South Korea ADS Business Conditions Index

The ADS business-conditions index is a coincident index that summarizes into a single number the one common business-cycle signal running through many real-activity indicators such as employment, production, and sales. It separates a single latent business-conditions factor from a panel of mixed-frequency real-activity variables through a state-space dynamic factor model and is computed monthly (Aruoba, Diebold, and Scotti 2009).

The standardized observation vector $y_t$ decomposes into a common component that loads on the single factor and an indicator-specific idiosyncratic component:

where $f_t$ is the latent business-conditions factor published as KRADS, $\Lambda$ the loading coefficients, and $\varepsilon_t$ the indicator-specific idiosyncratic component.

The factor is extracted by combining the log first differences of the input indicators, namely their growth rates, so it measures the flow that is the rate of change of real activity and does not measure its level. Standardization sets the mean to zero, and the orientation is fixed by aligning the sign with GDP growth so that positive values denote above-average activity momentum. Larger positive values denote better-than-average business momentum and larger negative values worse-than-average business momentum, with zero drawn on the chart as the reference line marking near-average conditions. This index is therefore not a level-deviation object of the sort the HLW output gap is, and should be read as a rate-of-change gauge.

With the latent factor $f_t$ and the standardized observation vector $y_t$, the model is

and the quarterly real-GDP growth rate is modeled through the Mariano-Murasawa aggregation as a weighted moving average of unobserved monthly growth rates.

Estimation is by the EM algorithm of the mixed-frequency state-space factor model with the following specification. (1) Inputs. The four trending Korean monthly indicators (payroll employment, all-industry production, retail sales, services production) enter as log first differences in percent. The quarterly real GDP series already ships as a quarter-on-quarter growth rate and is therefore passed to the quarterly slot without further differencing. (2) Standardization. Standardization rescales each observed variable to in-sample mean zero and unit standard deviation, so the factor carries no scale unit. (3) EM fit and smoothing. The specification uses one factor with AR(1) factor dynamics and AR(1) idiosyncratic disturbances, and quarterly GDP is supplied separately as the quarterly input. EM is iterated to convergence under a 500-iteration cap, after which the latent factor is recovered by the Kalman smoother. (4) Sign orientation. The smoothed factor is correlated with quarterly GDP growth expanded onto the monthly grid; if the Pearson correlation is negative the factor is reflected, fixing positive values to mean above-average activity.

The sample begins in January 2000, the first month for which both all-industry production and services production are available, and the full sample is re-fit on every pipeline tick. EM and the Kalman smoother are deterministic functions of the data, and the PCA-based initialization is deterministic too. The matrix-decomposition step is nevertheless BLAS-sensitive, so the model is run with OPENBLAS_NUM_THREADS=1, and the cross-platform snapshot harness gates the factor series against BLAS drift.

The ADS framework treats real activity as a noisy observation of a latent business-conditions variable and extracts that latent variable as a single factor (Aruoba, Diebold, and Scotti 2009). On U.S. data this latent variable is a coincident momentum signal that takes negative values throughout NBER recessions and stays near zero in expansions. The construction sits inside the dynamic-factor coincident-index lineage that Stock and Watson (1989) established and the mixed-frequency literature that Mariano and Murasawa (2003) opened, whose quarterly-flow temporal aggregation makes a quarterly GDP growth rate readable through an unobserved monthly factor. The nowcasting tradition that Giannone, Reichlin, and Small (2008) and Camacho and Perez-Quiros (2010) developed is what made the framework practical at the country level, and the Mariano and Murasawa (2010) restatement of the aggregation result extends the same temporal-aggregation logic to richer panels.

The estimation engine implements the Bańbura and Modugno (2014) EM extension and handles factor models with arbitrary missing-data patterns without dropping observations. This matters operationally because the four real-activity indicators of the Korean monthly panel and the quarterly GDP series run at different release lags, and the EM treatment lets every observation enter the likelihood at its natural timing, so there is no need to truncate the start to the slowest series and force a balanced panel.

