## _wp09214

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---

### Model purpose and scope
- GPM (Global Projection Model) is a family of small quarterly macroeconomic models used for forecasting and monetary policy simulations, focusing on inflation, output and/or the unemployment rate, a short-term interest rate, and the exchange rate.
- Typical specification: 4 or 5 behavioral equations determining:
  - the inflation rate (an inflation expectations augmented Phillips curve);
  - the exchange rate; and
  - the interest rate (a monetary policy rule).
- Practical advantages: ease of updating forecasts and examining alternative assumptions; mainstream economic logic facilitating policy analysis and communication.

### Key nonlinearities incorporated
- zero bound to interest rates;
- amplified bank lending tightening effect beyond a specified threshold;
- asymmetric output gap effect on inflation—excess demand raises inflation by more than excess supply reduces it;
- adjustment mechanism to avoid implosive disinflation in the event of very large negative output gaps.
- These nonlinearities produce asymmetric model-derived confidence bands, especially far from long-run equilibrium.

### Estimation and simulation methods
- Bayesian estimation methods are used, blending theoretical priors and the data to estimate models with plausible dynamics and to estimate unobservables (e.g., potential output).
- Confidence intervals are calculated by simulating the model with multiple drawings from estimated distributions of disturbance terms.
- Computationally efficient Latin hypercube sampling is employed to draw from the large sample space.

### U.S.-specific features
- The U.S. model includes an empirically constructed bank lending tightening (BLT) index from the Federal Reserve Board’s Senior Loan Officer Survey; the index subtracts the percentage of “eased” responses from the percentage of “tightened.”
  - A BLT in excess of 50 percent means an exceptional degree of tightening.
  - In 2008 the index rose above 80 percent.
- For the United States, the model imposes a nonlinear BLT effect: the marginal impact on the output gap doubles when the index is above a 50 percent threshold, to capture financial accelerator effects when lenders cut off credit and borrowers lack alternatives.
- The model respects the zero lower bound to the interest rate.

### Variable representation and equilibrium concepts
- Variables are expressed as deviations from equilibrium (gap format); inflation is measured as the annualized quarterly change in the CPI (400 times the first difference of the log of the CPI).
- The model specifies underlying equilibrium or steady-state values and stochastic processes for:
  - the log of the real equilibrium price of oil in real U.S. dollar terms (,US t RPOIL);
  - the growth rate of real oil prices in U.S. dollar terms (,RPOIL g US t) with disturbances ,RPOIL g US t ε and adjustment coefficient 1−g US ρ;
  - the gap between the current real oil price and its equilibrium (rpoil us t) following a damped process with parameter rpoil ρ;
  - potential output (Y it) following a stochastic trend with growth rate Y it g, steady-state growth rate ss Y i g, and adjustment coefficient i τ;
  - NAIRU (U it) with level and growth-rate shocks and parameters UU g ε and U Ug α;
  - equilibrium real interest rate (it R) as a mean-reverting process around ss i R with coefficient 1−R it ρ;
  - equilibrium real exchange rate (it Z) following a random walk (Z it = Z it−1 + ε it).
- Output gap dynamics include leads and lags, the real interest rate gap, BLT (U.S. only), trading-partner output gaps, the real exchange rate gap, and a disturbance—plus a state-dependent amplification of negative demand shocks via the product yneg US t ε ⋅ yneg US t y.

### Inflation and monetary policy treatment
- Inflation is modeled via an augmented Phillips curve where inflation depends on:
  - inflation expectations (weighted average of past and model-consistent future rates);
  - the lagged output gap;
  - the change in the real exchange rate;
  - current and lagged increases in the real price of oil;
  - a disturbance term.
- The Phillips curve contains a mechanism that sets the marginal negative effect on inflation to zero for output gaps wider than -3 percent of potential output to avoid implosive disinflation in large contractions.
- Monetary policy is represented by a Taylor-type, forward-looking policy rule for the short-term interest rate (Federal Funds rate for the United States), including smoothing (own lag) and responding to expected inflation deviations and output gaps.
- The model assumes a Fed target of 2.5 percent for current purposes.
- The long-run FOMC forecasts for the consumption deflator (PCE) in February 2009 were 1.7-2.0 percent; in terms of the CPI, the FOMC range would imply a long-run target of about 2 ¼ to 2 ½ percent, given historical differences between PCE and CPI.

### Cross-equation correlations in disturbances
- Three cross-equation correlations are included:
  - potential output and inflation disturbances: negative covariance (supply shocks; e.g., a positive shock to potential output reduces current inflation);
  - potential output growth and output gap disturbances: positive covariance (anticipated income effects producing short-run excess demand);
  - potential output growth and BLT disturbances in the United States: negative covariance (higher potential growth eases bank credit; anticipatory spending can produce a positive output gap).

### Sampling challenge and Latin hypercube solution
- Routine Monte-Carlo methods are infeasible due to the number of interacting variables in GPM and the dimensionality of the underlying joint distribution. The dimensionality is represented as MN⋅, where N is the number of periods, and M is the minimum of state variables and exogenous shocks. In our exercise MN⋅ is equal to 7203620=×.
- To reduce required simulations and ensure representative coverage of the random terms, Latin hypercube sampling is employed (McKay, Beckman and Conover, 1979).
- The LHSA algorithm (Fang, Li and Sudjianto, 2006) is used to generate Latin Hypercube samples that evenly spread a given number of points in a D-dimensional unit hypercube.
- For the experiment, we set 4800=K.

