## 5. A Simple Asymmetric Model

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---

### I. Introduction
- Objective: analyze determination of inflation in Sub-Saharan Africa (SSA), focusing on the output gap and the real money gap as measures of inflationary pressures.
- Approach: recover unobserved structural gaps using panel cointegration and economic theory, then test their role in inflation via panel GMM.
- Key preview: both measured gaps contain robust and considerable information regarding the evolution of inflation.

### II. Inflation determination (framework)
- Four frequently cited sources of inflation in developing countries:
  - Demand pressures: output gap (difference between output and potential output).
  - Fiscal and monetary policies: real money gap (difference between real money stock and equilibrium real money stock).
  - Supply shocks: e.g., terms of trade changes, drought, conflict.
  - Inertia: dynamic component from expectations or wage contracts.
- Augmented Phillips curve specification used:
  - πt+1 = α + α1πt + α2(yt − y*t−1) + α3(m*t+1 − mt−1) + α4z t+1 + εt (equation (1) in source).
- Expected inflation modeled as lagged inflation: π^e t+1 = πt−1.

### III. Measuring the output and real money gaps (method)
- Output gap:
  - Long-run trend identified through panel cointegration.
  - Postulated production function: two-factor Cobb-Douglas with tfp (equation (2)).
  - Short-term deviation around estimated long-term output interpreted as output gap.
  - Also compared with Hodrick-Prescott (HP) filter (λ=1600).
- Real money gap:
  - Long-run money demand estimated; specification disregards opportunity cost of holding money due to data limits (equation (3)).
  - Log money demand: ln(M/Y/P) = κ + θt + α ln Y + εm,t, with θt capturing financial technology changes.
  - m_t^ε interpreted as the real money gap.
- Trend extraction:
  - Trend total factor productivity and θi,t obtained by smoothing cointegration residuals.
  - Polynomial of order 3 fitted to allow flexible idiosyncratic TFP and financial-technology paths.

### IV. Data and methodology
- Data: annual panel of African countries, maximum span 1960 to 2003; subset of 19 countries due to investment data limitations.
- Variables: real output, total population (labor proxy), gross capital formation, inflation, M2 (money plus quasi-money), terms of trade, rainfall, war index.
- Sources and construction:
  - World Development Indicators (most series).
  - Nehru and Dhareshwar (1995) capital stock extended with gross fixed capital formation and depreciation rate of 10 percent.
  - FAO Clim 2.0 rainfall (1960–1998).
  - PRIO war index (1960–2003; observations with ≥25 battle deaths coded as 1).
- Panel methods:
  - Pedroni heterogeneous panel cointegration (group-mean FMOLS) to estimate long-run production and money-demand relationships.
  - Im, Pesaran and Shin (2002) panel unit root tests applied to levels of output, capital, labor, and real money; unit roots cannot be rejected for these variables.

