Estimating the trend in US real GDP using the ℓ₁ trend filtering

ABSTRACT

Using non-Gaussian state-space models, Perron and Wada (2009, Journal of Monetary Economics, 56, 749–765) obtained a nearly piecewise linear trend estimate of US real gross domestic product such that its slope changed around 1973. Such a trend may be regarded as a result of occasional permanent shocks to the growth rate. This article shows that the trend filtering, which is quite similar to the Hodrick–Prescott filtering and is a type of the recently popular lasso regression, yields almost the same trend estimate, and discusses the reason why this occurs.

KEYWORDS:

• Trend-cycle decomposition
• l₁ trend filtering
• Hodrick–Prescott filtering
• lasso regression
• double-exponential distribution
• fat-tailed distribution

JEL CLASSIFICATION:

• E32
• C22

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ See also Tibshirani and Taylor (2011) and Tibshirani (2014).

² For further explanations, see Yamada and Jin (2013) and Yamada and Yoon (2014, 2016).

³ See, e.g., Tibshirani (1996).

⁴ We set in Equation 1 because Perron and Wada (2009) note that the results from the HP filtering with are basically equivalent to their trend-cycle decomposition with a trend that has a single shift in slope in 1973.

⁵ We thank Professors Perron and Wada for providing their data set and Matlab codes. The Matlab codes were used to produce Fig. 2.

⁶ See Figure 7 of Tibshirani (1996).

Additional information

Funding

This work was supported by JSPS KAKENHI [grant number 15K13010].