South Korea DB Core CPI 3-Month Inflation-at-Risk

Inflation-at-Risk applied to Korean DB core CPI, with the conditional distribution of the 3-month-ahead average annualized inflation rate as the headline object (López-Salido and Loria 2020). With $\pi_{t\to t+h}$ the average annualized inflation rate from month $t$ to month $t+h$, this ticker is the $h=3$ horizon, and its headline reports the two-sided quantile set $\{Q_{0.05},\,Q_{0.50},\,Q_{0.95}\}$ in percent. The long-run anchor of the conditioning variable $\pi^{\text{long}}_t$ combines the Bank of Korea inflation target, a Stock and Watson (2007) UCSV trend, and a survey-expectations component.

The quantile fan from $q_{05}$ to $q_{95}$, the two-sided expected shortfalls, and the downside-minus-upside asymmetry $(q_{50}-q_{05})-(q_{95}-q_{50})$ are provided as auxiliary series for reading where in the distribution current conditions are acting. The 3-month horizon is the companion to the 12-month headline and responds more sensitively to shorter-cycle disturbances, so a fan that widens upward indicates current conditions are acting more strongly on the upper part of the distribution and a fan that widens downward indicates they are acting on the lower part. The indicator is a characterization of this conditional distribution, not a point-estimate forecast.

(1) Quantile-regression fit. The same gap-form quantile regression as KRIAR12M is fit at the $h=3$ horizon,

where

, $\pi^{\text{long}}_t$ is the long-run anchor described in the definition, $\text{gap}_t$ is the one-sided HLW Kalman-filtered output gap (Holston, Laubach, and Williams 2023), and $\text{FCI}_t$ is the level of KRFCI. The gap form imposes the implicit coefficient $1-\lambda_\tau$ on $\pi^{\text{long}}_t$ in the manner Faust and Wright (2013) identify as the standard discipline for any modern Phillips-curve specification, and the level quantiles are recovered by adding $\pi^{\text{long}}_t$ back in the same way. Per-row monotonicity across $\tau$ is enforced by the Chernozhukov, Fernández-Val, and Galichon (2010) rearrangement.

(2) Estimation and standard errors. Coefficients are estimated at monthly frequency on the target's natural cadence, with Newey-West HAC standard errors at lag

to absorb the overlapping 3-month targets. Monthly inputs are projected onto the business-day grid by step interpolation stamped at the official CPI publication lag, and KRFCI enters as its daily level.

(3) Seasonality handling. The 3-month horizon uses 3-month windows for both the trailing regressor and the forward target and is therefore seasonality-sensitive. Each pipeline run performs the seasonal-and-trend decomposition of Cleveland, Cleveland, McRae, and Terpenning (1990) on the log level and substitutes the deseasoned reconstruction when the Wang, Smith, and Hyndman (2006) seasonal-strength statistic exceeds 0.20.

(4) Expected shortfall and asymmetry. The two-sided expected-shortfall and asymmetry layers are computed by the same procedure as KRIAR12M. A four-parameter Azzalini-Capitanio skew-t (Azzalini and Capitanio 2003) is fit per month on the monthly-fit-evaluated level quantiles, the shape $(\alpha, \nu)$ is carried forward to daily, and the daily $(\xi, \omega)$ are recovered in closed form by ordinary least squares against the standardized quantiles, which mirrors the daily location-scale recovery pattern Korobilis (2017) lays out for daily quantile forecasting. The two-sided expected shortfalls are fixed at the QR-direct $q_{05}/q_{95}$ thresholds with hard clips, and the asymmetry is defined as $(q_{50}-q_{05})-(q_{95}-q_{50})$ and follows the KRGAR sign convention.

This series is an indicator of the short-term inflation tail distribution that rotates on Korea's funding-refinancing cycle. At the 3-month horizon the same anchored quantile Phillips construction of López-Salido and Loria (2020) applies unchanged, built on the Vulnerable Growth template of Adrian, Boyarchenko, and Giannone (2019) and the Koenker and Bassett (1978) quantile regression with an accelerationist sum-to-one anchor, so the framework is identical to the 12-month sibling and only the horizon differs. Korean financial intermediaries roll FX swaps and short-dated commercial paper on roughly the 3-month cadence, so the quarterly horizon is where the currency, swap, and credit conditions carried into KRFCI under the Hatzius, Hooper, Mishkin, Schoenholtz, and Watson (2010) construction transmit fastest into near-term consumer-price prints. Reading the 3-month line and the 12-month line together separates a transient funding-stress spike from a persistent at-risk shift, mirroring the funding-cadence companion role KRGAR3M plays for the growth tails. Mean inflation forecasts are extremely hard to beat as at the 12-month panel (Atkeson and Ohanian 2001), and the value of the quantile decomposition lies in surfacing the at-risk signal the mean averages away, in line with the inflation-forecast literature Faust and Wright (2013) survey.

