concordance#

cultivars.spectral.cycles.concordance(first, second)[source]#

Harding-Pagan concordance of two chronologies.

\[I = \frac{1}{T} \sum_{t=1}^{T} \bigl[ S_t S'_t + (1 - S_t)(1 - S'_t) \bigr],\]

the share of dates on which two contraction indicators \(S\) and \(S'\) agree. Each argument is a TurningPointsResult or a (T,) array: booleans are taken as the contraction indicator, and a probability series – a Markov-switching model’s smoothed recession probability – is thresholded at one half.

Parameters:
Return type:

float

Returns:

The share of dates on which the two indicators agree.

Raises:

Note

One half is the expected concordance of two independent chronologies that each spend half their time in contraction; in general it is \(p p' + (1 - p)(1 - p')\) for contraction shares \(p\) and \(p'\), so two chronologies that are mostly expansion agree by chance well above one half, and a reading should be set against that base rate rather than against one half. The threshold is strict: a probability of exactly one half reads as expansion. An integer 0/1 series is read as probabilities and works as expected; a series outside [0, 1] is not refused and is thresholded like any other.

See also

References

Harding, D., & Pagan, A. (2002). Dissecting the cycle: A methodological investigation. Journal of Monetary Economics, 49(2), 365-381.

Harding, D., & Pagan, A. (2006). Synchronization of cycles. Journal of Econometrics, 132(1), 59-79.

Example

>>> import numpy as np
>>> a = np.array([True, True, False, False, False])
>>> concordance(a, np.array([0.9, 0.6, 0.4, 0.1, 0.2]))
1.0
>>> concordance(a, ~a), concordance(a, [1, 1, 0, 0, 1])
(0.0, 0.8)
>>> concordance(a, np.zeros(4))
Traceback (most recent call last):
    ...
cultivars.exceptions.DimensionError: chronologies must align; got lengths 5 and 4.
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