*4.3. The β<sup>E</sup> Coefficient*

The β<sup>E</sup> coefficients are calculations over 9 months. Figure 2a shows that the β<sup>E</sup> coefficient is generally greater than 1.0 before a recession and then becomes less than 1.0 for a period of 4 to 52 months (21 ± 17 months) after the recession. Table 1 shows the values of the β<sup>E</sup> *coefficient* when it has its lowest value around the recession periods. In Figure 3c, the result is embedded in the "map" of the US economy. The distribution of the blue lines suggests that the period with the β<sup>E</sup> coefficient less than 0.5 is mostly associated with recession periods. Figure 3d shows that time windows where both EM leads GDP and βE(9) is less than 0.5 are concentrated around the recessions. The 2020 recession is an exception.

### *4.4. US Economy Results*

The "map" of the US economy seen in the loading plot on Figure 3b provides a reasonable stylized picture of an economy. There are some obvious relations. EM and UE point in opposite directions and align roughly with the x-axis. FF and IP point in opposite directions and align roughly with the y- axis. GDP, EM, WH and TRE all have high values at the same time. However, the close connection between IP and GIN may not be intuitive. A PCA loading plot of cyclic series will have some other characteristics relative to series with Gaussian distributions. If two identical and perfect sine functions are shifted a quarter of a cycle length (λ/4) relative to each other, their symbol representation will be connected to the origin with lines that are at 90o to each other (Seip and Grøn 2019). Thus, FF and EM may have similar cycle periods, but will be shifted λ/4 relative to each other.

Recessions appear to occur in all parts of the "map", but they are appearing after sharp bends in the trajectories. For the 1980, 1981 and 1990 recessions, trajectories are counterclockwise, whereas the trajectories rotate clockwise for the 2000, 2007 and 2020 recessions. The two series showing that EM → bGDP and βE(9) < 0.5 coincide in 34 months out of 547 months (Figure 3d). The 34 months partly preceded and partly followed five of six NBER recession dates, providing a probability of ≈0.0002 to coincide with the recessions by chance.

#### *4.5. Labor Productivity*

Labor productivity is generally regarded as a leading variable to the business cycle, (Abel et al. 1998, p. 321). Figure 4a and Table 1 show that the detrended LP is declining one year before most recessions, except the 2008 recession. On a decennial time scale, LP shows an overall decrease during the period 1977 to 1997 (the stagflation period under Volcker ≈ 1975–1985 and "the great moderation" period under Greenspan, ≈1985–1997, (McNown and Seip 2011) then increases until 2010 and decreases again until 2018. For the 12 months preceding a recession, the β<sup>E</sup> coefficient is zero per definition for linearly detrended series, but it is on average (minus) 0.10 ± 0.22, and all β<sup>E</sup> coefficients are either negative or close to zero. These values can be compared to the average 12-month increase of 0.28 in LP during the period 1996 to 2010.

We compared the five recession characteristics: recession depth and duration, LP depth and duration, and LP. However, we have only six sets of observation series, suggesting that outliers may play a dominant role. Figure 4b shows that outliers actually play a role for jobless recovery depth versus recession depth. Excluding the recessions in 2007 and 2020, it appears that jobless recovery depth is associated with recession depth, but the regression is not significant, and neither were the other five ((4 × 3/2) − 1) regressions.

**Figure 4.** Jobless recovery and recession depth. (**a**) Time series for linearly detrended labor productivity, GDP, and employment. Red droplines show recessions. (**b**) Jobless recovery depth as a function of recession depth. Separate regressions for all recessions and a subset where the two last recessions, 2007 and 2020, are excluded.

#### **5. Discussion**

There is no firm conclusion about how the LL relations between GDP, UE or EM should be. However, changes in LL relations between GDP, UE and EM may be associated with the recession periods. Not all recessions are similar, and our study shows that the recent 2020 recession caused by the COVID-19 pandemic was different. The time window around a recession can been divided into four periods: first an expansion in GDP, then two periods during the downturn in GDP, and then a second expansion period where the economy recovers.

