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James M. Nason
Federal Reserve Bank of Atlanta
August 18, 2009
In my analysis, output is measured by (log) U.S. real gross domestic product (GDP) in billions of 2005 chained dollars divided by total U.S. population. The estimation sample runs from 1954Q4 to 2009Q2. The five models I use are attempts to separate the trend (or long-term) movements in output from the cyclical (short-run) movements, with the deviations of output from trend representing a statistical measure of the output gap. These five models are
In the UC-HP model the quarter-to-quarter change in the trend in output is assumed to be a random walk, with the ratio of the uncorrelated shocks to the random walk and the cyclical components fixed at 1600 to replicate the Hodrick-Prescott (HP) filter.
The UC-0 and UC-BN models posit that output is the sum of a random walk trend and a cyclical component that is specified as a stationary second-order autoregression. Shocks to the trend and the cycle are uncorrelated in the UC-0 model. In the UC-BN model, shocks to the trend and cyclical components can be correlated. The estimated correlation equals -0.88 with a standard error of 0.11 in the UC-BN model. (The interested reader should consult Morley, Nelson, and Zivot 2003 for details about estimating the UC-0 and UC-BN models.)
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Chart 1 shows that linearly detrended output is more volatile and more persistent over time than either 4Q/4Q output growth or the smoothed UC-HP output gap. As a measure of the output gap, linearly detrended output indicates that the U.S. economy has been below trend since 2001Q4, and the gap has been increasing. In contrast, 4Q/4Q output growth and the UC-HP estimates indicate that a negative output gap emerged during 2008. As of 2009Q2, linearly detrended output places the output gap at -10.7 percent, while 4Q/4Q output growth is -4.9 percent and the UC-HP gap is -3.7 percent. On a historical basis all these estimates suggest the output gap (deviation from trend) is currently large.
Of note in Chart 2, the UC-BN trend is much more volatile than the UC-0 trend estimate because the UC-BN model allows for nonzero correlation between the shocks to the trend and the output gap. The estimated correlation of -0.88 implies that the UC-BN trend is more volatile compared to the trend of the UC-0 model. As a consequence, the UC-BN model attributes more of the variation in output to movements in trend.
The top right panel of Chart 2 supports this view. The smoothed UC-BN trend peaks in 2006Q4, but actual per capita output does not peak until the end of 2007. The implication is that the output gap is positive and rising during this period. Chart 2 also shows that the UC-BN places greater weight on trend shocks to explain recent output declines. Linear detrending, the 4Q/4Q growth rate, the UC-HP model, and the UC-0 model all primarily attribute the current recession to large transitory shocks.
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The two bottom panels of Chart 2 show that although the persistence of the UC-0 and UC-BN gaps are similar, the smoothed UC-BN output gap has about one-third of the volatility of the smoothed UC-0 output gap. The chart also shows that there is a noticeable difference in the 2009Q2 output gap estimates produced by the UC-0 and UC-BN models. The latter estimate is -0.54 percent, but the former is -4.9 percent. The UC-0 output gap peaks in 2006Q1 and has been negative since 2008Q3. The UC-BN output gap peaks in 2008Q2 and turned negative only in 2009Q2. The more recent peak in the UC-BN output gap suggests that this gap anticipated subsequent output growth movements. Nelson (2008) presents regression evidence supporting the hypothesis that the BN decomposition predicts that U.S. output must fall to return to trend after a transitory expansion signaled by a peak in the BN output gap. The reason is that the BN trend absorbs more of the recent volatility in U.S. output growth than does the BN output gap. Greater volatility in the BN trend compared to the BN output gap distinguishes the BN decomposition of output into trend and the output gap from many other univariate models of the output gap.
References
Beveridge, Stephen, and Charles R. Nelson. 1981. A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the "business cycle." Journal of Monetary Economics 7, no. 2:151–74.
Hodrick, Robert J., and Edward C. Prescott. 1997. Postwar U.S. business cycles: An empirical investigation. Journal of Money, Credit, and Banking 29, no. 1:1–16.
Morley, James C., Charles R. Nelson, and Eric W. Zivot. 2003. Why are the Beveridge-Nelson and unobserved-components decompositions of GDP so different? Review of Economics and Statistics 85, no. 2:235–43.
Nelson, Charles R. 2008. The Beveridge-Nelson decomposition in retrospect and prospect. Journal of Econometrics 146, no. 2:202–6.
Watson, Mark W. 1986. Univariate detrending methods with stochastic trends. Journal of Monetary Economics 18, no. 1:49–75.
