South Korea Time-Varying Employment Elasticity of Growth

KREMPEL is the time-varying employment elasticity of growth for Korea, defined as the ordinary-least-squares slope $\hat{\beta}_w$ from a regression of quarterly employment growth on quarterly output growth estimated over a rolling 40-quarter window and dated to the window's final quarter. The regressand is the log change in seasonally-adjusted employment and the regressor is the log change in real output, so $\hat{\beta}_w$ is the dimensionless within-window elasticity, namely the percent employment response to a one-percent change in output (Islam and Nazara 2000; Kapsos 2005). Each published point carries a per-window 95% confidence band built from a Newey-West heteroskedasticity- and autocorrelation-consistent standard error (Newey and West 1987; Newey and West 1994).

The estimand is the regression slope and nothing else. It is deliberately not the per-quarter arithmetic ratio $\Delta \ln E_t / \Delta \ln Y_t$, which the employment-intensity literature rejects as an unstable object because its near-zero-capable denominator makes it heavy-tailed (Kapsos 2005; Marsaglia 2006), and it is not a two-sided-smoothed latent state, since the only smoothing present is the estimation window itself. The regression form is the first difference of the levels log-log elasticity regression used across the cross-country literature (Crivelli, Furceri, and Toujas-Bernaté 2012; Kapsos 2005), the same move by which the Okun literature derives its changes specification from the levels relation (Okun 1962; Ball, Leigh, and Loungani 2013), and it follows the rolling-window design of the Okun-coefficient literature in both window length and window-end dating (Knotek 2007; Owyang and Sekhposyan 2012). The series is quarterly, its inputs are seasonally adjusted at source, and its published path begins in the third quarter of 2009 (Choi 2021).

The regression frame is built from three seasonally-adjusted inputs and estimated on a rolling window.

(1) Quarterly aggregation. Each monthly seasonally-adjusted employment level is aggregated to a quarterly value by the three-month mean $E_q = \tfrac{1}{3}(E_m + E_{m-1} + E_{m-2})$, following the rolling-Okun convention for mixing monthly labor data with quarterly output (Owyang and Sekhposyan 2012; Knotek 2007). Mean aggregation induces a first-order moving-average term in the quarterly difference, which is priced into the lag rule of step (4) rather than avoided (Working 1960).

(2) Growth transforms. Employment growth is the log difference $\Delta \ln E_t = \ln E_t - \ln E_{t-1}$ and output growth is recovered from the published quarter-on-quarter real-GDP growth rate $g_t$ in percent as $\Delta \ln Y_t = \ln(1 + g_t/100)$, the differenced counterpart of the elasticity slope (Islam and Nazara 2000; Okun 1962).

(3) Per-window regression. For each window of 40 consecutive quarters the published elasticity is the ordinary-least-squares slope $\beta_w$ of the differenced regression

which is the first difference of the levels log-log elasticity regression and returns the growth elasticity directly (Crivelli, Furceri, and Toujas-Bernaté 2012; Kapsos 2005; Ball, Leigh, and Loungani 2013). Differencing rather than a levels regression is adopted because the employment and output log-levels are individually integrated and their cointegration is not established on this sample, so a levels regression would carry spurious-regression risk (Stock 1987).

(4) HAC band. The slope standard error is a Newey-West estimator (Newey and West 1987; Andrews 1991) at the deterministic truncation lag $L(W) = \lfloor 4\,(W/100)^{2/9} \rfloor$ (Newey and West 1994), equal to 3 at the 40-quarter window and covering the moving-average-plus-omitted-lag error structure. The published band

is a per-window 95% confidence interval on a realized coefficient rather than a forecast band.

(5) Rolling and dating. The window advances one quarter at a time and each estimate is dated to the window's last quarter, a one-sided assignment that never revises a past point when new data arrive (Knotek 2007; Owyang and Sekhposyan 2012). The headline path runs unadjusted through the 2020 pandemic quarters, and window-end dating carries an average turning-point latency of about half the window length, roughly five years at the 40-quarter window.

Jobless growth, whether a given rate of output expansion still generates proportionate employment, is a recurring question in Korean macroeconomic debate, and the employment elasticity of growth is the quantity that debate turns on (Kapsos 2005; Crivelli, Furceri, and Toujas-Bernaté 2012). The rolling path answers it descriptively. Over the published span the elasticity has a median near 0.17 and moves within a range of roughly 0.03 to 0.47, consistent with the below-one short-run reading the differenced specification is designed to deliver (Crivelli et al. 2012; Choi 2021). The most recent window reads about 0.366, a rise over the post-2020 span that says growth has become somewhat more employment-intensive at the margin, though the window-end dating means the reading reflects the whole trailing decade rather than the latest quarter alone. Cross-country work frames such movement as a genuine elasticity-regime question rather than measurement noise (Ball, Leigh, and Loungani 2013), and the German labor-market experience of the Great Recession is the canonical case of an employment-output relation shifting within a single episode (Burda and Hunt 2011).

