Long-Memory and Level Shifts in the Volatility of Stock Market Return Indices

There has been interest in the possibility of confusing long memory and structural changes in level, and studies showed that when a short-memory process is contaminated by level shifts the estimate of the fractional differencing parameter is biased upward and the autocovariances decay slowly. We analyze the properties of the autocorrelation function, the periodogram, and the log periodogram estimate of the memory parameter when the level shift component is a mixture model. Our results explain many findings reported and uncover new features. We confront our theoretical predictions using log-squared returns as a proxy for the volatility of some assets returns, including daily S&P 500 returns over the period 1928–2002. The estimates follow patterns that would obtain if the process was short memory with level shifts instead of fractionally integrated. A simple test is also proposed, which reinforces this conclusion.