Tuesday, November 29th @ 11:00-12:30 PM (639 Evans Hall)
PCA with Model Misspecification
Robert M. Anderson, CDAR Co-Director
In this project with UC Berkeley Ph.D. Candidate Farzad Pourbabaee, Principal Component Analysis (PCA) relies on the assumption that the data being analyzed is IID over the estimation window. PCA is frequently applied to financial data, such as stock returns, despite the fact that these data exhibit obvious and substantial changes in volatility. We show that the IID assumption can be substantially weakened; we require only that the return data is generated by a single distribution with a possibly variable scale parameter. In other words, we assume that return is R t = v t φ t, where the variables φ t are IID with finite variance, and v t and φ t are independent. We find that when PCA is applied to data of this form, it correctly identifies the underlying factors, but with some loss of efficiency compared to the IID case. Now assume that the scale parameter is set by a continuous mean-reverting process, such as the volatility of return in the Heston Model. We use an exponentially weighted standard deviation of historical returns, as an estimate of v t. It is standard practice to estimate risk measures such as Value at Risk (VaR) and Expected Tail Loss (ETL) from the estimated volatility t by assuming that the returns are Gaussian. These Gaussian estimates systematically under forecast VaR and ETL in the presence of variable volatility, excess kurtosis, and negative skew. We propose the “historical method,” using the empirical distribution of R t+1 / t, as a more robust method for estimating VaR and ETL. In the simulation, we find the historical method provides accurate forecasts of both VaR and ETL in the presence of variable volatility and excess kurtosis, and accurate forecasts of VaR in the presence of negative skew.