The U.S. method check is the first stage, confirming that the specification is consistent with the original ADS. Applying the identical specification to U.S. nonfarm payrolls, industrial production, real personal income less transfers, real manufacturing and trade sales, and real GDP produces a factor that correlates at Pearson 0.94 over 713 common months with the monthly aggregate of the Aruoba-Diebold-Scotti Business Conditions Index (ADSBCI) that the Federal Reserve Bank of Philadelphia publishes. The Mariano and Murasawa (2003) quarterly-flow aggregation is the load-bearing piece that ties the monthly factor to quarterly GDP, and the U.S. method check is the test that this piece is implemented correctly.

The second stage, the Korean coherence check, is run against the Statistics Korea (KOSTAT) coincident composite cyclical component and the official KOSTAT business-cycle reference dates. Because KRADS is a growth-rate factor and the KOSTAT cyclical is a detrended level cycle, the two relate as a derivative to its integral and run roughly a quarter-cycle out of phase. The raw contemporaneous correlation of $-0.04$ is therefore the predicted outcome of comparing a flow with a level rather than evidence of missing signal. Once the dimension match is restored the coherence is clear. Reducing the flow to a level deviation through the cumulative sum that corresponds to its integral raises the correlation with the KOSTAT cyclical to 0.65, and the correlation with the 12-month rolling mean of the factor rises to 0.55, showing that the two series express the same Korean business cycle as a rate of change and as a level. The decisive evidence is that the factor reaches a trough within the (peak, trough] window in every one of the five official contractions that overlap the sample, namely 2000–2001, 2002–2005, 2008–2009, 2011–2013, and 2017–2020. The major events of the Korean macro record are captured consistently as extrema of the factor, with $-2.86$ at the February 2003 credit-card crisis, $-3.70$ just before the December 2008 global financial crisis, and $-5.03$ at the March 2020 COVID peak. The expansion mean of $+0.13$ and the contraction mean of $-0.26$ also satisfy the most basic sign-separation invariant required of a coincident gauge. The two checks aim at a consistency check rather than a basis-point reproduction of the published series.

KRADS measures activity momentum relative to average as a rate of change and does not measure a level gap. A negative reading is below-average activity momentum and a positive reading above-average activity momentum, which is consistent with the growth-rate transformation of the underlying indicators inside the ADS framework. The U.S. ADS that the Federal Reserve Bank of Philadelphia publishes is sometimes read as a level gap in popular writing, but the construction is a flow factor in both implementations, so KRADS is best read as a momentum gauge rather than the same kind of object as the HLW output gap.

Computing KRADS monthly rather than daily is a choice forced by data constraints. The U.S. ADS reaches daily resolution because of the weekly initial-claims hard anchor that the U.S. Department of Labor reports, and the Korean panel has no hard real-activity series of comparable cadence. The Camacho and Perez-Quiros (2010) European nowcaster works around the same constraint by mixing in financial-market data, but KRADS does not follow that path because its goal is a hard-real-activity gauge. KRADS is therefore computed monthly and charted monthly. The series is a coincident gauge, not a forecast, and it conveys different information from risk-distribution estimates such as KRGAR and KRIAR.

The factor reads most naturally as a central-tendency input to macro risk assessment. Positive readings indicate above-average activity and thus a favorable short-run-momentum environment for cyclically sensitive assets, negative readings the opposite, and readings near zero average activity conditions, where zero is the reference line on the chart. The magnitude of the factor admits a standard-deviation interpretation, so that $\pm 1$ corresponds to about one in-sample standard deviation and readings beyond $\pm 2$ mark the crisis extrema. The Aruoba, Diebold, and Scotti (2009) framework carries this interpretation in both the U.S. and the Korean implementations.

