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5. Conclusion
This paper uses the time-varying transition probability Markov switching model to examine structural breaks in the VIX index and understand whether the VIX index can be identified as two states, one is low mean value and smooth volatility, which is named tranquility state in this paper; the other is high mean value and intense volatility, which is named crisis state. The empirical findings are as follows: the VIX index can really be distinguished as two states described above, and state switching indeed exists in the VIX index during the sample period.
Another purpose of this paper is to investigate whether the TVTP model which incorporates exogenous financial variables performs better in identifying state switching than the fixed transition probability model which has no extra information added. This paper applies statistical and practical ways to examine those models. In TVTP model, this paper inputs some financial indicators to check whether they can help driving the VIX index going from one state to another. Seven representative financial indicators on the U.S. markets are used: SPDR S&P 500, U.S. Dollar Index, 3-month T-bill rate, 10-year term spread, 5-year term spread, 10-year credit spread, and 5-year credit spread. Through LR tests, the models respectively with SPDR S&P500, 10-year credit spread, or 5-year credit spread are statistically significant, which means the TVTP models with these three exogenous variables are statistically significant better than the FTP model. And the financial implication of these three significant variables is consistent with how investors realize their features.
After generalizing the nine significant TVTP models from statistical view, this paper examines and compares their practical performance during the same period in order to test whether the models incorporating significant variables performs practically better than the model without extra information even though the statistic
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results have showed the significance in the TVTP models. The findings are as follows:
in the nine TVTP models, there are six which have higher yearly rates of return than the FTP model, and the best one is the model with spot information of 5-year credit spread due to the highest yearly rate of return of 7.81% while the return of FTP is 7.16%. Thus, this paper finds six TVTP models which include three variables such as SPDR S&P500, 10-year credit spread, or 5-year credit spread are statistically significant and practically perform better than FTP model. However, the findings also indicate three models with lower return than FTP, which suggests applying the model practically for double checking is important even though the statistic results are significant. This paper also finds that credit spread is the most useful indicator to observe changes of VIX index, especially the 5-year credit spread.
This paper examines the effectiveness of models not only from statistical significance test, but also from practical return comparison, and finds some conflicts exist between them. Therefore, it is more comprehensive to measure model specification both statistically and practically, and this paper provides a case which may be a reference for future studies. In addition, credit spread plays an essential role in identifying state switching in VIX index, which is an important finding in this paper because not many literatures have considered it when identifying state shifts.
Finally, the data frequency used in this paper is daily, which is very different from the past literatures because most of them choose macroeconomic indicators as variables which are almost seasonal or monthly frequency. VIX index has abundant market information due to the day-by-day trading activity, so it may be better and decrease the loss of information to select the financial variables which also have daily data.
There are some restrictions in the research. First, data period this paper selects plays an important role for the empirical results. If researchers change sample period, selection of variables must also be changed to match the period, and the TVTP
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models should be reconstructed. On the other hand, this paper sets 0.5 as the probability threshold when identifying which state each trading day is. And the results of performance comparison among models rely on the assumption. Because this paper does not consider different criteria for identifying tranquility or crisis state, setting 0.5 as threshold is appropriate. If researchers would like to detect the crisis state more sensitively, lowering the probability threshold for crisis state below 0.5 may be feasible, and which may cause different return performance of each model. When calculating return performance, this paper does not consider any transaction cost and dividend payment due to the difficulties for data accessing. Thus, all of the return do not include the cost and dividend impacts. If considering the costs and dividends, the results of the performance comparison among models may be different.
For future research, additional factors especially variables which have daily data can be added into the TVTP model for improvement in the state identification process.
In-sample and out-of-sample tests can also be implemented to examine the effectiveness of the TVTP models in different time periods. In addition, without considering the transaction costs and dividend payments, the practical performance of TVTP model can surpass the U.S. market benchmark in this paper. Future studies may explore and compare models’ return performance further considering costs and dividends to examine the effects of TVTP models more practically, and even develop a complete and profitable investment strategy based on the TVTP model. On the other hand, this paper finds that the models with spot information of exogenous variables are almost better than the ones with lag information. High market efficiency is the inferred possible reason. Most of information is reflected in the VIX index or prices of other assets very timely and quickly, so using the spot data instead of lag information can explain the most and decrease the loss of information, and that’s why this paper uses daily data to follow the quick information flow in VIX. However, high
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market efficiency causes the unobservable effects of forecast in advance through TVTP models even using daily data. Therefore, future researchers can focus on the issues of forecast by using TVTP model and data with much shorter time interval, such as intra-day or shorter, and construct a dynamic forecasting model to identify the timing of state switching in advance so that investors can adjust their investment or trading strategies timely by following the model indications and improve their performance.
