## 1. Time-Invariant NAIRU

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---

### I. Introduction
- Purpose: Derive Poland’s output gap for 1995–2008 using a Cobb-Douglas production function methodology and obtain the natural rate of unemployment (NAIRU) by augmenting a Kalman decomposition with a Phillips-curve application.
- Motivation: Production-function-based methods preferred over purely statistical filters (e.g., HP filter) for emerging markets subject to large output fluctuations and structural change.
- Key high-level findings:
  - During the pre-crisis boom years Poland was growing above potential, consistent with inflation behavior and the estimated unemployment gap.
  - After the 2008 downturn, Poland is not expected to experience a sizable and persistent negative output gap relative to peers.

### II. The Production Function and Parameter Estimates
- Model specification:
  - Cobb-Douglas with constant returns to scale (CRS): Y_t = A_t K_t^α L_t^β with 1 = β + α.
- Data and measurement:
  - Labor input: number of employees from Polish Labor Force Survey (LFS).
  - Capital stock: perpetual inventory method K_t = (1 − δ) K_{t-1} + I_t with δ = 0.05.
  - Initial benchmark for capital: Q_{1995} = (i/ I) + δ (formula as in source).
  - Estimation uses seasonally adjusted quarterly data for the period 1995Q1―2008Q4; variables in natural logarithms.
- Time series properties and cointegration:
  - Unit root tests indicate GDP, capital and labor are non-stationary (Table 1).
  - Johansen cointegration tests suggest existence of one long-run relationship (Table 2).
  - Johansen test statistics reported: Trace p-value = 0.014; Max Eigenvalue p-value = 0.0004.
- Estimation methods and results:
  - OLS and Dynamic OLS (DOLS) used given small-sample concerns.
  - Sum of unrestricted DOLS coefficients statistically close to one; CRS not rejected.
  - Restricted DOLS coefficients broadly consistent with prior studies for Poland.
  - Recursive stability tests (Kalman filter based) suggest considerable parameter constancy (Figure 1).
  - Selected numeric outputs: restricted model R^2 = 0.9788; number of leads and lags in DOLS = 4.
  - ADF test statistics reported for Output, Labour input, Capital input with p-values shown in Table 1.

### III. Estimating Potential Inputs
- Total factor productivity (TFP):
  - TFP obtained as Solow residual: A_t = Y_t / (K_t^α L_t^{1−α}).
  - Trend TFP approximated by HP filter with λ = 40 following Gradzewicz and Kolasa (2005).
- Capital utilization:
  - Full utilization of existing capital stock assumed due to lack of capacity utilization series.
- Potential employment and NAIRU estimation overview:
  - Two-step approach:
    - 1) Kalman decomposition of the unemployment rate into trend (U_t) and cyclical (G_t) components.
    - 2) Identification of the cyclical component via a Phillips-curve relationship to obtain NAIRU-consistent equilibrium unemployment.

### III.A First step: Kalman decomposition of unemployment
- Unemployment decomposition and stochastic specification:
  - Observed unemployment: U_t = trend component + cyclical component.
  - Trend follows a local linear trend model: U_t = U_{t-1} + μ_{t-1} + η_t with μ_t = μ_{t-1} + ξ_t.
  - Cyclical component modeled as stationary AR(2): G_t = φ_1 G_{t-1} + φ_2 G_{t-2} + ψ_t.
- Smoothness prior:
  - σ_η set to 0.1 (smoothness prior consistent with Gordon (1996)); when σ_η = 0 the NAIRU is time-invariant.
- Estimation:
  - Kalman filter/smoother used to extract trend and cyclical components (Table 4, first column).

