4. Empirical Results
4.2 Process of Model Construction and Selection
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3 3-month T-bill rate T-bill A short-term debt obligation backed by the U.S.
government with a maturity of 3 months
4 10-year term spread Term10Y the spread between 10-year Treasury bond yield and the 3-month T-bill rate
5 5-year term spread Term5Y the spread between 5-year Treasury bond yield and the 3-month T-bill rate
6 10-year credit spread Credit10Y the spread between 10-year BBB corporate bond yield and 10-year Treasury bond yield
7 5-year credit spread Credit5Y the spread between 5-year BBB corporate bond yield and 5-year Treasury bond yield
The VIX index, SPDR S&P 500 and Dollar index are calculated as ln
261, where 𝑋𝑡 is the original index or price level of each variable at time t. The transformation indicates that this paper focuses on the changes in these kinds of indexes or prices during the past one year. The other variables are measured in level.
In next part, the software-Eviews will be used to help analyzing the data, and the Matlab is used to estimate the TVTP MS model.
4.2 Process of Model Construction and Selection
Before adopting the VIX index into the process of model estimation, this paper does the Augmented Dickey Fuller (ADF) Unit Root Test to check whether VIX index is a unit root series data. A time series data is not stationary when it has a unit root, and a non-stationary time series data will cause the consequence of spurious regression and ineffective inference if researchers don’t pay attention to the presence of unit root. From Table 4, it is significant to infer the VIX index (after calculation mentioned above) is stationary because it rejects the null hypothesis of unit roots existing. This paper also decides the lags of autoregression in the VIX index with
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Schwartz information criterion (SIC), and it suggests two periods of lag is the best.
Table 4. Augmented Dickey Fuller Unit Root Test
Method Statistic Prob.**
ADF - Fisher Chi-square 17.1582 0.0002
ADF - Choi Z-stat -3.55639 0.0002
In Figure 3, the stationary time series of changes of VIX is also observed clearly.
Figure 3. Changes of VIX index
Since the VIX index is stationary, the seven financial variables can be added into the TVTP model to examine whether the TVTP model incorporating exogenous information can significantly improve the fit of the data over FTP model which is without any exogenous variable. First, this paper adds the variables one by one, and calculates the SIC and likelihood ratio (LR) test statistics to estimate the significance of the TVTP model compared with the FTP model. In addition to the spot data, the lag data of each variable added into the TVTP model are from 1 to 3 periods, and this paper also examines the variation by turns. The estimated results are listed below.
-1.5
500 1000 1500 2000 2500
LN(VIX(T)/VIX(T-261))
500 1000 1500 2000 2500
LN(VIX(T)/VIX(T-261))
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Table 5. Significance test and SIC of models
Group Variable Lag SIC Log likelihood LR test p. value
Note: ***, ** and * indicate significance at 1%, 5% and 10% levels respectively.
The last columns indicate the significance for TVTP model compared with FTP model of each variable. This paper finds that the models respectively with variable SPY, Credit10Y, or Credit5Y are significant over all lag periods. It means that the
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TVTP models with any one of the above three variables are statistically significant better than the FTP model which doesn’t incorporate any exogenous variable.
Therefore, those three variables may be statistically useful to help distinguish the state switching in VIX index. Dollar, T-bill, Term10Y and Term5Y are not significant over all lag periods. In other words, these four variables may not be statistically helpful using in TVTP model to distinguish the state switching in VIX index. I also select the best specification by each model’s SIC among different periods of lag within every group. SIC is very useful to decide what period of lag is more appropriate for the variable added in the model. Smaller SIC means the variable’s lag period is better for the model specification than other periods of lags. For example, in the three significant groups of SPY, Credit10Y and Credit5Y, the SIC values of no lag models are smaller than other models within the same groups respectively, therefore, this paper infers that the models with spot information are more appropriate than others with lagged information whether the exogenous information is from SPDR S&P500, 10-year credit spread or 5-year credit spread.
It makes sense that the information contained in the SPDR S&P500, 10-year credit spread and 5-year credit spread is useful to identify the structural breaks in VIX index. I have mentioned in the previous sections that the S&P500 index is highly negatively correlated with the VIX index, and the empirical results of this paper also show that the SPDR S&P500 tracing and having a high correlation with the S&P500 index is helpful to identify the state switching in the VIX index. Thus, the result in this paper is consistent with past literatures. On the other hand, credit spread can indicate the atmosphere in the market or investors’ sentiment for the market condition.
