Methodology

Every number on the board is produced by the same process, run country by country. It is a live-data replication of the Advanced-Market track of Chen & Svirydzenka (2021). The six steps below run in order.

1. Assembling the series

Each economy contributes three series: seasonally-adjusted real GDP, a share-price index, and CPI. Monthly CPI is averaged to quarters, then used to deflate share prices into real equity prices.

2. Removing the drift

Both series grow over the long run, and that steady growth would otherwise be read as part of the cycle. So it comes out first: the average growth per quarter is measured across the whole sample, and the straight line it describes is subtracted, leaving a series that ends at the level it began. Everything after works on that drift-adjusted series.

The drift adjustment

Call the raw log series w_t. Its average growth per quarter, μ̂, is read off the two end points, and is then removed cumulatively: nothing at the first observation, one μ̂ at the second, two at the third, and so on to the end of the sample.

μ̂ = w_T − w₁T − 1 x_t = w_t − (t − 1)μ̂

Behind that estimate is an assumption that the raw series is a random walk with drift, each quarter's value being the last one plus a fixed step μ plus a zero-mean disturbance u_t.

w_t = μ + w_t−1 + u_t

Because the series is in logs, the straight line it describes is steady growth. This is Christiano & Fitzgerald's own drift adjustment, set out in footnote 7 on page 4 of The Band Pass Filter.

3. Choosing the band

The filter has to be told which cycle lengths to keep. Those bands are taken unchanged from Chen & Svirydzenka (2021): equity 8–22 quarters, GDP 3–51 quarters. A single band covers the whole advanced-market group, so every market on the board is filtered identically. The derivation is in its Tables 1 and 2 (see References).

Converting the band to frequencies

The band is a pair of periods, p_l (shortest) and p_u (longest), in quarters. The filter works in angular frequencies, so they are converted:

a = 2πp_u b = 2πp_l

4. Applying the filter

The Christiano–Fitzgerald band-pass filter separates each series into two parts. The cyclical swing is the movement whose length falls inside the band above. Everything else is the slow trend: drift too slow to fit in the band, along with wobble too fast for it.

A trend sits best through a point when the series can be seen on both sides of it, but in real time the future side does not exist yet. So the filter is run in its one-sided form, which uses only the data up to the quarter being measured and leans on an end-weight to stand in for the rest.

The filter weights, and the two-sided and one-sided forms

The filter is a set of weights applied to neighbouring quarters:

B₀ = b − aπ B_j = sin(jb) − sin(ja)πj ,j ≥ 1

With hindsight (two-sided), the cycle at quarter t is a weighted sum of the data on both sides of it:

ŷ_t = B₀x_t + ΣT−1−tj=1 B_jx_t+j + B̃_T−tx_T + Σt−2j=1 B_jx_t−j + B̃_t−1x₁

The one-sided form is the same filter with the forward terms dropped at the latest quarter T. The weight on the current observation halves and the end-weight takes up the remainder:

ŷ_T = 12B₀x_T + ΣT−2j=1 B_jx_T−j + B̃_T−1x₁

The end-weight B̃ is what stands in for the data past the edge of the sample. It is the sum of every weight that would have fallen beyond that edge, applied in a single lump to the boundary observation. Since the full set of weights sums to zero, that leftover tail can be written using only the weights you have:

B̃_k = −12B₀ − Σk−1j=1 B_j

Where k is the boundary observation's distance in quarters. Christiano & Fitzgerald set this out in footnote 3 on page 2 of The Band Pass Filter, with the one-sided case B̃_T−1 in footnote 4 on page 3.

5. The gap

The cyclical swing, multiplied by 100, is the gap: how far a series sits from its own calculated trend, in percent. This is the number plotted on every chart.

gap_t = 100 × ŷ_t

6. From gap to signal

Each gap is then compared with the threshold the paper calibrated for it: equity +6.7%, output +1.3% giving.

Symbols used above

w_t	the log, inflation-adjusted series at quarter t (i.e. log real equity price)
x_t	the same series with the drift line removed. What is applied to the filter
μ	the drift: the series' true average growth per quarter, in log points
μ̂	the estimate of that drift, read off the first and last observations
u_t	the zero-mean disturbance in quarter t
t, T	the quarter being evaluated, and the total number of quarters in the sample
ŷ_t	the extracted cyclical component at t, i.e. the gap before ×100
p_l, p_u	shortest / longest cycle length kept, in quarters (equity 8 / 22; output 3 / 51)
a, b	the band edges as angular frequencies
B₀	filter weight on the quarter being evaluated
B_j	filter weight on the quarter j steps away
B̃_k	end-of-sample correction weight on a boundary observation k quarters away

Which readings the site shows

Every reading on this site is the one-sided read for its quarter: the value an observer would have computed at the time, using only the data that existed then.

Because each cell uses only the data available in its own quarter, no cell ever changes. A new quarter of data adds a cell and leaves the earlier ones untouched. The cost of that is the end-weight: standing in for data that had not happened yet pulls a real-time reading toward zero, so a boom still building registers milder than its full size.

Source: Christiano & Fitzgerald, The Band Pass Filter, eqs (1.2)–(1.4); bands and thresholds from Chen & Svirydzenka (2021). Full citations under References.

Limits of the calibration

The cycle durations, filter bands, thresholds and weights have been determined by the paper's 1960–2014 sample and is held fixed. That freeze keeps every market comparable against the same crisis-calibrated bar.

The bands can drift out of date. The filter only extracts swings inside the band; if advanced-market cycles have lengthened since 2014, a boom building more slowly than 22 quarters would be absorbed into "trend" and the equity gap would understate it. A bias toward missed warnings rather than false alarms.

The thresholds cannot be re-checked. The +6.7% / +1.3% bars minimize missed crises versus false alarms over a small set of historical episodes with no new advanced-market systemic banking crisis to test against. (see References)

Treat a threshold crossing as a prompt to investigate the economy further and weigh persistent deviations more heavily than a single reading.