Variance estimation and goodness-of-fit test in a high-dimensional strict factor model

In the classic setting where the dimension p is small compared to the sample size n, an asymptotic likelihood estimation theory is well-known for the factor model by letting n tending to infinity while keeping p fixed. This theory is however no more valid for high-dimensional data where typically the dimension p is large compared to the sample size. In this paper, we develop new asymptotic results under the high-dimensional setting in a strict factor model with homoscedastic noise variance. For the maximum likelihood estimator of the noise variance, first we identify the reasons of a widely observed downward bias of the estimator. Second, a bias-corrected estimator is proposed using this knowledge. Third, we establish an asymptotically normal distribution for this corrected estimator under the high-dimensional setting. The second contribution of the paper concerns the correction of the likelihood-ratio statistic of the goodness-of-fit test to make it adapted to high-dimensional observations. The corrected statistic is proved asymptotically normal. Throughout the paper, Monte-Carlo experiments are conducted to assess the finite-sample behaviour of the methods. An application to returns of S&P 500 stock prices is also proposed.