We shall refer to this regime as "regime 1". In the second regime, equity returns have higher conditional means, lower volatility and are less correlated. Our regimes definitely correspond to periods of low and high volatility, but the evidence for significantly different conditional means and correlations across regimes is not as strong.
Table 5 presents p-values of Wald Tests for parameter equality across regimes for the various RS n particular see Hansen and the discussion in Ang and Bekaert However, joint tests fail to reject the null of constant conditional means for both countries. The Wald test for the Earnings Model rejects that the parameters of the conditional mean ,a st , 3 st are jointly equal across regimes. Evidence is also extremely strong for different regime-dependent volatilities for the Beta Models.
The evidence of different correlations across the regimes is not particularly strong. We cannot reject that correlations for the UK and Germany are constant across regimes. Similarly, for the Beta Models, where covariances are implied by the 3's of the individual assets, and for the Earnings Model, we cannot reject equality of the correlation across regimes.
To concentrate on the effect of changing covariances we estimate models where u1 and i2 are im- posed to be equal across states but different across assets. Table 6 shows that these models cannot be rejected when comparing them against their unconstrained specification. Moreover, regime classifica- tion, as measured by the Ang-Bekaert Regime Classification Measure RCM , generally improves slightly when this restriction is imposed. For intuition we specifically discuss this model in Section 5.
The turbulent equity returns of the OPEC oil shocks in the mid's are picked up, as is the crash. From the mid's onwards, the implied ex-ante probabilities place the economy almost definitely in the second regime.
The expected duration of the first regime is 6. The stable probabilities implied by the transition matrix are 0. Lower RCM values indicate better regime classification. I is the ex-ante probability of being in the first regime. We see in the top panel of Figure 2 that as Pt increases, the conditional mean of the US and UK become negative, but the standard error of the estimates also increases markedly. This reflects the increased uncertainty about the parameter estimates in the first regime.
The conditional volatilities in the middle panel likewise increase substantially when p increases. The plots also show that the conditional means and variances of the US and UK move in tandem, as is true in any one-factor model. In our Simple RS Models this factor is the ex-ante regime probability. The bottom panel shows the conditional US-UK correlation. The plots clearly show that the higher volatility regime is also associated with higher correlations.
In both cases we fail to reject the Simple Model. Moreover, the parameters A and. This lends support to the simple, but parsimonious DGP of the Simple RS Model: the US and UK face the same regime shifts and the stochastic volatility generated by the simple RS Model suffices to capture the time variation in monthly equity return volatilities.
The higher volatility in the first regime is driven by three parameters in this model. First, world volatility is higher in the first regime. Second, the 3's are invariably higher in the first regime. Third, the idiosyncratic volatilities are higher in the first regime. It is never possible to reject that the 3's are significantly different from 1 in the first regime, but they are often significantly below 1 in the second regime, which is more influenced by the idiosyncratic shocks.
The 3's of the unhedged returns are larger than the 3's of the hedged returns, reflecting a positive currency return! Our model implies that, conditional on the regime, the currency premium is constant. In Table 8 we report the state-dependent and unconditional currency premiums and the volatilities of the currency returns. The unconditional premium is approximately 1. In our model, the actual premium varies over time with the regime probability and can potentially change signs.
However, we estimate the premiums to be positive in both regimes, implying that US investors are always compensated for taking foreign exchange risk. For regime 1, the currency premium is very small and insignificant.
The smaller currency premium, combined with the larger volatility may contribute to home bias in this regime. Economically, the second regime corresponds to "normal" periods where interest rates are low and equity excess returns are posi- tive. Interest rates behave like a random walk, perhaps because of the monetary policy smoothing efforts of the US Fed. The first regime corresponds to "turbulent" periods of high monetary uncertainty with very volatile negative equity returns.
The parameter estimates of 3 St in equation 30 have very large standard errors in both regimes. The themselves standard errors on other parameters are also much larger than in the Basic Model. The probability coefficients in equation 31 are significant, and a constant probability version is rejected by the Basic Model.
