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Consumption and Fractional Differencing: Old and New Anomalies

Consumption depends on income, so testing theories of consumption involves testing theories of income. A prominent recent example is the work by Campbell and Deaton (1989), which uncovers a paradox. They model income as having a unit root instead of as a fluctuation around a trend, and so they find that consumption looks too smooth: the permanent-income hypothesis does not hold. Like some previous researchers, they find that a difference-stationary process fits the data better than a trend-stationary process.

The choice between a difference-stationary process and a trend-stationary process, however, ignores the intermediate class of fractionally differenced processes. Since fractional processes exhibit long-term dependence, they are often classified as having a unit root rather than as trend stationary. This makes permanent income seem rougher than it really is, while consumption, which responds to the true, fractional income, looks too smooth. Specifying consumption correctly removes the paradox.

This paper reviews the techniques of fractionally differenced stochastic processes, calculates the stochastic properties of consumption when income follows a fractional stochastic process, and shows how this may explain the excess-smoothness results.

Suggested citation: Haubrich, Joseph, 1990. “Consumption and Fractional Differencing: Old and New Anomalies,” Federal Reserve Bank of Cleveland, Working Paper no. 90-10.

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