8 Financial Data
8.3 Dynamic Factor Analysis
In this section, we will evaluate the performance of the new mixed-state factor model de-veloped in section 3 using objective measures and different specifications. We trained factor models which differ by their hidden volatility structure on the 600 observations of the in-ternational exchange rate returns data from 03/05/1991 to 07/05/1993. The key point for initialization is to start with a good segmentation of the data set where, by segmentation, we mean a partition of the data, such that each part is modeled by a particular dynamic factor model. For our model, the best strategy of initialization consists in implementing a standard EM algorithm by supposing that the factors are homoscedastic6. Thereafter and given the output of the EM algorithm, we can use the posterior probabilities p(St = j/Y) in order to obtain the optimal sequence of hidden states. At the second step, a particular simple conditionally heteroscedastic factor model is initialized for each segment. For this, one can use the empirical covariance matrices as estimates of the idiosyncratic variance ma-trices Ψj and the empirical means as estimates of the means θj. The parameters of the
6In practice, 20 iterations of the EM algorithm are largely sufficient.
conditionally heteroscedastic variances are initialized by applying a GQARCH(1,1) model to each segment of data. Finally, the elements of the transition matrix P, can be initialized by counting the number of transitions from state i to state j, (i, j = 1, ..., m), and dividing by the number of transitions from state i to any other state.
In order to identify the number of common factors and discrete states in the model, we estimate different models, assuming that each state variable (discrete and continuous) can take one to three states and choose the model with the lowest criterion presented in section 7.3. For simplicity we have assumed that the coefficients of the conditionally heteroscedastic variances are constant for all t.
Our proposed model must also be further constrained to define a unique model free from identification problems. A first constraint is that Xt be of full rank k, ∀ t, to avoid identification problems arising through invariance of the model under location shifts of the factor loading matrix (e.g., Geweke and Singleton [19]). Second, we must further constrain the factor loading matrix to avoid overparametrization - simply ensuring that the number of free parameters at time t in the factor representation does not exceed the q(q + 1)/2 parameters in an unrestricted Σt. Finally, we need to ensure invariance under invertible linear transformations of the factor vectors (Press [32], chapter 10). On this latter issue, our work follows Geweke and Zhou [20], among others, in adopting the "hierarchical" structural constraint in which the loadings matrix has the form:
Xj= immediately ensures that Xj is of full rank k.
Table 8 provides summaries of the various information criteria for assessment of the number of factors as well as the number of hidden markovian states. Figure 11 shows the evolution of the log-likelihood function of the different specifications during the EM iterations. The overall suggestions is that k = 2 and m = 3 is strongly favored. The summaries of the empirical analyses with k = 2 and 3 conditionally heteroscedastic common factors within a 3 state hidden markovian structure are, respectively, given in tables 9 and 10 and figures 12 to 19. In the first case (when k = 2), the transition matrix and the initial state probabilities are given by:
P=
0.9773 0.0227 0.0000 0.0000 0.9698 0.0302
0 50 100 150
−5000
−4500
−4000
−3500
−3000
−2500
−2000
The EM iterations
Log−likelihood
k=1 m=2 k=2 m=2 k=3 m=2 k=1 m=3 k=2 m=3 k=3 m=3
Figure 11: Likelihood of different specifications with conditional heteroscedasticity.
Table 8: Values of the log-likelihood, AIC and BIC statistic for the chosen factor models estimated on the period March 05, 1991-July 05, 1993.
Criterion k = 1 k = 2 k = 3
m = 1
log-likelihood (-) 3904.5 (3921.2) 4321.6 (3755.9) 4279.4 (3759.6) AIC 7865.0 (7890.3) 8699.2 (7575.9) 8638.8 (7599.2) BIC 7988.1 (7995.8) 8822.3 (7716.6) 8814.7 (7775.1)
m = 2
2648.7 (2660.4) 2512.6 (2531.6) 2404.7 (2506.3) 5413.5 (5428.8) 5145.1 (5203.2) 4961.3 (5184.6) 5668.5 (5666.2) 5408.9 (5511.0) 5295.5 (5562.8)
m = 3
2381.2 (2400.7) 2202.7(2225.7) 2207.5 (2253.4) 4938.4 (4969.3) 4631.4(4715.3) 4685.0 (4722.8) 5325.3 (5338.7) 5128.3(5295.7) 5278.6 (5197.7) . Models with conditional heteroscedasticity, (.) Standard Models
Using this specification, figure 12 shows how the model is capable of accurately detecting abrupt changes in the DEM time series structure and, in particular, the severe disruption by the violent storm which hit the European currency markets in September and October 1992, following the difficulties over ratifying the Maastricht Treaty in Denmark and France.
We can clearly see from figure 13 that the third model is responsible for the high volatility segments, the second model is mainly responsible for the time period before August 1992, and the first one for the lower volatility segments after October 1992. Figure 12 shows also that the average duration stay in the first regime is about 37.8 weeks versus 76 in the second and 6.2 in the third. Some other interesting points arise from this analysis:
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2 3 4
Absolute values of the DEM returns
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The crisis period
Figure 12: Graphics 2,3,4: Posterior probabilities of the three hidden states Mt/n(j) given by the smoothing algorithm.