The 3-month horizon carries about four times the effective sample size of the 12-month horizon. With roughly seventy-five effective independent observations against the 12-month horizon's roughly nineteen, it is the better-identified calibration read on this sample, which is why the out-of-sample block here is the one that bears more statistical weight when the two horizons diverge. The look-ahead-free one-sided Kalman-filtered HLW output gap (Holston, Laubach, and Williams 2023) and the Bank of Korea 2 % point target anchor that Stock and Watson (2007) report as the persistence-modelling discipline for advanced-economy inflation and that Mertens and Nason (2020) carry across countries enter at this horizon exactly as they do at 12 months. The structural underpinning of a well-behaved anchored conditional distribution at either horizon is the expectations-anchoring channel Coibion and Gorodnichenko (2015) identify, and the way the conditional median tracks the era-level anchor at the 3-month horizon is of the same kind Korobilis (2017) reports for analogous direct-quantile inflation models elsewhere.

The Korean 3-month sample does not statistically identify an FCI asymmetry direction either. The conditional FCI slope $\hat\theta_\tau$ runs over a narrow band with absolute values below roughly six basis points and mixed signs across $\tau$, no individual $t$-statistic crosses one, the diagonal-HAC joint Wald test on equality across $\tau$ does not reject at any conventional level, and all pairwise contrasts carry $p$-values above 0.90. The López-Salido and Loria (2020) lower-tail channel for the U.S. and the emerging-market upper-tail rising channel that Banerjee, Contreras, Mehrotra, and Zampolli (2020) record in samples where currency depreciation lifts the upper inflation quantile are both consistent with the point profile without being asserted as Korean results.

Out of sample the 3-month picture is materially milder than at 12 months. The conditional pinball ratio against the unconditional benchmark sits near one in the lower tail and at the median and slightly above one in the upper tail, the upper-tail PIT runs about five percentage points below nominal rather than nine, and an ex-crisis recomputation that excludes the 2008–2009 and 2020 windows leaves these ratios essentially unchanged. The 3-month out-of-sample miss in the upper tail is therefore sample-limited rather than episode-driven, in contrast to the 12-month horizon where the miss concentrates in the crisis windows. The 2009 Q1 and 2020 Q1 episodes remain the periods where this horizon descriptively transmits a KRFCI shock to both sides of the inflation distribution, in line with the official CPI vintages.

For near-term inflation surveillance the headline reads on both sides, with the weight on each tail calibrated to what the 3-month out-of-sample block shows. The upper-tail pinball ratio slightly above one against the unconditional benchmark and the upper-tail PIT close to but still below nominal mean the upper-tail read carries less weight than the lower-tail read, though the miss at this horizon is markedly milder than the 12-month horizon's. Korean financial intermediaries roll FX swaps and short-dated commercial paper on roughly the 3-month cadence, so a shift in either 3-month tail is a signal to read alongside the funding-line cycle the Bank of Korea Financial Stability Report documents for swap-line and short-dated paper markets.

The two horizons are meant to be read together. The 3-month line moves first and amplifies acute KRFCI moves, while the 12-month line carries the structural cycle on which monetary-policy transmission completes, so when the two diverge the source of the divergence is itself informative. A two-sided-tail blow-out that appears at 3 months alone points to a transient cost or demand event, and a synchronized move across both horizons points to a structural turn. The fan and the two-sided expected shortfalls at this horizon are the same descriptive objects they are at 12 months, with the lower-tail expected shortfall the more coherent input for deflation-scenario design and the upper-tail object treated as a description of how current conditions are pricing the upper quantile rather than a quantitative ceiling on future risk.

The pairing that matters operationally is with KRFCI and KRGAR3M. How the growth tails KRGAR3M draws and the two-sided inflation tails this indicator draws respond differently to the same KRFCI event is the first-order near-term policy variable. The parallel-horizon framing Adrian, Boyarchenko, and Giannone (2019) provide for the growth tails and López-Salido and Loria (2020) provide for the inflation tails translates into a coherent near-term dashboard, and Hatzius, Hooper, Mishkin, Schoenholtz, and Watson (2010) supply the financial-conditions link that closes the loop.