In particular, b2 is negative and highly significant, so in normal periods, as the short rate increases a transition to the first regime becomes increasingly likely. Comparing the ex-ante probabilities of the Basic Model in Figure 3 to the Simple US-UK Model, we see that the Short Rate Basic Model classifies the early 's as regime 1, which is driven by the highly turbulent short rates during the monetary targeting period.
The implied conditional correlations for the Basic Short Rate Model are presented in the bottom panel of Figure 3. Interestingly, short rates and equity returns are more negatively correlated in regime 1 than regime 2. In addition to the low conditional means of equity, this implies that holding equity is even more unattractive in the first regime.
As short rates are high during this period risk-averse investors will want to hold mostly cash. In particular, in the second regime, b2 in equation 31 is significant and negative, so as the earnings yield increases a transition to the first regime becomes more likely.
The predictability coefficients 13 in the conditional mean of equity returns are significant in both regimes for the US, with a strong effect in regime 1, and significant for the UK in the first regime. In Table 7 , a likelihood ratio test for this model versus the null of no predictability gives a p-value of 0. There is borderline significance for each case of predictability through the conditional means and through the transition probabilities.
In the first regime, earnings yields have higher conditional means, are more mean-reverting and have higher conditional volatility. The average earnings yield conditional on regime 1 2 is Lower earnings yields on average are associated with normal periods as higher prices relative to earnings push down the earnings yield.
The Earnings Yield Model, like the other RS models, has average equity returns conditional on the regime being lower and more volatile in the first regime.
For the US the average equity returns in regime 1 2 are Similarly for the UK, the average equity returns in regime 1 2 are We will first discuss general results across all the models and tabulate results for risk aversion levels of '' 5 and The portfolio weights, with statistical tests, are presented in Tables 9 to Tables 13 to 17 present the economic compensation required under various sub-optimal strategies.
The US, because of its lower volatility in the first regime compared to overseas equity, becomes a "safer" asset. Risk averse investors choose to hold more of the US at the expense of international equity during the downturn state. Portfolio weights as a function of y are shown in Figure 4. The more risk-averse the investor, the greater the proportion of the US held in both states. Table 13 presents the "cents per dollar" compensation required for an investor with an all-equity portfolio to hold only the US instead of investing optimally with overseas holdings.
Table 13 shows that at a 1 month horizon, the costs of holding only US equity are small and, as expected, grow with the horizon. At one year we need a compensation of 1. This is because of the particular covariance structure in regime 1. We might expect that as correlations are higher in state 1, the costs of no international diversification in that state will be less than in state 2.
This is generally not true. In the three country model the optimal holdings of both US and Germany rise, making diversification more valuable in this regime. Figure 5 shows that even for the US-UK, the benefits of diversification for state 1 may be greater than for state 2 for small -'.
The bottom panel of Figure 5 shows that because of the benefits of holding Germany in state 1, the costs of no international diversification are uniformly higher in state 1 than in state 2. In the bottom panel of Table 13 , costs for various levels of the earnings yield for not holding overseas equity are presented.
In the second regime, the cost of not diversifying is on average smaller, being only 0. As the earnings yield increases the cost of not diversifying increases in both regimes. In this model no international diversification refers to holding neither hedged nor unhedged foreign equity. To obtain a measure of the benefits of currency hedging, we need to obtain the optimal portfolios under the restriction that only investment in unhedged equity can be made. This shows that the costs of not using currency hedging, like the costs of not internationally diversifying, are relatively large.
We can compare the two panels in Table Currency hedging contributes about half of the total benefit of no international diversification under the RS Beta Models.
Table 11 shows the asset allocation weights for the RS Beta Models. We also list the proportion of the portfolio covered by a forward contract position, which is unrestricted. In the RS Beta Models, short positions in the forward contracts hedge the currency risk of the foreign equity position. These positions are statistically significant. The Tables also list hedge ratios, which are the value of the short forward position as a proportion of the foreign equity holdings.
