3. Methodology
3.2 Time-varying Transition Probability (TVTP) Model
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𝑦𝑡− 𝜇𝑆𝑡 = ∑ 𝜑𝑖(𝑦𝑡−𝑖− 𝜇𝑆(𝑡−𝑖)) + 𝜖𝑡
𝑝
𝑖=1
𝜖𝑡~𝑁(0, 𝜎2) 𝑆𝑡 = , 𝑆𝑡−𝑖 =
, ,
P(𝑆𝑡= |𝑆𝑡−1 = ) = 𝑝𝑖𝑗
Where 𝑆𝑡 and 𝑆𝑡−𝑖 are the unobserved regime variables that take the values of 1 or 2 and 𝑝𝑖𝑗 indicates the probability of switching from state j at time t-1 into state i at t.
The transition probabilities between regimes are governed by a first order Markov process as follows:
P(𝑆𝑡 = |𝑆𝑡−1= ) = 𝑝11 P(𝑆𝑡 = |𝑆𝑡−1= ) = 𝑝12= − 𝑝22 P(𝑆𝑡 = |𝑆𝑡−1= ) = 𝑝21 = − 𝑝11
P(𝑆𝑡 = |𝑆𝑡−1= ) = 𝑝22 𝑝11+ 𝑝21= 𝑝22+ 𝑝12 =
Usually these probabilities are grouped together into the transition matrix, and they keep constant in the time series. I transfer the symbols and use p and q to represent the transition probabilities.
P = [𝑝11 𝑝12
𝑝21 𝑝22] = [ 𝑞 − 𝑝 − 𝑞 𝑝 ]
3.2 Time-varying Transition Probability (TVTP) Model
I use the TVTP MS model Filardo (1994) developed as main methodology in this
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paper, where the transition probabilities vary across time. Due to the asymmetric response to the bull and bear market in the volatility of VIX index I mentioned before, TVTP MS model can reveal this kind of characteristics. I distinguish VIX index into two states: tranquil and crisis, but both states are unobserved in advance. However, this unobserved binary state variable can pick up different phases of VIX index that corresponds to current state of tranquility or crisis. The evolution of the unobserved state measures the mean value and volatility of VIX index across the series, and will depend on available information represented in the time series of exogenous variables.
In other words, the state of VIX index depends on not only the level of previous conditions, but also some exogenous financial variables. Therefore, I implement the time-varying transition probability (TVTP) Markov switching model as the main methodology in this paper. This methodology allows for identification of not only VIX index’s states, but also of financial variables which have statistically significant information about the changing of VIX.
First, I start from the introduction of basic idea of TVTP MS model.
In this paper, I am interested in examining the characteristics of VIX index, 𝑦𝑡 , = , , , , so I use the TVTP MS model described as below (see also Filardo and Gordon (1998)).
𝑦𝑡 = 𝜇0+ ∑𝑝𝑖=1𝜑𝑖(𝐿)(𝑦𝑡−𝑖− 𝜇𝑆(𝑡−𝑖)) + 𝜖𝑡 if state 0 = 𝜇1+ ∑𝑝𝑖=1𝜑𝑖(𝐿)(𝑦𝑡−𝑖− 𝜇𝑆(𝑡−𝑖)) + 𝜖𝑡 if state 1
Here 𝜑𝑖(𝐿) = 𝛿1𝑖+ 𝛿2𝑖𝐿 + ⋯ + 𝛿𝑑𝑖𝐿𝑑−1 is a lag polynomial, 𝜖𝑡~𝑁(0, 𝜎2), p is lag periods, and 𝑆𝑡 {0, }; thus the state-dependent mean is 𝜇𝑆𝑡 = 𝜇0+ 𝜇1𝑆𝑡. The state variable 𝑆𝑡 is governed by a 2-state Markov-chain with transition probability matrix,
(1)
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P(𝑆𝑡= 𝑠𝑡|𝑆𝑡−1= 𝑠𝑡−1, 𝑋𝑡) = [ 𝑞(𝑋𝑡) − 𝑝(𝑋𝑡) − 𝑞(𝑋𝑡) 𝑝(𝑋𝑡) ]
Where the exogenous financial variables is 𝑋𝑡= {𝑥𝑡, 𝑥𝑡−1, … }. Compared with the transition matrix in FTP, it is obvious to observe that the transition probabilities in TVTP are the function of the exogenous financial driving covariates 𝑋𝑡. The two states in this paper are named as tranquil state (state 0) and crisis state (state 1), respectively. Here I use 0 and 1 to replace the state 1 and 2 in the above part of FTP.
