Dependent mixture model - Scenario generation

Scenario generation

6.5 Dependent mixture model

The MSAR framework is often applied towards modelling a single time series [23] [24] [21] [25] [29] [30], yet as the purpose of this analysis is to generate scenarios for asset allocation, the correlation between the indices is of great importance and cannot be ignored.

The dependent mixture model, as applied in this section, models the in-dices simultaneously. As such it is able to t the transition model while allowing for time dependent covariance matrices. Since it was concluded

6.5 Dependent mixture model 67 in the previous section that when applying regimes, the AR eect dis-appears (in all but one instance), now both regimes are modelled as a multivariate normal distribution. In the multivariate normal distribution the mean and variance structure in each regime is determined by

µ_R=E(Y_R)

Σ_R=Cov(Y_Ri,Y_Rj), i= 1,· · · ,9, j= 1,· · · ,9

where YR is a vector of observations belonging to regime R. µR is a vector of nine elements and Σ_Ris a9×9covariance matrix.

The fundamental assumption of the dependent mixture model is that at any time point, the observations are distributed as a mixture with R regimes, and that time dependencies between the observations are due to time-dependencies between the mixture components, that is transition probabilities between the components. These latter dependencies are as-sumed to follow a rst-order Markov process on the nite space(1,· · ·, R) [27].

The parameters in the dependent mixture model are estimated by the EM algorithm also mentioned in section 6.4. Expression (6.2) gives the joint log-likelihood of the observed data and the estimated regimes, given the parameters. This is the same method as described above, with a modi-cation to the parameter vector as well as the the observation densities, b_j(y_t) which are now given by the expression

bj(yt) =−1

2log|Σ_j| −1

2^T_tΣ⁻¹_j t

which is the likelihood function for the multivariate normal distribution, where in the usual notation

t=yt−E(ˆyt|ψt−1,Θ) =yt−µj

Referring to (6.3), (6.4) and (6.5) the observations, Y, is now a matrix with nine rows andy_t is a vector of nine observations of returns at time t. In the parameter vectorΘ,θ_1,j and θ_2,j remain unchanged relative to the previous specication, but θ3,j is updated in the dependent mixture model to (µ_j,Σ_j), that is the model parameters of the relevant model.

6.5.1 Scenario generation

As in the previous frameworks, a model is tted to data each four weeks, every time using all past observations. Following the model ts the es-timated parameters are extracted for scenario generation. As mentioned the obtained model in each regime is very simple, as all dependence on past observations has been removed. Thus predictions are made by gen-erating one random multivariate normally distributed return, with mean µ_R and covariance structure Σ_R in each regime. Additionally the pre-diction start-o regime as well as the transition probabilities between regimes are needed.

µ_R,Σ_RandPare extracted from the tted models. The start-o regime is the regime which the model t has deemed the most likely regime of the last observation. This probability, along with corresponding probabilities for all previous observations, is also extracted from the model.

From the extracted parameter values, predictions are made. The rst regime is sampled with probabilities (p,1−p) of being regime 1 respec-tively 2 where p is equal to the probability of the tted model ending in regime 1, as extracted from the model. The initial regime is used to deter-mine the parameters used to generate the rst prediction. Subsequently the following regime is sampled as a function of the current regime, now using the estimated transition probabilities, also extracted from the tted model. The new regime is used to make the next prediction, etc.

In gure 6.17the estimated regime sequence for 9 iso-distant models are shown. The gures depict the evolution in the regime sequence as more observations are added to the model tting. Each plot is 32 observa-tions apart, corresponding to approximately eight months. The black line shows the estimated regime sequence in each time step, and the red and green line shows the estimated probability of being in either regime at each time step. Starting in the upper right corner, going from graph 1 to graph 2 the certainty of current regime between observations 50 and 150 is increased, and thus the number of shifts is reduced. Adding addi-tional 32 points from graph 2 to graph 3 does not make a dierence to the estimated regime shifts.

6.5 Dependent mixture model 69 From graph 6 and onwards it is evident that the regime corresponding to calm waters is easily determined while the other regime, corresponding to nancial instability, is a lot less obvious, resulting in periods with numerous regime shifts in very short intervals. The explanation for this is found, at least in part, by referring to gure6.12, where it was seen that the indices change regime at very dierent points in time. Exemplied by ELR one regime is well determined, while the other regime is somewhat undetermined and the state probabilities jump back and forth. The plot illustrates the process of estimating the regimes as more observations are included. Note in the last plot that the regimes have been shifted, so the green regime is now e.g. high volatility where it was low volatility in the previous models. This has no inuence on the estimation.

