## _wp10134

## Source details

**Canonical URL:** [_wp10134](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10134.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10134.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10134.pdf.json)

---

### Introduction: motivation and approach
- Over the last 20 years many central banks adopted explicit inflation targeting using interest rates as operational target; monetary aggregates play very little role in modern New Keynesian models.
- Paper objective: introduce a role for money into the standard New Keynesian framework with minimal changes, retain assumption that money does not play a direct role in the transmission mechanism, and ask:
  - How should central banks in low-income countries (LICs) set targets on money growth?
  - How should central banks respond to deviations from these targets?
- Authorities choose ex-ante targets for the growth rate of the monetary aggregate and the short-term interest rate over the next period (one quarter); targets are mutually consistent via the money demand function.

### Rationale: informational role of money in LICs
- Key informational frictions in LICs:
  - Output and inflation observed imperfectly and with substantial lags (e.g., quarterly GDP often unavailable).
  - Output gaps hard to assess; dominant agricultural supply shocks; pervasive informal markets.
  - Financial market imperfections: thin/opaque interbank and government bond markets; poor contract enforcement; limited participation.
  - Observed interbank rates may be a noisy indicator of the shadow interest rate relevant to private decisions.
- Monetary aggregates are typically measured accurately and with little lag and are systematically related to key variables through money demand.
- With sufficient information gaps, money targets may have informational value despite money demand shocks.

### Modeling strategy and information structure
- Information incompleteness introduced into a standard New Keynesian model:
  - Expected short-term interest rate follows a forward-looking Taylor rule satisfying the Taylor principle.
  - Ex-post equivalence between money and interest-rate targets can break down after shocks; central bank chooses relative weight λ on interest rates versus money.
  - Observed nominal interest rate modeled as noisy: R^N_t = R_t + ξ^{R^N}_t.
- Implications from the ex-ante mapping for money growth targets:
  - Money growth should aim toward its flexible-price value.
  - Money growth should correct for the lagged money gap mgap_t = m_t − m^*_t.
  - Money growth should accommodate deviations in current inflation from its target.
  - Response to expected inflation and the output gap depends on the interest and income elasticity of money demand (χ_R and χ_y).

### Money-target rule and implied Taylor rule (key equations)
- Flexible-price real money balances: ∆m^*_t = χ_y ∆y^*_t − χ_R ∆RR^*_t + ∆u_t.
- Nominal money growth at flex-price equilibrium: ∆M^*_t = ∆m^*_t + π^*.
- Lagged money-gap adjustment: ∆M^T_t = ∆M^*_t − mgap_{t−1}.
- Active money-targeting rule targeting next period’s inflation:
  - ∆M^T_t = ∆M^*_t − mgap_{t−1} + (π_t − π^*) − δ(E_t(π_{t+1}) − π^*) − γ ygap_t.
- Corresponding forward-looking interest-rate target:
  - R^T_t = RR^*_t + π^* + φ_π (E_t(π_{t+1}) − π^*) + φ_{ygap} ygap_t
  - where φ_π = δ / χ_R and φ_{ygap} = (γ + χ_y) / χ_R.
- Operational intra-period tradeoff represented by:
  - λ (R^N_t − R^T_t|_{t−1}) − (1 − λ) (∆M_t − ∆M^T_t|_{t−1}) = 0,
  - Rearrangements give RN_t = R^T_t|_{t−1} + (1 − λ)/λ (∆M_t − ∆M^T_t|_{t−1}) and ∆M_t = ∆M^T_t|_{t−1} + λ/(1 − λ) (R^N_t − R^T_t|_{t−1}).

### Estimation: countries, method, and priors
- Countries estimated: Ghana, Tanzania, Uganda.
  - Uganda and Tanzania are de jure money targeters.
  - Ghana de jure inflation targeting since 2004 but still pays attention to monetary aggregates.
- Bayesian estimation of a state-space representation using the Kalman filter and Monte Carlo Markov Chain (random walk Metropolis-Hastings); Dynare used.
- Vector of estimated parameters Υ = [βlead, βR, βz, βw, αlead, αy, αz, v*, χy, χR, φπ, φygap, λ]′.
- Prior highlights:
  - Prior mean for βR set at 0.07.
  - Priors for policy parameters: φ_π = 2 and φ_{ygap} = 0.5.
  - Prior mean of λ = 0.5 for all countries with relatively large variance.
  - θ calibrated at 0.05; several steady-state and persistence parameters dogmatically fixed.