### V. Panel estimation and key empirical results
- Panel cointegration tests (Table 1):
  - CRS production function (1960–2004, N=17) and real money demand (1970–2004, N=19).
  - Panel statistics for CRS production function (examples reported):
    - panel v-stat 16.64**, panel rho-stat -1.81, panel pp-stat -1.73, panel adf-stat -4.8**, group rho-stat -0.53, group pp-stat -1.17, group adf-stat -4.42**.
  - For money demand 6 out of 7 tests reject null of no cointegration.
- Cointegration regression estimates (Table 2):
  - Panel group FMOLS capital share (CRS production function, 1960–2004): 0.28** (t-stat 21.62).
  - Panel group FMOLS income elasticity of money demand (1970–2004): 1.02** (t-stat 30.36).
  - When CRS not imposed: capital share 0.27, labor share 0.78; cannot reject sum to one.
- Measured gaps:
  - Output gaps (imposing capital share 0.28) typically in the range of plus or minus 0.5 percent of GDP.
  - Real money gap (imposing money-demand elasticity 1.02) fluctuates around plus or minus 1.5% of GDP and is economically larger than the output gap.
- Stationarity of gaps (Table 3) — selected statistics:
  - Y_gap Im, Pesaran and Shin W-stat: -10.54** (without trend) and -7.14** (without/with trend).
  - Hadri Z-stat for Y_gap: -2.79 (without trend) and 0.10 (without/with trend).
  - Y_gapHP Im, Pesaran and Shin W-stat: -11.55** and -9.67**; Hadri Z-stat -1.89 and 1.17.
  - M_gap Im, Pesaran and Shin W-stat: -10.65** and -7.82**; Hadri Z-stat -2.89 and -0.45.
- Phillips curve estimates (Panel GMM, 3 lags instruments; Table 4) — lagged inflation and gap coefficients:
  - Lagged inflation coefficient (Π-1) ≈ 0.50 and always significant:
    - Regression (1): 0.53*** (5.74)
    - Regression (2): 0.48*** (6.90)
    - Regression (3): 0.51*** (7.79)
    - Regression (4): 0.52*** (8.57)
    - Regression (5): 0.41*** (6.06)
    - Regression (6): 0.53*** (6.66)
  - Output gap and real money gap significance:
    - Regression (1) (Y_gap only): Y_gap-1 = 0.28** (2.48).
    - Regression (2) (M_gap only): M_gap = 0.34*** (4.47).
    - Regression (3) (both): Y_gap-1 = 0.19*** (3.35); M_gap = 0.29*** (3.91).
    - Regression (4) (HP-filter Y_gap): Y_gapHP-1 = 0.23*** (2.97); M_gap = 0.29*** (3.99).
    - Regression (5) (controls: ∆TOT, rainfall, war, CFA devaluation): M_gap = 0.37*** (3.65); ∆TOT coefficient -1.84e-4 (0.19).
    - Regression (6) (sample 1980–2003): M_gap = 0.33*** (3.11).
- Economic significance (using regression (5) estimates):
  - A 1 percent increase in the output gap → total excess inflation of 0.5 percentage points over the future.
  - A 1 percent increase in the real money gap → total excess inflation of 0.6 percentage points over the future.
- Interpretation:
  - The real money gap plays a larger role in SSA inflation than the output gap, reflecting that demand stimuli are often financed via monetary financing.
- Robustness:
  - Controlling for supply shocks (terms of trade, rainfall, war, CFA devaluation) does not alter main conclusions.
  - Restricting sample to 1980–2003 does not change results.

### VI. Asymmetry in the Phillips curve
- Asymmetric specification tested (equation (4)): separate positive-part variables for output and money gaps (y_pos, m_pos).
- Table 5 results (GMM) — selected coefficients:
  - Lagged inflation around 0.5 and significant:
    - Regression (7): Π-1 = 0.51*** (5.24)
    - Regression (8): Π-1 = 0.46*** (6.36)
    - Regression (9): Π-1 = 0.55*** (8.57)
  - Example coefficients and standard errors:
    - Regression (7): Y_gap-1 = 0.41** (2.25); Y_gap-1pos = -0.29 (1.32).
    - Regression (8): M_gap = 0.24*** (2.66); M_gappos = 0.13 (0.77).
    - Standard errors of regression: 0.15, 0.13, 0.14 respectively.
- Conclusion on asymmetry:
  - No significant evidence of asymmetry.
  - Point estimates hint that the real money gap may play a more important role in inflation buildup than the output gap, but differences are not statistically significant.

### VII. Conclusions and policy implications
- Methodological conclusion:
  - Panel cointegration techniques permit recovery of structural gaps despite short data spans and structural changes in SSA.
- Key estimates:
  - Capital share ≈ 0.28.
  - Money-demand income elasticity ≈ 1.02.
- Main empirical conclusions:
  - Structural output and real money gaps contain considerable information about inflation dynamics in SSA.
  - The real money gap plays a larger role in inflation processes in SSA than the output gap, consistent with frequent monetization of fiscal imbalances and monetary financing.
  - No statistically significant evidence of an asymmetric Phillips curve, though excess money appears more important in inflation buildup.
- Policy implication:
  - Targeting monetary aggregates in SSA can provide an effective anchor to control inflation, especially relevant for IMF-supported programs.

*Source: _wp0732 - 5. A Simple Asymmetric Model (IMF PDF chapter).*

### References..............................................................................................................

### _wp0732 - References..............................................................................................................

