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 filtering. 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 read off the Kalman filter, so each month's value uses the observations only up to that month. (4) Sign orientation. The filtered 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 filter are deterministic functions of the data, and the PCA-based initialization is deterministic too. Their numerical results nevertheless depend on the linear-algebra implementation, so the published values are computed on the production platform with single-threaded linear algebra.

The EM parameters and the filter's initial state, which the EM step updates, are estimated on the full sample in every tick, so each month's value is conditioned on full-sample quantities even though no later observation enters it directly, and the series is not a real-time estimate (Orphanides and van Norden 2002). The first months lean on that full-sample initial state, and every month is revised when the model is re-fit. No smoothed factor is stored or published, and the factor is a filtered latent state of the model, not observed data.

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 0.93 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 published index is a two-sided smoothed estimate while KRED's factor is one-sided, so this correlation checks the specification rather than reproducing the published index like for like. 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 0.63, and the correlation with the 12-month rolling mean of the factor rises to 0.53, 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 $-2.71$ at the February 2003 credit-card crisis, $-3.60$ just before the December 2008 global financial crisis, and $-4.93$ at the March 2020 COVID peak. The expansion mean of $+0.13$ and the contraction mean of $-0.24$ 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 both the published index and KRADS are flow factors, the former two-sided smoothed and the latter one-sided filtered, 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 is read against its in-sample standard deviation of 1.18, so $\pm 1$ is somewhat less than one standard deviation, and while the 2003, 2008 and 2020 crisis troughs all lie beyond $\pm 2$, single months beyond that line also occur outside crises, so one such reading is not by itself a crisis signal. 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 is re-fitted at every scheduled update through the Bańbura and Modugno (2014) EM step. The Giannone, Reichlin, and Small (2008) nowcasting construction is the natural lens for this update logic, although the factor is not a real-time estimate, because every re-fit re-estimates the parameters and the filter's initial state on the full sample (Orphanides and van Norden 2002).

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 move whenever new data shift the full-sample EM parameters and initial state, even though no later observation enters a month's filtered value directly, 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 as a two-sided smoothed index, whereas KRADS is a one-sided filtered factor, so the 0.93 correlation of the KRADS U.S. method check anchors the specification and is not a like-for-like replication of the published index.

KRADS is the single latent factor of KRED's Aruoba-Diebold-Scotti mixed-frequency dynamic factor model on Korean data, the one-sided Kalman-filtered factor and never a smoothed one, so the statistical object is a persistence summary of the filtered path and not an integration order of the data. The factor's first-order autoregressive dynamics are imposed by the model's state equation rather than estimated as a property of the economy, and a one-sided filter's gain dominates the serial correlation of the extracted factor (Aruoba, Diebold, and Scotti 2009).

The augmented Dickey and Fuller (1979) regression with the lag augmentation of Said and Dickey (1984), the nonparametric Phillips and Perron (1988) test and the KPSS stationarity test of Kwiatkowski et al. (1992) were deliberately not run on this path, because a verdict from them would test the autoregressive factor dynamics the model assumes rather than a property of the data, the ground the matrix gives for every filtered latent state whose persistence the filter sets (Stock and Watson 1998). The GLS-detrended test of Elliott, Rothenberg, and Stock (1996) was not run for the same reason, and neither was the modified-statistic battery of Ng and Perron (2001), since both confirm an integration reading this archetype does not take. The level break search of Bai and Perron (1998) was likewise not run, because a break in the mean of a filtered factor reflects the filter's response to its inputs rather than a change in the economy (Perron 1989). A level portmanteau such as Ljung and Box (1978) is not applied either, since on a filtered path it reads the filter gain rather than evidence about the data-generating process.

What the matrix mandates in their place is a descriptive persistence summary and a statement about revision. Over the 316 months from 2000-01-01 to 2026-04-01 the first-order autoregressive coefficient of the filtered factor is −0.232, so the implied half-life is undefined, because a negative lag-one coefficient means that consecutive months tend to move in opposite directions and there is no monotone decay to measure. The figure is a property of the filtered path and never of Korean business conditions, and like any first-order estimate on a finite sample it carries small-sample bias (Andrews 1993). The revision cell records that the expectation-maximization parameters and the filter's initial state are re-estimated on the full sample at every update and that each load replaces the whole factor history, so the value at a fixed past month moves across updates although no smoothing enters it, and no dated vintage panel exists from which the size of that movement could be measured (Orphanides and van Norden 2002).

Three limits are stated rather than implied. No order of integration is assigned, by ruling and not by an inconclusive test (Hamilton 2018). The factor is one-sided but it is not a real-time estimate, because its parameters and initial state are estimated on the full sample, which makes it a quasi-final estimate in the sense of Orphanides and van Norden (2002). And the method check that applies the same specification to United States data correlates KRED's one-sided factor with the published benchmark index at 0.935, but that index is not built as a one-sided filtered factor, so the correlation checks the specification rather than replicating the index like for like, and it says nothing about the integration order of the Korean factor.