Data and Estimation Results

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mechanism, and the only distinction between these two rules is the way they treat historical prices in forecasting future price movement. The market volatility ratio is a function of past historical prices, therefore, it makes sense that it can exert effects on the generation of the SMA rule’s trading signals.

We redefine d_tin the FTP and TVTP Markov-switching model as p_t− ma_t, where ma_tis the logarithm of SMA value at time t of a particular n-day SMA rule and n is selected as 5, 20, 40, 75, 100 and 250, which are one-third of the settings for n. The same information of market volatility is chosen as the VRtfrom the best VMA rule in the full sample, σ_tⁿ/σ_t^ref with n = 15 and ref = 30. Similarly, z_tin the TVTP Markov-switching model are VR_t−1, dVR_tand dVR_t−1. AR lag in d_t and the elements in z_t are chosen according to the suggestions by both AIC and SC. Lastly, we make comparison between the profitability of trading signals generated from the FTP Markov-switching model and that from the TVTP Markov-switching model. The trading signals from these two types of Markov-switching model are obtained as follows. Using the full-sample smoothing algorithm of Kim (1994), we can get the smoothing probability of state 0 (rising market, the price line above the SMA line) and state 1(falling market, the price line below the SMA line). We use the smoothing probabilities to infer the rising and falling markets by simply taking 0.5 as the cut-off value for S_t = 0 or 1. If the smoothing probability of State 0 at time t is greater (less) than 0.5, it is more likely to be a rising (falling) market at time t. After indexing the full data set as rising-market state or falling-market state, we can get the trading signals through the transition between these two states. A buying signal occurs if the regime of dt switches from the falling-market state to the rising-market state. Contrarily, the generation of a selling signal comes from the transition from the rising-market state to the falling-market state.

3.4.1 Data and Estimation Results

Table 15 reports the summary statistics of d_t for six n-day SMA rules. We observe that the mean, the maximum, the minimum and the standard deviation of the d_t series get smaller as n is smaller (i.e. the shorter-period SMA rule). On the other hand, the level of kurtosis get rising

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Table 15: Descriptive Statistics And Unit Root Tests: d_tFrom Six SMA Rules

This table reports summary statistics and the unit root tests results for six dtin the full period. One dtis defined as ptminus mat, where ptand matare the logarithm of the DJIA closing price and the SMA value from a particular n-day SMA rule at time t. n is chosen as 5, 20, 40, 75, 100 and 250. For unit root tests, ADF and PP are augmented Dickey-Fuller and Phillips-Perron test statistics.

In both tests, the test equation includes the intercept term and the null hypothesis is that the series has a unit root. Test critical values for ADF and PP are -3.443834 (1%), -2.867379 (5%), and -2.569943 (10%). Lags in ADF tests are chosen by Schwartz Bayesian information criterion (SC).

n 5 20 40 75 100 250

Descriptive Statistics

Mean 0.0004 0.0017 0.0035 0.0067 0.0090 0.0221

Median 0.001 0.004 0.008 0.013 0.017 0.038

Max 0.133 0.294 0.355 0.401 0.386 0.488

Min -0.251 -0.356 -0.442 -0.516 -0.521 -0.745

Standard Deviation 0.013 0.030 0.043 0.060 0.070 0.116

Skewness -0.998 -1.141 -1.009 -1.005 -1.052 -1.393

Kurtosis 20.728 14.878 13.041 10.184 8.760 8.146

Unit root tests

ADF -43.436 -30.506 -21.286 -14.084 -12.177 -7.195

PP -58.207 -26.958 -20.254 -15.472 -13.220 -7.579

as n decreases. In other words, the d_tdistribution is more concentrated around the mean. The above results come from that as n decreases, the weight of the current price (i.e. p_t) in forming the SMA will be greater. Greater weight on the current price will lessen the gap between the price line and the SMA line and then lead to the dt series with smaller mean, less standard deviation and higher kurtosis. In addition, in order to implement the estimation of the Markov-switching models, we have to make sure whether these six dtseries are stationary or not. Thus, the conventional unit root tests, ADF test and PP test, are conducted. Clearly, as shown in Table 15, the unit root process is rejected for each series.¹⁴

Table 16 presents the estimation results for the FTP Markov-switching models. The

positive-14The two types of test equations in the ADF and PP test are conducted as follows. One only puts the intercept term in the model. The other contains not only the intercept term but also the trend term. The hypothesis of unit root process is rejected for each series no matter what model of test equation is used.

