Forecasting Higher-Order Moments for Asset Allocation
Summary
This question explores an asset allocation strategy for four equity indices using excess-return forecasts from separate ARMA(1,0)-GARCH(1,1) models. The researcher wants portfolio weights to reflect expected returns and second-, third-, and fourth-order co-moment matrices. For covariance, they propose combining univariate volatility forecasts with a constant correlation estimate from standardized residuals, giving a conditional covariance forecast for the next period.
The central issue is that next-period standardized residuals depend on returns that have not yet occurred, while the portfolio weights must be selected in advance. The document raises, but does not resolve, how to estimate the needed higher moments when each return series uses a different fitted distribution. It provides no empirical results or recommended estimator. Its value is in framing the distinction between forecasts available at decision time and innovations observed only afterward, while leaving the multivariate dependence model and higher-moment construction open.
Key ideas
- The proposed allocation method uses expected returns and co-moments through the fourth order.
- Univariate GARCH forecasts can supply conditional volatilities for a constant-correlation covariance forecast.
- A future standardized residual cannot be calculated from an unobserved future return at portfolio formation time.
- The question leaves open how to combine distinct marginal distributions into multivariate higher-order co-moments.
Tags
Full text
# Asset allocation and GARCH models
# Asset allocation and GARCH models
I am trying to solve an asset allocation problem and I am having some troubles grasping the concept. I am working with excess returns on 4 stock indices and I am obtaining the excess returns forecasts for period $t+1$ as the conditional mean forecasts from ARMA(1,0)-GARCH(1,1). For each excess returns series I am using a different distribution, one that suits best.
Ideally I would like to use an asset allocation strategy that is taking the expected mean and the second, third and fourth co-moment matrices into consideration. And here is my problem, since I am building the strategy at time $t$ how can I obtain the co-moment matrices of innovations necessary for my asset allocation strategy.
For the covariance matrix my idea was to bind my univariate GARCH models in to a CCC-GARCH model and obtain the covariance matrix for time $t+1$ as $H_{t+1}=D_{t+1}RD_{t+1}$ where $H$ is the covariance matrix, $R$ is the unconditional correlation matrix of standardized residuals up to time $t+1$ and $D$ is a matrix with conditional standard deviation forecasts at the diagonal.
However, all the times I am encountering the same problem: how to obtain the standardized residuals for period $t+1$, which are necessary for calculating the 2nd, 3rd and 4th co-moment matrices? I need to choose the weights for each index at time $t$ and therefore cannot obtain the residuals as observed returns minus the forecast, as I do not yet know the observed return. Could you give me some advice on how to develop such a strategy? Some papers are obtaining the co-moment matrices directly from the parameters of various multivariate distributions, however, I would like to use different ones for each series. Any help on this matter would be greatly appreciated.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.