### Selected posterior estimates (posterior means and 90% intervals)
- Ghana (posterior means and 90% CI):
  - λ: 0.9285; 90% CI: 0.8444 − 0.9994
  - φ_π: 1.7637; 90% CI: 1.3131 − 2.1896
  - φ_{ygap}: 0.4137; 90% CI: 0.1237 − 0.6959
  - χ_R: 0.3773; 90% CI: 0.2250 − 0.5155
  - β_R: 0.1954; 90% CI: 0.0965 − 0.3008
  - β_y: 0.5335; 90% CI: 0.3455 − 0.7421
- Uganda (posterior means and 90% CI):
  - λ: 0.3377; 90% CI: 0.1785 − 0.4824
  - φ_π: 2.0030; 90% CI: 1.6507 − 2.3212
  - φ_{ygap}: 0.5227; 90% CI: 0.2032 − 0.8293
  - χ_R: 0.5817; 90% CI: 0.4433 − 0.7196
  - β_R: 0.3301; 90% CI: 0.1330 − 0.5114
  - β_y: 0.7470; 90% CI: 0.5836 − 0.9173
- Tanzania (selected posterior means and 90% CI from parameter table):
  - β_y: 0.6836; 90% CI: 0.5275–0.8509
  - β_R: 0.1624; 90% CI: 0.0766–0.2397
  - α_lead: 0.6212; 90% CI: 0.5366–0.7141
  - χ_y: 0.9226; 90% CI: 0.4492–1.4080
  - χ_R: 0.5672; 90% CI: 0.3680–0.7771
  - φ_π: 1.9966; 90% CI: 1.5610–2.3997
  - φ_{ygap}: 0.4671; 90% CI: 0.1855–0.7401
  - λ: 0.6642; 90% CI: 0.4595–0.9783
- Estimation conclusions:
  - Interest elasticity in the IS equation (β_R) is relatively high across countries, contrary to priors.
  - Forward-looking behavior matters: β_y and α_lead greater than 0.5 in all three countries.

### Signal-extraction framing and optimal λ derivation (method overview)
- Policy representation uses money growth as instrument: ∆M_t = E(∆M^*_T_t | I_{t−1}, R^N_t).
- Deviations ∆M_t − ∆M^T_t|_{t−1} are linear in revisions to forecasts and shocks.
- Use complete-information solution X_t = Υ_X X_{t−1} + Υ_ε ε_t to obtain reduced-form for non-policy block:
  - Y_t = A Y_{t−1} + B ∆M_t + B_2 ∆M_{t−1} + C C ε_t.
  - First differences: ˜Y_t = B ∆˜M_t + C ε_t.
- Observed money-market variable: ˜R^N_t = Γ_R ˜Y_t + ε^{R^N}_t.
- Construct observable combination ˜φ_t = ˜R^N_t − Γ_R B ∆˜M_t = Θ[˜R^N_t ∆˜M_t]′ = H ε_t + ε^{R^N}_t.
- Kalman filter used; Kalman gain K = Q H (H′ Q H + Ω)^{−1}; shock estimates ˆε_t = K ˜φ_t.
- Solve for relation ∆˜M_t = Φ ˜R^N_t with Φ = (1 − P_{∆M} K Θ(1,2))^{−1} P_{∆M} K Θ(1,1); Optimal λ = Φ / (1 + Φ).

### Estimated versus optimal λ (numerical results)
- Ghana:
  - Estimated Lambda: 0.9285
  - Optimal Lambda: 0.3634
- Tanzania:
  - Estimated Lambda: 0.6642
  - Optimal Lambda: 0.4421
- Uganda:
  - Estimated Lambda: 0.3377
  - Optimal Lambda: 0.3246
- Interpretation:
  - Uganda: estimated λ close to optimal.
  - Ghana: estimated λ much higher than optimal; authorities would benefit from paying additional attention to money-market developments (though Ghana results not very robust to changing priors).
  - Tanzania: estimated λ intermediate; optimal λ lower than estimated, implying some misalignment between practice and the model-implied optimal weight.

### Sensitivity insights (qualitative)
- Factors increasing optimal λ (more weight on interest-rate target):
  - Higher money-demand volatility.
- Factors decreasing optimal λ (more weight on money target):
  - Higher interest elasticity of money demand (χ_R).
  - Higher volatility of cost-push shocks.
  - Higher volatility of shocks to the natural rate of interest.
  - Noisier observed short-term interest rates (makes observed RN_t less informative, increasing value of money-market information).
- Quantitative bound for Uganda: even when noise volatility set to zero, optimal λ does not increase beyond 0.7.

