**3. Methodology**

We use a relatively novel technique for calculating high resolution LL relations between cyclic time series. The method relates to a dual representation of the time series, x(t) and y(t), first as a series depicted as a function of time and second as depicted in a phase plot with one series on the x-axis and the other series on the y-axis. Time in the phase plot is then shown as the trajectories between points. For a quick intuitive illustration, see: https://en.wikipedia.org/wiki/Lissajous\_curve#/media/File:Lissajous\_phase.svg (accessed on 15 February 2022).

Seip et al. (2018) describe the method in detail, but recently Krüger (2021) has described an LL method that is based on wavelet techniques and on the same dual representation of paired time series. Figure 1a shows two sine functions with identical cycle lengths, but the dashed curve is shifted a few time steps forward, and we have added a small amount of stochasticity to make the example a little more realistic. The bold curve that is to the left of the target sine function (dashed curve) is a leading series (Figure 1a). With the leading series (bold) on the x-axis and the lagging series (dashed) on the y-axis, the trajectories in the phase plot rotate counterclockwise (positive per definition, Figure 1b). Thus, we can identify LL relations by the way that trajectories rotate in the phase space. For time series normalized to unit standard deviation, the trajectories will form an ellipse-like curve with the major axis either in the 1:1 direction or the −1:1 direction. For shifted perfect sine functions with common cycle periods, the minor axis will show the phase shifts between the sine functions.

The rotational direction represented by the angle θ between two successive vectors, **v**<sup>1</sup> and **v**2, through three consecutive observations in the trajectory is calculated with Equation (2):

$$\theta = \operatorname{sign}(\mathbf{v}\_1 \times \mathbf{v}\_2) \cdot \operatorname{Arccos}\left(\frac{\mathbf{v}\_1 \cdot \mathbf{v}\_2}{|\mathbf{v}\_1||\mathbf{v}\_2|}\right) \tag{2}$$

The vectors are calculated as (yi − yi−1)/(xi − xi−1) with i = 2, 3, . . . . We define a measure of LL strength as

$$\text{LL} = (\text{N}\_{\text{+}} - \text{N}\_{\text{-}}) / (\text{N}\_{\text{+}} + \text{N}\_{\text{-}}) \tag{3}$$

where N+ and N<sup>−</sup> is the number of positive and negative angles, θ, in a set of n consecutive observations in the two series. Using n = 9 and with N+ = 9 and N<sup>−</sup> = 0, we obtain LL = (9 − 0)/(9 + 0) = 1. The number 9 is a trade-off between the goal of identifying LL relations for short time windows and the goal of identifying a confidence band. In the time series mode, it means that one series leads the other for nine consecutive observations. In the phase representation, it means that when the two series are plotted as trajectories in the phase plot, the trajectories will rotate persistently in one direction.

In Figure 1c is the angles, and θ(3) in Figure 1b is shown as a function of time (light blue bars) with the LL(9) relations as dark blue bars. The angles are measured in radians (range −π, π) and the LL relations are in the interval [−1 to 1]. Note, for example, that the first angle is negative, showing a clockwise rotation. The dark blue bars are all negative, showing that the added noise is not sufficient to impair the overall clockwise rotation.

Pro-cyclic and counter-cyclic relations: If the OLR β coefficient is positive, the two series are pro-cyclic. If the β coefficient is negative, the two series are counter-cyclic. Figure 1d shows that an LL relation can be positive both for pro-cyclic and counter-cyclic series. Since a phase plot with GDP on the x-axis and EM on the y-axis will show an ellipsoid with its major axis in the 1:1 direction, the average β<sup>E</sup> coefficient is 1 per definition. However, β<sup>E</sup> coefficients based on short time windows of the series may deviate from 1 and thus give rise to interpretations of how coefficients change with the economy, e.g., expansions or recessions.

Figure 1e shows two sine functions where one function has a constant argument, a = sin (0.25t), whereas the other function, b = sin (0.25t + φ), has a variable argument, φ, that varies from a positive to a negative value. Figure 1b shows the ratio of the slopes a(t)/b(t) as a function of time (time window: nine time steps) compared to the average of the two sine functions. The growth rate of a/b is Δa/a − Δb/b (Abel et al. (1998, p. 658)). It is seen that at the downturn side of the sine function the ratio a/b first shows a positive peak and then a negative peak. At the upturn side it first shows a negative peak and then a positive peak.

Confidence interval. The 95% confidence interval (CI) is based on the probability that two uniformly stochastic series will show a persistent rotation in one direction. It is calculated with Monte Carlo simulations, applying Equations (2) and (3) to two uniformly stochastic series, and the confidence limits are the asymptotic values for 1000 replicates. Values of LL < −0.32 and LL > 0.32 suggest that for time series longer than nine time steps, the LL values are significant at the 95% level. When time series are smoothed with a smoothing algorithm, the probability that consecutive angles will have the same sign increases, so the CI does not strictly apply to smoothed series. However, by comparing LL relations for smoothed series with LL relations for unsmoothed series, the confidence in the LL relations for the smoothed series may be enhanced.

Detrending. Since we want to study interannual to decennial time windows corresponding to the typical duration of recessions (Burns and Mitchell 1946) we detrended the 12 US economy macroeconomic data series with a linear or a quadratic function, depending upon the large-scale form of the time series. Several detrending methods are available, but linear detrending is simple and will not introduce anomalies that have no economic relevance.

**Figure 1.** (**a**) Example: Calculating leading-lagging (LL) relations and LL-strength. Two sine functions: the smooth curve is a simple sine function, sin (0.5t), the dashed curve has the form sin (0.5t + φ × RAND()) where φ = +0.785 (**b**) In a phase plot with sin (0.5t) on the x-axis and the sin (0.5t + φ RAND()) on the y-axis, the time series rotates counterclockwise; θ is the angle between two consecutive trajectories. The wedge suggests the angle between the origin and lines to observations 6 and 7. (**c**) Angles between successive trajectories (light blue bars) and LL strength (dark blue bars). Dashed lines suggest confidence limits for persistent rotation in the phase plot and persistent leading or lagging relations in the time series plot. (**d**) LL relations and pro cyclic/counter cyclic relations between two cyclic series as a function of the phase shift between them. (**e**) Two sine functions; sin (0.25t + φ), the blue function has φ = 0 and the red function has φ shifted gradually from a positive to a negative value, thus the two functions shift in being a leading function. (**f**) The ratio a/b = sin (0.25t)/sin (0.25t + φ) as a function of time and the average of the two sine functions. (**a**–**d**) are redrawn after Seip and Grøn (2019) and Seip and Wang (2022).

Smoothing. We use the LOESS-smoothing algorithm. The algorithm has two parameters: the parameter (f), which shows how large fraction of the series is that is used as a moving window; and the parameter (p), which shows the polynomial degree used for interpolation. We always use p = 2. With 540 months ≈ 45 years of observations and f = 0.1, the moving time window is ≈50 time steps. We use the LOESS-smoothing algorithm as implemented in SigmaPlot, but the algorithm is implemented in many statistical packages. Since we always use the parameter p = 2, we use the nomenclature LOESS(f) for LOESS smoothing.

Principal component analysis (PCA). Both the data series for the US economy and the data that were extracted for time windows around the recession periods were analyzed with PCA and presented in PCA loading and score plots. However, we only have six recession periods, so we present the main results also as scattergrams to see if outliers affect the regression results. All data for the US economy were LOESS(0.1)-smoothed to avoid sharp peaks that could cause failures in the PCA algorithm. The PCA calculates new variables that are orthogonal.