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mean stable and negative-mean volatile state in dt are still labeled as the rising markets and falling markets, respectively. Similar to the results of d_t from the best VMA rule, σ₁ > σ₀ shows that the SMA rule no matter what n is chosen can not trace the price line closely in the falling markets, compared to in the rising markets. This might be because, the market price in the falling markets is more volatile than that in the rising markets. Moreover, we observe that dtfrom the 5-day SMA rule gets smaller σ1 and σ0. This result coincides with what we find in Table 15. Except for dtfrom the 5-day SMA rule, dtfrom other SMA rules in two regimes are quite persistent because we get higher bβ_0,1and bβ_1,1. Lastly, we find that the regime persistence the VMA and six SMA rules expect is alike. Both the VMA and the SMA rules anticipate the rising-market state will persist for 111 days. The regime persistence for the falling-market state the SMA rules expect ranges from 25 days to 28 days, while the VMA rule’s expectation is 26 days.

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Table 16: The FTP Markov-Switching Models: The SMA Rules

This table presents the estimation results of the Fixed-Transition-Probability (FTP) Markov-Switching model in the full sample for six n-day SMA rules. The value of n includes 5, 20, 40, 75, 100 and 250. The dependent variable in the FTP Markov-Switching Model is dt, pt− mat. Where matis the logarithm of SMA value at time t of a particular n-day SMA rule. The model is dt= µS_t+ βS_t,1dt−1+ twith (µ0, σ²₀, β0,1) in regime 0 and (µ1, σ²₁, β1,1) in regime 1. βS_t,1is the AR1 coefficient in each regime. p⁰⁰denotes the transition probability of staying in regime 0 while p¹¹denotes the transition probability of remaining in regime 1. The entries in brackets are the standard errors. LogLik represents the value of log-likelihood function.

n 5 20 40 75 100 250

µ0 0.0003 0.0003 0.0004 0.0004 0.0004 0.0004

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

µ1 -0.0008 -0.0009 -0.0012 -0.0013 -0.0014 -0.0015

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

β_0,1 0.698 0.928 0.964 0.981 0.985 0.996

(0.006) (0.003) (0.002) (0.002) (0.001) (0.001)

β_1,1 0.649 0.920 0.959 0.976 0.981 0.990

(0.012) (0.006) (0.005) (0.003) (0.003) (0.002)

σ₀ -0.006 -0.007 0.007 0.007 -0.007 0.007

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

σ₁ -0.018 -0.021 0.022 0.022 0.022 0.022

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

p⁰⁰ 0.991 0.991 0.991 0.991 0.991 0.991

(0.001) (0.001) (0.001) (0.001) (0.001) (0.001)

p¹¹ 0.964 0.963 0.960 0.961 0.961 0.961

(0.004) (0.004) (0.004) (0.004) (0.004) (0.004)

LogLik 71155.60 68006.30 67549.80 67338.30 67294.60 67197.90

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The results for the TVTP Markov-switching models reported in Table 17 are very similar to the results in Tables 12. The information of market volatility, dVR_t and dVR_t−1 resulted from the VR_tseries of the best VMA rule, have the same effect on these six SMA rules’ trading signals. Firstly, bθ₁ > 0, bθ₂ < 0 andbγ₂ > 0 are still found. bγ₁ = 0 for the 20-day and 100-day SMA rules whilebγ₁ < 0 for the others. Secondly, the estimated values of θ₁, θ₂and γ₂all tend to be around the values obtained from the VMA rule. Although we get strongerbγ1 in the case of the SMA rules, the economic significance of its effect on the probability of buying signals generated is still slight. In addition, it is reasonable that we get similar results for the VMA rule and SMA rules because of using the same information of market volatility.