### III.B Second step: Economic identification — Phillips curve
- Phillips-curve specification and estimation:
  - Phillips-curve relates changes in inflation to unemployment gap, lagged inflation changes, import price inflation changes (Z_t), and other terms (equation (7) in source).
  - Estimation requires non-linear methods; initialized with OLS using cyclical component from step one.
  - Cyclical component re-estimated within the Phillips-curve specification under the AR(2) restriction (Table 4, third column).
  - A shift dummy for years after 2006 included to account for changes in inflation (splitting sample 1995–2005 vs 2006–2008).

### III.C NAIRU estimates and unemployment components
- Annual-average series (1995–2008) reported for observed unemployment, trend components (step one and step two), HP-filter trend, and cyclical components (Table 5).
- Selected annual values (observed unemployment rate series, years in sequence as in table):
  - Observed unemployment: 13.4, 12.4, 11.2, 10.6, 13.8, 16.1, 18.3, 19.9, 19.6, 18.9, 17.7, 13.8, 9.6, 7.1.
  - Trend component — Step two (Phillips curve): 12.5, 12.5, 12.8, 13.3, 14.7, 15.9, 17.0, 17.7, 17.6, 16.9, 15.7, 13.5, 11.1, 8.8.
  - Cyclical component — Step two (Phillips curve): 0.8, 0.2, -1.2, -2.7, -1.6, 0.2, 0.9, 2.0, 2.1, 2.1, 2.1, 0.9, -1.2, -1.8.
- HP-filter smoothing parameter for unemployment: λ = 1600.
- Dynamics:
  - The estimated unemployment gap follows Poland’s post-reform business cycle: trough around 1998 (Russia crisis), rise through 2001–02 global recession, decline after EU membership, bottoming during the 2008 global financial crisis.
- Empirical relationship:
  - The estimated equilibrium unemployment rate (NAIRU) relates closely to inflation behavior (Figure 4), consistent with literature on disinflation and increases in the natural rate of unemployment.

### III.D Potential employment
- Potential employment definition and estimation:
  - Potential employment computed as: 1t* tt* NAIRUPRactiveL  (equation (8) in text), where tactive is the working age population and * tPR is the trend (or equilibrium) participation rate.
  - Proxy for equilibrium participation rate obtained by regressing actual activity rate on a constant, the unemployment rate and a time trend; fitted values used as potential participation rate (Figure 5).
  - The unemployment rate enters the regression with the expected negative sign (-0.049).
- Advantages:
  - Produces a relatively smooth potential employment series that accounts for changes in the working age population, trend participation rate, and NAIRU.
- Empirical observation:
  - The increase in unemployment during 1998–2004 is consistent with a downward trend in the participation rate.
  - Poland’s actual and estimated potential employment are depicted in Figure 6.

### Box 1 — Time-Invariant NAIRU (methodology and estimation)
- Expectational augmented Phillips Curve:
  - Standard relation: (a) ( ) ttt e tt vZUU  11 where e t1  is expected inflation for t+1.
- Inflationary expectations modeled as random walk: t e t  1 so that 11  ttt .
- Autoregressive Phillips specification to accommodate serial correlation:
  - (b) ( ) ( ) ( ) ( ) ttttt ZLUU LL  1 where ( ) L , ( ) L and ( ) L are lag polynomials and t  is non-serially correlated.
- Time-invariant NAIRU simplification and OLS estimation:
  - (c) ( ) ( ) ( ) ( ) ttttt ZLULL  1 with ( )    1i i UU L and    1i i /U .
- Estimation particulars reported:
  - No lags for unemployment; 12 lags for inflation and changes in the commodity price index.
  - Regression includes a dummy for changes in inflation over the sample.

### Potential output estimates (production function approach) — main results
- Potential output from Cobb-Douglas: t * t * tt KLAY  with exponent notation as in text (    1).
- Key empirical findings:
  - During the boom years preceding the recent financial crisis, Poland was growing above its potential, with an output gap of 2 percent by early 2008.
  - HP-filter based output gap peaked earlier and turned negative by end-2008.
  - The production-function (PF) output-gap series exhibits a more gradual reversal, indicating the Polish economy was at a level above potential as late as the fourth quarter of 2008.
- Labor evidence supporting unsustainability:
  - Annual growth rate of potential employment slowed from about 3 percent in early-2008 to 2 percent by the fourth quarter of 2008.
  - The growth rate of actual employment remained above 3 percent throughout the year.
  - These patterns provide evidence that Poland’s rapid output and employment growth pre-crisis was unsustainable.