As with KRGAR and KRFXAR the gauge is descriptive rather than predictive, the FCI asymmetry direction is not statistically identified in this sample, and the point that the upper tail carries less out-of-sample weight than the lower tail holds even though the 3-month upper-tail miss is mild compared with the 12-month one. The model re-estimates on the full sample every run, so the most recent published day is the least settled and history revises as the panel grows.

KRIAR3M is an estimated conditional quantile path of three-month-ahead inflation, $\hat{q}_t(\theta)=x_t^{\top}\hat{\beta}(\theta)$ from the gap-form quantile regression, so its persistence is inherited from the conditioning variables and the evaluation object is the violation-indicator sequence, not the path. Unit-root, structural-break, and level Ljung and Box (1978) batteries on the path were deliberately not run, because coverage is a property of the joint forecast-outcome distribution that a univariate path diagnostic cannot reach (Christoffersen 1998; Engle and Manganelli 2004; Murphy and Winkler 1987), and a bounded estimated quantile cannot be an unbounded unit-root process on which standard inference over-rejects (Cavaliere 2005; Cavaliere and Xu 2014). The appropriate evidence is coverage backtesting at the two pinned extreme tails, the fifth and ninety-fifth conditional quantiles.

At a three-month horizon the violation sequence overlaps across adjacent months, so no size-correct unconditional-coverage test exists on the full sample. A literal Kupiec (1995) likelihood-ratio test on the overlapping sample and a Newey and West (1987) autocorrelation-robust coverage statistic constructed to repair it were both found materially oversized in a pre-registered Monte-Carlo size check recorded in the model specification, rejecting a correctly-covered null far above the nominal level, so a formal unconditional-coverage verdict is withheld on the full sample and the full-sample record is reported descriptively.

Two coverage panels appear because they measure different things. The reused validation panel is the in-sample probability-integral-transform hit rate over the estimation panel, the empirical frequency that the realized outcome falls at or below the fitted conditional quantile, so it targets the quantile level itself and reads 0.060 at the fifth-percentile quantile and 0.944 at the ninety-fifth over 234 months, the upper figure being cumulative coverage near its ninety-five-percent target rather than a breach rate. The pinned monthly-sampled backtest of 235 observations from 2006-09-30 to 2026-03-31 instead counts tail exceedances directly, the breaches below the fifth-percentile quantile and above the ninety-fifth, recording eleven lower-tail exceedances at a rate of 0.047 and eight upper-tail exceedances at 0.034, both near the five-percent tail mass.

The one size-correct formal check is the Kupiec test on the offset-zero non-overlapping subsample, where independent Bernoulli sampling is restored by construction, and on that subsample of seventy-eight observations with an expected 3.9 exceedances it records three exceedances at each tail, 0.237 at p = 0.627, and fails to reject correct coverage. The subsample check is exact under the hypothesis of correct conditional coverage, which renders the spaced violation indicators independent Bernoulli draws (Christoffersen 1998), and it does not claim size control for exceedances of a fixed unconditional quantile of a persistent process, where dependence can cross the non-overlapping window boundaries. The exceedance rate ranges from 0.038 to 0.051 at the lower tail and from 0.013 to 0.051 at the upper tail across the three admissible offsets, reported to preclude offset selection.

The Christoffersen (1998) independence and conditional-coverage tests reject at both tails, the independence statistics 25.715 at p = 0.000 and 11.153 at p = 0.001 and the conditional-coverage statistics 25.760 at p = 0.000 and 12.532 at p = 0.002, which is the clustering that the overlapping three-month target induces in the hit sequence rather than a coverage failure and is consistent with the verdict withheld on the full sample. The Engle and Manganelli (2004) dynamic-quantile test is run in its constant-only binomial-score form because the overlap removes the eligible instruments, and it fails to reject at both tails, 0.050 at p = 0.822 and 1.260 at p = 0.262, with the full instrument set deferred. The cross-quantile monotonicity check confirms a non-crossing fan (Chernozhukov, Fernandez-Val, and Galichon 2010).

The limits are stated as such. The overlapping three-month target drives the independence and conditional-coverage rejections, so at this horizon the non-overlapping subsample Kupiec test and the reused validation coverage are the only formal reads, and both are underpowered by the effective number of independent windows rather than the 235 sampled points, so a non-rejection is weak evidence for calibration (Kupiec 1995; Berkowitz 2001; Diebold, Gunther, and Tay 1998; Rosenblatt 1952). Because these are estimated quantiles whose parameter uncertainty is not absorbed into the reference distribution, the coverage reading is descriptive of how current conditions place the inflation distribution rather than a validated forecast (Campbell 2005; Escanciano and Olmo 2010).