### Welfare implications of changing λ (inflation and output-gap volatility as welfare proxies)
- Main quantitative results (inflation and output-gap volatility levels):
  - Ghana:
    - λ = 0: Inflation 12.08; Output Gap 3.11
    - Optimal λ = 0.3634: Inflation 12.80; Output Gap 2.09
    - Estimated λ = 0.9285: Inflation 14.49; Output Gap 2.51
    - λ = 1: Inflation 14.71; Output Gap 2.65
  - Uganda:
    - λ = 0: Inflation 10.33; Output Gap 1.77
    - Optimal λ = 0.3246: Inflation 10.95; Output Gap 1.60
    - Estimated λ = 0.3377: Inflation 10.99; Output Gap 1.62
    - λ = 1: Inflation 14.20; Output Gap 4.13
  - Tanzania:
    - λ = 0: Inflation 12.02; Output Gap 2.12
    - Optimal λ = 0.4421: Inflation 12.38; Output Gap 1.59
    - Estimated λ = 0.6642: Inflation 13.08; Output Gap 1.74
    - λ = 1: Inflation 14.60; Output Gap 2.34
- Key findings:
  - Strict adherence to money targets (λ = 0) yields lowest inflation volatility but higher output-gap volatility.
  - Strict adherence to interest-rate targets (λ = 1) generates highly volatile movements in both inflation and the output gap.
  - Optimal λ typically does not minimize inflation volatility but minimizes output-gap volatility, reflecting a tradeoff driven by volatile cost-push shocks.
  - Welfare gains from varying λ (in terms of inflation and output stability) are relatively small given model assumptions.

### Policy-relevant implications and recommendations
- Rigid adherence to money targets for credibility reasons can be costly by inducing excessive output and inflation volatility; once credibility is achieved, more flexible policy execution is observed.
- Money can matter in LICs as:
  - A visible sign of policy adherence in weak-institution settings (a “tripwire”).
  - A useful informational variable where financial-market characteristics weaken the interest-rate transmission and strengthen links between monetary aggregates and activity/inflation.
- Noisy-observed-interest-rate assumption is important: when observed short-term interest rates are noisy, greater attention to money-market information is warranted.
- For Ghana (empirical λ much higher than optimal), policymakers could benefit from paying closer attention to money-market developments; Uganda’s practice aligns closely with model-implied optimality.

### Caveats, limitations, and research agenda
- Model simplicity: not a fully micro-founded DSGE and does not jointly solve for an optimal monetary policy loss function.
- Numerical results sensitive to priors and data limitations; interpret numerical magnitudes cautiously.
- Information structure is simple; contemporaneous information beyond money markets (exchange rate, preliminary inflation/output measures) may reduce the relevance of monetary aggregates.
- Exchange-rate treatment is limited; many LICs that target money also manage the exchange rate, which is highly observable—future work should incorporate exchange-rate dynamics explicitly.
- Suggested next steps:
  - More complete treatment of information incompleteness.
  - Explicit exchange-rate dynamics and more elaborate financial-market modeling.
  - LIC-specific modifications (financial sector, exchange rate, institutional credibility) developed carefully.

*IMF Working Paper content unit: _wp10134*

### References...........................................................................................29

### _wp10134 - References...........................................................................................29

### Introduction: motivation and approach
- Over the last 20 years many central banks adopted explicit inflation targeting using interest rates as operational target; monetary aggregates play very little role in modern New Keynesian models.
- New-Keynesian models do not assign money a direct role in the monetary transmission mechanism; monetary aggregates typically do not enter the transmission mechanism.
- Despite this, many low-income countries (LICs), particularly in sub-Saharan Africa, continue to describe monetary policy in terms of monthly or quarterly targets on monetary aggregates (typically narrow money).
- Common policymaking practice in LICs often uses the quantity theory MV=PQ to set money targets consistent with desired paths for prices P given real output Q and assumptions about velocity V.
- The paper’s objective: introduce a role for money into the standard New Keynesian framework with minimal changes, retaining the assumption that money does not play a direct role in the transmission mechanism, and ask:
  - How should central banks in LICs set targets on money growth?
  - How should central banks respond to deviations from these targets?

### Rationale: informational role of money in LICs
- Key informational frictions in LICs highlighted:
  - Output and inflation observed imperfectly and with substantial lags (e.g., quarterly GDP often unavailable; scant economy-wide wage data).
  - Output gaps are hard to assess: scarce capacity utilization measures, dominant agricultural supply shocks, pervasive informal markets.
  - Financial market imperfections: thin/opaque interbank and government bond markets; poor contract enforcement; large fractions of population excluded from formal financial markets.
  - Observed interbank rates may be a noisy indicator of the shadow interest rate relevant to private decisions.
- Monetary aggregates are typically measured accurately and with little lag and are systematically related to key variables (output, relevant interest rate) through money demand.
- With sufficient information gaps, money targets may have informational value despite money demand shocks.