### Tables

- 1. Panel Cointegration Tests ......................................................................................................8
- 2. Cointegration Regressions .....................................................................................................8
- 3. Stationarity Tests for Measured Gaps....................................................................................9
- 4. Phillips Curve Panel GMM Estimates .................................................................................11

*Source: _wp0732 - References..............................................................................................................*

### 5. A Simple Asymmetric Model ...........................................................................................

### 5. A Simple Asymmetric Model

### I. Introduction
- Objective: analyze determination of inflation in Sub-Saharan Africa (SSA), focusing on the output gap and the real money gap as measures of inflationary pressures.
- Approach: recover unobserved structural gaps using panel cointegration and economic theory, then test their role in inflation via panel GMM.
- Key preview: both measured gaps contain robust and considerable information regarding the evolution of inflation.

### II. Inflation determination (framework)
- Four frequently cited sources of inflation in developing countries:
  - Demand pressures: output gap (difference between output and potential output).
  - Fiscal and monetary policies: real money gap (difference between real money stock and equilibrium real money stock).
  - Supply shocks: e.g., terms of trade changes, drought, conflict.
  - Inertia: dynamic component from expectations or wage contracts.
- Augmented Phillips curve specification used:
  - πt+1 = α + α1πt + α2(yt − y*t−1) + α3(m*t+1 − mt−1) + α4z t+1 + εt (presented in the source as equation (1) with the same variables and timing conventions).
- Expected inflation modeled as lagged inflation: π^e t+1 = πt−1.
- Rationale: data limits and lack of expectation surveys in SSA lead to inclusion of lagged inflation.

### III. Measuring the output and real money gaps (method)
- Output gap:
  - Long-run trend identified through panel cointegration.
  - Postulated production function: two-factor Cobb-Douglas with tfp (equation (2)).
  - Short-term deviation around estimated long-term output interpreted as output gap.
  - Also compared with Hodrick-Prescott (HP) filter (λ=1600).
- Real money gap:
  - Long-run money demand estimated; specification disregards opportunity cost of holding money due to data limits (equation (3)).
  - Log money demand: ln(M/Y/P) = κ + θt + α ln Y + εm,t, with θt capturing financial technology changes.
  - m_t^ε interpreted as the real money gap.
- Trend extraction:
  - Trend total factor productivity and θi,t obtained by smoothing cointegration residuals.
  - Polynomial of order 3 fitted to allow flexible idiosyncratic TFP and financial-technology paths.

### IV. Data and methodology
- Data: annual panel of African countries, maximum span 1960 to 2003; subset of 19 countries due to investment data limitations.
- Variables: real output, total population (labor proxy), gross capital formation, inflation, M2 (money plus quasi-money), terms of trade, rainfall, war index.
- Sources: World Development Indicators (most series), Nehru and Dhareshwar (1995) capital stock extended with gross fixed capital formation and depreciation rate of 10 percent, FAO Clim 2.0 rainfall (1960–1998), PRIO war index (1960–2003; observations with ≥25 battle deaths coded as 1).
- Panel methods:
  - Use Pedroni heterogeneous panel cointegration (group-mean FMOLS) to estimate long-run production and money-demand relationships allowing heterogeneous dynamics, cointegrating vectors, and endogeneity.
  - Im, Pesaran and Shin (2002) panel unit root tests applied to levels of output, capital, labor, and real money; unit roots cannot be rejected for these variables.

