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Factor Analysis and Principal Components Analysis 237 yt, t = 1,... , T of the scalar dependent variable and by T observations [ ...

238 The Basics of financial economeTrics Explicit form. We can write a factor model explicitly as a set of N equations: yabf bf ...

Factor Analysis and Principal Components Analysis 239 where =                 Y yy yy yy N tNt TNT 11 1 1 1   ...

240 The Basics of financial economeTrics Alternatively, we could subtract the mean from the data and write fac- tor models in te ...

Factor Analysis and Principal Components Analysis 241 content. We will define different categories of factor models, but common ...

242 The Basics of financial economeTrics between the monthly returns series for the 500 stocks and some macroeco- nomic variable ...

Factor Analysis and Principal Components Analysis 243 N × q matrix of factor loadings B by the matrix inverse of T, we obtain a ...

244 The Basics of financial economeTrics Estimation of Factor Models After defining the two fundamental classical factor models— ...

Factor Analysis and Principal Components Analysis 245 Obviously this is a mathematical abstraction because no real market is inf ...

246 The Basics of financial economeTrics taBLe 12.1 A Sample of Stock Daily Returns for the Period December 2, 2011, to December ...

Factor Analysis and Principal Components Analysis 247 We can thus estimate the factor loadings and the residual variances under ...

248 The Basics of financial economeTrics Assume that the standardized data X can be represented with four fac- tors. Hence, we c ...

Factor Analysis and Principal Components Analysis 249 introduce the concept of the Frobenius norm of a matrix. Given any matrix ...

250 The Basics of financial economeTrics Because xBtt=+f εt, the vector []xftt has a normal distribution with mean zero and the ...

Factor Analysis and Principal Components Analysis 251 If we compute the covariance matrix of F, we obtain: ΣF= − 0.9975 0.0000 0 ...

252 The Basics of financial economeTrics and therefore are observable while factors are generally non observable variables. Ther ...

Factor Analysis and Principal Components Analysis 253 matrix ΣX using the eig function of MATLAB. We obtain the following matric ...

254 The Basics of financial economeTrics where each column is the product of the data X and the corresponding eigenvector. The c ...

Factor Analysis and Principal Components Analysis 255 That is, we can exactly obtain the data multiplying principal components b ...

256 The Basics of financial economeTrics FIGUre 12.1 Plot of Principal Components and Eigenvalues −4 0 5 10 15 20 −2 0 2 4 6 8 T ...