#1




Interpreting and applying results PCA
Dear madame, sir,
I have a problem with executing the methodology I use in my paper (Master thesis at the University of Amsterdam). I have the following data:  Returns of seventeen countries per year (1993 untill 2010)  Returns of investable indices (about 1994 till 2010) The methodology is as follows. A covariance matrix for the returns of the seventeen countries has to be constructed for every year in the data sample. The data sample reaches from 1993 till 2009, because the investable indices do not originate earlier than 1994. Using the covariance matrices the eigenvectors can be calculated. The eigenvectors are used to calculate the principal components of the next years returns. For example, the eigenvectors of 1994 are used to calculate the principal components of the returns of 1995. This calculation will be done for the entire time horizon, which delivers 16 years with principal components. The amount of principal components which will be retained should be about ten (according to the paper I’m replicating). This is a principal component analysis, which I have done using SPSS (so not walking through the steps mentioned before, but just using the method provided by SPSS). But I have problems to understand and to use the output of the principal component analysis. The multifactor model will be regressed on the investable indices, which are described in the data section. The explanatory variables will be constructed out of the principal components. Every investable index will be regressed yearly, which will result in sixteen regression with a Rsquared as output. The Rsquared will be the measure for market integration. Now I have the output for the years 1993 until 2009 and the output looks a bit like this . I have a Rotated Component Matrix, with a RAW section and a SCALED section, with ten components (which I retained). Rescaled Component 1 2 3 4 5 6 7 8 9 10 AUSTRALIA ,059 ,108 ,931 ,038 ,164 ,087 ,072 ,014 ,022 ,193 AUSTRIA ,785 ,099 ,026 ,099 ,003 ,094 ,134 ,123 ,026 ,003 BELGIUM ,784 ,007 ,035 ,062 ,026 ,062 ,095 ,123 ,194 ,056 S&P/TSX COMPOSITE INDEX ,115 ,845 ,200 ,063 ,007 ,014 ,029 ,011 ,032 ,121 MSCI DENMARK ,221 ,012 ,014 ,010 ,029 ,030 ,142 ,962 ,013 ,034 FRANCE ,561 ,274 ,067 ,132 ,079 ,073 ,041 ,038 ,650 ,082 DAX30 ,743 ,012 ,300 ,111 ,061 ,088 ,071 ,005 ,359 ,191 HANG SENG ,181 ,095 ,255 ,056 ,333 ,064 ,027 ,045 ,064 ,876 IRELAND ,724 ,324 ,154 ,080 ,070 ,211 ,153 ,059 ,263 ,022 ITALY ,181 ,042 ,040 ,976 ,006 ,011 ,003 ,011 ,086 ,042 TOPIX ,173 ,008 ,070 ,002 ,083 ,010 ,965 ,140 ,007 ,023 NETHERLANDS ,704 ,213 ,091 ,220 ,207 ,042 ,008 ,113 ,329 ,111 SINGAPORE ,151 ,072 ,169 ,006 ,915 ,025 ,095 ,035 ,060 ,269 SOUTH AFRICADS MARKET ,175 ,073 ,076 ,010 ,019 ,970 ,009 ,029 ,045 ,046 SWITZ ,801 ,145 ,017 ,152 ,140 ,059 ,039 ,029 ,004 ,078 UK ,463 ,479 ,005 ,181 ,159 ,168 ,045 ,049 ,416 ,007 S&P 500 COMPOSITE ,029 ,798 ,059 ,028 ,059 ,082 ,008 ,027 ,142 ,024 But I have no Idea how to interpret this results and how to apply these on the original returns in such a way to do the multifactor model. I also have different outputs if needed. But my question is, how do I apply these results to construct the explanatory variables I need for my multifactor model. Do I simply construct the components by multiplying the numbers with returns of the countries in that specific year? I hope somebody can help me since my supervisor is not giving me any help in this. Your effort is really appreciated. Best regards, Bas ter Stege University of Amsterdam 
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