Dear Tom:

Recently, I read Fernandes et al.(2003), “Modelling output gaps in the Euro Area with structural breaks: The COVID-19 recession”. I read again the several papers about UC model with structural break, including, Morley et al. (2003), Perron & Wada(2009), Luo & Startz(2014), Grant & Chan(2017).

I have two questions for your help.

1. In tradition UC decomposition, the log GDP, y is sum of trend component tau and cyclical component c, not including irregular component e. However, Morley et al. (2003) said “sometimes an additional irregular term is added, although these changes have little influence on the estimated cycle component for U.S. GDP”. These papers all wrote before Covid-19 pandemic. When sample including the 2020Q2, the estimates is different while UC model with or without irregular component. On the other hand, Harvey always incorporates the irregular component in his series papers.

My question is whether or not standard UC model includes the irregular component?

2. Perron & Wada (2009) used exogenous break date, whereas Luo & Startz (2014) used endogenous break date. They did not allow the trend growth rate drift as random walk, i.e, local linear trend. Allowing for a break significantly reduces estimates of trend variance. I guess the multi breaks reduce much. Further, local linear trend allowing time-varying the drift is ideal.

Perron & Wada (2009) said “The reason is that any positive variance [of irregular] would imply changes in the slope occurring at every period, though of different magnitudes each time, and the real GDP series would then be I(2), a feature not supported by the data”. However, in measuring the natural rate of interest (working paper), Laubach and Williams (2001) use the local linear trend, and argued in p5 “The hypothesis that log real GDP is I(2) is typically rejected by an ADF test. However, as Stock and Watson (1998) point out, when the disturbance to the growth rate component has small variance, such a test statistic has a high false-rejection rate”

My question is, should trend component be local linear trend or local lever with multi-breaks?

Best Regard

Hardmann

Reference

Morley, J.C., Nelson, C.R., Zivot, E., 2003. Why are the Beveridge-Nelson and unobserved-components decompositions of GDP so different? Rev. Econ. Stat. 85 (2), 235–243.

Perron, P., Wada, T., 2009. Let’s take a break: Trends and cycles in US real GDP. J. Monetary Econ. 56 (6), 749–765.

Luo, S., Startz, R., 2014. Is it one break or ongoing permanent shocks that explains U.S. real GDP? J. Monetary Econ. 66, 155–163.

Grant, A.L., Chan, J.C.C., 2017. Reconciling output gaps: Unobserved components model and Hodrick–Prescott filter. J. Econom. Dynam. Control 75, 114–121.

Harvey,A.C., 1985. Trends and cycles in macroeconomic time series. J. Bus. Econ. Stat. 3(3), 216–227.

Harvey, A.C., Jaeger,A., 1993. Detrending, stylized facts and the business cycle. J. Appl. Econom. 8(3), 231.

Harvey, A.C., Trimbur, T.M., VanDijk, H.K., 2007. Trends and cycles in economic time series: a Bayesian approach. J. Econom. 140, 618–649.

## question about UC model

### Re: question about UC model

UC models attempt to decompose a single time series into two, three, four or more (if seasonal) components based solely on the time series properties of those components. Doing that successfully usually requires some rather strong assumptions---orthogonality isn't enough in practice, particularly in distinguishing the "trend" from other components. And the assumptions required to get "decent" results over one time span may fail when applied to a different one, or similar but not same data set. (The Laubach-Williams model tends to fail rather spectacularly when the sample is extended). You've run through a whole set of papers which are all trying to figure out what (minimal??) set of assumptions are required to make a UC model work in practice and come up with different sets of "solutions" even when applied to the same data set (US GDP). The choice may also depend upon how the model is intended to be used. A model with hard breaks might be fine for looking at a set of historical data, but isn't really capable of being extended out of sample since you need substantial data to determine if you are moving into a new trend regime.