### Contributions to potential growth (Box 2)
- Decomposition framework:
  - Potential GDP growth decomposed into contributions from labor, capital, and TFP; contributions computed as year-on-year percentage changes and sum to potential GDP growth rates.
- Empirical evolution:
  - Following the 2001–02 recession, contribution of factor productivity growth rose steadily through 2004; remained positive until 2007, then turned negative through late-2008—coinciding with trend-reversal in potential output growth.
  - Contribution of capital was steadily increasing but insufficient to prevent decline in potential output growth throughout 2008.
  - Labor contribution rose in recent years, largely reflecting a decrease in the NAIRU from 2004.
- Potential labor decomposition:
  - Most of the increase in potential labor force is attributed to a corresponding decline in the NAIRU.
  - Rate of growth in Poland’s active population held roughly constant since 2004.
  - Participation rate has been decreasing at a constant rate with a negligible effect on the growth of the equilibrium employment rate.

### Forecasts through end-2010 and comparative outlook
- Forecast extension through Q4 2010:
  - The measure of the output gap turned negative in the first quarter of 2009 and is expected to remain negative throughout 2010.
  - The output gap is projected to bottom out at just around minus 1 percent, during the second quarter of 2010, vs. minus 2 percent in the 2001–02 downturn.
  - The output gap is expected to close in 2011.
- Modeling and sample notes:
  - Parameter estimates from the 1995:I-2008:IV sample left unchanged; forward-looking NAIRU modeled consistent with a relatively stable inflation outlook.
  - Footnote specification in text:  11 50   tttt NAIRUNAIRU .NAIRUNAIRU .

### Conclusion — summary of analytical findings
- Methodological choice:
  - The production-function methodology, combined with a NAIRU-based potential labor estimate, provides a robust alternative to HP-filter approaches and helps to better identify recent boom-bust turning points.
- Main conclusions:
  - Pre-crisis period showed Poland’s output growing above its potential; confirmed by employment behavior relative to equilibrium.
  - Pre-crisis decline in TFP coincided with deceleration in potential output growth.
  - Rising capital contribution suggests the rapid investment-led output expansion in 2006–07 was unsustainable and driven less by fundamentals.
  - In the aftermath of the global crisis, Poland is not expected to experience a sizable and persistent negative output gap.

*Source: _wp1015 - 1. Time-Invariant NAIRU (IMF PDF chapter content provided).*

### 1.  Time-Invariant NAIRU ...............................................................................................

### 1. Time-Invariant NAIRU

### I. Introduction
- Purpose: Derive Poland’s output gap for 1995–2008 using a Cobb-Douglas production function methodology and obtain the natural rate of unemployment (NAIRU) by augmenting a Kalman decomposition with a Phillips-curve application.
- Motivation: Production-function-based methods preferred over purely statistical filters (e.g., HP filter) for emerging markets subject to large output fluctuations and structural change.
- Key high-level findings:
  - During the pre-crisis boom years Poland was growing above potential, consistent with inflation behavior and the estimated unemployment gap.
  - After the 2008 downturn, Poland is not expected to experience a sizable and persistent negative output gap relative to peers.

