South Korea Real GDP 3-Month Growth-at-Risk

KRGAR3M is Growth-at-Risk applied to Korean real GDP, headlined by the 5th conditional percentile of the one-quarter-ahead annualized growth rate. With $y_{t\to t+h}$ the average annualized growth rate of real GDP from quarter $t$ to quarter $t+h$, this series is the $h=1$ horizon (3 months), and its headline is the conditional quantile $Q_{0.05}(y_{t\to t+h}\mid x_t)$ in percent. The object is the whole conditional growth distribution given current conditions and the current secular growth state, and the fan from $q_{05}$ to $q_{95}$, the lower-tail expected shortfall, and the downside-minus-upside asymmetry are exposed as auxiliary series for reading where in the distribution current conditions are biting.

The conditional distribution is estimated by implementing the Vulnerable Growth framework (Adrian, Boyarchenko, and Giannone 2019) for Korea with Koenker-Bassett quantile regression (Koenker and Bassett 1978), conditioning on the macro-orthogonalized financial-conditions index $\text{KRFCI}^{\perp}$, current annualized growth as a level control, and the one-sided HLW filtered trend-growth state (Holston, Laubach, and Williams 2023). Conditioning on the trend state lets the whole conditional distribution ride Korea's secular path rather than the full-sample intercept. This 3-month horizon is the companion to the 12-month surveillance-standard headline, and it responds more sensitively to shorter-cycle disturbances.

A lower conditional 5th percentile means a deeper downside risk gauged when financial conditions tighten, while the headline level itself moves with where its centre settles as Korea's trend evolves. The series is a characterization of the conditional distribution, not a point forecast.

Computed in four steps that combine conditional quantile regression with a skew-t layer.

(1) Conditional Quantile Regression. Conditional quantiles come from quantile regression fit separately at each $\tau$ (Koenker and Bassett 1978),

where the regressors are

The level control $g_t$ is current annualized quarter-on-quarter growth, $\text{KRFCI}^{\perp}$ is the macro-orthogonalized financial-conditions index, and $g^{f}_t$ is the one-sided trend-growth state from the HLW model (Holston, Laubach, and Williams 2023). $g^{f}_t$ is the quarter-end Kalman-filtered estimate rather than the two-sided RTS-smoothed estimate the HLW step publishes as its headline, and the filter at $t$ uses only data through $t$, so $g^{f}_t$ carries no look-ahead. Coefficients are estimated at quarterly frequency on the target's natural cadence, with Newey-West HAC standard errors at lag $h-1$ to absorb the overlapping $h$-quarter targets.

(2) Per-Row Evaluation and Rearrangement. The estimated intercepts and slopes are evaluated at each quarter-end on the at-period regressor vector

to produce seven per-row conditional quantiles, and per-row monotonicity across $\tau$ is enforced by the Chernozhukov, Fernández-Val, and Galichon (2010) rearrangement. The series is quarterly, not a quarterly model projected onto a daily grid, so no publication-lag forward-fill is applied.

(3) Expected Shortfall and Asymmetry. The expected-shortfall and asymmetry layers come from an Azzalini-Capitanio skew-t (Azzalini and Capitanio 2003) fitted to the rearranged conditional quantiles. At quarterly cadence the panel is roughly eighty rows, so rather than fitting the shape parameters at quarter-end and forward-filling, the full four-parameter $(\xi, \omega, \alpha, \nu)$ skew-t is fit per quarter directly, and the fit is carried out by seeded multistart least squares against the closed-form skew-t quantile function. The expected shortfall is $E[y\mid y < q_{05}]$ under the per-quarter skew-t with the threshold pinned to the QR-direct $q_{05}$, which keeps $\text{ES}\le q_{05}$ by construction. Quarters where the skew-t fit fails to converge or $\nu$ pins at its upper bound are handled by a skew-normal fallback, and residual grid-interpolation noise is absorbed by a hard clip $\text{ES}\le q_{05}$. The asymmetry is

so positive values mark a fatter conditional downside than upside.

(4) Full-Sample Re-estimation. Coefficients are re-estimated on the full sample at every quarterly run, so recent quarters revise as the HLW filter incorporates later data, and the latest published quarters are the least settled.

At the one-quarter horizon the same Vulnerable Growth construction (Adrian, Boyarchenko, and Giannone 2019) reads as a near-term downside-risk gauge that turns over faster than the 12-month headline. The quantile regression applied to the lower tail continues the conditional-quantile time-series tradition Engle and Manganelli (2004) opened with the CAViaR family. The gauge is more sensitive to acute shifts in financial conditions than to the slower structural signal, so reading the two horizons together separates a transient KRFCI spike from a persistent widening of the four-quarter-ahead distribution. The one-quarter horizon is also the horizon on which funding-rollover risk is priced, where currency, swap, and credit conditions transmit fastest into the near-term real economy. The 12-month sibling is the standard surveillance horizon the International Monetary Fund (2017) Global Financial Stability Report uses, and the calibration debate Plagborg-Møller, Reichlin, Ricco, and Hasenzagl (2020) and Brownlees and Souza (2021) shape applies in the same form here, namely that the framework is bounded by what specification and predictor choices can claim out of sample on a short panel.

