The Selection Of The Number Of Factors In The Near Factor Model Based On Fourier Transform

These years,approximate factor models have become increasingly important in both finance and macroeconomics.The well-known arbitrage pricing theory and capi-tal Asset pricing model are factor models applied in finance.These models use a small number of common factors to explain the co-movements of a large number of time se-ries.For example,asset returns are often modeled as a function of a small number of factors.But with the increase of the complexity of the data,people find that the dat not only have serial correlation in time dimension,but also have cross-correlation in cross-country variations,which enables the approximate factor model are more effective in applications.However,as the common factors are often unobserved,it is natural to ask how many factors should be included in practice.This paper proposes a new method for determining the correct number of factors in approximate factor models.We use discrete Fourier transforms,which is normally used in generalized in dynamic factor model,in approximate static factor model.We utilize the transformed data to obtain corresponding eigenvectors,together with the original matrix x,we can calculate approximate eigenvalues called?.With discrete Fourier transform,?have such property:the first r?which is the real factor number?eigenvalues tend to infinity while the rest are bounded when N and T tend to infini-ty,much better,the gap between of the rth eigenvalue and?r+1?th eigenvalue is larger than before,which is beneficial for our method.However,unlike previous methods,we do not calculate the eigenvalues of the variance matrix.We compute the discrete Fourier transforms in our static factor model and calcu-late the sum of residual squares?hereafter this text will be abbreviated as SSE?for different factor numbers,in general from 1 to k_max.Thanks to DFT,then it is clear that SSE?k?will decrease with the increase of k?the factor number?.However,be-cause increase of the factor number k cannot provide more effective information when k is greater than r,the true factor number.Then the difference between SSE?r?and SSE?r+1?goes to zero while the gap between SSE?r-1?and SSE?r?is still large,at least far greater than zero.Then we obtain the estimator of the number of common factors by ratio of difference between of adjacent values of SSE,we choose the val-ue k corresponding to the ratio's maximum value as our final estimator.Under some mild conditions,the resulting estimator can be proved to be consistent.Monte Car-lo simulation study shows that the new estimator has more desired performance than other methods in most cases,it's obvious that it still performs well even there exists a dominant factor.Finally,we use our new estimators to estimate the common factors in AIDS children all over the world and American market economy respectively.