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Mueller (2015, working paper) says that an HP filter with smoothing parameter 900,000 (for monthly data) corresponds to a smoothing parameter of 100,000 for quarterly data.

How does one do this exact calculation? If I were to calculate the corresponding annual parameter, how would I proceed?

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Morten O. Ravn and Harald Uhlig (2002)

This paper complements these insights using two different analytical approaches. The first approach uses the time domain and focuses on the ratio of the variance of the cyclical component to the variance of the second difference of the trend component: this ratio is often used for calculating the smoothing parameter. For a particular benchmark stochastic process, it is shown that time aggregation changes this ratio by the fourth power of the observation frequency. The second approach uses the frequency domain and investigates the transfer function of the HP filter, thereby obtaining a general result. Again, a change-of-variable argument shows that one should adjust the HP parameter with approximately the fourth power of the frequency change. Both approaches therefore yield a value of approximately $1600 / 4^4 = 6.25$ for annual data, which is close to the value of 10 given by Baxter and King (1999).

Source: Notes On Adjusting the HP-Filter for the Frequency of Observations

Mueller's result is not obvious to me given this rule. By that rule of thumb, a monthly parameter given an quarterly parameter of 900,000 is $900,000 / 3^4 \approx 11,111$ (because there 3 months in a quarter). It also gives a yearly parameter of $900,000 / 12^4 \approx 43$ (12 months in a year).

The following quote seems to support Mueller.

On suboptimality of the Hodrick–Prescott filter at time series endpoints Hodrick and Prescott (1997) proposed, on somewhat subjective grounds, a value λ = 1600 for quarterly data. However, it is desirable to adjust this value when observations of different frequencies are subject to the filter. Backus and Kehoe (1992) suggested an adjustment of the value by multiplying the standard value of 1600 with the square of the frequency of observations relative to quarterly data. For example, the relative frequency is 3 for monthly data and 1/4 for annual data. Hence, the corresponding values of the smoothing parameter is $\lambda$ = 100 and 14,400 for annual data and monthly data, respectively. This suggestion has been also used in commercial packages such as EVIEW. We shall use these values throughout the paper. With regard to the choice of the smoothing parameter, it is worth noting that, in research that has gone largely unnoticed in this field, Akaike (1980), while further allowing a seasonal component in the decomposition, proposed precisely the HP approach together with a data-dependent Bayesian procedure for the choice of $\lambda$.

So just as $1,600 (quarterly) \rightarrow 3^2 \cdot 1,600 = 14,400 (monthly)$ so to would $100,000 (quarterly) \rightarrow 3^2 \cdot 100,000 = 900,000 (monthly)$

Misea, Kimb, and Newboldc (2005)

But the cited Backus and Kehoe (1992) result seems to be superseded by the Ravn and Uhlig (2002) result, so this may not reflect the state of the art thinking on this matter.

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