### V. Panel estimation and key empirical results
- Panel cointegration tests (Table 1):
  - CRS production function (1960–2004, N=17) and real money demand (1970–2004, N=19).
  - Panel ADF-stat and group ADF-stat strongly reject null of no cointegration for output; for money demand 6 out of 7 tests reject null.
  - Reported sample statistics include panel v-stat 16.64**, panel rho-stat -1.81, panel pp-stat -1.73, panel adf-stat -4.8**, group rho-stat -0.53, group pp-stat -1.17, group adf-stat -4.42** for CRS production function (note: (**) indicates significance at the 5% level).
- Cointegration regression estimates (Table 2):
  - Panel group FMOLS capital share (CRS production function, 1960–2004): 0.28** (t-stat 21.62).
  - Panel group FMOLS income elasticity of money demand (1970–2004): 1.02** (t-stat 30.36).
  - When CRS not imposed: capital share 0.27, labor share 0.78; cannot reject sum to one.
- Measured gaps:
  - Output gaps (imposing capital share 0.28) typically in the range of plus or minus 0.5 percent of GDP.
  - Real money gap (imposing money-demand elasticity 1.02) fluctuates around plus or minus 1.5% of GDP and is economically larger than the output gap.
- Stationarity of gaps (Table 3):
  - Y_gap Im, Pesaran and Shin W-stat: -10.54** (without trend) and -7.14** (without/with trend).
  - Hadri Z-stat for Y_gap: -2.79 (without trend) and 0.10 (without/with trend).
  - Y_gapHP Im, Pesaran and Shin W-stat: -11.55** and -9.67**; Hadri Z-stat -1.89 and 1.17.
  - M_gap Im, Pesaran and Shin W-stat: -10.65** and -7.82**; Hadri Z-stat -2.89 and -0.45.
  - Note: (**) indicates significance at the 5% level; H0 W-stat: individual unit root; H0 Z-stat: no common unit root.
- Phillips curve estimates (Panel GMM, 3 lags instruments; Table 4):
  - Lagged inflation coefficient (Π-1) ≈ 0.50 and always significant:
    - Regression (1): 0.53*** (5.74)
    - Regression (2): 0.48*** (6.90)
    - Regression (3): 0.51*** (7.79)
    - Regression (4): 0.52*** (8.57)
    - Regression (5): 0.41*** (6.06)
    - Regression (6): 0.53*** (6.66)
    - Note: (***) denotes high significance as reported in tables.
  - Output gap and real money gap significance:
    - Regression (1) (Y_gap only): Y_gap-1 = 0.28** (2.48).
    - Regression (2) (M_gap only): M_gap = 0.34*** (4.47).
    - Regression (3) (both): Y_gap-1 = 0.19*** (3.35); M_gap = 0.29*** (3.91).
    - Regression (4) (HP-filter Y_gap): Y_gapHP-1 = 0.23*** (2.97); M_gap = 0.29*** (3.99).
    - Regression (5) (controls: ∆TOT, rainfall, war, CFA devaluation): M_gap = 0.37*** (3.65); ∆TOT coefficient -1.84e-4 (0.19).
    - Regression (6) (sample 1980–2003): M_gap = 0.33*** (3.11).
  - Economic significance (using regression (5) estimates):
    - A 1 percent increase in the output gap → total excess inflation of 0.5 percentage points over the future.
    - A 1 percent increase in the real money gap → total excess inflation of 0.6 percentage points over the future.
  - Interpretation: real money gap plays a larger role in SSA inflation than output gap, reflecting that demand stimuli are often financed via monetary financing.
- Robustness:
  - Controlling for supply shocks (terms of trade, rainfall, war, CFA devaluation) does not alter main conclusions.
  - Restricting sample to 1980–2003 does not change results.

### VI. Asymmetry in the Phillips curve
- Asymmetric specification tested (equation (4)): separate positive-part variables for output and money gaps (y_pos, m_pos).
- Table 5 results (GMM):
  - Lagged inflation around 0.5 and significant across regressions:
    - Regression (7): Π-1 = 0.51*** (5.24)
    - Regression (8): Π-1 = 0.46*** (6.36)
    - Regression (9): Π-1 = 0.55*** (8.57)
  - Coefficients:
    - Y_gap-1: e.g., 0.41** (2.25) in regression (7); Y_gap-1pos = -0.29 (1.32) (not significant).
    - M_gap: e.g., 0.24*** (2.66) in regression (8); M_gappos = 0.13 (0.77) (not significant).
  - Standard errors of regression: 0.15, 0.13, 0.14 respectively.
- Conclusion on asymmetry:
  - No significant evidence of asymmetry.
  - Point estimates suggest opposite signs for money gap (positive) and positive output-gap component (negative in some specs), hinting that real money gap may play a more important role in inflation buildup than the output gap—though differences are not statistically significant.

