4.0 PROPOSED METHODOLOGY
4.1.2 Multivariate Markov-switching Market Equilibrium Model
4.1.2.2 Multi-chain Markov-switching vector equilibrium model (MCMS-VEM)
Under this specification we relax the assumption of co-breaking of the state variables in the system so that overlapping regime processes can directly be inferred from the model as in PBM of Barrett and Li (2002) in the following;
( ) 1
φ
−δ
=
t+ +
t ct ct t
R R u
4.18As in equation (4.14 & 4.15), (Rt ) of equation (4.18) defines observed series (
=2 in our case- rent and trade variables; and
T
is the length of the series) and all other parameters remain as already defined. The only difference here is that, unlike the MS-VEM, the state variable at time t is a vector of length , which implies multiple chains( ). The MS-VEM, which inherently assumes co-breaking can only capture MI representation as demonstrated in Barrett and Li (2002) by increased number of states, specifically six if no symmetric structure is imposed on the system (see the illustration, above). With the MCMS-VEM however, each variable’s state mechanism is explicitly defined to accommodate any consistent theoretical view point (see Gallo and Otranto 2006).x
T N N
ct N N Ct
In this case the unrestricted six-state system of MI within PBM specification is represented by three- and two-state regime switching processes on the rent and trade variables respectively.
While the MS-VEM can be seen as a restricted form of MCMS-VEM with observationally similar form, it is not nested in the later as the functional structure of the latent variables Ct
differs significantly (see Otranto 2005). If we define
C
t ≡{ c c
1m, 2m,...,c
NM}
then represents the state associated with variablec1m ( )
R
t nm , where Mis number of states in a chain. The first subscript of represents the variable number in the time series vector (impliedly, with rent and trade,c1m (1)
R
t andR
t(2) obtains). Given the transition probability matrix A=Pr⎡⎣C Ct t−1⎤⎦, if we impose symmetry on the system to simplify the MI space to be two-state for each variable, for the sake of clarification (say rent is either equal to zero or otherwise; and tradability defined by with or without trade flow), then N = M = 2. In this case the state vector can assume four different valuesCt
{
11, , ,12 21 22}
C
t ≡c c c c
and the matrix A is a 4 x 4 matrix which correspond directly to say perfect integration ( combining either ), which reads, rent is zero at expectation with trade equals zero or not ; imperfect integration ( and ) and segmented market conditions ( and ).c11 c21 or c22
c12 c22
c12 c21
The rationale behind the Multi–Chain Markov Switching model is the flexibility to specify multivariate process such that the switching mechanism across regimes makes it possible to express the state for one variable dependent on the lagged states of all the variables in the system. Gallo and Otranto (2006) utilise the MCMS to test for Asian stock markets interdependencies, contagion and independence. From our conceptual settings, the functional dependence structure of the regime dynamics and more importantly the state overlaps of rent-trade state level relations makes the MCMS likely candidate to correct potential misleading conclusions of calibrating MI via the six-regime market conditions implied by the PBM.
Since the properties of the MCMS are founded on same theoretical views as those of standard Markov switching models, Otranto (2005) suggests filtering and smoothing procedures described by Hamilton (1989) and Kim (1994). Because of computational complications associated with this type of modelling, as noted in Krolzig (1997), some restrictions are required
on the general model (4.18) in order to make it tractable, and also to retain consistent interpretation of the results according to the specific application at hand.
4.2.0 Summary and Concluding Remarks
In this section, we have demonstrated that given the flexibility of hidden Markov models and the fact that market equilibrating processes fall within a complex time series system, HMMs (Markov switching in particular) methods can directly be adopted in market integration analysis.
It has been argued from the basic structures of HMMs representations and existing regime switching models of MI measurement that Markovian framework is consistent for MIA based on the dynamics and nonlinearity of markets inter-relations as implied by spatial market equilibrium and tradability concepts, and their resultant arbitrage conditions.
Specifically, from spatial equilibrium and tradability theories, since market integration can be assessed by;
(a) arbitrage conditions (outcomes), i.e. no arbitrage, arbitrage failure or autarky ruling, or
(b) periods with or without trade.
HMMs are flexible regime switching tools for MI assessment. It is also demonstrated that since in typical PTE the basic representation of market integration is described by the adjustment parameters (especially of the ECT), which naturally implies arbitrage process, with relatively high frequency data, rent and or trade dynamics can directly be reflected in the equilibria representations along TAR settings without assuming a priori market integration in switching framework. In effect, two variants of HMMs have been proposed for MIA, defined within two major lines, by taking into account both short- and long-run processes and roles of various market data:
1) Markov switching equilibrium model MS-(V)EC
2) Markov-switching multi-chain model MSMC.
Thus, MI dwells not just on whether the two price series are inter-related, but more importantly how they differ conditional on the transactions cost component. The models above combine these two tenets of MI notions in equilibrium framework. Put differently, given the transactions cost, prices dynamics defined by adjustments in rent series and switching equilibrium conditions are represented. Our choice for Markovian framework is based on its flexibility.
It is however obvious, as it is always in economic issues, that the models outlined here are more or less specific given ones knowledge and underlying theoretical assumptions about the markets in question. Although given the strong growing evidence of non-linear time series dynamics in market economic systems our proposed models can be seen as a benchmark for integrated and robust tools for MI analysis. The basic models, as defined above can be extended to take into accounts all sorts of conceptually consistent notions of market integration- asymmetry and more importantly imposing variational-restrictions on the TC component to account for a particular policy effects.
In the next section we implement Multivariate MS models by analysing an ideal market data along side classical MI (PBM and b-TAR) tools with a synthesized series.
SECTION FIVE
The aim of this section is to implement our proposed MS-VEM model in the previous section and to evaluate how they can identify market integration patterns from empirical perspectives. We approach this by using synthesised ideal market data (prices, trade flow volumes and transactions cost) where crucial inter-markets conditions are imposed under guided assumptions. These assumptions are based on market equilibrium and arbitrage theories that drive classical approaches as outlined in the previous sections. Our focus is to evaluate how MS-(V)EM can be used to infer the very insights PBM and TAR models generate in complex non-linear market equilibrium conditions.