We will look at the effects of regime- dependent weights first. The weights reported in Tables 9 to 1. Nevertheless, the standard errors are often large and we cannot reject the null that the portfolio weights are constant across regimes for many cases.
The economic costs of ignoring RS range from fairly small to substantial at high levels of risk aver- sion. When investors ignore regime switching they are assumed to hold myopic weights implied by fitting an lID multivariate normal distribution to the equity returns. These weights are an approximate average of the regime-dependent weights. The lID weights give a reasonable approximation to the optimal weights in each regime, especially in regime 2 which has the longest duration.
Finally note that the cost of ignoring RS is higher in state 1 than state 2. This is in accordance with intuition, since in the normal regime, conditional means and variances will be closer to their uncondi- tional counterparts, than they are in state 1. The markedly different behavior in state 1, which can persist for several periods, makes the costs of ignoring RS higher in this regime.
Figure 5 plots the costs of ignoring regime switching for the Simple RS Models as a function of -y. Hedging demands are never significant, and the p-values are generally very large.
Brandt also cannot reject myopia in his non-parametric estimate of domestic asset allocation weights. The Tables also show that the convergence of the portfolio weights is extremely fast.
After 3 years, the portfolio weights are constant. The convergence is even faster than in Brandt , who finds convergence after 15 years. His setting however, incorporates instrument predictability and rebalancing at intervals greater than I month. With only regime changes and monthly rebalancing horizon effects become even smaller. The economic costs of myopia are effectively zero. Table 16 lists the compensation required for an investor to hold myopic portfolio weights instead of the optimal T horizon weights.
The numbers are astoundingly small for all models. This evidence suggests that investors lose almost nothing by solving a myopic problem at each horizon, rather than solving the more complex dynamic programming problem for longer horizons. Let us examine the portfolio weights in Table 9. In regime 1, the point estimates show that investors hold more US equity.
The US acts as a "safer" asset in this regime. The large standard errors, though, mean that statistically there is weak evidence that the true portfolio weights change across regimes. Looking at Table 9 , notice that as the horizon is increased the point estimates of the holdings of US equity increase with horizon. That is, with increasing horizon, investors want to hold more of the less risky asset.
Samuelson works with two assets, cash and and a risky asset. The risky asset follows a Markov chain where the returns can be "low" or "high". He defines a "rebound" process, or mean-reverting process, as having a transition matrix which has a higher probability of transitioning to the alternative state than staying in the current state. That is, under rebound, long horizon investors are more tolerant of risky assets than short horizon investors.
Our setting is the opposite of a rebound process. Samuelson calls such a process a "momentum" process: it is more likely to continue in the same state, rather than transition to the other state. Under a momentum process, 21A full list of parameter estimates is shown in Table A-I as part of the supporting Table Appendix.
Even with the much more negative conditional mean for the US in state 1. In the case of the Short Rate Model Figure 6 the "less risky" asset is cash. Note that in the Earnings Yield Model, the change in US equity holdings depends on the prevailing earnings yield. Intuitively, the long-run volatility is smaller under a rebound process than under a momentum process with the same short-mn volatility.
In our setting, the risky asset is overseas equity, and the safer asset is US equity. The persistence of our regime probabilities means that investors with longer horizons hold less foreign equity, so long-horizon investors are less tolerant of holding more risky overseas equity than short-horizon investors. However, Section 5. From Section 4. Portfolio weights as a function of the short rate and regime are presented in Figure 6. The top two plots show the asset allocation weight for US and UK equity in regime 1 and 2 and the remainder of the portfolio is held in cash.
The Figure shows that the hedging demand is small, and is only visible for the first regime. In regime 2, as the short rate increases investors hold less equity, but in regime I there is almost no effect of the short rate on the portfolio allocations. This is driven by the non-linear predictability in the probability coefficients.
The portfolio holdings in state 1 are flat because the excess returns are constant and no significant short rate predictability b1 is insignificant drives the transitions from this regime. In the second regime b2 is highly significant and negative. As the short rate increases, a transition to regime 1 becomes increasingly likely. As the first regime has much higher equity volatility, investors seek to hold less equity to mitigate the higher risk.