### VII. Conclusions and policy implications
- Panel cointegration techniques permit recovery of structural gaps despite short data spans and structural changes in SSA.
- Key estimates:
  - Capital share ≈ 0.28 (close to one third typical in developed economies).
  - Money-demand income elasticity ≈ 1.02 (unity).
- Main findings:
  - Structural output and real money gaps contain considerable information about inflation dynamics in SSA.
  - The real money gap plays a larger role in inflation processes in SSA than the output gap, consistent with frequent monetization of fiscal imbalances and monetary financing.
  - No statistically significant evidence of an asymmetric Phillips curve, though excess money appears more important in inflation buildup.
- Policy implication:
  - Targeting monetary aggregates in SSA can provide an effective anchor to control inflation, especially relevant for IMF-supported programs.

*Source: _wp0732 - 5. A Simple Asymmetric Model (IMF PDF chapter).*

### References

### _wp0732 - References (Appendix excerpts and empirical findings)

### Country sample
- Countries included: Botswana, Cameroon, Cote d’Ivoire, Ethiopia, Ghana, Kenya, Madagascar, Malawi, Mali, Mozambique, Niger, Nigeria, Rwanda, Senegal, Sierra Leone, Uganda, Zambia, Zimbabwe.
- South-Africa was removed from the sample because it is considered a middle-income country that differs substantially from the rest of the developing SSA countries.

### Data properties and unit-root/cointegration testing
- ADF tests on levels (without/with trend; SIC lag length) for Y, K (est2), L, M2r show many cases where the null of a unit root cannot be rejected; note: (*) indicates significance at the 10% level and (**) at the 5% level, H0: series has a unit root.
  - Example entries (levels, without/with trend):
    - Côte d’Ivoire Y: -3.23*/-1.55
    - Ghana Y: 1.10/-0.53
    - Kenya Y: -3.29*/-0.04
    - Mozambique Y: 3.14/-0.52
    - Nigeria Y: -0.78/-2.46
    - Uganda Y: 0.14/-1.00
  - Example K (est2) entries:
    - Côte d’Ivoire: -2.05/-0.39
    - Mozambique: -3.02**/-2.68
    - Nigeria: -2.52/-2.68
- Im, Pesaran and Shin W-stat (levels / first-order difference; all variables in log):
  - Y: 2.22/-0.06 (levels) ; -17.19**/-16.90** (first-diff)
  - K (est2): -0.34/3.17 ; -6.10**/-6.87**
  - L: 5.83/2.17 ; -6.65**/-3.83**
  - M2r: -0.65/-0.35 ; -17.78**/-16.70**
  - Π: -6.24**/-4.91** (levels); tot: -4.95**/-3.93** (levels)
  - Note: (**) indicates significance at the 5% level, H0: individual unit-root process.

### Production function and cointegration (time-series analysis for six countries)
- OLS Cobb-Douglas production function estimates (Table A2a) — coefficients and residual ADF t-test:
  - Côte d’Ivoire: K 0.24 (SE 0.04); L 0.51 (SE 0.03); ADF t-test -3.30
  - Ghana: K 0.32 (0.08); L 0.79 (0.07); ADF t-test -2.15
  - Kenya: K 0.54 (0.10); L 1.05 (0.03); ADF t-test -2.04
  - Mozambique: K 0.19 (0.13); L 2.37 (0.17); ADF t-test -2.26
  - Nigeria: K 0.16 (0.04); L 0.84 (0.06); ADF t-test -2.06
  - Uganda: K 0.54 (0.02); L 0.40 (0.04); ADF t-test -1.82
  - Note: SE in parentheses, ADF t-test: (*) and (**) indicates significance at the 10 (t=3.74) and 5% level, H0: no cointegration.
- Findings:
  - There is no strong evidence of a long-run relationship for the Cobb-Douglas production function across the six countries; the null of no cointegration cannot be rejected in all cases.
  - In two cases (Mozambique and Kenya), coefficient estimates are described as unrealistic.
  - Johansen cointegration tests yielded the same conclusions; cointegrating vector estimates were very sensitive to lag choice and sample period.