Time-varying transition probabilities can gauge the persistence of the phases of tranquil state and crisis state, and allow the probability of exiting a particular phase to vary with economic information such as the financial driving covariates 𝑋𝑡. The variables I choose from the past literatures will input 𝑋𝑡 in the empirical process.
The selection of functional form of 𝑞( ) and p( ) is typically probit or logistic type, and I will use the probit type in this paper. Here 𝑞( ) and p( ) indicate the same meanings as used in the transition matrix in the above part of FTP. In TVTP model, the parameters in Eq.(1) and the transition probability parameters in Eq.(2) are jointly estimated. The conditional joint density distribution, , with AR dynamics of order r, is:
(𝑦𝑡|𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡) = ∑ … ∑ ̂(𝑦𝑡|𝑆𝑡= 𝑠𝑡, … , 𝑆𝑡− = 𝑠𝑡− , 𝑦𝑡−1, … , 𝑦𝑡− )
1
=0 1
=0
× P(𝑆𝑡 = 𝑠𝑡|𝑆𝑡−1= 𝑠𝑡−1, 𝑋𝑡)
× P(𝑆𝑡−1= 𝑠𝑡−1, … , 𝑆𝑡− = 𝑠𝑡− |𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡−1)
And the log-likelihood function is:
L(θ) = ∑𝑇𝑡=1𝑙𝑛[ (𝑦𝑡|𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡 )]
(2)
(3)
(4)
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Here denotes the parameter vector. The probability density function (𝑦𝑡|𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡) makes a link with the exogenous variables contained in 𝑋𝑡 with regards to how they feature in the procedure for a Markov switching model for series 𝑦𝑡 via the transition probabilities. The information affects the system directly and indirectly through the inference of the past states. The information in 𝑦𝑡 and its lags directly affects the likelihood through the normal density, ̂, and the lags of 𝑦𝑡 indirectly influence the likelihood through the past states P(𝑆𝑡−1 = 𝑠𝑡−1, … , 𝑆𝑡− = 𝑠𝑡− |𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡−1) . The exogenous financial variables directly affect the transition probabilities, P(𝑆𝑡= 𝑠𝑡|𝑆𝑡−1= 𝑠𝑡−1, 𝑋𝑡), and the distribution of the states, P(𝑆𝑡−1 = 𝑠𝑡−1, … , 𝑆𝑡− = 𝑠𝑡− |𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡−1) indirectly.
Next, I would like to tell some details about calculating the transition probability.
In this paper, I use a latent variable version of probit model referring to Filardo and Gordon’s (1998) methodology to measure the transition probability matrix at each time ,
P(𝑆𝑡= ) = (𝑆𝑡 0) P(𝑆𝑡= 0) = (𝑆𝑡 0)
where 𝑆𝑡 is a latent variable defined by
𝑆𝑡 = 0+ 𝑥𝑡+ 𝑠𝑡−1+ 𝑡
𝑡~𝑁(0, )
The latent variable evolves due to a set of explaining variables, and Eq.(6) shows the relation between them. The standard normal distributional assumption will simplify the calculation of the transition probabilities. For instance, calculating transition probabilities at each time is done by evaluating the conditional cumulative (5)
(6)
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distribution function for u. Let the function | ( (𝑥)) represent a conditional standard normal cumulative distribution function (CDF) such that:
| ( (𝑥)) = ∫− ̅( )√2 1 (−12 2) ~ (0, )
Where ̅(𝑥) is the upper limit of the integration and is determined by Eq.(3) and (4).
