South Korea Shadow Short Rate

The shadow short rate is the latent short-term interest rate that would prevail absent the zero lower bound (ZLB) constraint on nominal interest rates. Unlike the observed policy rate, which is floored at the effective lower bound $r_{lb}$, the shadow rate $s_t$ can take negative values, providing a more accurate measure of the monetary policy stance during periods of near-zero rates.

Formally, the observed short rate is determined as the larger of the shadow rate and the effective lower bound:

When $s_t > r_{lb}$, the constraint is not binding and the shadow rate equals the observed rate. When $s_t < r_{lb}$, the central bank would prefer to lower rates below the bound but cannot, and the shadow rate captures the degree of additional accommodation it would provide if unconstrained. A decline in the shadow rate indicates a deepening of monetary accommodation delivered through unconventional policy tools, while a rise indicates a retreat from that accommodative stance.

Estimated using the Wu-Xia (2016) first-order ZLB approximation applied to the ACM Gaussian affine term structure model.

(1) Gaussian ACM Estimation. The standard five-step ACM pipeline is run first to obtain the Gaussian affine parameters. This pipeline consists of PCA, a VAR(1), Bauer-Rudebusch bias correction, risk price estimation, and affine recursion, and its outputs are the state dynamics $(\mu, \Phi, \Sigma)$, the risk prices $(\lambda_0, \Lambda_1)$, the short rate equation $(\delta_0, \delta_1)$, and the affine coefficients $(A_n, B_n)$. Exact evaluation of the shadow rate bond price

requires simulation or global nonlinear optimization, but applying the Wu-Xia first-order approximation keeps the likelihood function analytically tractable.

(2) Forward Rate Decomposition. The Gaussian log bond price

is decomposed into one-period forward rates:

The $j = 0$ rate is the current shadow short rate, a known value, while the $j \geq 1$ rates incorporate the convexity adjustments embedded in the affine coefficients.

(3) ZLB Option Adjustment. Each forward rate is adjusted for the ZLB floor using the expected value of $\max(s, r_{lb})$ when $s$ is normally distributed:

where $z_j$ is defined as:

and the per-period lower bound $\tilde{r}$ is:

Here $\Phi(\cdot)$ and $\phi(\cdot)$ are the standard normal CDF and PDF, respectively. The function

satisfies

for all $z$, ensuring $f^{SR}_j \geq \tilde{r}$. As $z \to +\infty$, namely when rates lie far above the ZLB,

so $f^{SR}_j \to f^G_j$. Conversely, as $z \to -\infty$, namely when the shadow rate lies far below the ZLB,

so $f^{SR}_j \to \tilde{r}$.

The conditional standard deviation $\sigma_j$ of the shadow rate at horizon $j$ is:

The initial condition $V_0 = 0$ reflects that the current rate is known, and $\Phi^*$ is the Q-measure transition matrix:

$V_j$ is computed iteratively through the recursion:

where $P_k$ is given by:

The total cost is $O(N \cdot K^2)$ for $N = 120$ maturities and $K = 5$ factors.

(4) Shadow Yield Reconstruction. The ZLB-adjusted yield at maturity $n$ is obtained by summing the adjusted forward rates and annualizing:

where $\tau_t(n)$ is the Actual/Actual year fraction for maturity $n$ months from date $t$. The shadow short rate reported in this series is the annualized Gaussian one-month yield

, which equals the model-implied short rate before ZLB adjustment and can take negative values.

The shadow short rate provides a more accurate gauge of the monetary policy stance when the policy rate approaches or reaches the effective lower bound (Black 1995; Krippner 2013). During such periods, the central bank provides additional stimulus through unconventional tools such as quantitative easing, forward guidance, and yield curve control, so the observed short rate alone does not fully reflect the actual degree of monetary accommodation.

Wu and Xia (2016) demonstrate that the shadow federal funds rate declined to approximately $-3$% during the Federal Reserve's zero-rate period (2009–2015), reflecting the substantial easing delivered through Large-Scale Asset Purchases. This shadow rate is used as a direct substitute for the policy rate in Taylor rule analysis, regression-based macro models, and monetary policy shock identification when the observed rate is constrained.

Korea's policy rate has not yet reached the zero lower bound. The Bank of Korea's record-low base rate was 0.50% in May 2020, and the shadow rate could become increasingly useful if rates approach zero in a future easing cycle.

The shadow rate is compared with the HLW natural rate ($r^*$, KRNR series) to assess the policy stance (Holston, Laubach, and Williams 2017). A shadow rate below $r^*$ indicates accommodative policy, while a shadow rate above $r^*$ indicates restrictive policy.

The shadow rate is a key input for bond valuation models that must respect the non-negativity constraint on nominal interest rates. Standard Gaussian affine models can produce negative model-implied yields, which are economically implausible for nominal bonds (Kim and Singleton 2012). Black (1995) recognized that the option-like nature of the ZLB means nominal bonds embed a put option on interest rates struck at zero, and introduced the shadow rate concept. The shadow rate framework corrects this by ensuring that all model-implied yields remain at or above the effective lower bound (Wu and Xia 2016).

For fixed income portfolio managers, the shadow rate provides a cleaner signal for duration positioning when rates are near zero. Near the ZLB, an asymmetric distribution arises in which rate increases produce larger price declines than rate decreases produce price gains, and this asymmetry is captured by the shadow rate model but not by the standard Gaussian framework (Kim and Singleton 2012).

The shadow rate also serves as a gauge of the effectiveness of forward guidance. When the shadow rate is significantly below the observed rate, it suggests that the market is pricing in a substantial probability that rates will remain near the lower bound for an extended period (Christensen and Rudebusch 2015).

This is a model-derived series, the output of the Wu-Xia shadow-rate term structure, so the unit-root reading describes the fitted rate rather than a directly observed price, and the serial-correlation and break diagnostics run on the first difference. The series is tested over its own span from 2000-12-18 to 2026-06-09.

The shadow short rate is integrated of order one on the level, with the Dickey and Fuller (1979) test in the Said and Dickey (1984) form not rejecting at p = 0.5017, the Phillips and Perron (1988) test concurring at p = 0.6126, and the Kwiatkowski et al. (1992) test rejecting stationarity. On the first difference the Ljung and Box (1978) portmanteau rejects white noise at lags 10 and 20, Q = 66.56 and Q = 126.77 at p = 0.000 and p = 0.000, and the automatic portmanteau of Escanciano and Lobato (2009) concurs with a statistic of 13.68 at p = 0.000. The Bai and Perron (1998, 2003) procedure finds no break in the mean, a reading consistent with the parameter-instability inference of Andrews (1993) on the differenced object (Perron 1989).

This is a daily model output with no low-integer seasonal period, so the seasonal-unit-root and seasonal-stationarity machinery of Hylleberg et al. (1990) and Canova and Hansen (1995) is deliberately not run (Beaulieu and Miron 1992; Ghysels and Osborn 2001).