Perron (1991). of lagged differences to be included can be determined by the standard t-test of significance on the last lagged difference term,and is also reported in Table II.The hypothesis of a unit root is rejected for all three volatility series at the 5 percent level,whether a deterministic time trend is allowed or not,and regardless of the treatment of the 1987 crash. Given these results,we proceed to analyze the volatility series in levels rather than first differences.We report some descriptive statistics and trend regressions in Table III.The top panel presents results for annualized vol- atility series based on daily returns and the two following panels report results for annualized volatility series based on weekly and monthly re- turns,respectively.Consider first the absolute magnitudes of the volatility components in our benchmark sample based on daily returns.The annual- ized mean of MKT is about 0.015,which implies an annual standard devi- ation of 12.3 percent.IND has a slightly lower mean of 0.010,implying an annual standard deviation of about 10 percent,whereas FIRM is on average substantially larger than both MKT and IND,with a mean of 0.064 implying an annual standard deviation of 25 percent.These numbers imply that over the whole sample the share of the total unconditional variance that is due to the market variance,or the R2 of a market model,is only about 17 percent. Thus industry and particularly firm-level uncertainty are important com-
of lagged differences to be included can be determined by the standard t-test of significance on the last lagged difference term, and is also reported in Table II. The hypothesis of a unit root is rejected for all three volatility series at the 5 percent level, whether a deterministic time trend is allowed or not, and regardless of the treatment of the 1987 crash. Given these results, we proceed to analyze the volatility series in levels rather than first differences. We report some descriptive statistics and trend regressions in Table III. The top panel presents results for annualized volatility series based on daily returns and the two following panels report results for annualized volatility series based on weekly and monthly returns, respectively. Consider first the absolute magnitudes of the volatility components in our benchmark sample based on daily returns. The annualized mean of MKT is about 0.015, which implies an annual standard deviation of 12.3 percent. IND has a slightly lower mean of 0.010, implying an annual standard deviation of about 10 percent, whereas FIRM is on average substantially larger than both MKT and IND, with a mean of 0.064 implying an annual standard deviation of 25 percent. These numbers imply that over the whole sample the share of the total unconditional variance that is due to the market variance, or the R2 of a market model, is only about 17 percent. Thus industry and particularly firm-level uncertainty are important comTable II Unit Root Tests Raw Data Downweighted Crash MKT IND FIRM MKT IND FIRM Constant r-test 2328 2103 280.3 2175 288.5 246.5 t-test 212.17 24.59 23.98 28.55 24.28 23.29 Lag order 2 5 5 145 Constant & trend r-test 2330 2125 2145 2177 291.7 279.1 t-test 212.24 25.60 26.35 28.60 24.36 24.34 Lag order 1 3 2 145 Note: This table reports unit-root tests for monthly volatility series constructed from daily data. MKT is market volatility constructed from equation ~17!, IND is industry-level volatility constructed from equations ~18! and ~19!, and FIRM is firm-level volatility constructed from equations ~20!–~22!. All measures are value-weighted variances. The columns denoted “downweighted crash” replace the observation in October 1987 with the second-largest observation in the respective series. The unit-root tests are based on regressions that include a constant, or a constant and time trend. The 5 percent critical values for the Dickey-Fuller r-test are 28.00 when a constant is included in the regression and 221.5 when a constant and a linear trend are included. The 5 percent critical values for the t-test are 22.87 with a constant and 23.42 with a constant and a trend. The number of lags is determined by the “general to specific” method recommended in Campbell and Perron ~1991!. Have Individual Stocks Become More Volatile? 15