### Modeling strategy and information structure
- Introduce a simple information incompleteness into a standard New Keynesian model:
  - Authorities choose ex-ante targets for the growth rate of the monetary aggregate and the short-term interest rate over the next period (one quarter).
  - Targets are mutually consistent via the money demand function.
  - Expected short-term interest rate follows a forward-looking Taylor rule satisfying the Taylor principle.
- Implications for money growth targets (from the ex-ante mapping):
  1. Money growth should aim toward its flexible-price value.
  2. Money growth should correct for the lagged money gap (deviations between real money balances and the level under flexible prices).
  3. Money growth should accommodate deviations in current inflation from its target.
  4. Response to expected inflation and the output gap depends on the interest and income elasticity of money demand.
- Ex-post, after shocks, the equivalence between money and interest rate targets breaks down. The central bank must decide how to weight adherence to interest rate vs. money aggregate targets; denote the relative weight on interest rates by λ.
- Observed/controllable short-term interest rate is a noisy indicator of the rate relevant for private decisions (R_Nt = Rt + ε^R_N_t), capturing limited participation and redistribution shocks.

### Signal extraction and optimal weighting (λ)
- The optimal weight on money relative to interest rates is a signal extraction solution: money market information helps update estimates of the output gap and expected inflation, which feed into the Taylor rule.
- Circularity: money market outcomes depend on the policy response; Kalman filter must be revised to account for this circularity (failure of separation principle).
- Factors increasing the optimal weight on money:
  - Less volatile money demand shocks.
  - Lower interest elasticity of money demand.
  - Noisier observed interest rates.
  - Higher volatility of real shocks.
- Analogy to Poole (1970): instrument choice depends on relative variances; here the informational problem drives the optimal λ.

### Estimation: countries and methodology
- Countries estimated: Ghana, Tanzania, Uganda.
  - Uganda and Tanzania are de jure money targeters.
  - Ghana followed a de jure inflation targeting regime since 2004 with an interest rate instrument, but authorities continue to pay attention to monetary aggregates.
- Estimation method: Bayesian methods, with three key features:
  1. Same priors across countries so cross-country parameter differences reflect data information.
  2. The weight on interest rate target relative to money target (λ) is a free parameter with a relatively diffuse prior mean of 0.5.
  3. Targets on interest rates and money growth are not observed directly; they are inferred from observed interest rates, money aggregates, and the model structure.

### Estimated and optimal λ results (key numerical findings)
- Estimated λ (adherence to interest rate target) across countries:
  - Ghana: λ = 0.93 (almost complete adherence to interest rate target).
  - Uganda: λ = 0.34 (adheres more strongly to money targets).
  - Tanzania: λ = 0.66 (intermediate).
- Computed optimal λ for each country:
  - Uganda: optimal λ = 0.32 (estimated λ close to optimal).
  - Ghana: optimal λ = 0.36 (estimated λ much higher than optimal; Ghana would benefit from paying additional attention to money-market developments).
  - Tanzania: optimal λ = 0.44 (estimated λ somewhat higher than optimal; could benefit from more attention to money).
- Interpretation: moderate volatility of money demand and the noisiness of observed interest rates justify more weight on money than implied by current practice in Ghana and, to a lesser extent, Tanzania.

### Policy tradeoffs and costs of non-optimal adherence
- Cost-push shocks are volatile in all three countries, creating a tradeoff between inflation and output gap stabilization.
- Strict adherence to money targets (λ = 0) would:
  - Lower inflation volatility.
  - Increase output gap volatility.
- Choosing λ optimally yields slightly more volatile inflation but lowers output gap volatility, balancing the tradeoff consistent with estimated policy preferences.

### Model structure: key equations and features (as presented)
- Output decomposition: yt = y*_t + ygap_t. (1)
- Potential output random walk with drift g/4: y*_t = y*_{t−1} + g/4 + ξ^{y*}_t. (2)
- Hybrid IS curve for output gap (parameters β_y, β_R, β_z, β_w): equation (3).
- Ex-ante real interest rate: RR_t = R_t − E_t(π_{t+1}); natural rate RR^*_t depends on shocks to potential output and equilibrium real interest rate. (4)
- Inflation dynamics: π_t = α_lead E_t(π_{t+1}) + (1−α_lead)π_{t−1} + α_y ygap_{t−1} + α_z (z_t − z_{t−1}) + ξ^π_t. (4)
- UIP-type open-economy relation: RR_t − RR^w_t = 4(z^e_{t+1} − z_t) + ξ^{RR}_t. (5)
- Expectation formation for expected real exchange rate: z^e_{t+1} = θ E_t(z_{t+1}) + (1−θ) z_{t−1}. (6)
- Money demand in levels for real money balances m_t: m_t = χ_y y_t − χ_R R_t + u_t. (7)
- Money demand shock as random walk with drift: u_t = μ + u_{t−1} + ξ^u_t. (8)
- Observed nominal interest rate noisy: R^N_t = R_t + ξ^{R^N}_t. (9)

### Scope and caveats
- The model is simpler than fully micro-founded DSGE models and abstracts from jointly solving for optimal monetary policy; focus is on the optimal use of money as an indicator variable under informational frictions.
- Authors note the simplicity of the information structure and data scarcity argue against taking numerical results too literally for policy without further corroboration.
- The remainder of the original paper (not summarized here) is structured: Section 2 presents the model; Section 3 estimates the model; Section 4 derives the optimal use of money market information; Section 5 discusses other rationales for the relevance of money in LICs; Section 6 concludes.

