A combined iteration algorithm for the implicit cycles of gold price and the US dollar index

The relationship between gold and US dollar series, which are non-stationary, is commonly known to be negative from a periodic perspective. Is this really the case? This paper established a combined iteration algorithm using the theory of spectral analysis after subtracting the trend using penalized B-spline functions to obtain the implicit cycles in gold and US dollar series. This algorithm accurately separates the trend terms and periodic terms of the two series to produce more precise and complete periodic information. The results show that both series share three common implicit cycles: two long periods and one short one. Both long-period terms are negatively correlated, whereas the short-period terms are positively correlated.