### II. The Production Function and Parameter Estimates
- Production function assumed: Cobb-Douglas with constant returns to scale (CRS):
  - Y_t = A_t K_t^α L_t^β with 1 = β + α.
- Data and measurement:
  - Labor input: number of employees from Polish Labor Force Survey (LFS).
  - Capital stock: perpetual inventory method K_t = (1 − δ) K_{t-1} + I_t with δ = 0.05.
  - Initial benchmark for capital: Q_{1995} = (i/ I) + δ (formula as in source).
  - Estimation uses seasonally adjusted quarterly data for the period 1995Q1―2008Q4; variables in natural logarithms.
- Time series properties and cointegration:
  - Unit root tests indicate GDP, capital and labor are non-stationary (Table 1).
  - Johansen cointegration tests suggest existence of one long-run relationship (Table 2).
- Estimation methods and results:
  - OLS and Dynamic OLS (DOLS) (Stock and Watson, 1993) used given small-sample concerns.
  - Sum of unrestricted DOLS coefficients statistically close to one; CRS not rejected.
  - Restricted DOLS coefficients broadly consistent with prior studies for Poland.
  - Recursive stability tests (Kalman filter based) suggest considerable parameter constancy (Figure 1).
- Selected numeric/statistical outputs (as reported):
  - ADF test statistics reported for Output, Labour input, Capital input with p-values shown in Table 1.
  - Johansen test: None* Trace p-value = 0.014; Max Eigenvalue p-value = 0.0004 (Table 2).
  - DOLS results summary: restricted model R^2 = 0.9788 (Table 3); number of leads and lags in DOLS = 4.

### III. Estimating Potential Inputs
- Total factor productivity (TFP):
  - TFP obtained as Solow residual: A_t = Y_t / (K_t^α L_t^{1−α}).
  - Trend TFP approximated by HP filter with λ = 40 following Gradzewicz and Kolasa (2005).
- Capital utilization:
  - Full utilization of existing capital stock assumed (consistent with literature) due to lack of capacity utilization series.
- Potential employment and NAIRU estimation overview:
  - Two-step approach:
    1. Kalman decomposition of the unemployment rate into trend (U_t) and cyclical (G_t) components.
    2. Identification of the cyclical component via a Phillips-curve relationship to obtain NAIRU-consistent equilibrium unemployment.

### III.A First step: Kalman decomposition of unemployment
- Unemployment decomposition:
  - U_t (observed unemployment) = trend component + cyclical component: U_t = U_t (trend) + G_t.
  - Trend follows local linear trend model: U_t = U_{t-1} + μ_{t-1} + η_t with μ_t = μ_{t-1} + ξ_t.
  - Cyclical component modeled as stationary AR(2): G_t = φ_1 G_{t-1} + φ_2 G_{t-2} + ψ_t.
- Smoothness prior:
  - σ_η set to 0.1 (smoothness prior consistent with Gordon (1996)); when σ_η = 0 the NAIRU is time-invariant.
- Estimation:
  - Kalman filter/smoother used to extract trend and cyclical components (Table 4, first column).

### III.B Second step: Economic identification — Phillips curve
- Phillips-curve specification (used to identify cyclical component):
  - Δπ_{t+1} + γ = [polynomials in lag operator] × unemployment gap + lagged Δπ terms + Z_t (import price inflation changes) + other terms (equation (7) in source).
  - Estimation requires non-linear methods; initialized with OLS using cyclical component from step one.
  - Cyclical component re-estimated within the Phillips-curve specification under the AR(2) restriction (Table 4, third column).
- Additional estimation detail:
  - A shift dummy for years after 2006 included to account for changes in inflation (splitting sample behavior 1995–2005 vs 2006–2008).