KRADS plays a different role from the other recession channels on this site. The recession probability (KRRECP) is a predictive channel that extracts a 3-month-ahead signal from the ACM-decomposed yield-curve slope and the near-term forward spread, while KRADS independently measures the current state of activity as a coincident factor. The Sahm gap (KRSAHM) is a narrow gauge that measures labor-market stress from the single variable of the unemployment rate, whereas KRADS is a broad coincident gauge that combines four real-sector series with quarterly GDP through the Mariano and Murasawa (2003) aggregation in the Stock and Watson (1989) tradition, and both are reported because the asymmetry between the labor market and broad real activity is informative.

KRADS provides a faster read than the official KOSTAT coincident index. The KOSTAT coincident index is the official benchmark for the Korean cycle but is published with a delay relative to the underlying input data, whereas KRADS uses the same monthly official inputs (payroll employment, all-industry production, retail sales, services production) and quarterly real GDP and re-fits hourly through the Bańbura and Modugno (2014) EM step. The Giannone, Reichlin, and Small (2008) nowcasting construction is the natural lens for this real-time-update logic.

Two operating caveats apply to KRADS. First, the sample-start constraint of the input series confines the KRADS sample to January 2000 onward, so earlier episodes such as the 1997 Asian financial crisis cannot be evaluated with this index. Second, because the full sample is re-fit on every pipeline tick, historical factor values can move marginally when the EM fit settles on a slightly different stable point, and the full-replace loading discipline shared with KRSAHM, KRFCI, and KRGAR keeps the persisted series consistent with the latest fit at all times. The Federal Reserve Bank of Philadelphia ADSBCI is published at higher cadence under the same discipline, and the 0.94 correlation of the KRADS U.S. method check is the methodological-equivalence anchor between the two implementations.

KRADS is the single latent factor of the Aruoba-Diebold-Scotti mixed-frequency dynamic factor model, a monthly business-conditions state extracted by a two-sided Kalman smoother. Over 316 monthly observations from 2000-01-01 to 2026-04-01, its measured serial correlation is dominated by the smoother's gain rather than by the data-generating process, so the object framed here is a persistence summary of the smoothed path and not an integration order of the data.

The integration-order battery is therefore deliberately not run. The augmented unit-root regression of Dickey and Fuller (1979) with the lag augmentation of Said and Dickey (1984), the semiparametric Phillips and Perron (1988) test, the KPSS stationarity test (Kwiatkowski et al. 1992), the efficient GLS-detrended test of Elliott, Rothenberg, and Stock (1996), and the modified M-tests of Ng and Perron (2001) are all excluded, because a symmetric two-sided filter manufactures the persistence an integration test reads and the random-walk-versus-constant character of the latent state is not point-identified by the likelihood (Stock and Watson 1998; Orphanides and van Norden 2002). The level mean-break search of Bai and Perron (1998) is likewise not run, since its asymptotics require a stationary object and the filter-persistent path spuriously segments (Perron 1989).

The matrix-mandated replacement is a descriptive persistence summary labeled as a property of the smoothed series. The lag-one autocorrelation is −0.185 and the implied half-life is undefined, since the lag-one coefficient is negative, a description of how slowly the smoothed path decays that carries no integration-order claim. Because the mixed-frequency factor is smoothed over the full sample and the loader rewrites the state history on every re-run, the factor at any fixed past month revises across vintages, and no stored vintage panel exists to quantify the revision magnitude (Orphanides and van Norden 2002).

No order of integration is assigned, by ruling rather than by an inconclusive test. The lag-one autocorrelation is itself a small-sample and filtering artifact of the two-sided smoother rather than evidence about the process, its median-unbiased treatment would follow the grid of Andrews (1993), and a portmanteau such as that of Ljung and Box (1978) on this level would read the same filter gain. The model-implied content is honest as an assumption rather than a discovery, namely a single mean-zero latent factor with autoregressive dynamics estimated by a mixed-frequency dynamic factor model, whose method reproduces the benchmark United States business-conditions index at a correlation of 0.943, a replication check rather than a statement about the Korean state's integration order (Stock and Watson 1998).