*IMF Working Paper content unit: _wp10134 - References...........................................................................................29*

### 1.    Money targets

### 1.    Money targets

### Flexible-price money targets and the money gap
- Flexible-price real money balances (star variables) are defined as:
  - ∆m∗t = χy ∆y∗t − χR ∆RR∗t + ∆ut
- Nominal money growth at the flex-price equilibrium:
  - ∆M∗t = ∆m∗t + π∗
- Authorities should accommodate exogenous changes in money demand (ut); identifying and forecasting velocity changes is a key challenge.
- Adjustment for the economy not being at flex-price equilibrium uses the lagged money gap:
  - mgapt = mt − m∗t
  - ∆MTt = ∆M∗t − mgapt−1

### Active money targeting: responding to inflation and the output gap
- Money growth targets should respond to expected inflation and the output gap. Targeting next period’s inflation yields:
  - ∆MTt = ∆M∗t − mgapt−1 + (πt − π∗) − δ(E t(πt+1) − π∗) − γ ygap t
- Replacing the money-target rule into the money demand/interest relation produces a forward-looking interest-rate target:
  - RTt = RR∗t + π∗ + φπ (E t(πt+1) − π∗) + φygap ygap t
  - where φπ = δ / χR and φygap = (γ + χy) / χR
- Implications:
  - A money-targeting rule consistent with these principles is equivalent to a forward-looking Taylor rule under complete information.
  - The interest elasticity of money demand (χR) is crucial: a low χR increases the required interest-rate response to inflation; thus a mild response of money growth to expected inflation can still satisfy the Taylor principle.
  - For the output gap, the implied interest-rate response compounds with the money-target response; if χR is sufficiently low and χy moderately high, authorities might set γ < 0 (i.e., accommodate a positive output gap) when setting money targets.

### Monetary policy under incomplete information and the operational tradeoff
- Targets are announced based on information available at t−1. The announced money growth target is:
  - ∆MTt|t−1 = ECB t−1(∆MTt)
- The corresponding interest-rate target consistent with the money target is:
  - RTt|t−1 = ECB t−1(RR∗t) + π∗ + φπ (ECB t−1(πt+1) − π∗) + φygap ECB t−1(ygapt)
- Assuming the central bank’s information at t−1 is complete:
  - ECB t−1(.) = Et−1(.)
  - Et−1(RNt) = RTt|t−1
- When time t arrives, the central bank observes RNt and ∆Mt and must choose whether to adhere to the interest-rate target or to make money growth adhere to its pre-announced target. The central bank’s commitment is represented by:
  - λ (RNt − RTt|t−1) − (1 − λ) (∆Mt − ∆MTt|t−1) = 0
- Rearrangements highlight two interpretations:
  - Interest-rate revision after observing money market:
    - RNt = RTt|t−1 + (1 − λ)/λ (∆Mt − ∆MTt|t−1)
    - Suggests λ should be lower when money growth provides valuable forecast information.
  - Adjusting money growth after observing interest rates:
    - ∆Mt = ∆MTt|t−1 + λ/(1 − λ) (RNt − RTt|t−1)
    - Useful when money is the operational target.
- λ captures the weight in between policy meetings (intra-period adjustments) and can be treated as a free parameter to be estimated; optimal λ is discussed in section 4 (not included here).

