“Accuracy” measures to what degree a positive/negative movement in IP follows a positive/negative movement in the leading index. “Timing” is the time before a movement in the leading variable is reflected in a corresponding movement in IP. The timing is a function of the series’ cycle length, CL, which ideally are identical for the leading index and its target IP.

With leading indexes, the forecasting is just to quote the value of the leading index. With sentiment-based indexes, the forecast timing is given by the time between the collection of the sentiments and the stated forecasting horizon in the sentiment questionnaire. Since the sentiments are expressed as an index, the range of variation for the index could be normalized to unit standard deviation corresponding to a similar normalization for the target series. We evaluate the forecasting skill by reporting the LL strength over a given period. With the nomenclature used here, a perfect leading index to IP has a LL strength value close to −1 and a perfectly lagging index to IP has a LL strength value close to +1. Visually, the peak (trough) of the leading index will come before the peak (trough) of the target series, but less than ½ cycle length. Trajectories in the phase plot with IP on the x-axis and the candidate leading index on the y-axis would always rotate clockwise.

#### 2.2.2. We Explain the LL Method in Four Steps

Step. 1. Detrending and smoothing. We detrended the target variable, IP, the leading indexes, and the lagging index by calculating the residuals after removing a linear regression against time. To remove high-frequency variations, we smoothed the variables using the LOESS locally weighted smoothing algorithm by SigmaPlot©. The smoothing algorithm has two variables. The first, f, shows how large the fraction is of the series that is used for calculating the rolling average. The second, p, is the order of the polynomial function used to make interpolations.

To find a reasonable degree of smoothing, we used the time series 1994 to 2014. We used four fractions of the series as rolling average windows: f = 0.02, 0.06, 0.1 and 0.2, and we always interpolated with a second order polynomial function, p = 2. The detrending and smoothing of the indexes are intended to mimic numerically the visual processes that are used in real life applications. The smoothing algorithm is described further in

Section 3.2.

Step 2. Rotational directions in phase space. We then calculated the angles

θ between two successive vectors

**v**_{1} and

**v**_{2} through 3 consecutive observations:

5 The rotational direction for the paired series in

Figure 2a is shown in as grey positive bars (counter-clockwise rotations) and as grey negative bars (clockwise rotations) in

Figure 2c.

Step 3. The strength, LL strength, of the mechanisms that cause two variables to either rotate clockwise or counter-clockwise in a phase portrait is measured by the number of positive rotations (as sign(θ) > 0) minus the number of negative rotations (as sign(θ) < 0), relative to the total number of rotations over a certain period.

This means that we can assess the persistence of the rotational direction. We use the nomenclature: LL(x, y) $\in $ [−1, 1] for leading–lagging strength: LL (x, y) < 0 implies that y leads x, y→x; LL(x, y) > 0 implies that x leads y, x→y. In a range around LL(x, y) = 0 no LL relations are significant.

Significance levels were calculated with Monte Carlo simulations for the LL strength measure. We found the 95% confidence interval for the mean value (zero per definition) to be ±0.32 for n = 9, that is, in a phase plot the series rotate persistently clockwise or persistently counter-clockwise. This corresponds to significant leading–lagging signatures for the series,

Figure 2c, black bars.

Step 4. The cycle length (CL) of two paired series that interact, can be approximated as:

θ_{i−}_{1,I,i+1} is the angle between two consecutive vectors determined by three consecutive observations. The number of angles that close a full circle corresponds to the cycle length. With two perfect sines (no random component added, and series normalized to unit standard deviation as in

Figure 2b), we found CL = λ = 6.30, which is close to the design cycle length of λ = 2π ≈ 6.28. With a phase shift of λ/4, the trajectories form a closed circle, and the average angle is −1.00 ± 0.00 radians. With a phase shift of λ/2, the average angle is −1.07 ± 0.48, that is, the rotational pattern is an ellipse. We obtain the same average angle, but with greater standard deviation. The wedge in

Figure 2b suggests that the cycle time corresponds to the number of time steps, 1, 2, … n required to fill the ellipse with wedges.

Step 5. The timing (TL). The regression slopes, s, or the β-coefficients, will for cyclic series give information on the shift, or time lag, between the series. For a linear correlation applied to paired time series that are normalized to unit standard deviation, the regression coefficient (r) for the paired series and the β-coefficient (the slopes) will be identical. If the two series co-vary exactly, their regression coefficient will be 1, and the time lag will be zero. If they are displaced half a cycle length, the series are counter-cyclic, and the correlation coefficient is r = −1. Lead or lag times, TL, are estimated from the correlation coefficient, r, for sequences of 5 observations, TL (5). With λ as the cycle length, an expression for the time lag between two cyclic series can be approximated by:

The method is implemented in Excel and requires only the pasting of new datasets into two columns. The data set and all calculations are available from the authors.

For the whole period, we first calculate the LL relations for 3 consecutive months and then calculate the rolling average LL relations for 9 months. The LL strength for the whole period 1991 to 2016 is the average LL strength of the 308 observations calculated with Equation (2).

Since many of the leading indicators aim at finding turning points in the economy (

Banerji et al. 2006;

OECD 2012;

Ulbricht et al. 2017), we found the LL strength for the periods before and a little into the recessions and the recoveries. We examined the leading relationship for 9 months, with respectively 6 months before and 3 months after the recession peak, and correspondingly for the recovery trough.