### III.C NAIRU estimates and unemployment components
- Reported annual-average series (1995–2008) for observed unemployment, trend components (step one and step two), HP-filter trend, and cyclical components (Table 5).
- Selected annual values (reported values based on annual averages):
  - Observed unemployment rate series (1995–2008): 13.4, 12.4, 11.2, 10.6, 13.8, 16.1, 18.3, 19.9, 19.6, 18.9, 17.7, 13.8, 9.6, 7.1 (each year in sequence as in table).
  - Trend component — Step two (Phillips curve): 12.5, 12.5, 12.8, 13.3, 14.7, 15.9, 17.0, 17.7, 17.6, 16.9, 15.7, 13.5, 11.1, 8.8.
  - Cyclical component — Step two (Phillips curve): 0.8, 0.2, -1.2, -2.7, -1.6, 0.2, 0.9, 2.0, 2.1, 2.1, 2.1, 0.9, -1.2, -1.8.
  - Hodrick-Prescott smoothing parameter for unemployment: λ = 1600.
- Dynamics of the unemployment gap:
  - The estimated unemployment gap follows Poland’s post-reform business cycle: trough around 1998 (Russia crisis), rise through 2001–02 global recession, decline after EU membership, bottoming during the 2008 global financial crisis.
- Empirical relationship:
  - The estimated equilibrium unemployment rate (NAIRU) relates closely to inflation behavior (Figure 4), consistent with literature on disinflation and increases in the natural rate of unemployment.

### III.D Potential employment
- Potential employment derived by combining NAIRU-based equilibrium unemployment with labor force/participation measures (section introduction indicates attention to equilibrium employment estimates; detailed numbers and procedure follow in subsequent text not included in this unit).

### Figures and Tables referenced (for context within the unit)
- Figures: Recursive Estimates of Cobb-Douglas Coefficients; Actual, Equilibrium, and Trend Component of Employment; Unemployment Gap (Phillips Curve Estimation); Year-on-Year Inflation versus NAIRU Estimates; Observed and Potential labor Market Participation Rate; Actual and Equilibrium Employment Level; Production Function Estimates.
- Tables: ADF Test Statistics For Variables' Stationarity; Johansen's Cointegration Test; Static and Dynamic Least Squares Estimation; Cyclical Component and Phillips Curve Estimates; Trend and Cyclical Components of Unemployment.

*Source: _wp1015 - 1.  Time-Invariant NAIRU (IMF PDF chapter content provided).*

### 14.      Given the long-run unemployment rate estimates, Polish potential employment level

### 14.      Given the long-run unemployment rate estimates, Polish potential employment level

### Potential employment: definition and estimation
- Potential employment is computed as:
  - (8) 1t* tt* NAIRUPRactiveL 
    - where tactive is the working age population and * tPR is the trend (or equilibrium) participation rate.
- Estimation approach:
  - A proxy for the equilibrium participation rate is obtained by regressing the actual activity rate on a constant, the unemployment rate and a time trend. The resulting fitted values are used as the measure for the potential participation rate (Figure 5).
  - The outcome of the OLS regression is not reported. The unemployment rate enters with the expected negative sign (-0.049).
- Advantages of equation (8):
  - Produces a relatively smooth potential employment series that accounts for changes in the working age population, the trend participation rate, and the structural unemployment rate (NAIRU).
- Empirical observation:
  - The overall increase in unemployment during the period 1998–2004 is consistent with a downward trend in the participation rate.
  - Poland’s actual and estimated potential employment are depicted in Figure 6.

### Box 1 — Time-Invariant NAIRU (methodology and estimation)
- The standard NAIRU model is based on an expectational augmented Phillips Curve relation (Greene, 2003):
  - (a) ( ) ttt e tt vZUU  11
    - where e t1  is the expected inflation rate for period t+1.
- Inflationary expectations modeled as a random walk (Staiger, Stock and Watson, 1991):
  - t e t  1 so that 11  ttt .
- To accommodate serial correlation, the Phillips relation is estimated in an autoregressive specification:
  - (b) ( ) ( ) ( ) ( ) ttttt ZLUU LL  1
    - where ( ) L , ( ) L and ( ) L are lag polynomials and t  is a non-serially correlated error term.
- Time-invariant (textbook) NAIRU simplification:
  - Assuming tU is constant, equation (b) can be reformulated and estimated by OLS:
    - (c) ( ) ( ) ( ) ( ) ttttt ZLULL  1
      - with ( )    1i i UU L . It is straightforward to derive    1i i /U .
- Estimation particulars reported:
  - Estimates are computed by using no lags for unemployment, 12 lags respectively for both inflation and changes in the commodity price index.
  - The regression includes a dummy accounting for the changes in inflation on the overall sample period.