### Estimation strategy and selected results
- Estimation approach:
  - Bayesian estimation of a state-space representation using the Kalman filter and Monte Carlo Markov Chain (random walk Metropolis-Hastings) methods; Dynare is used.
  - Measurement equation: Ot = Λ Xt with Ot = [πt, yt, zt, Rt, ∆Mt, ygapwt, RRwt]′.
  - Dogmatic priors imposed for several parameters (steady-state values π∗, g∗, rr∗ and persistence ρygapw, ρRRw). θ is calibrated at 0.05.
  - Vector of estimated parameters:
    - Υ = [βlead, βR, βz, βw, αlead, αy, αz, v∗, χy, χR, φπ, φygap, λ]′
- Priors and calibration highlights:
  - Prior mean for βR set at 0.07 to reflect possible weakness of the interest-rate channel in low-income countries.
  - Prior means for χy and χR informed by empirical money-demand studies for Ghana and Uganda.
  - Priors for policy parameters: φπ = 2 and φygap = 0.5.
  - Prior mean of λ = 0.5 for all countries with relatively large variance.
- Selected posterior results (based on Tables presented for Ghana and Uganda, posterior maximization):
  - Ghana (Table 2) — Posterior means and intervals include:
    - λ posterior mean: 0.9285; 90% Confidence Interval: 0.8444 − 0.9994
    - φπ posterior mean: 1.7637; 90% Confidence Interval: 1.3131 − 2.1896
    - φygap posterior mean: 0.4137; 90% Confidence Interval: 0.1237 − 0.6959
    - χR posterior mean: 0.3773; 90% Confidence Interval: 0.2250 − 0.5155
    - βR posterior mean: 0.1954; 90% Confidence Interval: 0.0965 − 0.3008
    - βy posterior mean: 0.5335; 90% Confidence Interval: 0.3455 − 0.7421
  - Uganda (Table 3) — Posterior means and intervals include:
    - λ posterior mean: 0.3377; 90% Confidence Interval: 0.1785 − 0.4824
    - φπ posterior mean: 2.0030; 90% Confidence Interval: 1.6507 − 2.3212
    - φygap posterior mean: 0.5227; 90% Confidence Interval: 0.2032 − 0.8293
    - χR posterior mean: 0.5817; 90% Confidence Interval: 0.4433 − 0.7196
    - βR posterior mean: 0.3301; 90% Confidence Interval: 0.1330 − 0.5114
    - βy posterior mean: 0.7470; 90% Confidence Interval: 0.5836 − 0.9173
- Key estimation conclusions highlighted in the text:
  - Contrary to priors, the interest elasticity in the IS equation (βR) is relatively high across all countries, suggesting aggregate demand sensitivity to the (shadow) interest rate.
  - Forward-looking behavior matters: βy and αlead (parameters describing how expectations of the future affect the output gap and inflation) are greater than 0.5 in all three countries referenced.

*Source: IMF working paper chapter "1.    Money targets" (source PDF content provided).*

### 4. Prior and posterior distributions for parameters and volatilities: Tanzania

### 4. Prior and posterior distributions for parameters and volatilities: Tanzania

### Prior and posterior parameter estimates (selected)
- βy (Beta): Prior mean 0.3; Prior Std. dev. 0.15. Posterior mean 0.6836; 90% Confidence Interval 0.5275–0.8509
- βR (Gamma): Prior mean 0.07; Prior Std. dev. 0.1. Posterior mean 0.1624; 90% Confidence Interval 0.0766–0.2397
- βz (Gamma): Prior mean 0.1; Prior Std. dev. 0.05. Posterior mean 0.0696; 90% Confidence Interval 0.0158–0.1218
- βw (Gamma): Prior mean 0.25; Prior Std. dev. 0.2. Posterior mean 0.4683; 90% Confidence Interval 0.0194–0.9197
- αlead (Beta): Prior mean 0.4; Prior Std. dev. 0.05. Posterior mean 0.6212; 90% Confidence Interval 0.5366–0.7141
- αy (Gamma): Prior mean 0.25; Prior Std. dev. 0.2. Posterior mean 0.9535; 90% Confidence Interval 0.0016–1.5946
- αz (Gamma): Prior mean 0.2; Prior Std. dev. 0.1. Posterior mean 0.1637; 90% Confidence Interval 0.0338–0.2964
- χy (Gamma): Prior mean 0.7; Prior Std. dev. 0.2. Posterior mean 0.9226; 90% Confidence Interval 0.4492–1.4080
- χR (Gamma): Prior mean 0.4; Prior Std. dev. 0.1. Posterior mean 0.5672; 90% Confidence Interval 0.3680–0.7771
- υ* (Gamma): Prior mean 0.45; Prior Std. dev. 0.2. Posterior mean 0.5281; 90% Confidence Interval 0.1722–0.8689
- φπ (Gamma): Prior mean 2; Prior Std. dev. 0.2. Posterior mean 1.9966; 90% Confidence Interval 1.5610–2.3997
- φygap (Gamma): Prior mean 0.5; Prior Std. dev. 0.2. Posterior mean 0.4671; 90% Confidence Interval 0.1855–0.7401
- λ (Beta): Prior mean 0.5; Prior Std. dev. 0.2. Posterior mean 0.6642; 90% Confidence Interval 0.4595–0.9783

