  This paper deals with the time-varying high dimensional covariance matrix
estimation. We propose two covariance matrix estimators corresponding with a
time-varying approximate factor model and a time-varying approximate
characteristic-based factor model, respectively. The models allow the factor
loadings, factor covariance matrix, and error covariance matrix to change
smoothly over time. We study the rate of convergence of each estimator. Our
simulation and empirical study indicate that time-varying covariance matrix
estimators generally perform better than time-invariant covariance matrix
estimators. Also, if characteristics are available that genuinely explain true
loadings, the characteristics can be used to estimate loadings more precisely
in finite samples; their helpfulness increases when loadings rapidly change.