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Figure 13: Volatility of the different series using three states and two conditionally het-eroscedastic factors.
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Figure 14: Two factor model: Proportion of the time series variances explained by each of the factors (common and specific), from 03/05/1991 to 07/05/1993.
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1 1.005
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Figure 15: Three factor model: Proportion of the time series variances explained by each of the factors (common and specific), from 03/05/1991 to 07/05/1993.
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Figure 16: Two factor model: Common factors means and their volatilities, the diagonal elements of Ht (03/05/1991 to 07/05/1993).
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Figure 17: Three factor model: Common factors means and their volatilities, the diagonal elements of Ht (03/05/1991 to 07/05/1993).
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Figure 18: Two factor model: Time series co-dependence structure from 03/05/1991 to 07/05/1993.
Figure 19: Two factor model: Autocorrelation functions of the residuals.
Table 9: Two factor model with conditional heteroscedasticity
θ X diag(Ψ) φ1 φ2
0.0127 0.7044 0.0000 0.0010 0.0860 0.0826 -0.0082 0.6778 0.0178 0.0869 0.1071 0.1504 -0.0414 0.1107 0.5790 0.0157 0.0826 0.1919 State 1 -0.0467 0.0693 0.6167 0.0817 0.8742 0.6332
-0.0548 0.0790 0.4661 0.3194 -0.0457 0.0953 0.6159 0.0007 0.0915 0.4798 0.2731 0.3692 0.0120 0.6997 -0.0054 0.0015
-0.0035 1.0375 0.0000 0.0004 0.0860 0.0826 -0.0159 0.9688 0.0237 0.0402 0.1071 0.1504 0.0188 0.0030 0.3176 0.0031 0.0826 0.1919 State 2 0.0116 -0.0347 0.3163 0.0725 0.8742 0.6332
0.0131 0.0211 0.3039 0.0134 0.0212 -0.0419 0.3289 0.0037 0.0129 0.5367 0.0277 0.1599 -0.0009 1.0302 0.0044 0.0032
0.0492 0.5576 0.0000 0.0034 0.0860 0.0826 0.0042 0.5914 -0.0706 0.1615 0.1071 0.1504 0.0290 0.0833 1.4265 0.0318 0.0826 0.1919 State 3 0.0562 0.1109 1.1904 0.4202 0.8742 0.6332
-0.3031 0.0384 1.0897 1.7904 0.0067 0.0654 1.4910 0.0348 0.1477 0.4625 0.2418 0.2974 0.0315 0.5788 0.0009 0.0400
1. From figure 14, It appears, that the second factor is responsible for time evolving movements in the variances of the European currencies, FRF, CHF, ITL and DEM.
Figure 15 shows that the third factor seems to provide little or nocontribution to the fit of the model.
2. Looking at the common factor variances (figures 16 and 17), it can be argued that the first two factors are indeed more important than the third one. Figure 13 shows marked changes and increased volatility in the European currencies across the board towards the end of 1992 when Britain and Italy withdrew from the European exchange rate agreement. The impact of this event is evident in the estimated trajectories of both the posterior volatilities of currencies and in the corresponding variances of the common factors.
3. Notice also that the end-1992 volatility changes impact across all factors, highlighting the apparent dependencies in factor trajectories across the entire time period. This indicates the need for dependence structure in modeling latent volatility processes in dynamic factor analyses, as is allowed in the theoretical framework described in section 3. For the two common factors, the sum of the estimated αiand δi is slightly less than
Table 10: Three factor model with conditional heteroscedasticity
θ X diag(Ψ) φ1 φ2 φ2
0.0226 0.6805 0.0000 0.0000 0.0011 0.0692 0.5184 0.0788 0.0001 0.6541 0.0090 0.0000 0.0875 0.0825 0.1122 0.1312 -0.0379 0.2243 0.2386 -0.0894 0.0032 0.0605 0.2956 0.1586 State 1 -0.0381 0.1490 0.2396 0.4421 0.0732 0.9090 0.5884 0.1866
-0.0514 0.2467 0.1897 0.0148 0.3175 -0.0396 0.2054 0.2434 0.2207 0.0018 0.1041 0.4463 0.1039 0.3130 0.3588 0.0219 0.6764 -0.0024 0.0008 0.0013
-0.0033 0.9921 0.0000 0.0000 0.0004 0.0692 0.5184 0.0788 -0.0176 0.9312 0.0076 0.0000 0.0415 0.0825 0.1122 0.1312 0.0202 0.0039 0.1181 0.0173 0.0032 0.0605 0.2956 0.1586 State 2 0.0106 -0.0260 0.1130 0.7365 0.0125 0.9090 0.5884 0.1866
0.0151 0.0185 0.1139 -0.0002 0.0135 0.0229 -0.0400 0.1224 0.0121 0.0037 0.0110 0.5178 0.0059 0.2120 0.1573 -0.0006 0.9847 0.0022 -0.0155 0.0032
0.1452 0.8805 0.0000 0.0000 0.0024 0.0692 0.5184 0.0788 0.1197 0.9175 -0.0775 0.0000 0.1673 0.0825 0.1122 0.1312 0.0855 0.2714 0.6790 0.1739 0.0578 0.0605 0.2956 0.1586 State 3 0.1267 0.3729 0.6019 -0.6542 0.3393 0.9090 0.5884 0.1866
-0.2990 0.1667 0.5007 3.4690 0.6658 0.0571 0.2660 0.7232 0.0800 0.0101 0.2352 0.7487 0.1151 0.2919 0.2972 0.1249 0.9215 -0.0026 -0.1860 0.0394
one. It indicates strong GARCH effects and persistence in the volatility of exchange rates.
