3.3 Trading Signals and Market Volatility Ratio
3.3.3 Empirical Results
leptokurtic. For these three variables, we also conduct unit root tests to investigate whether these series are stationary. The results of the augmented Dickey-Fuller (ADF) test and the Phillips-Perron (PP) test show that the hypothesis of unit root process is rejected for each series in the full and all sub-sample periods.11
3.3.3 Empirical Results
Table 12 reports the estimation results for two types of the Markov-Switching model, the fixed-transition-probability (FTP) MS model and the TVTP MS model. First of all, it is obvious that the FTP MS model with AR lag 1 in dt yields a quite higher value of the likelihood function than that with no AR lags; therefore, AR lag 1 in dtis chosen in Eq. (19).
The FTP MS-AR(1) model, where the process is allowed to switch between regimes, iden-tifies two regimes. The Regime 0, with a positive mean (bµ0 > 0), represents the situation that the VMA line is below the rising price line. While the Regime 1 stands for the opposite case that the VMA line is above the falling price due to its negative mean (bµ1 < 0). The estimated AR1 coefficients, bβ0,1 and bβ1,1, indicate that dt in both regimes is quite persistent. The higher variance in dtin Regime 1 displays that the VMA line can not trace the price line more closely in the case that the price descends.
We label the positive-mean stable and negative-mean volatile states in dt as the rising mar-kets with the upward direction of price movements, and falling marmar-kets with the downward price direction, respectively. As dtswitches from a rising-market state to a falling-market state, a selling signal will be suggested by the VMA rule. Conversely, as the regime shifts from a falling-market to a rising-market, the VMA rule will generate a buying signals. Finally, the transition probabilities in the FTP MS-AR(1) model exhibits that both rising-market and falling-market states are highly persistent. The rising-falling-market regime (Regime 0) persists on average for 1/(1 −pb00) = 111 days while it is expected that the falling-market regime (Regime 1) will
11We conduct two types of test equations in the ADF and PP test. One includes the intercept term in the test equations. The other contains the intercept and trend terms in the models. The results in these two types of test models are the same. Thus, we just report results of the unit root tests in which the intercept term is included in the models.
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Table11:DescriptiveStatisticsAndUnitRootTests Thistablereportssummarystatisticsandtheunitroottestsresultsfordt,VRt−1anddVRtinthefullandsub-sampleperiods.dtisdefinedasptminusvmat,whereptandvmatarethelogarithmof theDJIAclosingpriceandtheVMAvalueattimet.VRt−1isthemarketvolatilityratioattimet-1anddVRtisdescribedasVRtminusVRt−1.Thevaluesofvmat,VRt−1anddVRtcomefrom applyingthebestVMAruleintheDJIAdailyindex.ThefullsampleperiodisbetweenNov14,1928andJune28,2010,with20,496observations.Ourthreesub-samplesarebuiltinordertodistinguish the1930sgreatdepressionperiod,pre-October1987crashperiods,andpost-October1987crashperiods.Theyare1928/11/14−1938/12/30,1939/01/01−1987/10/18and1987/10/19−2010/06/28with 2,528,12,246and5,722observations,respectively.Forunitroottests,ADFandPPareaugmentedDickey-FullerandPhillips-Perronteststatistics.Inbothtests,thetestequationincludestheintercept termandthenullhypothesisisthattheserieshasaunitroot.TestcriticalvaluesforADFandPPare-3.443834(1%),-2.867379(5%),and-2.569943(10%).LagsinADFtestsarechosenbySchwartz Bayesianinformationcriterion(SC). dtVRt−1dVRt FullSub.1Sub.2Sub.3FullSub.1Sub.2Sub.3FullSub.1Sub.2Sub.3 DescriptiveStatistics Mean0.005-0.0100.0070.0070.7390.7290.7350.7520.000010.000100.00004-0.00008 Max0.2750.2750.1270.0981.3831.3521.3831.3800.4500.3790.4500.417 Min-0.407-0.407-0.207-0.3470.0810.1280.0810.156-0.559-0.325-0.559-0.368 StandardDeviation0.0460.0900.0340.0380.2690.2550.2740.2630.0740.0720.0750.074 Skewness-1.434-0.657-0.676-1.7570.1370.1920.1350.1250.1830.1850.2060.132 Kurtosis11.5544.6264.8879.8182.1642.2192.1132.2494.7724.6384.7664.830 Unitroottests ADF-17.015-5.793-13.743-10.728-36.072-12.679-28.146-18.564-36.607-17.632-33.374-23.200 PP-17.550-5.539-13.301-12.954-8.629-12.278-8.640-10.666-97.568-18.397-47.320-29.272
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persist for 1/(1 −pb11) = 26 days. In other words, the long positions suggested by the VMA rule will keep for 111.11 days on average. But the short positions will keep for a shorter period, 26.32 days.
The last column in Table 12 presents the results for the TVTP MS model. The elements in zt, dVRt and dVRt−1, are chosen according to the suggestions by both Akaike’s information criterion (AIC) and Schwarz’s criterion (SC).12 Using dVRtand dVRt−1 enables us to see how the dynamic of the VRtaffecting the generation of the VMA’s trading signals. The generation of crossovers between the price line and the VMA line is mainly due to two conditions, based on an important premise that the direction of the price line and the VMA line before a trading signals generated should be opposite. The first condition is whether the value of the VRtis large enough to make the VMA line approaching the price line. The second one is the difference between the Pt−1 and VMAt−1. If the VMA line is near to the price line at time t-1 due to the big VRt−1, a trading signal might be generated at time t even the value of VRt is not very large. In this study, dVRt and dVRt−1 can be used to measure the conditions of the crossovers. If the dVRt
is increasing, we expect that it might result from a higher value of VRt. While the increase in the dVRt−1 might be explained as due to a increase in VRt−1. Higher VRt−1 will make the difference between the price line and the VMA line become narrower.