### Prior and posterior volatilities (selected)
- σπ (Inverted Gamma): Prior parameters 2, Inf. Posterior mean 6.6332; 90% Confidence Interval 5.0164–8.1436
- σy* (Inverted Gamma): Prior parameters 0.5, Inf. Posterior mean 2.4231; 90% Confidence Interval 1.7926–3.0961
- σRR* (Inverted Gamma): Prior parameters 0.5, Inf. Posterior mean 1.2086; 90% Confidence Interval 0.1247–3.3236
- συ (Inverted Gamma): Prior parameters 4, Inf. Posterior mean 6.5969; 90% Confidence Interval 4.5062–9.0519
- σR* (Inverted Gamma): Prior parameters 4, Inf. Posterior mean 6.8166; 90% Confidence Interval 1.1170–10.2908
- σRR (Inverted Gamma): Prior parameters 5, Inf. Posterior mean 14.3811; 90% Confidence Interval 11.0792–17.4901
- σygap w (Inverted Gamma): Prior parameters 0.25, Inf. Posterior mean 0.3777; 90% Confidence Interval 0.2931–0.4639
- σRR w (Inverted Gamma): Prior parameters 0.3, Inf. Posterior mean 0.6930; 90% Confidence Interval 0.5347–0.8504

### Empirical interpretations and comparisons across Ghana, Tanzania, Uganda
- Posterior assignment for λ:
  - The posterior distribution assigns a much larger probability density to the higher values of λ for Ghana (0.99).
  - Uganda: estimated λ much lower (0.34), consistent with de jure money-targeting role.
  - Tanzania: estimated λ somewhat in the middle (0.66), suggesting more flexible adherence to the money-targeting framework than Uganda.
- Volatility comparisons:
  - The estimated standard deviation of money demand shocks (συ) is larger in Tanzania and smaller in Uganda.
  - The estimated volatility of interest rate noise is larger in Uganda and smaller in Ghana.
- Interpretation: money demand shocks (interest rate noise) make it more (less) costly to adhere to money targets; assessing how shocks and parameter values affect optimal target adherence requires the formal analysis that follows.

### Deriving optimal λ (method overview)
- Policy representation uses money growth as the left-hand-side policy instrument: ∆Mt = E(∆M*Tt | It−1, RNt)
- Signal-extraction framing: deviations ∆Mt − ∆MTt|t−1 are linear in revisions to forecasts of target variables and shocks (see equation (22)).
- Use of complete-information solution Xt = ΥX Xt−1 + Υε εt to obtain reduced-form for non-policy block:
  - Yt = A Yt−1 + B ∆Mt + B2 ∆Mt−1 + C C εt (equation (24))
  - First differences yield: ˜Yt = B ∆˜Mt + C εt (equation (25))
- Observed money-market variable: ˜RNt = ΓR ˜Yt + εR N t (equation (26))
- Construct observable combination ˜φt = ˜RNt − ΓR B ∆˜Mt = Θ[˜RNt ∆˜Mt]′ = H εt + εR N t (equation (27))
- Kalman filter:
  - Kalman gain K = Q H (H′ Q H + Ω)−1
  - Shock estimates: ˆεt = K ˜φt = K Θ[˜RNt ∆˜Mt]′ (equation (28))
- Solve for policy deviation and relation between ∆˜Mt and ˜RNt:
  - ∆˜Mt = Φ ˜RNt with Φ = (1 − P∆M K Θ(1,2))−1 P∆M K Θ(1,1)
  - Optimal λ = Φ / (1 + Φ)

### Optimal λ: Results for Ghana, Tanzania and Uganda (Table 5)
- Ghana
  - Estimated Lambda 0.9285
  - Optimal Lambda 0.3634
- Tanzania
  - Estimated Lambda 0.6642
  - Optimal Lambda 0.4421
- Uganda
  - Estimated Lambda 0.3377
  - Optimal Lambda 0.3246
- Interpretation:
  - For Uganda and Tanzania, the optimal λ is close to the empirical estimate, suggesting near-optimal extraction of money-market information.
  - Tanzania could benefit from higher adherence to money targets (optimal λ > empirical? note: empirical 0.6642 vs optimal 0.4421 implies empirical adherence is already above optimal; the text states Tanzania would benefit from higher adherence — the supplied numbers show empirical λ 0.6642 and optimal 0.4421; the narrative in the source indicates Tanzania would benefit from higher adherence to their money targets).
  - For Ghana, optimal λ is much lower than the empirical estimate, suggesting authorities would benefit from paying closer attention to monetary aggregates (results for Ghana are noted as not very robust to changing priors).

### Sensitivity analysis (focus on Uganda; qualitative results similar for Ghana and Tanzania)
- Effects on optimal λ:
  - Optimal lambda increases as money-demand volatility increases.
  - Lambda decreases as the interest elasticity of money demand increases (cost of policy mistakes from adhering too strictly to money targets is smaller).
  - Optimal lambda declines as the volatility of cost-push shocks increases.
  - Optimal lambda declines as the volatility of shocks to the natural rate of interest increases (equivalent to IS-curve shocks in Poole framework).
  - Noisiness of short-term interest rates reduces optimal lambda: as observed short-term interest rates become a noisier indicator, authorities pay less attention to interest-rate movements and adhere more to previously-announced money targets.
- Quantitative bound noted:
  - Even when noise volatility is set to zero, Uganda’s optimal lambda does not increase beyond 0.7, implying it remains preferable to let interest rates fluctuate somewhat relative to their target.

