**4. Results**

We first show the initial preparation of the observed data, and thereafter the LL results for GDP, EM, and UE. Third, we show the results of embedding the portion of the time windows where GDP leads EM, and the portion of the time windows where β<sup>E</sup> (GDP/EM) is less than 0.5 in a "map" of the US economy. Recall that with the time series centered and normalized to unit standard deviation, graphs for x = GDP and y = EM will form an elliptic form with the major axis in the 1:1 direction. Therefore, we evaluate the anomalies from the β<sup>E</sup> = 1.0 line. Last, we examine how labor productivity relates to recession characteristics.

#### *4.1. Data Preparation*

To extract information from GDP and UE, GDP will most often have to be detrended. Mukoyama and Sahin (2009) use the Hodrick–Prescott (HP) filter to extract a trend. Ziegenbein (2021) uses a quadratic deterministic trend for GDP and EM. For the LL and βE-analysis, we did not detrend the data since the LL method we use can be applied to time series with similar trends. However, we center and normalize the data to unit standard deviation. With the high-resolution LL method, we can identify LL relations over three synoptic observations in the paired series (nine time steps to obtain the confidence interval). Other studies use either the whole series or time windows that are considerably longer than ours, e.g., decadal scales, (Cazes et al. 2013; Donayre and Panovska 2021).

### *4.2. LL Relations*

In Figure 2a, the upper two series show GDP and EM time series centered, normalized and slightly LOESS(0.1)-smoothed. The lower line shows the βE(9) coefficients for an OLR over a rolling 9-month time window. The β<sup>E</sup> coefficient shows characteristic peak anomalies for paired cyclic curves where one curve is sifted in time relative to the other. The droplines show when the NBER recessions starts. Figure 2b shows the LL relations for the GDP and EM series in Figure 2a. The dark grey bars show θ(3) and the light grey bars show LL(9). GDP leads EM pseudo-significantly 58% of the time and GDP leads EM pseudosignificantly 34% of the time. Table 1 shows the number of months during a 12-month period where θ(3) < 0. Figure 2c shows the residual time series for LOESS(0.8)-smoothed GDP. This series and the UE series are, in addition, slightly LOESS(0.1)-smoothed to avoid sharp peaks.

**Figure 2.** Lead-lag relations between GDP, Employment, EM and unemployment, UE. (**a**) GDP and EM, both LOESS(0.1)-smoothed. Drop lines show NBER recession. (**b**) LL relations between GDP and EM, both LOESS(0.1)-smoothed (grey) and raw, unsmoothed (dark grey). Numbers show percentage "pseudo significant" LL relations (see text). Drop-lines show the beginning of recessions. OLR between grey and black bars give R = 0.30, *p* < 0.001. (**c**) GDP-LOESS (0.8) residual and UE (%) both series LOESS(0.1)-smoothed. (**d**) LL(GDP LOESS (0.8) residual, UE), both series LOESS(0.1)-smoothed and normalized to unit standard deviation. GDP leads UE during the periods 1997–1991; 1993–1995, 1997–2008; 2011–. Red horizontal lines are recession periods. Droplines shows beginning of recessions. (**e**) Phase plot for GDP and EM, not detrended, 1989–1993M3, dropline show the beginning of the 1990 recession. (**f**) Same as (**e**), but with GDP and UE.

Figure 2d shows the LL relations between the two series. GDP leads UE pseudosignificantly 35% of the time and GDP lags UE pseudo-significantly 55% of the time. The LL(GDP,UE) relation shows that UE may lead GDP both before and after a recession.

The LL relations for GDP and EM could in principle be the inverse of the LL relations for GDP and UE. To see how the differences would show up in a phase plot, we plotted the GDP/EM series and the GDP/UE series for time windows around the 1990 recession in two phase plots for the years 1989M1 to 1993M4 (Figure 2e,f). The drop lines show the timing of the July 1990 recession. The graph shows both associations and rotational directions. The results for the GDP/EM pair are shown embedded in a "map" of the US economy in Figure 3a. The blue lines show that the recessions, with the 2020 recession as an exception, are associated with a leading relation for EM to GDP (EM → GDP), but EM also leads GDP during other periods. Figure 2f shows that higher GDP generally leads to lower UE. However, there is no discontinuity around the 1990 recession as it was with the GDP/EM pair.

**Figure 3.** Principal component plots of US Economy 1977 to 2022. (**a**) Score plot for US economy. Blue lines show when employment leads real GDP: Red dots shows US NBER recessions. (**b**) Loading plot for US economy. (**c**) Same as a, but βE(9) < 0.5. See text. (**d**) same as (**a**), but both EM leads GDP and βE(9) < 0.5. UE = unemployment, FDE = federal debt, EXP = federal expenditures, FF = fed's interest rate, M2 = monetary supply, M2, INF = inflation (we use the consumer price index), GDP = gross domestic product, TRE = tax receipts, GIN = Gini's index, but reconstructed from Mukoyama and Sahin (2009), IP = industrial production, WH = working hours, EM = employment. βE(9) coefficient is for running GDP/EM.

3&
