Overlapping Observations

A final topic concerns the analysis of returns series constructed on moving windows of historical returns. It is common, for example, to smooth the noise in observed returns by constructing moving averages. An example is Shiller's 10-year moving average of trailing earnings on the S&P 500 that he uses to filter noise in earnings. This smoothed earnings number is used to estimate the p/e ratio. Three-year or five-year moving averages of monthly returns or volatilities, for example, are often found in presentations of portfolio performance. A three-year moving average will typically construct the current month smoothed return, using a sample of 36 months of trailing returns. A time series of three-year moving averages means that adjacent months have 35 observations in common. Observations on the moving average that are two months removed have 34 observations in common and so on. Since the moving averages overlap in this fashion, they constitute a serially correlated time series, and this property has implications concerning the information content of the newly constructed series as well as its statistical properties.

Let's think about the information content first. The reason the assumption of independence of returns is so attractive is that each return can be thought of as a truly novel byte of information. Independence implies that there are N distinct bytes of information in a sample of size N. So, for example, if returns were independently distributed ...

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