*Source: _wp10134 - 4. Prior and posterior distributions for parameters and volatilities: Tanzania*

### 2.    Welfare implications of changingλ

### 2.    Welfare implications of changing λ

### Assessment approach
- The analysis evaluates implications of various λ by assessing inflation and output gap volatility as implicit measures of welfare, given policymakers follow a simple Taylor Rule rather than a formally specified loss function.
- Two alternative evaluation approaches noted (not used here): comparing mean square errors from central bank forecasts; assessing the policymaker loss function directly.

### Main quantitative results (Table 6)
- Ghana
  - λ = 0: Inflation 12.08; Output Gap 3.11
  - Optimal lambda (λ = 0.3634): Inflation 12.80; Output Gap 2.09
  - Estimated lambda (λ = 0.9285): Inflation 14.49; Output Gap 2.51
  - λ = 1: Inflation 14.71; Output Gap 2.65
- Uganda
  - λ = 0: Inflation 10.33; Output Gap 1.77
  - Optimal lambda (λ = 0.3246): Inflation 10.95; Output Gap 1.60
  - Estimated lambda (λ = 0.3377): Inflation 10.99; Output Gap 1.62
  - λ = 1: Inflation 14.20; Output Gap 4.13
- Tanzania
  - λ = 0: Inflation 12.02; Output Gap 2.12
  - Optimal lambda (λ = 0.4421): Inflation 12.38; Output Gap 1.59
  - Estimated lambda (λ = 0.6642): Inflation 13.08; Output Gap 1.74
  - λ = 1: Inflation 14.60; Output Gap 2.34

### Key findings and interpretation
- Strict adherence to interest rate targets (high λ) generates highly volatile movements in both inflation and the output gap across all three countries.
- Strict adherence to money targets (λ = 0) generates the smallest movements in inflation but at the cost of larger movements in the output gap.
- Choosing the optimal λ:
  - Does not produce the lowest inflation volatility in these cases.
  - Does produce the lowest output gap volatility.
- The pattern is rationalized by the presence of relatively volatile cost-push shocks:
  - Under strict money targeting (λ = 0), such shocks induce automatic increases in interest rates, tightening policy, reducing inflation volatility, but contracting output more (larger output gap movements).
- Estimated coefficients in the Taylor rule imply policymakers prefer a smaller interest rate response than strict adherence to money-market information would imply; when λ is chosen optimally, the policy response tends to be more muted.
- Strict adherence to interest rate targets is similarly undesirable.
- Excluding extreme cases, welfare gains (in terms of inflation and output stability) from varying λ are relatively small.

### Modeling caveats and limitations
- The model assumes uncertainty is resolved at the end of period t, so information incompleteness has no long-lasting effects; this moderates potential welfare gains from changing λ.
- The money targets used are model-derived and may differ from actual authorities’ targets; actual target-setting methods could justify larger ex-post deviations than the model suggests.
- The model assumes contemporaneous information comes only from the money market:
  - This may understate the role of other contemporaneous observables (nominal exchange rate, preliminary/noisy measures of inflation and output) which could weaken the relevance of monetary aggregates.
- Observed interest rates are modeled as a noisy measure of the relevant interest rate for private sector decisions; this channel is quantitatively important for optimal adherence to money targets.
  - Qualitative results persist even if interest rates are perfectly observed.
- The treatment of the exchange rate is a noted weakness; many LICs that target money also conduct managed-floating exchange rate regimes, and the exchange rate is observable at high frequency.

### Policy-relevant implications
- Rigid adherence to money targets for credibility reasons can be costly, implying excessive output and inflation volatility; once credibility is achieved, more flexible policy execution is observed in practice.
- Money may matter in LICs for multiple reasons:
  - As a visible sign of policy adherence in weak-institution settings (a “tripwire”).
  - Because financial market characteristics (thin markets, dependence on bank finance, poor contract enforcement) may weaken interest-rate channels and strengthen links between monetary aggregates and activity/inflation.
- The noisy-observed-interest-rate assumption is a first-stage representation of LIC financial sector characteristics; more elaborate modeling of financial markets is needed.

### Research agenda and next steps
- A more complete treatment of information incompleteness is left for future work.
- Incorporate exchange rate dynamics explicitly given its high-frequency observability and role in LICs that target money.
- Further LIC-specific modifications of core model relationships (financial sector, exchange rate, institutional credibility) should be developed carefully.
- The paper is positioned as part of a broader research program to extend modern analytic frameworks to LIC macroeconomic policy challenges.

*Source: IMF working paper section "2.    Welfare implications of changing λ"*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10134.pdf_