In the TVTP MS model, the estimation results in dt equation (i.e., Eq. (19)) is similar to those in the FTP MS-AR(1) model. We still have a clear identification of the rising-market and the market states. Since the Regime 0 and 1 represent the rising-market and the falling-market states respectively, the estimates of θi and γi for i = 1, 2 measure how the information of market volatility affects the generation of a selling signal and a buying signal for the VMA rule, respectively. At the 5% significant level, it is found that bθ1 > 0, bθ2 < 0, bγ2 > 0 butbγ1 is not significantly different from zero. Here,bγ2 > 0 means that the increase of dVRt−1raises the probability of generating a buying signal at time t. The insignificance ofbγ1 denotes that no matter how the price fluctuates from time t-1 to time t, the generation of a buying signal will not
12We have tried different kinds of ztsuch as dVRt, VRt−1, dVRt−1, (dVRt, dVRt−1) and (dVRt, VRt−1) and found that (dVRt, dVRt−1) was the best one.
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be affected. dVRt and dVRt−1 have different effect on the probability of generating a selling signal. The increase of dVRt or the decrease of dVRt−1 makes the probability of generating a selling signal higher, because we get bθ1 > 0 and bθ2 < 0.
We can further investigate the economic significance of the effect of the change of the market volatility ratio on the transition probability from one regime to another. They are calculated by
∂p01t /∂zit = bθi pb01t [1 −pb01t ] and ∂p10t /∂zit = γbi pb10t [1 −pb10t ], where pb01t = p01t ( bθ0, bθ1, bθ2, zit), pb10t = p10t (γb0,γb1,γb2, zit) and zit = (dVRt, dVRt−1).13 dVRtand dVRt−1are the mean of dVRt
and dVRt−1, respectively. Every 0.1 increase in dVRtincrease the probability of generating a selling signal with ∆p01t = 0.728 but it will not affect the probability of generating a buying signal. However, an 0.1 increase in dVRt−1 reduces the probability of a selling signal with
∆p01t = −0.349 but it enlarges the probability of a buying signal with ∆p10t = 0.268.
The reason why we get bγ1 = 0 is probably because that some cases in the sample show that dVRt has positive effect on the probability of a long position signaled at time t, whereas others display that it has negative effect. The two types of cases make bγ1 insignificant. In the cases of negative effect, the increases in dVRt−1 dominates that in dVRt, since VRt from t-2 to time t-1 increases a lot to force the value of the VMAt−1 to near to the value of the Pt−1. Therefore, the possibility that the best VMA rule generates a buying signal at time t is higher no matter how big the variation of the market price was from time t-1 to time t, even the value of VRtis less than that of VRt−1. According to the results in the economic significance of the effect of dVRtand dVRt−1, the probability of a buying signal will be raised by 0.537 and 0.805 (∆p10t = 0.537, 0.805) as dVRt−1 increases by 0.2 and 0.3 unit, respectively. Once dVRt−1
increases by 0.3 unit, it means that there is a quite big variation in price from time t-2 to time t-1 (i.e., the value of VRt−1is quite larger than that of VRt−2). In the cases of positive effect, the crossovers are determined by both the increase of dVRt and dVRt−1. Here, the strength of the increase in dVRt−1is less, compared to that in the cases of negative effect.
Moreover, if we set the significant level as 10%, bγ1 < 0 and every 0.1 increase in dVRt
13The details ofpb01t andpb10t are as follows:
pb01t = exp{bθ0+ bθ1dVRt+ bθ2dVRt−1}/1 + exp{bθ0+ bθ1dVRt+ bθ2dVRt−1}, pb10t = exp{γb0+bγ1dVRt+bγ2dVRt−1}/1 + exp{bγ0+bγ1dVRt+γb2dVRt−1}.
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will make the probability to signal buying the asset at time t lower by 0.032 (∆p10t = −0.032).
However, this effect is relatively little. Finally, bγ1 < 0 andbγ2 > 0 illustrate that the cases of negative effect has higher frequency in the sample. On average, the main factor causing the VMA rule to signal a long position at time t is that the market price varies a lot from time t-2 to time t-1, since there might be an important information in the market at time t-1.
For the selling signals, bθ1 > 0, bθ2 < 0 and their results in economic significance show that the main factor affecting the generation of the selling signals is dVRt. The strong effect of a 0.1 increase in dVRt on the possibility of a selling signal generated at time t and bθ2 < 0 implies that the value of VRt−1 and VRt−2 are not small. If the value of VRt−2 is large and dVRt−1 increases, it makes senses that the probability of a selling signal at time t will be reduced (i.e., θb2 < 0), since in this case time t-1 will be considered to be a better time to have a short position. Furthermore, bθ1 > 0 with stronger effect and bθ2 < 0 can be used to measure how the market price behaves before a selling signal, high VRt−2 and the decrease of dVRt−1 (i.e., the difference between VRt−2 and VRt−1 is less). Before time t-2, the recent market price is more volatile than the reference period due to a reflection of recent information (i.e., high VRt−2), and the variation in market price becomes moderate from time t-2 to time t-1 causing the decrease of dVRt−1. However, a negative shock surged at time t-1 in the market and the downward price jump will take place making the increase of dVRt. As a result, the probability of a selling signal will be increased in order to take advantage of the downward price movement.
In summary, we use dVRt−1and dVRtto detect whether there is any influential information or new shock in the market, based on the premise that the variation in market prices is due to the reaction to the information in the market. Once the information emerged in the market is more influential, the VMA rule will response to it to make the change in dVRt−1or dVRt. Therefore, it will raise the probability of taking better advantage of the price movement.