My background is more in traditional econometrics, and I have recently come across "principal component analysis". I would just like to confirm with anybody whether I have the concept straight in my head.Points I believe to be true:1. PCA finds a linear combination of variables that captures the majority of the variation across all of the variables. The coefficients of this combination is called the "first principal component".2. PCA then finds a second linear combination that captures as much variation as possible, subject to the condition that it is orthogonal, (or has zero correlation) with the first linear combination. This is called the "second principal component".3. The third PC is calculated in the same manner as the second etc etc.4. The coefficients on the principal components can then be interpreted for any possible economic meaning .Applying what I know:I'm interested in applying PCA to stock indices around the world to find the major source of variation or volatility in the global equity markets. For example I could calculate daily percent changes for broad stock indices from the US (S&P 500, or the Dow), the UK (Footse), Germany (Dax), Japan (Nikkei) , Australia (ASX 200) etc. Lets say for arguments sake that the first principal component captures 70% of the total variation and it looks something like this: PC1US: 8UK: 1Germany: 2Japan: 2Australia: 0.2Can I interpret the above to mean that the major source of variation in global equity markets comes from the US equity markets? Any takers?Thanks in advance!

I had done the same with European indices. I can recall that i had split my sample into two subsamples and i had found 3 significant components in the first subsampe and 4 significant components. I had also found that the total variation explained by the significant components was higher.Thus i concluded that markets are getting a stronger comovement For the purpose of your study i think that a VAR model with Granger causality tests, impulse responses, and Variance Decomposition can do the job.A few minutes work in EviewsAnthis

I'll brush up on my VAR and give it a go. Thanks.

QuoteOriginally posted by: MrQwertyCan I interpret the above to mean that the major source of variation in global equity markets comes from the US equity markets? Any takers?It's dangerous to infer causality from a PC, or any regression-type, analysis. Suppose the Australian stock index is 100 times as volatile as the US index, then the 0.2 from Australia is actually more contribution to the first PC than the 8 from the US. For another, it could be possible that Japanese bank stocks drive the US and the rest of the world, but not Japanese non-banks. In that case the US index looks important because it tracks the Japanese bank stocks better than the Japan index. Finally, the US market could dominate the first PC if the US market is the largest part of the global market.If you standardize your data first (for each market, subtract from each daily percent price change the mean for that market, and then divide by the standard deviation for that market) the component weights are meaningful. However it's safer to say that the biggest factor affecting the world stock market is most highly correlated with the US market. Whether the US market is the cause of those changes or just the most faithful indicator, cannot be determined by the PC weights.You should also decide if you want to apply PC analysis to the daily percentage index changes (standardized or not) or to the covariance matrix (standardized into a correlation matrix or not).

E-GARCH and VAR-GARCH have also been used to test for volatility spillovers among a set of indices

Thanks Aaron. I have been away from this forum for a couple of days...."Suppose the Australian stock index is 100 times as volatile as the US index, then the 0.2 from Australia is actually more contribution to the first PC than the 8 from the US. "Yes, I failed to standardize my data (major major boo boo). I'll give that a go.This forum is excellent.

Hi,1) I did some PCA analysis on Interest rate markets (Indian Risk free securities). Even first 5 components giving only about 72% variation. When I did similar analysis on US market, first 3 pca explains 98% variation. Could anyone explain the reasons poor results in the first case?2) The return correlations between 9 year and 11 year security is in the region of 50% and similar weak correlatiosn exhibit throughout? Is it due to illiqudity? Correlations on prices series is about 97%. Please note Indian risk free market operate on prices. Is this a reason for poor correlation on yield returns?Is there any other way to reduce the dimension like PCA does (instead of PCA)? I need to simulate large portfolios. Or can we improvise PCA process itself?Thank you for your help.Nivas

I think you have poor quality data. When you say the price correlation is 97%, I suspect you mean on the undifferenced prices. When you take first differences (daily changes in price) you should get similar correlations to yield, since price and yield have close to a fixed linear relation over daily-sized interest rate increments.My guess is that some bonds are quoted frequently while other prices are stale. Or it could be simpler, perhaps some data points are wrong. With least squares, even one error can throw off the entire analysis, say if one decimal point is misplaced. However those type of errors usually lead to artificially high correlations.There are more robust techniques for dimensional reduction, cointegration for example. You can also try using a longer time interval (or overlapping intervals). Another approach is to smooth your data before fitting.

Dear Aaron,Thank you very much.I used undifferenced price series for correlation calculation since the PCA on normalised returns produced very low correlations between 9yr ytm and 11yr ytm.Data quality could be problem. I checked two different data sets on same market but both produce similar poor correlations.Can I use cointegration for Yield curve simulation? Do I have to ignore the white noise terms? Could you please brief further how to use cointegration in simulation.Thank you once again for your assistance.

Yes, you can use cointegration. But I think it makes sense to investigate your data first. Cointegration is complicated and may not be the right approach.There are three common problems that would cause the low correlation between the 9 and 11 year bonds. First is noisy data, if there are a lot of errors, correlation computations will be unreliable. The second is that one or both of the quotes are often stale. Third is that there is some conversion error, say the 9 and 11 year bonds are quoted differently or have some important financial difference.jYou need some data analysis before you go to far in modeling.

Thank you Aaron. I will clean the data and will perform my analysis again