The first expansion is the one that triggers a monetary or financial response because the economy is overheating, e.g., a rise in FF (Taylor and Williams 2010). The first recession period is characterized by an uncertainty in the reality of the recession. During the third period, the recession is a certainty. The fourth period is the expansion period that terminates in "natural" growth and employment rates.

#### *5.1. Lead-Lag Relations*

A leading role for a causal effect is a prerequisite, but not a sufficient, criterion for causation. However, the leading role is often offered as a strong argument for a causal effect (Sugihara et al. 2012). We examined the LL relations between the two pairs: real GDP and EM, and GDP (detrended) and UE (percentages). Our results for EM support Hamilton's (2018, p. 838) finding that the cyclical component of EM starts to decline before the NBER business cycle peak for essentially every recession (here, before 1981, 1990, 2001 and 2007 recessions). We also found that whereas EM leads GDP before a recession, it lags GDP after the recession.

Figure 2e,f showed two characteristics for paired time series during the period 1989 to 1993, that is, around the 1990 recession. The first shows how EM and UE vary with GDP (slopes in the graph), and it shows which of the variables are leading the others (rotational directions). The graphs confirm the LL—the results are in Figure 2b,d. In the following discussion, we use traditional "stylized facts" to explain the time series movements we observe. Following that, we will discuss possible causal factors examined in the literature.

Before July 1990M7, the first expansion, EM increases as GDP increases, and the rotational direction is clockwise (CW), but not easy to identify visually (Figure 2e). The graph confirms the LL results in Figure 2b. The interpretation could be that firms fill vacant positions as production or demand for services increases, but the overall US labor productivity anomaly decreases (from a linear trend) during the period one year before the 1990M7 recession. We did not find a first recession period where business was slow to cut workers. (Note that EM increases before 1990M1 (Figure 4a), whereas Figure 3a,b shows a movement towards higher UE at the same time). However, there are 12 variables in the PCA plot that show a movement towards a higher UE, so there are probably other variables that are responsible for the movement, e.g., working hours, as suggested by Obst (2020, p. 229) Following the trajectories in Figure 2e, it is seen that during the whole recession period from 1990M7, both GDP and EM decrease, but whereas GDP increases after March 1991M3, EM increases just slowly after November 1991M11 and is not up to pre-recession values before November 1992M11. Thus, for one year there is a "jobless recovery" during the second expansion period. During this expansion (the recovery), firms are slow to rehire as they use current workers more intensively. Rotations are counterclockwise, showing that GDP is leading EM from 1990M9 to 1993M4 (outside graph: to 1994M10). The labor production anomaly increases from 1991M1 to 1992M7. We have not discussed the effects of an increase in the underground economy during a recession, nor its potential impact on the recovery process. However, it may contribute a significant amount in countries with large underground economies, e.g., Seip and Orsi (2022).

For the GDP–UE pair (Figure 2f), GDP decreases and UE increases, rotations are clockwise, and UE leads GDP. However, it is the trough in unemployment that would imply a peak in GDP, thus, the interpretation should be that (minus) UE apparently leads GDP or GDP leads UE, i.e., it is business growth that creates hiring.

Our results are based on EM and UE data downloaded in September 2021, however Ahn and Hamilton (2022) conclude that current unemployment measures underestimate the number of people that are unemployed, and the magnitude of the bias is larger when the true unemployment rate is higher). Ahn and Hamilton (2022)' s revised UE series running from 2001 to 2020 showed similar LL(GDP, UE) relations as in Figure 2d, except that from 2006 to 2010 GDP was leading UE. Mimicking the verbal Ahn and Hamilton assessments for the whole period of 1977 to 2020 and replacing UE with UE1.1, the results for the period did not change the LL(GDP, UE) patterns appreciably.

In contrast to the LL relations for GDP and EM, Elhorst and Emili (2022) found that for the Netherlands, output growth leads UE. However, the results reported here apply to employment and not to unemployment, and Figure 2e,f and Appendix A show that EM and UE provide different results when put into similar contexts.