4. From figure 14 it can be observed that the first common factor explains at least 95%
of the USD, CAD and HKD currency’s variances at all times (and at least 99% before the 1992 crisis for the USD and HKD). This factor explains also 70% of the Japanese currency variances before August 1992 and 50% after this date. The contribution of the second factor in the variance of these currencies is negligible, except for the JPY where the contribution is about 10% after August 1992. As a matter of fact, the variance dynamics of the JPY are determined at 50% (at most) by its idiosyncratic variances after the 1992 crisis.
5. From this figure it can also be observed that the second factor is responsible for about 90% of the variances of the FRF, ITL and DEM currencies before the 1992 crisis. For the CHF, the contribution of the second factor is about 80% after August 1992 and 40% before this date. The contribution of the first factor in the variances of these European currencies is in particular meaningless before August 1992.
Broadly, the results show that all the correlations between the European currencies have increased just after August 1992 (figure 18). This is the effect of financial contagion that can be defined as a significant increase in comovement of financial prices such as the exchange rate, interest rates, and stock prices experienced by a group of countries, after controlling fundamentals (domestic news, ...) and common shocks, following a crisis elsewhere. The first factor represents the value of sterling relative to a basket of currencies in which the HKD, USD and CAD are dominant. Table 9 shows that the USD, CAD and HKD are roughly equally weighted, which is expected as CAD and HKD rates are heavily determined in international markets by USD rates. This first factor may be termed the North American factor. The second factor may be similarly termed the European Union factor. It represents a restricted basket of currencies dominated by the European currencies, with a relatively reduced weighting on JPY. USD, HKD and CAD are practically absent from this factor with x1,2j, x2,2j and x8,2j for j = 1, 2, 3 indicating very small values. Inferences about idiosyncratic variances strengthen and extend these conclusions. Those of USD and DEM are very small, indicating that these two currencies play determining roles in defining their sector factor. CHF and ITL have larger idiosyncratic variances (during the crisis period), indicative of their departures from their sector factors. Finally, the sample autocorrelation functions of the residuals reported in figure 19 show no autocorrelation. The Ljung-Box statistic for the serial correlation of the squared residuals does not also reject the null hypothesis of uncorrelated squared residuals. Hence, all the covariance or correlations between the different exchange rate returns are explained by the common and specific factors.
9 Conclusion
Two powerful and popular tools of modeling conditionally heteroscedastic financial data characterized by dynamic changes over time (which can be attributed to various reasons, such as: wars, changes in economic policy and business cycles), in a multivariate framework, are linear factor models and HMM’s. In this article we proposed a model that combines these tools and we called it Conditionally Heteroscedastic Factor Analysed Hidden Markov Model.
We formulated the model and developed maximum likelihood estimates for its parameter.
An iterative method based on the generalized EM principle, was adopted to circumvent the statistical inference problem inherent to the modeling of regime switching within the conditionally heteroscedastic structure of the common latent factors.
Our simulation experiments have demonstrated promising results in classification of the volatility behavior. Using two model selection criteria, we demonstrated accurate discrimi-nation between specifications characterized by different hidden structures. We showed that switching conditionally heteroscedastic factor models provide more robust tracking perfor-mance than simple standard models or conditionally heteroscedastic models without regime switching. The fact that these models can be learned from data may be an important advantage in financial applications, where accurate on-line predictions of the time varying covariance matrices are very useful for dynamic asset allocation, active portfolio management and the analysis of options prices.
From a broader viewpoint, this study illustrates the usefulness of Markov regime switch-ing models in the analysis of process that exhibit only local homogeneity. Such complex process can be found in a variety of scientific fields, and we believe the ideas presented here can be successfully applied in many such contexts. The analysis in this paper can be also extended in several ways. First, our model can be generalized to one where we allow the idiosyncratic variances to be a stochastic function of time. Secondly, we can also think of the case where the state transition probabilities are not homogeneous in time, but depend on the previous state and the previously observed covariates levels. The study of such models would provide a further step in the extension of hidden Markov models to dynamic factor analysis and allow for further flexibility in applications.