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Advanced estimates

Stambaugh combined-sample estimates for means and covariances


This technique consists in obtaining maximum-likelihood estimates of the model parameters for assets with different start dates and common final date.

In the event of portfolio components having different depths of historical data standard parameters estimates, considered in the previous section, result in a loss of information. The covariance matrix of portfolio components, calculated based on a historical dataset with different start dates is not necessarily positive-definite. Hence, the analyzed dataset is truncated traditionally to the date corresponding to the asset with the shortest history. With such an approach, the loss of the information contained in the initial dataset set is inevitable.

The method suggested in
[Stambaugh; 1997], consists in the consecutive use of the regression of assets with a shorter history on assets with a longer history. This time all the information contained in the data is used.

Note. It is worth mentioning that the use of the entire dataset improves estimates for longer-history assets as well as estimates for shorter-history assets.

For more detailed information on the discussed method see
[Stambaugh; 1997].


About shrinkage estimators


The shrinkage estimator is a statistical tradeoff between the bias and the estimation error. For the first time the shrinkage estimator appeared in [Stein; 1956], where it was shown that “shrinking” sample means of multivariate normal distribution to an appropriate common constant improves estimation accuracy.

In general, the shrinkage estimate is obtained via a “shrinkage” of sample unbiased estimate towards some biased target with lower estimation error.

Shrinking sample excess Mu towards excess Mu of GMV portfolio
Shrinking sample covariance matrix towards constant correlations covariance matrix
Shrinking sample estimates towards values, implied by asset pricing model


Shrinking sample excess Mu towards excess Mu of GMV portfolio


Shrinkage estimator for was obtained in [Jorion; 1986]. Based on Bayes approach, estimate implies shrinking os sample Mu  towards a common constant, equal to sample excess Mu  for global minimum variance portfolio. Calculation formulae for  are presented below:




where  is vector of ones;
is number of assets;
is number of observations.



Shrinking sample covariance matrix towards constant correlations covariance matrix


This estimate is a shrinkage estimator for covariance matrix. It was proposed in [Ledoit, Wolf; 2004].
Final estimate is obtained via shrinkage of the sample covariance matrix towards the covariance matrix, produced by averaging correlations across asset pairs.


where
– sample covariance matrix,
– constant correlations covariance matrix.

Details, including the formulas for the optimal shrinkage intensity can be found in
[Ledoit, Wolf; 2004].


Shrinking sample estimates towards values, implied by asset pricing model


The given method of the estimation of parameters  and , first suggested in [Pastor, Stambaugh; 1999], consist in applying Bayes approach, which assumes a certain degree of investor's confidence in that the market satisfies the selected factor-based asset pricing model (such as CAPM or Fama-French 3-factor model).

The degree of investor’s confidence in the chosen factor model is determined by the parameter , taking values in the range form 0 to 1. Value means that the investor completely ignores predictions of the factor model, using as estimates for and their sample counterparts, while means, that the estimates for and coincide with
model-implied estimates  and .

Analytical formulas in general case are presented in
[Wang; 2005].


Joint estimator for means and covariances based on Missing Factor approach


This method described in [MacKinlay, Pastor; 2000] can be applied in a situation when there are reasons to believe that the dynamics of the portfolio assets is determined by influence of one factor only, however the factor itself is "unobservable", i.e. the structure of the latter is unknown.

Note. By an unobservable factor one may imply the unknown structure of the market portfolio.

Let vector denote simple
rates of return for assets on the interval , where is small enough. By we shall designate corresponding simple rate of return of an unobserved factor.

Suppose that the dynamics of the portfolio components satisfies the one-factor model




where denotes vector of betas and vector contains regression residuals.

The additional assumption made in
[MacKinlay, Pastor; 2000] is that the covariance’s matrix of the residuals is proportional to the identity matrix:
 
The above assumption, basically, is not burdensome if portfolio components belong to the same asset class (big-cap stocks, small-cap stocks, bonds etc.).

With the assumptions made it is possible to establish a connection between a covariance matrix and a vector , as a result one obtains estimates  and . Analytical formulas used in calculation of  and are given in
[Kan, Zhou, 2005].












 
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