Given the estimated timing of the structural change, we compare means and covariances before and after the structural change. Furthermore, we perform the CHP test which is widely used for detecting an existence of Markov switching.2 The result is reported in Table 2, where the test is performed sector by sector. Regarding the timing of the irreversible structural change, Figure 3 reports the estimated probability of being at regime Dt = 0.
This confirms that the multi-sector estimate is a key to infer the timing of the irreversible structural change. Given the estimated period of the structural change, we then compute mean vectors and covariance matrices for each regimeSt before and after the structural change. The irreversible structural change thus does not have a statistically significant impact on means and volatilities.
Before the irreversible structural change, the correlation structure of the recession regime is slightly tilted towards the positive direction than the boom regime. Moreover, the correlation structure of the booming regime has also shifted in the positive direction after the irreversible structural change. These results suggest that the irreversible structural change has strengthened the comovation of asset returns in both boom and recession regimes.
To confirm the above observations, we test for each boom and recession regime whether the irreversible structural change has increased or decreased the correlations.
4 Asset Allocation Problem
After the irreversible structural change, the same table also shows that 80% of correlations are higher in the recession regime, while none of the correlations are higher in the boom regime. See Tables 14 and 15 in the Appendix for the hypothesis test results on correlation coefficients before and after the structural change, based on which we created the summary table (this is Table 7). The result after the irreversible structural change also contrasts strongly with that of hypothesis testing of correlations without the irreversible structural change, where only 22% of correlations are higher in the recession regime at the 10% significant level.
Only regime information (ignore structure) investor's portfolio is ϕS and only structure information (ignore regime) investor's portfolio is ϕD. We have assumed that the initial wealth of investors, W0A, A = F, S, D is equal to 1 and that the investors cannot sell short. In practice, we use the estimated parameters of three models – the full model (0< q00<1, full information investor's assumption), only recursive regime switching model (q00= 1, only regime information investor's assumption) and only structural change model (structural information investor's assumption only, see Appendix 6.2, Model D1) – to calculate the moments of the industry indices considered by the investors.
We generate log-return processes using the Monte Carlo method using the estimated parameters of the full model. Investors react to their available information and rebalance the minimum variance and tangency portfolios. For example, if the regime is extreme (St = 0) and the structure at time 1 (Dt = 1), then the investor of the first type chooses the optimal portfolio for one month with the information St = 0 and Dt = 1, but the investor of the second type chooses the optimal portfolio with information St = 0 and Dt = 0, since it only has information about the market regime and assumes that Dt = 0.
The only investor with structure information does not change her portfolio, regardless of the value of the initial state variable S0. If the investors use the global minimum variance portfolio strategy, when the Sharpe ratio is compared, the negative effect of ignoring structure D (using only the information from the recursive regime S) is greater than ignoring the recursive regime S (using only the information from structure D). On the other hand, the negative effect of ignoring the recursive regime is greater than ignoring the structure when the tangent portfolio strategy is adopted.
We thus conclude that both the recursive state and structure are important information about the investment.
5 Concluding Remarks
6 Appendix
The EM Algorithm
Hamilton [1990] shows that if we repeat these steps ad infinitum, then the sequence of parameters obtained by this algorithm converges to maximum likelihood estimators. These updating formulas are obtained from the first-order maximization conditions of Q(Θ; Θ(0)) with respect to Θ. Let's assume we've evaluated the iteration up to kth and explain how to update at (k+ 1) evaluations.
Forward calculation: We further assume that we have obtained P(St−1 =s|Ft−1; Θ(k)), then we have, for the next stept,. By inserting (6.1a) and (6.1b) into the recursive parameter estimation formulas above, we have updated for iteration (k+1). Note that the smoothed probabilities, which are used in the updating formulas for parameter estimation, have rich information since they estimate the probabilities of being in a given regime at time using the complete observations.
Model Specification
But, we cannot reject the hypotheses that the means of log returns are identical across the structure for the entire sector. However, differences in means and variances are not clearly identified, as is the full model. In conclusion, our proposed model - as structural change with recursive regime switching model is suitable to distinguish recursive market states (dynamics of individual means and variances) and irreversible structural change (dynamics of correlations).
Tables
