add Baxter King filter

[Shunsuke-Hori]

test is added too

[codecov-io]

Codecov Report

Merging #223 into master will decrease coverage by 0.01%.
The diff coverage is 100%.

@@            Coverage Diff             @@
##           master     #223      +/-   ##
==========================================
- Coverage   95.87%   95.86%   -0.02%     
==========================================
  Files          25       25              
  Lines        1067     1063       -4     
==========================================
- Hits         1023     1019       -4     
  Misses         44       44

Impacted Files	Coverage Δ
src/filter.jl	100% <100%> (ø)	⬆️
src/util.jl	98.14% <0%> (-0.07%)	⬇️
src/markov/markov_approx.jl	98.37% <0%> (-0.04%)	⬇️
src/markov/ddp.jl	98.63% <0%> (-0.02%)	⬇️
src/arma.jl	100% <0%> (ø)	⬆️

---

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[sglyon]

Hi @Shunsuke-Hori thanks for submitting!

I googled for some references on the BK filter equations and found conflicting evidence.

Two references are

• http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.197.8691&rep=rep1&type=pdf
• http://help.prognoz.com/en/mergedProjects/Lib/02_time_series_analysis/uimodelling_baxterkingfilter.htm

A few differences:

• the html page only divides b[2:end] by (2K+1), but keeps b[1] without dividing
• The pdf has a negative sign in front of theta, while the html page keeps it positive.

your code seems to more closely follow the pdf, but I'm not sure.

What reference did you use? How do we know which is correct?

[Shunsuke-Hori]

@sglyon Thank you for the review.

I first looked the html but since the output was not same as python's, I switched to the pdf.

I checked the original paper and it seems the pdf is consistent with the paper (although, honestly, I mindlessly trusted the result of `bk_filter' of python when I wrote the code)

[andi3141]

Hi @Shunsuke-Hori
I had a look at the code. It seems perfectly in line with
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.197.8691&rep=rep1&type=pdf
The second reference
http://help.prognoz.com/en/mergedProjects/Lib/02_time_series_analysis/uimodelling_baxterkingfilter.htm
seems not to be online any more. Further the original reference
https://www.mitpressjournals.org/doi/abs/10.1162/003465399558454?casa_token=pfn7A97wi0QAAAAA:78AlfSOyiz__a6_snyH7jwydLu8uGZV-U3ZVY5Vo3dTi0r7da5uuVtUyUR87-uZBAuarQAYMiywJ76o
is not publicy available (MIT press). I try to get my hands on that at some point.
The implementation gives the same result as the python implementation in
https://www.statsmodels.org/stable/generated/statsmodels.tsa.filters.bk_filter.bkfilter.html#statsmodels.tsa.filters.bk_filter.bkfilter

So I think, it is fine.

[jstac]

Should be leads and lags

[jstac]

Many thanks @andi3141 for your review. This looks good to me too (aside from very minor comment above).

[codecov-commenter]

Codecov Report

Merging #223 into master will decrease coverage by 0.01%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master     #223      +/-   ##
==========================================
- Coverage   95.87%   95.86%   -0.02%     
==========================================
  Files          25       25              
  Lines        1067     1063       -4     
==========================================
- Hits         1023     1019       -4     
  Misses         44       44              

Impacted Files	Coverage Δ
src/filter.jl	100.00% <100.00%> (ø)
src/util.jl	98.14% <0.00%> (-0.07%)	⬇️
src/markov/markov_approx.jl	98.37% <0.00%> (-0.04%)	⬇️
src/markov/ddp.jl	98.63% <0.00%> (-0.02%)	⬇️
src/arma.jl	100.00% <0.00%> (ø)

---

Continue to review full report at Codecov.

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Δ = absolute <relative> (impact), ø = not affected, ? = missing data
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[Shunsuke-Hori]

Thank you for reviewing, @andi3141 and @jstac

I fixed the grammar.

[jstac]

Looks good to me. Over to you @sglyon