3.3 Trading Signals and Market Volatility Ratio
3.3.1 The TVTP Markov-Switching Model
we will get a selling signal. Therefore, we can define a new variable, the price value minus the VMA value at each time, and use a simple two-state mean Markov-switching model of this new variable (positive and negative mean) to describe the signal generating process of the VMA rule.
It is, foremost, the time-varying-transition-probability in the Markov-switching model allows us to explore how the market volatility ratio affects the dynamics of switching in states because the probability of switching between states can be designed to depend on the market volatility ratio.
3.3.1 The TVTP Markov-Switching Model
Let dt = pt− vmat, where pt and vmat are the logarithm of the DJIA closing price and the VMA value at time t, respectively.10 In other words, dtmeasures the gap between the logarithm of the price line and that of the VMA line. Consider a two-state Markov-switching model of dt:
dt= µSt +
k
X
j=1
βSt,jdt−j+ t, t ∼ i.i.d.N (0, σ2St). (19)
where term µSt and σS2t are respectively the state-dependent mean and variance of dt. dt−j
is allowed to have different impacts on dt across different states, βSt,j. The unobserved state variable Stis a latent dummy variable equaling either 0 or 1, which indicates that the price line is above/below the VMA line, i.e. µSt is either positive or negative. It is assumed to follow a two-state Markov process with time-varying transition probability matrix:
P (t) =
Therefore, it is obvious that the time-varying transition probability matrix in Eq. (20) displays how the two different regimes shift over time. p01t and p10t measure the probability of a switch from State 0 to State 1 and from State 1 to State 0 at time t. As dt switches from State 0
10The trading signals generated by the price value and the VMA value are the same as that generated by their logarithms.
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line crosses the VMA line from above. On the contrary, a buying signal will be expected as dt changes from State 1 to State 0, due to the price line crossing the VMA line from below.In this study, the probability of regime switching is assumed to vary with the evolution of the information of market volatility. In other words, the information of the market volatility will affect the signal generation of the VMA rule.
The functions of the transition probabilities are then specified as follows:
p01t (zt) = exp{θ0+ zt0θ} and γi, respectively. Therefore, the estimates of θi and γi show how the information of market volatility influences the probability of the generation of trading signals in the VMA rule.
In this study, we adopt VRt−1, dVRtand dVRt−1as the candidate for zt. VRt−1is the market volatility ratio at time t-1, while dVRt is defined as VRt minus VRt−1 in order to measure whether the market volatility ratio is increasing from time t-1 to time t. According to the definition of VRt, σtn/σreft with ref > n, the increase of the market volatility ratio from time t-1 to time t (i.e., dVRt> 0) can imply σtn> σt−1n based on the premise that σreft ' σt−1ref as the reference period is longer enough. Here, σtn> σt−1n denotes that the price variation in the period from time t-1 to time t is larger than that from time t-n to time t-n+1. We explain the market experiencing a larger price movement within the period from time t-1 to time t is due to there
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might be an influential or new information surged in this period. On the contrary, dVRt < 0 illustrates the situation that the price variation from time t-1 to time t is less, compared with that from time t-n to time t-n+1. Less variation in price from time t-1 to time t may denote that the information in this period is relatively less influential than that the period from time t-n to time t-n+1. Similarly dVRt−1, defined as VRt−1minus VRt−2, measures whether there exists a larger or smaller change in price from t-2 to time t-1. As a result, dVRtand dVRt−1are used to see whether there are any new information or influential shock appearing from time t-2 in the market.
3.3.2 Data
dt examined in this paper comes from the difference between the DJIA daily index and the value of the best VMA rule applied in the period of 1928/10/1 to 2010/6/28. Since the best VMA rule needs 30 days to generate trading signals, we have 20,496 observations for dtfrom 1928/11/14 to 2010/6/28. This paper focuses on the U.S. stock market and investigates the role of market volatility ratio in the generation of signals in moving averages both in full sample and sub-samples. The reason to conduct sub-sample analysis is to serve as the robustness check of our empirical results.
We split the full sample into the following three sub-samples: 1928/11/14−1938/12/30, 1939/01/01−1987/10/18 and 1987/10/19−2010/06/28. These sub-samples are chosen for two common reasons. The first sub-sample includes the turbulent times of the 1930s great depres-sion. The second and the last sub-samples are divided based on the most important economic event (change in price in percentage terms) to recently affect the DJIA index, the October crash of 1987. Thus, the second one comprises the World War II and the pre-October 1987 crash periods, and the last one covers the post-October 1987 crash period.
Table 11 reports the summary statistics and the unit root tests results for dt, VRt−1and dVRt both for the full and sub-samples. We observe that the dt distribution exhibits left skewness and excess kurtosis for the full and all sub-samples. Both VRt−1 and dVRt have a long right tail in the sample distributions; however, the VRt−1 distribution is platykurtic but dVRt’s is
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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