The effect of predictability seems much weaker in our model than in the predictability models ana- lyzed by Brennan, Schwartz and Lagnado , Kandel and Stambaugh , and Barberis These models have linear predictability 0 in the conditional mean rather than the non-linear pre- dictability in the probability coefficients and much longer rebalancing intervals than 1 month.
Table 17 presents the economic compensation required for an investor not to hold the UK. To obtain the first panel in the Table a constrained optimization problem must be solved where investors are only permitted to hold cash and US equity.
In this setting, the cost of not holding the UK is only very modest, and higher in regime 2 where US-UK correlations are lower.
The main effect of introducing the short rate as a predictor is the benefit of holding cash. The bottom panel of Table 17 shows that the costs of holding only equity and ignoring regime switching is substantial. In the top panel portfolio weights for different horizons are presented, which shows that the intertemporal demands from this model are very small. In regime 1, as the earnings yield increases, US investors seek to hold more risky UK equity.
In regime 2, this same effect is repeated at higher earnings yields, but a small hump is seen at lower earnings yields. The Samuelson effect of a small increasing exposure to the safer US asset with increasing horizon can also be seen.
These are large, but are smallest at the average value of the earnings yield conditional on each regime. In regime 1 2 , the confidence bands are smallest at Myopic weights and the null of constant portfolio weights across regimes can definitely not be rejected. The first panel lists the costs of ignoring RS and predictability where an investor holds Samuelson lID portfolio weights. The middle columns for each regime list the costs associated with the average earnings yield conditional on the regime.
The other numbers are rep- resentative "high" and "low" earnings yields conditional on the regime. The lID weights provide a good approximation to the optimal weights at most earnings yield levels making the costs to ignoring both RS and predictability small.
The bottom panel of Table 18 lists the costs of ignoring predictability but taking into account regime-switching. In this case, the constrained portfolio weights are those implied by the Simple RS US-UK Model with j jt , and are quite dissimilar from the RS Earnings Yield weights in the first regime at conditional average yield levels. Generally, this produces higher costs in the first regime relative to the lID case, which ignores both predictability and changing regimes.
For example, for a 1 year horizon in regime 1, the costs are 0. The reason for this is because at average values of the earnings yield conditional on each regime, the lID portfolio weights are better approximations to the optimal weights while the portfolio weights implied by the Simple RS Model tend to over under state in the optimal weights in regime 1 2.
The optimal myopic US weights at the conditional average earnings yield in regime 1 2 are 0. The lID portfolio weights 0. In fact, it is striking that at the average earnings yields the home bias in the first regime disappears. Looking at the estimated moments conditional on the regime it is clear why this happens. In the second regime the opposite effect is true but it is brought about by differences in the conditional means and volatility, whereas the correlation is estimated to be about 0.
In Section 6. Finally, in Section 6. Although the point statistics suggest this relationship, the standard errors on the conditional means in regime 1 are large.
This in turn may dampen the potential asset allocation effects of the high volatility regime. In order to examine this further, we re-estimate the Simple RS models constraining the conditional means to be equal across countries, but different across regimes.
These models cannot be rejected in favor of the alternative of unconstrained means p-value 0. The resulting portfolio weights are largely unchanged, with almost the same economic costs and significance levels for the statistical tests.
Consequently, our focus on time-varying covariances seems justified. The plots are very intuitive. As P increases, holdings of the safer US asset increase in both states as the expected duration of regime 1 increases.
As P1 increases the diversification benefits of holding UK equity decrease, so holdings of the US increase. Note that it is only for P1 greater than 0. Finally, as o increases the US becomes less "safe" and the proportion allocated to the UK increases.
Overall, since the correlations are similar across regimes there is little difference in the regime-dependent portfolio weights and the main effect is to alter the amount of the US held in each regime. Figure 8 suggests that of the parameters affecting the conditional distribution of returns in regime 1, the biggest effects on the regime-dependent weights come from conditional correlations and the relative difference in means.