Conditioning on the HLW filtered trend changes the level the 3-month gauge sits at as well. The 3-month median descends with Korea's trend the same way the 12-month median does. At the tails, the 3-month $\tau=0.05$ pinball ratio improves to about 1.15 against the expanding-window benchmark while the upper-tail ratio comes in below one at 0.93. On the Christoffersen (1998) coverage frame and the broader quantile-evaluation toolkit Galbraith and van Norden (2019) lay out, the PIT hit rates stay within roughly two percentage points of nominal across all seven $\tau$. The headline at the funding-rollover horizon then sits on a level that respects the secular environment rather than a full-sample average.

The honest caveats are the same in shape and arguably sharper in magnitude. The quarterly sample is short, the lower-tail bands are wide, and the slope of $\text{KRFCI}^{\perp}$ at $\tau=0.05$ stays imprecisely estimated. The 2009 Q1 trough is the sharpest reading in the sample. The conditional $q_{05}$ priced a severe one-quarter contraction at the depth of the global financial crisis, when KRFCI was still at its $+10$ SD readings (Baba and Shim 2010) and the basis carried Korea's dollar-funding stress through to the growth distribution within the quarter. The realized one-quarter print landed near the conditional $q_{05}$ and did not fall below it, which is a result the framework allows.

For near-term financial-stability monitoring the headline $q_{05}$ reads as a one-quarter-ahead Value-at-Risk for real activity at the horizon on which funding rollover is priced. Korean financial intermediaries roll FX swaps and short-dated CP on roughly that cadence, so a widening one-quarter downside is a signal that maps directly onto the funding-line cycle. The expected shortfall at this horizon sizes the conditional tail bound that a stress scenario for a funding squeeze should clear, and the asymmetry shows whether current conditions are pulling the distribution down on one side or fattening both tails.

This series should be read together with the 12-month headline, not on its own. The two horizons separate transient funding-stress spikes from persistent at-risk shifts. The 3-month line moves first and amplifies acute KRFCI moves, while the 12-month line carries the structural cycle that central-bank surveillance reports on (Adams, Adrian, Boyarchenko, and Giannone 2021). When the two diverge the source of the divergence is informative. A 3-month-only spike points to a funding episode, while a synchronized move points to a structural turn.

The pairing that matters operationally is with KRFCI and the cross-currency swap basis. Korea's near-term tail risk is largely a dollar-funding story, and the basis is where that story prints first. Overlaying the 3-month $q_{05}$ on the 1-year basis therefore yields a fast-moving stress dashboard for KRW funding markets. As at the 12-month horizon, the level of $q_{05}$ now rides Korea's filtered trend, so the dashboard reading is anchored to where the secular environment sits, and a slightly positive recent print in a low-trend regime is the calibrated reading rather than an under-statement.

KRGAR3M is an estimated conditional quantile path of the form $\hat{q}_t(\theta)=x_t^{\top}\hat{\beta}(\theta)$ fit by quantile regression of growth on the orthogonalized financial-conditions index, so the path's persistence is inherited from its conditioning variables and is not a property of the growth signal itself. The object that carries forecast quality is therefore the violation-indicator sequence, not the path, and unit-root, structural-break, and level Ljung and Box (1978) batteries on the path were deliberately not run, since a coverage question is a property of the joint forecast-outcome distribution that a univariate path diagnostic cannot address (Christoffersen 1998; Engle and Manganelli 2004; Murphy and Winkler 1987). A bounded estimated quantile also cannot be an unbounded unit-root process, on which standard unit-root inference would over-reject toward spurious stationarity without shrinking in the sample size (Cavaliere 2005; Cavaliere and Xu 2014). The evidence appropriate to a quantile object is coverage backtesting at the two pinned extreme tails, the fifth and ninety-fifth conditional quantiles.

Coverage is read first from the reused validation panel, which 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 and so targets the quantile level itself, reading 0.069 at the fifth-percentile quantile and 0.940 at the ninety-fifth over 116 quarters, the upper figure being cumulative coverage near its ninety-five-percent target rather than a breach rate (Diebold, Gunther, and Tay 1998; Rosenblatt 1952).

The three-month horizon steps by a single quarter, so its violation sequence is non-overlapping and the Kupiec (1995) unconditional-coverage test applies directly, recording four lower-tail and two upper-tail exceedances over 78 quarters, 0.003 at p = 0.959 and 1.177 at p = 0.278, and failing to reject correct coverage.

The Christoffersen (1998) independence and conditional-coverage tests read against the same sequence. No exceedances are consecutive at either tail, so the independence test is uninformative and conditional coverage fails to reject, 0.445 at p = 0.801 and 1.180 at p = 0.554. Because only a handful of quarterly exceedances occur, too few for the full dynamic-quantile regression, the Engle and Manganelli (2004) test is run in its constant-only binomial-score form, and it fails to reject at both tails, 0.003 with p = 0.959 and 0.974 with p = 0.324, with the full instrument set deferred until the sample supports it. The cross-quantile monotonicity check confirms a non-crossing fan at this horizon (Chernozhukov, Fernandez-Val, and Galichon 2010).

Two limits are stated as such. The seventy-eight quarterly observations carry limited power against a correctly specified tail, so a non-rejection at three months is weak evidence for calibration rather than strong evidence of it (Kupiec 1995). The reused validation coverage and the descriptive hit rate are the honest content alongside the direct unconditional test (Diebold, Gunther, and Tay 1998; Rosenblatt 1952; Berkowitz 2001). Because these are estimated quantiles whose parameter uncertainty is not folded into the reference distribution, the coverage reading is descriptive of how current conditions place the growth distribution rather than a validated tail forecast (Campbell 2005; Escanciano and Olmo 2010).