### Real money demand cointegration (Table A2b)
- OLS real money demand with time trend and controls (tot, war):
  - Côte d’Ivoire M2r 1.46*** (0.20); residual ADF t-test -3.27*
  - Ghana M2r 3.77*** (0.25); residual ADF t-test -3.32*
  - Kenya M2r 0.61*** (0.12); residual ADF t-test -4.41**
  - Mozambique M2r 1.36* (0.76); residual ADF t-test -3.42*
  - Nigeria M2r 0.89*** (0.21); residual ADF t-test -4.19**
  - Uganda M2r 3.23*** (0.93); residual ADF t-test -2.90
  - Note: SE in parentheses, (*) and (**) indicates significance at the 10 (t=3.04) and 5% level, H0: no cointegration.
- Findings:
  - The null of no cointegration for long-run real money demand can be rejected at 10% in 5 out of 6 cases; Uganda is almost significant at 10%.
  - Some coefficient estimates are surprising for Ghana and Uganda.

### Panel FMOLS for production function (1960-2004) (Table A4)
- Panel group FMOLS estimates (no common time trend; reported values distributed N(0,1); (**) indicates significance at the 5% level, H0: no cointegration for most countries):
  - K: 0.27** (t-stat 17.40)
  - L: 0.78** (t-stat 28.81)
- Interpretation: Panel FMOLS yields statistically significant long-run coefficients for capital and labor at the panel level.

### Phillips curve analysis — panel and country time-series results
- Panel Phillips curve OLS estimates (Table A5) — dependent variable: Π; sample variations and FE estimation:
  - Π_{-1} coefficients: 0.65*** (9.28), 0.66*** (9.50), 0.50*** (9.10), 0.67*** (8.59) across equations (1),(2),(3),(5).
  - Y_gap_{-1} coefficients: 0.23** (2.86) in eq (1); 0.19** (2.45) in eq (3); 0.29*** (2.99) in eq (5); Y_gapHP_{-1} 0.29*** (3.45) in eq (2).
  - M_gap coefficients: 0.30*** (4.03), 0.29*** (4.21), 0.34*** (3.49), 0.33*** (3.35).
  - ∆TOT: 2.27E-5 (0.05) in one specification.
  - Adj R^2: 0.63, 0.63, 0.66, 0.67 across reported FE specifications.
  - Note: t-stat in parentheses, White-period SE.
- Time-series Phillips curve regressions for six countries (Table A6) — OLS:
  - Π_{-1} coefficients:
    - Côte d’Ivoire 0.49** (3.95)
    - Ghana 0.26** (4.99)
    - Kenya 0.58** (3.17)
    - Mozambique 0.77** (7.06)
    - Nigeria 0.43** (3.32)
    - Uganda 0.62* (1.60)
  - Y_gapHP_{-1} coefficients (examples):
    - Kenya 0.56* (2.14)
  - M_gap coefficients:
    - Côte d’Ivoire 0.08 (0.98)
    - Ghana 0.54** (6.45)
    - Kenya 0.31** (2.14)
    - Mozambique 0.28* (1.59)
    - Nigeria 0.35** (4.37)
    - Uganda 0.53** (2.18)
  - Adj R^2 by country: Côte d’Ivoire 0.44; Ghana 0.86; Kenya 0.42; Mozambique 0.62; Nigeria 0.56; Uganda 0.79.
  - Note: t-stat in parentheses, White SE.
- Key empirical conclusions reported:
  - Lagged inflation is always significant with an average value of around 0.5.
  - The real money gap is significant in 5 out of 6 countries with a mean of 0.4.
  - Since variables are in logs, coefficients can be interpreted as elasticities: "a 1% deviation from long-run money demand leads to a significant and positive output deviation ranging from 0.28% to 0.54%."
  - The output gap shows no evidence of predictive power for future inflation in the presented results.
  - Because of absence of cointegration for output, regressions with the structural output gap on the left-hand side would be spurious; analysis therefore focuses on Phillips curves with HP-filtered output gaps and structural money gap.

### Additional empirical diagnostics and notes
- ADF t-test of residuals for production function and money demand reported; critical t-values and significance flags retained as in tables.
- Panel diagnostics note: "no common time trend" for FMOLS panel estimates; H0 refers to no cointegration for most countries.
- The text highlights caution: short data spans, structural changes, and sensitivity of results to lag length and sample period make many estimates fragile and difficult to interpret.

*References and appendix tables/graphs as provided in the source document.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp0732.pdf_