HP Filter Smoothing Parameter 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?
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.