Because according to the credit spread definition used in this paper, when the market becomes instability, even recession, investors will worry that the default possibilities of BBB corporate bonds increase, so they ask higher yields to compensate the higher
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risk. And they also invest more in the Treasury bond for safety, causing the Treasury bond yields decrease. As a result, the credit spread expands while the market is downward or crisis happens. Therefore, credit spread is very similar with VIX for detecting the market downturn, and the empirical results give supports as well.
To sum up, this paper chooses three significant TVTP models with appropriate lag period due to SIC, including the models respectively with SPY, Credit10Y or Credit5Y, and whose appropriate lag periods are all 0, namely no lag.
In next part, this paper only focuses on the above three significant variables, and mixes the periods of lag in each variable group to explore whether the models become better or worse after mixing different periods of lags. I take the spot data as a base for each variable, and mix 1, 2, or 3 periods of lag with the base respectively. The result is when mixing the periods of lag for Credit10Y, all of the results become insignificant.
Thus, the Credit10Y group won’t be considered in this part. The models based respectively on SPY and Credit5Y are still significant. According to SIC, the model with SPY mixed with no lag and 2 periods of lags data is the best in group 1 while the model with Credit5Y mixed with no lag and 1 period of lag data is the best in group 3.
Table 6. Significance test and SIC of models mixed with periods of lags
Group Variable Lag SIC Log likelihood LR test p. value
1 SPY
0+1 1109.261 -518.818 41.1654 0.0000***
0+2 1106.517 -517.448 44.5954 0.0000***
0+3 1108.068 -518.225 43.707 0.0000***
2 Credit10Y
0+1 1145.072 -536.724 5.3538 0.252883 0+2 1145.562 -536.971 5.5498 0.235387
0+3 1146.417 -537.4 5.3584 0.252459
3 Credit5Y
0+1 1139.787 -534.082 10.6388 0.030938**
0+2 1141.447 -534.913 9.6654 0.046457**
0+3 1141.01 -534.696 10.7658 0.029326**
Note: ***, ** and * indicate significance at 1%, 5% and 10% levels respectively.
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This paper also combines the three significant variables meanwhile or any two of them into the model to test model significance and whether the accuracy for identifying state switching get better if model contains more information by incorporating more variables. This paper also considers different periods of lags and mix them like above to select the best model in each group through SIC. The results are as follows. In group 1, the model with spot information of SPY, Credit10Y, and Credit5Y incorporated at the same time is selected due to its significance and the smallest SIC. In group 2, the model with spot and 2 periods of lag information of SPY and Credit10Y is relatively a better choice. In group 3, considering the smallest SIC, the model with spot and 3 periods of lag information of SPY and Credit5Y is chose.
In group 4, Credit10Y and Credit5Y are incorporated in the model, and the best one is with spot information.
Table 7. Significance test and SIC of models mixed with variables and lags.
Group Variable Lag SIC Log likelihood LR test p. value
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0+1+2+3 1230.433 -531.663 16.8324 0.396522 Note: ***, ** and * indicate significance at 1%, 5% and 10% levels respectively.The nine models selected above are summarized and showed the state-dependent mean value of the changes in VIX index and expected duration of states in each model specification in the below table. Observing from each model, the estimates of the state-dependent means, 𝜇1 and 𝜇2, are statistically significantly different. It is based on the assumption of the existence of two states. Moreover, in each model, the estimates of the mean in state 1, 𝜇1, is significantly negative and the estimates of the mean in state 2 , 𝜇2, is significantly positive at 1% significance level. The difference between the state-dependent means is consistent with the assumption that the VIX index can be identified as two states. The negative estimates of the mean in state 1 can be labeled as the tranquility state because the change in VIX is negative which means
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the VIX becomes smaller. In contrast, the positive estimates of the mean in state 2 can be labeled as the crisis state because the change in VIX is positive which means the VIX becomes larger and larger. The standard errors in state 2 is larger than in state 1, which is also consistent with the concepts that crisis state is more volatile than tranquility state. Therefore, the nine TVTP models concluded in this paper are proven to be effective to distinguish VIX index as two distinct states, and significant better than FTP model from statistical view when identifying state shifts.
Table 8. Estimates of state-dependent parameters of nine models
No. Variable Lag 𝝁𝟏 Note: ***, ** and * indicate significance at 1%, 5% and 10% levels respectively.
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