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There's more than one way to shrink a covariance matrix. You can think of shrinking a covariance matrix as part of general class of estimators that limit the norms of a matrix. You could alternately think of shrinkage as a form of Bayesian analysis. Given the broad set of techniques one could use, it can be more helpful to think in terms of techniques to ...
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When using the estimated covariance in the context of mean-variance optimization, then, yes, shrinking the covariance matrix is useful even when you have sufficient data.
A good reference is Golts and Jones, A Sharper Angle on Optimization, who discuss convariance shrinkage among other techniques and give two examples of the usefulness of shrunk covariance ...
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I thought I would answer the question of "what am I using." All shrinkage estimators map to a Bayesian estimator that differs only in the prior distributions. In other words, you get a point estimate that is indistinguishable from a Bayesian estimate except that the calculation rule determines the prior distribution. Stein estimators for the Gaussian are ...
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PYTHON
I have found this class from the statsmodels library for calculating Garch models. Unfortunately, I have not seen MGARCH class/library.
Below you can see the basic information about the garch models in mentioned class from the statsmodels. Probably you have to implement it by your own in python, so this class might be used as a starting point.
...
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Broadly speaking, as you probably already know, there are 2 approaches to estimating large covariance matrices:
1) Shrinkage Methods like Ledoit-Wolf that try to reduce the noise in a large matrix (N by N) that has been estimated using the conventional method.
2) Factor Models of Covariance as described in for example Connor Korajczik 2007 that assume that ...
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The question you asked can be explained by these two lines of the code
e means <- t(returns) %*% ones / T
z <- returns - matrix(rep(t(means), T), ncol=N, byrow=TRUE)
term.1 <- t(z^2) %*% z^2 e
Here returns is TxN which gives you matrix ${y_{nt}}$ where n has i and j elements ; means is TxN of matrix ${z_i}$ , same mean for each asset for the ...
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As an addition to the already rich answers, I would suggest you to read the following paper by Marcos L. De Prado on the computation of Forward-Looking Correlation Matrices.
Estimation of Theory-Implied Correlation Matrices
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3484152
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The Ledoit-Wolf estimate cited by @develarist can be quite good, but as you say you already knew about "shrinking". It takes the population of correlations observed as an effective Bayesian prior for any given correlation, so it sort of inherently assumes that all pairs are similar an some sense. It would not work well, say, with known block sets of highly ...
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Quantile regression is considered a robust procedure but lacks the quality of being fully differentiable. There are also regularized regression models like ridge regression, lasso regression and elastic net regression that implicitly consider the covariance of the data like OLS, but additionally reduce volatility in estimates through the introduction of bias....
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I have actually considered the problem that you are working on, though configured somewhat differently.
There isn't going to be a universal answer to your question. See, in particular,
Holland, Paul W. Covariance Stabilizing Transformations. Ann. Statist. 1 (1973), no. 1, 84--92.
Nonetheless, there are answers, some already mentioned. I would argue ...
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This is not a complete answer, more a different perspective to the answers already given. If you have some a-priori knowledge about the covariance structure and about the factors influencing it, you should try to reflect this in your statistical model. Three ideas:
Divide your sample into subpopulations with identical factor values and estimate separately. ...
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Slight correction: the package in R is called rmgarch, not mgarch. It works well with rugarch, which provides a variety of univariate GARCH models. Both packages allow for parallelized computation on local cluster and return a nice and full set of fitted parameters, model specs, etc. I provided some additional links in this post.
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this answer is on hold
first it used the fact that your function $y$ is symmetric around 0 (proof) can be found here, so i don't need to type everything.
then just expanding the summation $$lim_{n \rightarrow +\infty} n^{-1} \sum_{|h| < n} \left(|\gamma(h)| \right) = lim_{n \rightarrow +\infty} n^{-1} * \frac{(y(-n)+y(n))*2n}{2} $$ because h is from -n ...
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The implementation is explained in more detail in the Horse - Race of DeMiguel: see here
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Say that you did the calculations in the classic regression way. If you stick the returns of your 4 asset returns in a $(T\times 4)$ matrix $Y$, and your 3 factor returns in a $(T\times 3)$ matrix $X$, then your betas would solve the multiple regressions, collected in a $(3\times 4)$ matrix
$$Y = X\cdot \beta + \epsilon$$
You could also add a column of ones ...
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With this solution you have to split your covariance matrix somewhat, but it should give you a vector with betas based on you conditional covariances.
Example with two indexes, $x1$ and $x2$, and one asset $y$.
$$[\sigma_{y,x1}, \sigma_{y, x2}]\begin{bmatrix} \sigma_{x1}^2 & \sigma_{x1,x2} \\ \sigma_{x1,x2} & \sigma_{x2}^2 \end{bmatrix}^{-1}$$
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To close this question.
Steps used, in short :
get matrix N x M where N - number of assets, M - number of history
samples
normalize all samples using logarithms and mean to have returns instead of some asset specific values
obtain covariance matrix, or correlation, if you want to avoid
influence of volatility
solve eigenproblem using SVD and Jacobi's ...
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