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Appendix A: Estimated Parameters of Generalized TVTP Models
(1) Transformation for calculating transition probability (Ding, 2012)
For a k-state system, there are (k-1)k independent time varying probability components needed to be estimated:
𝑡= [
𝑞11,𝑡 ⋯ 𝑞1 ,𝑡 𝑞 −1,1,𝑡 ⋯ 𝑞 −1, ,𝑡]
For each probability cell (i, j) (where i=1,2…k-1, j=1,2…k ), the probability generating function is as follows:
𝑞𝑖𝑗,𝑡 = (𝑋𝑖𝑗,𝑡 𝑖𝑗)
Where () is the cumulative normal density function, 𝑋𝑖𝑗,𝑡 is the exogenous variable vector for cell (i, j), and 𝑖𝑗 is the parameters from state j to state i to be estimated. The exogenous variables can be different for different cells.
Next, we can generate an auxiliary matrix 𝑡 based on 𝑡 as below:
𝑡= [
− 𝑞11,𝑡 ⋯ − 𝑞1 ,𝑡
∏( − 𝑞𝑖1,𝑡)
−1
𝑖=1
⋯ ∏( − 𝑞𝑖 ,𝑡)
−1
𝑖=1 ]
The final time-varying transition probability matrix can be constructed as follows:
𝑡 = 𝑡‧ 𝑡= [
𝑝11,𝑡 ⋯ 𝑝1 ,𝑡 𝑝 1,𝑡 ⋯ 𝑝 ,𝑡]
‧
Where ‧ stands for elementwise matrix product. The final probability transition matrix 𝑡 can also be expressed as:
𝜇1: the mean of the changes in VIX index in tranquility state
𝜇2: the mean of the changes in VIX index in crisis state
𝑖𝑗,𝑙𝑎𝑔𝑡: the parameter for the variable with lag t data in the transition probability from state j to state i
𝑖𝑗,𝑙𝑎𝑔𝑡 : same as 𝑖𝑗,𝑙𝑎𝑔𝑡, but respective for the 𝑟𝑡ℎ variable in the model 𝜑1: the parameter of the autoregressive VIX with lag 1 in Eq.(1)
𝜑2: the parameter of the autoregressive VIX with lag 2 in Eq.(1)
𝜎2: the standard deviation of the error term in Eq.(1)
No. 1 2 3 4 5
Variable SPY Credit10Y Credit5Y SPY Credit5Y
Lag 0 0 0 0+2 0+1
‧
(Std. Error) (10.189493)
12,𝑙𝑎𝑔0
SIC 1124.376 1128.603 1126.364 1106.517 1139.787
Log likelihood
-534.333 -536.447 -535.327 -517.448 -534.082
LR test 9.5728 5.3452 7.5846 44.5954 10.6388
p. value 0.008342*** 0.069072* 0.022544** 0.0000*** 0.030938**
No. 6 7 8 9
‧
SIC 1150.989 1136.485 1134.816 1139.94
Log likelihood -531.722 -516.516 -515.684 -534.156
LR test 14.794 46.4588 48.7896 9.9258
p. value 0.021921** 0.0000*** 0.0000*** 0.041696**
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Appendix B: Details about Practical Performance Examination
(1) FTP model No. State switching
day
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No. State switchingday
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(3) TVTP model-Credit10Y (lag 0) No. State switching
day
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(4) TVTP model-Credit5Y (lag 0) No. State
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(5) TVTP model-SPY (lag 0+2) No. State switching
day
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(6) TVTP model-Credit5Y (lag 0+1) No. State switching
day
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(7) TVTP model-SPY+Credit10Y+Credit5Y (lag 0) No. State switching
day
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(8) TVTP model-SPY+Credit10Y (lag 0+2) No. State switching
day
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(9) TVTP model-SPY+Credit5Y (lag 0+3) No. State switching
day
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(10) TVTP model- Credit10Y +Credit5Y (lag 0) No. State switching
day
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Appendix C: Smoothed Probability Comparison
(1) TVTP model-Credit5Y (lag 0) and FTP model
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(2) TVTP model-SPY (lag 0+2) and FTP model
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(3) TVTP model-Credit5Y (lag 0) and TVTP model-SPY (lag 0+2)
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(4) TVTP model-Credit5Y (lag 0), TVTP model-SPY (lag 0+2) and FTP model