When considering the correlation matrices in the regimes in gure 6.18, it is apparent that the correlation is generally higher in one regime, thus implying the indices agree about where they are, whereas the other regime show less correlation. While there seem to be a general trend similar to that of gure 6.4, the correlation is lower in gure 6.18b. The correlation coecient change slightly along the course of the 79 periods, but the general trend as described remains. In the dependent mixture model, as mentioned, the model must determine one common point in time to change regime. Yet considering the very dierent behaviour of data, this apparently causes the model to move the regime shifts back and forth when new data is added to the estimation, causing the model to appear confused.

The result is that the models along the course of the 79 periods end in dierent regimes every other time. In gure 6.19is shown the estimated probabilities of the model ending in regime 1, through the course of the 79 periods. This is also the probability used to sample a starting regime for the scenario generation. Due to the high probabilities in almost all periods, this is with fair certainty also the start regimes for the scenario generation. To be fair there is a short period of consistency around period 20 and again between 30 and 40. These are the only periods of time where the nine indices have to a sucient degree agreed on an appropriate regime. this despite their highly correlated returns.

0 50 100 150 200 250 300

0.01.02.0

Trailing 2006−02−03

0 50 100 200 300

Trailing 2006−10−13

0 100 200 300 400

Trailing 2007−06−22

0 100 200 300 400

Trailing 2008−02−29

0 100 200 300 400 Trailing 2008−11−07

0 100 200 300 400 500 Trailing 2009−07−17

0 100 200 300 400 500 Trailing 2010−03−26

0 100 200 300 400 500 Trailing 2010−12−03

0 100 200 300 400 500 600 Trailing 2011−08−12

Figure 6.17: The red and green lines show the probability of being in regime 2 respectively 1. The black line show the chosen regime at time t. The plots illustrate the progression of the models attempts to determine the time of regime shifts as a compromise between the indices as more information is added.

Heatmap of numeric correlation matrix

DGT ELR FEZ GLD RWX STN STZ TOPIX XOP

DGTELRFEZGLDRWXSTNSTZTOPIXXOP

0.0 0.2 0.4 0.6 0.8 1.0

(a) Regime 1

Heatmap of numeric correlation matrix

DGT ELR FEZ GLD RWX STN STZ TOPIX XOP

DGTELRFEZGLDRWXSTNSTZTOPIXXOP

0.0 0.2 0.4 0.6 0.8 1.0

(b) Regime 2

Figure 6.18: Full dataset correlation matrices of the nine indices in respectively regime 1 and 2.

6.5 Dependent mixture model 71

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0 10 20 30 40 50 60 70 80

Estimated probability of regime 1 0.00.51.0

Figure 6.19: Sequence of scenario generation start regimes over the course of the 79 considered periods.

6.5.2 Results

Evaluation of the method is carried following the same method as previ-ously. Figure 6.20 shows that all of the trends apparent in the previous similar plots are also present in this case, however at a less pronounced level. The fractions are generally evenly distributed over the interval with-out apparent white spots in the plots. XOP is trumpet-shaped, and is only barely accepted as uniformly distributed. The variance is noticeably lower than the theoretic ideal of 0.0833. ELR is rejected as uniformly dis-tributed. There is a tendency that the fractions cluster towards the bot-tom of the [0,1] interval. Whereas TOPIX also seem to have the fraction points cluster, the fraction span most of the [0;1] interval, distinguishing it from ELR.

Figure 6.19shows that the regimes jump back and forth, but this is not reected in the variance of the scenarios, as depicted in gure 6.22. The reason is that while the regimes jump back and forth, the model param-eters do the same. The result is a realised mean return and variance of returns as illustrated in gure6.23. The plots show the model parameters related to each period assuming the scenarios are generated from the ex-iting regime of the model t. Supporting plots are attached in appendix Figure A.8.2.

Looking over gure 6.22 it is seen that there is a brief but consistent increase in scenario variance in periods 1517. There appear also to be a permanent increase in scenario variance starting in period 22, although

Time

Figure 6.20: Fractions of scenarios which exceed the observed return (second axis) in each period (rst axis).

0 5 10 15

−0.20.41.0 p=0.3353 var=0.0819

DGT p=0.0127

var=0.0705

ELR p=0.7413

var=0.0854 FEZ