Here we focus on the RS Simple US-UK and US-UK-GER Models and re-compute the eco- nomic costs of no international diversification, the economic costs of ignoring RS and the economic costs of myopic strategies for 1, alternative parameter values drawn from the asymptotic normal parameter distributions implied by the estimation.
We take the sample estimates to be "population values" and use the estimations where the conditional means are constrained to be equal across regimes. The distributions of the economic costs have means which are larger than their population values in Tables 13 and The median values of the economic costs are much closer to the population values. The economic cost computations can be viewed as non-linear transformations of the parameters. The transformations result in economic costs which are skewed to the right, especially for the costs of not diversifying internationally which are far more right skewed than the costs of holding lID weights.
This means that if we draw a particular set of realistic parameter values, we may likely find costs for not diversifying internationally that are substantially larger than the population values.
Finally, Table 19 shows the costs of using myopic weights are effectively zero when drawing from the asymptotic parameter distribution. We take the Out of sample period to be from January to December 11 years , which includes the crash. We use a fixed horizon of December time T and for each month t in the out of sample period we record the accumulated wealth from each strategy. At a given time t, we estimate the model up to time t. To find the appropriate portfolio weights we solve the dynamic programming problem for a horizon ofT — t.
We also find the portfolio weights for an investor using lID portfolio weights and using RS myopic weights. In the former case, these weights are estimated using the multivariate normal distribution with means and covariances estimated from data up to time t. The Table confirms that over this period there is very little difference from using RS weights and ignoring RS.
In Table 20 we see that the myopic RS strategies are almost identical since the intertemporal hedging demands implied by the Simple RS Models are very weak to the optimal RS strategy. As we increase -y, our returns become larger because more risk averse investors hold more of the "safer" US asset. The US produced the best returns over this period, which also explains the higher performance of holding only the US and UK relative to holding all three countries.
Of course, this sample includes the bull market of the 's and, apart from the few months following the crash. As the lID weights are much closer to the RS weights for regime 2, this may represent a very biased draw.
Regime-switching can potentially have a large impact in this setting by producing state-dependent portfolio weights and intertemporal hedging demands. However, the evidence on higher volatility is much stronger than the evidence on higher correlation. The regime-dependence of the means also has weak statistical significance, although the point estimates suggest that the high-volatility regime is associated with lower, and possibly negative, conditional means than the "normal" regime.
We consider a number of different settings from simple regime-switching multivariate normals to a model where short rates predict equities through their effect on the regime transition probabilities, but our main conclusions are robust across these models.
First, the existence of this high volatility regime does not negate the benefits of international diversification. When currency hedging is allowed these benefits are even greater.
Second, the costs of ignoring regime switching are small for moderate levels of risk aversion. The optimal behavior of a US investor is to switch towards US equity or cash, if available , at the expense of overseas equity when the high volatility regime is reached. It is the much higher volatility of overseas equity compared to the "safer" US equity which drives this result. However, it is not very costly not to switch, if an investor were to use lID portfolio weights even if the true data generating process were regime-switching.
Although the portfolio weights may be significantly different across regimes, the lID weights act as an "average" portfolio weight which diversifies risk well in both regimes.
This result continues to hold when currency hedging is allowed. Third, in common with the non-parametric results obtained by domestic dynamic allocation studies such as Brandt , the intertemporal hedging demands under regime switches are economically negligible and statistically insignificant. Investors have little to lose by acting myopically instead of solving a more complex dynamic programming problem for horizons greater than one period.
Our results are remarkably robust. When we draw random parameters from the estimated parame- ter distribution, the conclusions remain: for all equity portfolios, failing to diversify internationally is typically much more costly than ignoring the regimes, which, in turn, is more costly than ignoring the in- tertemporal hedging demands. However, our results remain premised on our assumptions, which include CRRA preferences, the absence of transactions costs and full knowledge on the part of the investors of the data generating process.