The published series deliberately departs from the arithmetic employment-elasticity figures that circulate in the press, which divide a year's employment growth by the same year's output growth. That arithmetic ratio is the object the platform refuses to publish, and an executed demonstration on Korean data shows why. Computed quarter by quarter over the sample, the arc ratio swings between −3.55 and +2.37 among its finite values, diverges outright in the quarters where output growth is essentially zero, and flips sign in 18 of the sample's quarters, so that output falling while employment rises reads as a spurious negative elasticity (Kapsos 2005; Gordon 2010). Its dispersion is roughly five times that of the rolling regression path it is not allowed to replace. This instability is exactly what ratio-distribution theory predicts, since a ratio whose denominator can sit near zero is heavy-tailed and need not possess a finite mean or variance (Marsaglia 1965; Marsaglia 2006; Hinkley 1969), the same reasoning by which the intensity-of-growth literature adopted regression over point elasticities in the first place (Islam and Nazara 2000; Owyang and Sekhposyan 2012).

For a rates or credit investor the employment intensity of growth is a gauge of the quality of an expansion and not only of its pace. A high elasticity means output gains are passing through to jobs, and hence to household income and consumption, which supports the demand side of the inflation outlook and shapes how the central bank is likely to lean (Okun 1962; Knotek 2007). A low elasticity describes growth that is thin on employment, the kind that leaves the labor market slack even as activity holds up, and that composition matters for the path of policy rates and therefore for duration and credit positioning (Crivelli, Furceri, and Toujas-Bernaté 2012; Ball, Leigh, and Loungani 2013). Reading the elasticity alongside output growth lets an investor separate an expansion that is broadening into the labor market from one that is not, and gauge the share of output gains that reaches employment (Islam and Nazara 2000).

The series is most useful as a business-cycle-quality lens rather than a timing signal. Because it is dated to the window end and moves slowly, it will not turn ahead of the cycle, but its level and drift tell whether the current growth mix is employment-rich or employment-poor, which is the same information the Okun-coefficient literature reads for the recession question (Knotek 2007; Owyang and Sekhposyan 2012). A falling elasticity during an apparent expansion is the jobless-recovery configuration that has repeatedly complicated the labor-market read in the United States and elsewhere (Gordon 2010; Burda and Hunt 2011), and in the Korean context the level sits in the historical 0.3 to 0.4 range that domestic studies report for the employment-output relation (Choi 2021; Woo 2023; Kapsos 2005).

KREMPEL is a rolling-window regression-coefficient path. The published object at each date is the ordinary-least-squares slope $\hat{\beta}_w$ of employment growth on output growth over the trailing 40-quarter window, dated to the window end and carrying a per-window 95% Newey-West heteroskedasticity- and autocorrelation-consistent confidence band. It is a sequence of realized in-sample coefficients rather than a forecast or a smoothed latent state.

Unit-root, persistence, and portmanteau diagnostics on the path itself were deliberately not run. Adjacent window-end estimates share 39 of their 40 observations, so an augmented Dickey-Fuller, Phillips-Perron, KPSS, or half-life read on the path is precluded, since it would measure that mechanical window overlap rather than any property of the economy, and the smoothness of the path is the estimation window's and not the data's (Hansen and Hodrick 1980). No integration order and no persistence figure is assigned to the path, by ruling rather than by an inconclusive test.

The native uncertainty statement is the per-window HAC band. The most recent 40-quarter window, the sixty-seventh on a path spanning 2009Q3 to 2026Q1, has a slope of 0.36626 with a HAC standard error of 0.133241 and a 95% interval from 0.105108 to 0.627412, computed with the Newey-West estimator (Newey and West 1987) at the deterministic truncation lag of 3 fixed by the Newey and West (1994) rule, and the path median is 0.170887. Full-sample stability of the underlying regression is the second leg. The sup-F test for a single slope break at an unknown date is 5.4697, below the 10% critical value of 7.12, so no significant break is found (Andrews 1993; Andrews 2003), and the procedure of Bai and Perron (1998), computed by the algorithm of Bai and Perron (2003), selects zero mean-shift breaks on the 106-quarter sample. The full-sample slope is 0.281839 with a HAC t ratio of 3.4078, significant and correctly signed. Re-estimating on the nearest-neighbour 48-quarter window, the coefficient path correlates with the 40-quarter headline at 0.945892, so the path shape is robust to the single free design parameter.

Three descriptive disclosures accompany the series. First, the 2020 pandemic quarters are retained in the headline, and of the 25 windows overlapping the four 2020 quarters, 24 are pandemic-dominated in the sense that dropping those quarters moves the coefficient by more than its own HAC standard error, with a maximum standardized divergence of 3.416 at the 2021Q1 window-end, so the headline is nonetheless left unadjusted because for a descriptive labor-market gauge the crisis quarters are signal rather than contamination. Second, the differenced specification is affirmatively the licensed one, since both the employment and the output log-levels are integrated of order one under the augmented Dickey-Fuller (Said and Dickey 1984), Phillips-Perron (Phillips and Perron 1988), and KPSS (Kwiatkowski et al. 1992) honest-agreement reading, the Engle-Granger residual test does not detect cointegration with a statistic of −1.9408 (Engle and Granger 1987; Phillips and Ouliaris 1990), and the reduced-rank system trace test fails to reject the no-cointegration null with a statistic of 10.97, the lone dissenting no-constant residual signal being outvoted under the platform's conservative Phillips-Perron veto (Ng and Perron 2001), so the levels log-log form cannot be licensed on this sample and the differenced elasticity is the required specification. Third, the differenced short-run slope of 0.281839 sits below the levels-based long-run Korean anchors of about 0.17 to 0.38 because the differenced form omits the long-run adjustment channel, which is expected rather than an error. The sample is short at 67 published estimates, window-end dating carries a turning-point latency of roughly half the window length, and source-data revisions re-estimate the whole path through the full-replace loader.