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Following @silencer's comment, your formula for variance is wrong. I would suggest that instead of trying to re-invent the wheel, you just use the formula that everyone else uses. So I'd replace your first indented line with $$ w^{*}\equiv argmin\left\{ \frac{1}{2}w'\varSigma w-\lambda\left(w'\mathbf{1}-1\right)\right\} $$ which will give you $$ ...


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Perhaps this paper by Hyun Woo Byun and coauthors is what you're looking for: Using a Principal Component Analysis to develop Multi-Currency Trading algorithms in the FX market They apply principal component analysis to a currency basket of 9 pairs with a 2 month rolling window. In a second step, various techniques (logistic regression, decision trees, ...


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If you look at changes of the points on the yield curve, then you probably find something stationary - right? Applying PCA on the covariance of these changes makes sense. E.g. you will find out that on PC describes a parallel shift (a change in the yield curve). Look at this question too: What do eigenvalues/eigenvectors of the yield/forward rates ...


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I am also interested in resolving this problem, although, decided not to create separate thread for it yet. This is kind of continuation of previous question below. http://stats.stackexchange.com/questions/34396/im-getting-jumpy-loadings-in-rollapply-pca-in-r-can-i-fix-it In factor analysis, specifically PCA, sign of the loadings does not mean anything, ...


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Sounds like PCA is not the approach you're looking for. If you're looking to transform a risk vector in terms of securities V into a risk vector in terms of securities W, then the basic approach would be to perform a linear regression of V against W. The resulting regression coefficients will form a matrix B which will give a change of basis between V and W. ...


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I just want to mention that it's highly prevalent to apply PCA to rate levels in rich/cheap analyses. Personally I prefer that... There's an old MS publication that discusses this very topic and the recommendation is to use level PCA for rich/cheap, and to use change PCA for risk management. There's a really good Salomon paper (Principles of Principal ...


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The literature on cointegration in large datasets or panels is really the only place where I've seen this sort of issue discussed. Breitung and Pesaran, among other places, talks about it. I would recommend applying the PCA to the rate changes (perhaps with some kind of zero lower bound adjustment). Then, take the cumulative sum of each of the factors. ...


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The choice of normalization depends on your data set: Without normalization : variable with high variance will have more impact on the PCA. You will have size effects. For exemple if you have one variable in meters and the other one in kilometers the one in meters will have way more impact. To avoid that you can normalize but now every variable will have ...


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The derivation is correct and given the formula you should get $w^{*'} \vec{1} = 1$. My guess is that the inversion of the $\Omega$ matrix is numerically badly conditioned. Instead of implementing the formula as it is, have you tried to calculate $\vec{1}^{'} \Omega^{-1}$ and $e^{'} \Omega^{-1}$ only once and rewrite: $$ w^{*\prime} = \frac{1}{2}\left[ ...


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The paper alternatives between using eigenportfolios and sector/industry ETFs for statistical arbitrage. For instance, sections 2.1-2 vs. 2.3. The trade in Section 4.1 is long some stock and short an appropriate amount of sector/industry ETFs. That being said Sections 5.3 and 5.4 discuss PCA strategies in a backtest, with relatively little additional ...


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Q #1: I'm not sure if you have the answer quite right. The signs for the loadings are arbitrary, but you cannot take the absolute value. You can multiply by -1. Q #2: It might be helpful to think about what PCA is actually doing. This paper might be helpful: http://arxiv.org/pdf/1404.1100v1.pdf (A Tutorial on Principal Component Analysis by Jonathon ...


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The first principle component of interest rates will not help you capture the term structure better at all. It will basically remove all term structure affects you are going to see. When we decompose the returns on interest rates you are going to get 3 PC's which explain 99.9% of the variance. PC1 - Level of the interest rates (~90% of variance) PC2 - ...


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Personally, I don't find any of the answers provided to be of that much help in answering the question. Factor analysis was developed for information collected for a single point in time. It's only been in the last few decades that extensions were made to, first, two or a few time periods, and then most recently truly longitudinal models have been proposed. ...



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