With transactions costs, or learning about the regime, it is even less likely to be worthwhile for investors to change their allocations when the regime changes. However, using differ- ent utility functions, for example First Order Risk Aversion Epstein and Zin could potentially cause regime switching to have much bigger effects than in the traditional CRRA utility case, and such preferences can be treated in the same dynamic programming framework considered in this paper.
We first fit a discrete Markov chain to the predictor instrument zt. From hereon, we use the word "state" to refer to the discrete states of the Markov chain which approximate the continuous distribution in each "regime state", or "regime".
The equity return shocks are correlated with the short rate, but the short rate states are the only driving variables in the system. We will show how to easily incorporate equity without expanding the number of states beyond those needed to approximate the distribution of rt. A-3 For any highly persistent process such as short rates, discretization is difficult because Pu are com- puted from a conditional distribution, and there is a different conditional distribution at each r and these may differ substantially from the unconditional distribution of Tt.
The high persistence requires a lot of states for reasonable accuracy. When a square root process is introduced, the asymmetry of the distribution and the requirement that the states be non-negative introduce further difficulties. To aid us in picking an appropriate grid for Tt in each regime we first simulate out a sample of length , from equations A-I and A-2 , with an initial pre-sample of length 10, to remove the effects of starting values.
During the simulation we record the associated regime with each interest rate. We record the minimum and maximum simulated points in each regime. For regime 1, which is the less persistent higher conditional mean regime, we take a grid over points 2. For regime 2, the "normal regime" with very low mean reversion, the persistence leads us to take a grid starting close to zero, to 2.
We also employ a strategy of "over-sampling" from the over-lapping range of the regimes. This is to aid in picking points where the discretized Markov chain is more likely to have non-zero probabilities in switching from one regime to another.
The rows of each flj To mix the 11j—. For example, for a state in the first regime, r, we calculate P11 r and P12 r' , Then the appropriate row in LI corresponding to r will consist of Pu rfl times the appropriate row corresponding to fl In particular, following Bekaert, Hodrick and Marshall , when a sample of , is simulated from the Markov chain and the RS process re-estimated, all the parameters are well within 1 standard error of the original parameters.
Also, the first two moments of the chain match the population moments of the RS process to significant digits. In a given regime, a Cholesky decomposition can be used to make a transformation from the uncorrelated normal errors u1 U2 into the correlated errors ei e2 e3 , with Pij denoting the correlation between e and e3.
Note that in this formulation only the short rate is the driving process, and is the only variable we need to track at each time t. To accomodate the equity states we can expand IT column-wise. We choose 3 states per equity, making an effective transition matrix of x where the rows sum to 1. Note, a full x transition matrix could also be constructed, but the 9 rows corresponding to a particular r would be exactly the same.
Each short rate state is associated with 9 possible equity states. The only modification we need in. These partial transition matrices can be mixed in the same manner as outlined before. This results from the regime-dependent distributions not being exactly unconditionally normally distributed in each regime from the presence of the square root term in the volatility of Tt, so Gaussian-Hermite weights wifl not be optimal in this setting.
Our final Markov chain matches means, variances and correlations to significant figures. In these ranges the portfolio weights are not as smooth as the plots that appear in Figure 6. At very high interest rates the portfolio weights also start rapidly increasing for regime 2. These do not affect any solutions in the middle range. The inaccuracies arise because at the end of the chains, the Markov chain must effectively truncate the con- ditional distributions on the left right at low high interest rates.
With experimentation we found that the inaccuracies at the end of the chain decrease as the persistence decreases. However, the approach given here is computationally more tractable. Hodrick, and D. Schwartz, R. Ingersoll, and S. Uppal, , "International Portfolio choice with Systemic Risk", working paper. What does the yield curve tell us about GDP growth more.
A lot, including a few things you may not expect. Previous studies find that the term spread forecasts GDP but these regressions are unconstrained and do not model regressor endogeneity. We build a dynamic model for GDP growth and yields The model does not permit arbitrage.
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