### Potential output estimates (production function approach) — main results
- Potential output estimated from Cobb-Douglas production function:
  - t * t * tt KLAY  with exponent notation as in text (    1 etc. ).
- Key empirical findings (Figure 7):
  - During the boom years preceding the recent financial crisis, Poland was growing above its potential, with an output gap of 2 percent by early 2008.
  - HP-filter based output gap peaked earlier and turned negative by end-2008.
  - The production-function (PF) output-gap series exhibits a more gradual reversal, indicating the Polish economy was at a level above potential as late as the fourth quarter of 2008.
- Labor evidence supporting unsustainability:
  - Annual growth rate of potential employment slowed from about 3 percent in early-2008 to 2 percent by the fourth quarter of 2008.
  - The growth rate of actual employment remained above 3 percent throughout the year.
  - These patterns provide evidence that Poland’s rapid output and employment growth pre-crisis was unsustainable.

### Contributions to potential growth (Box 2) — decomposition and insights
- Framework:
  - Production function framework decomposes potential GDP growth into contributions from labor, capital, and factor productivity (TFP).
  - Contributions are computed as year-on-year percentage changes; labor, capital and TFP contributions sum up to potential GDP growth rates (any discrepancy is due to rounding).
- Empirical evolution:
  - Following the 2001–02 recession, contribution of factor productivity growth rose steadily through 2004; it remained positive until 2007, then turned negative through late-2008—largely coinciding with the trend-reversal in potential output growth.
  - Contribution of capital was steadily increasing but was insufficient to prevent the decline in potential output growth throughout 2008.
  - Labor contribution rose in recent years, largely reflecting a decrease in the NAIRU from 2004.
- Potential labor decomposition:
  - Most of the increase in potential labor force is attributed to a corresponding decline in the NAIRU.
  - Rate of growth in Poland’s active population held roughly constant since 2004.
  - Participation rate has been decreasing at a constant rate with a negligible effect on the growth of the equilibrium employment rate.

### Forecasts through end-2010 and comparative outlook
- Forecast extension through the fourth quarter of 2010 (paragraph 17):
  - The measure of the output gap turned negative in the first quarter of 2009 and is expected to remain negative throughout 2010.
  - The output gap is projected to bottom out at just around minus 1 percent, during the second quarter of 2010, vs. minus 2 percent in the 2001–02 downturn.
  - The output gap is expected to close in 2011.
  - This outlook contrasts with many other European countries, many of whom have larger negative output gaps expected to persist for a number of years.
- Modeling and sample notes for forecasts:
  - For consistency, parameter estimates from the 1995:I-2008:IV sample are left unchanged, while the forward-looking NAIRU is modeled consistent with a relatively stable inflation outlook.
  - Footnote specification in the text:  11 50   tttt NAIRUNAIRU .NAIRUNAIRU .

### Conclusion — summary of analytical findings
- Methodological choice:
  - The production-function methodology, combined with a NAIRU-based potential labor estimate, provides a robust alternative to HP-filter approaches and helps to better identify recent boom-bust turning points.
- Main conclusions:
  - Pre-crisis period showed Poland’s output growing above its potential; confirmed by employment behavior relative to equilibrium.
  - Pre-crisis decline in TFP coincided with deceleration in potential output growth.
  - Rising capital contribution suggests the rapid investment-led output expansion in 2006–07 was unsustainable and driven less by fundamentals.
  - In the aftermath of the global crisis, Poland is not expected to experience a sizable and persistent negative output gap.

*Source: IMF World Economic Outlook and authors' computations, as presented in the provided content.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1015.pdf_