Therefore, 𝑝𝑡 and 𝑞𝑡 are:
𝑝𝑡= (𝑆𝑡= |𝑆𝑡−1= ) = ( 𝑡 − 0− 𝑥𝑡− ) = − | ( 0− 𝑥𝑡− ) 𝑞𝑡= (𝑆𝑡= 0|𝑆𝑡−1 = 0) = ( 𝑡 − 0− 𝑥𝑡) = | (− 0− 𝑥𝑡)
In this model, the unobserved state variable, 𝑆𝑡, is a nonlinear function of the observable exogenous variables. The timing of those states is influenced through 𝑆𝑡 by the vector of covariates, 𝑋𝑡. The probit type model computes the probability that the VIX index is in tranquil state or crisis state given a set of financial covariates, 𝑋𝑡.
After the simple introduction of model construction, now I will move to the method of parameter and state probability estimation.
Estimation is carried out via maximum likelihood (ML) methods. Once having the joint probability, I can calculate the likelihood estimates. The maximum likelihood estimates for is obtained by maximizing the likelihood function by updating the likelihood function at each iteration using the algorithm as follows.
This is the log-likelihood function:
L(θ) = ∑𝑇𝑡=1𝑙𝑛[ (𝑦𝑡|𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡 )]
(7)
(8)
(9)
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And I rewrite the conditional probability density function as:
(𝑦𝑡|𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡)
= ∑ (𝑦𝑡|𝑆𝑡 = , 𝑦𝑡−1, … , 𝑦𝑡− ) × P(𝑆𝑡 = |𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡−1)
𝑖=1
= ∑ (𝑦𝑡|𝑆𝑡 = , 𝑦𝑡−1, … , 𝑦𝑡− ) ∑ 𝑝𝑖𝑗P(𝑆𝑡−1= |𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡−1)
𝑗=1 𝑖=1
Then I can calculate the filtered probability:
P(𝑆𝑡−1 = |𝑦𝑡−1, … , 𝑦𝑡− , 𝑋𝑡−1)
= (𝑦𝑡−1|𝑆𝑡−1= , 𝑦𝑡−2, … , 𝑦𝑡− ) × P(𝑆𝑡−1= |𝑦𝑡−2, … , 𝑦𝑡− , 𝑋𝑡−1)
∑𝑖=1 (𝑦𝑡−1|𝑆𝑡−1= , 𝑦𝑡−2, … , 𝑦𝑡− ) × P(𝑆𝑡−1= |𝑦𝑡−2, … , 𝑦𝑡− , 𝑋𝑡−1)
I can infer the filtered probability of each state through the above algorithm.
Once estimating the model using the filtering algorithm, I can also make advanced inference of the probability on the state using all the information from the beginning to the end in the sample, and this kind of probability is called the smoothing probability. The filtered and smoothing probabilities help us decide which state 𝑦𝑡 is at each point of time. Generally, in most applications, filtered probabilities and smoothing probabilities will give a very similar conclusion. (Ismail and Isa, 2008)
Next, I would like to introduce the test of significance for TVTP models and the criterion of model selection in this paper.
If there is no statistically meaningful information existed in the evolution of the state of the financial market contained in 𝑋𝑡, the fixed transition probability (FTP) model is useful enough. In order to examine model’s effects, I will investigate the significance of the TVTP MS model using the likelihood ratio test (LR test). Formally,
(10)
(11)
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under the null hypothesis of no time varying transition probabilities, the likelihood ratio test statistic is given by
ψ = × [𝐿( ) − 𝐿𝑅( )]~𝜒(𝑀2 1+𝑀2),𝛼
where 𝐿𝑅( ) is the restricted log-likelihood, 𝑀1+ 𝑀2 is the number of restrictions on the test at significance level of 𝛼. LR test can be perceived as a test of the information of the exogenous financial variables in modeling and identifying the turning points in VIX index. Furthermore, the LR test can be implemented to select between alternative lag specifications for the conditioning variables. The Akaike information criterion (AIC) and the Schwartz information criterion (SIC) are usually used to gauge the appropriate order of the lags of the variables in TVTP.