Large Precision Matrix Estimation for Time Series Data with Latent Factor Model
published: May 6, 2009, recorded: April 2009, views: 4455
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Estimating a large precision (inverse covariance) matrix is diﬃcult due to the curse of dimensionality. The sample covariance matrix is notoriously bad for estimating the covariance matrix when the dimension p of the multivariate vector is comparable or even larger than the number of time points n observed. It is singular and hence cannot be inverted for the precision matrix. We use the factor model and procedure proposed by Pan and Yao (2008) for multivariate time series data to carry out dimension reduction when p ≈ n or even p > n. A version of the unknown factors and the corresponding factor loadings matrix are obtained. We show that when each factor is shared by O(p) cross-sectional data points, the estimated factor loadings matrix, as well as the estimated precision matrix for the original data, converge weakly in L2 -norm to the true ones at a rate independent of p. This striking result demonstrates clearly when the “curse” is cancelled out by the “blessings” in dimensionality. It is particularly useful in portfolio allocation in ﬁnance when the number of stocks p is large. Convergence rate in L2 norm for the precision matrix is directly related to the goodness of the estimated optimal portfolio, which converges weakly to the true one in the average squared L2 norm at a rate also independent of p as a result. We also show that the method cannot estimate the covariance matrix better than the sample covariance matrix, which coincides with the result in Fan et al. (2008) when factors are known. Simulations demonstrate a variety of eﬀects to the estimators when assumptions are not met. A set of real stock market data is analysed.
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