### LHSA algorithm mechanics (as described)
- Let D = MND⋅ be the dimension, and K be the number of samples.
- The hypercube is conceptually split into D K D-dimensional hypercubes by splitting each coordinate into K sections.
- Latin hypercube sampling selects K cubes such that in each coordinate hyperplane there is one and only one selection.
- LHSA achieves this by generating D random independent permutations D ππ,,1  of integers K,,1. Each selection is defined as )](,),(,[ 1 kk D ππ  for Kk,,1=.
- The Latin Hypercube samples are obtained as centers of the selected cubes perturbed by uniform D-dimensional distributions not exceeding the boundaries of the selected cubes.
- When samples are scaled by the length of edges of the hypercube, we obtain K draws from a D-dimensional uniform distribution.
- Illustration: for 2D= and 6=K the square is divided into 66× squares, two random permutations of integers 61 select 6 squares (one per column and row), and each selected center is perturbed by a uniform distribution to obtain 6 draws )](2),(1[iXiX for .61=i

### Converting LHSA points into model shock draws
- LHSA produces K points in the MN⋅ = MND⋅ dimensional unit hypercube.
- To obtain K series of N independent draws from ),0(VN (V is the variance-covariance matrix of the M exogenous shocks):
  - For each of the K points with MN⋅ coordinates, split the coordinates into N sections containing M numbers to obtain K series of N independent draws from M-dimensional uniform distributions.
  - Convert each uniform M-dimensional draw to represent a draw from ),0( M IN (M-dimensional Gaussian distribution).
  - Multiply each draw by a matrix A, where VAA=⋅ * , to obtain Gaussian shocks with the given variance-covariance matrix V.

### Decomposition of confidence intervals (counterfactual experiments)
- Confidence intervals can be decomposed by stepwise elimination of random components and recalculation of confidence intervals. Eliminating components generally results in narrower intervals (but not always, due to negative covariances).
- The sequence followed in the experiment (cumulatively removed) was:
  - first, foreign shocks other than the oil price
  - second, domestic supply shocks
  - third, oil-price shocks — leaving U.S. demand shocks as the only contributor to the confidence interval

### Key quantified model outcomes and contributions to uncertainty
- Oil price fan:
  - Baseline oil-price path assumed in March 2009 baseline rises from about $40 per barrel to almost $60 over 20 quarters.
  - Calculated confidence intervals for the oil price: $30 to $70 at the 4-quarter horizon, $30 to $90 at 8-quarters, and $25 to $115 at the 12-quarter horizon.
- U.S. CPI inflation:
  - Baseline inflation falls from an initial rate near zero to a trough of -3 percent in quarter 3; inflation becomes positive again at about quarter 8; after 20 quarters it remains short of 2.5 percent.
  - At 4 quarters, the All Shocks 95 percent confidence interval for the annual inflation rate is about -2.0 to +1.0 percent.
  - At 8 quarters, the interval is just inside the ± 2 percent range.
  - Decomposition effects:
    - Eliminating foreign (non-oil) shocks does not materially change the U.S. inflation confidence band.
    - Eliminating domestic supply shocks slices about 1 percent from each side of the All Shocks band at horizons of 4 quarters and beyond.
    - Eliminating oil-price shocks further narrows the band; domestic supply shocks and oil-price shocks account for almost all risk to the inflation forecast, leaving U.S. demand shocks with only a small contribution.
- U.S. interest rate:
  - Baseline policy-determined rate stays near zero for 8 quarters and rises slowly thereafter, remaining below equilibrium at 20 quarters.
  - All Shocks 95 percent confidence interval at 20 quarters goes from near zero to almost 7 percent.
  - Removal of foreign non-oil shocks significantly narrows the confidence interval for the interest rate—by at least ½ percentage point after 4 quarters.
  - Eliminating U.S. supply shocks has little effect on interest-rate variability; removing oil-price shocks produces major narrowing.
  - Domestic demand shocks account for a substantial part of the interest-rate confidence interval at all horizons.
- U.S. output gap:
  - No Shocks baseline: a 5-quarter decline with a trough near -6 percent; baseline output not back to potential until the very end of the 20-quarter horizon.
  - Asymmetry: at 8 quarters, the 95 percent confidence interval extends from about 4 percent below to 2 percent above the 5 percent midpoint (greater downside risk).
  - Eliminating foreign shocks has no material effect on output-gap uncertainty; removing U.S. supply shocks has virtually no effect.
  - Removal of oil price shocks halves the interval. Oil price and domestic demand shocks together account for almost the entire confidence interval around the forecast output gap.
- Bank lending tightening (BLT):
  - No Shocks forecast reverts from an extremely high starting point to near zero within 5 quarters; zero is high under current circumstances, and return to a normal level would require extended easing (negative BLT values).
  - Uncertainties are considerable: at the 8-quarter horizon the width of the All Shocks band is about 80 percentage points, and the interval is asymmetric with much more risk of unexpected tightening than easing.
  - Decomposition: oil price shocks dominate the confidence interval for BLT; domestic demand shocks (including BLT shocks) have a surprisingly small independent influence.

### Methodological and substantive conclusions
- The confidence-interval construction includes innovations: Bayesian estimation, nonlinearities (notably the zero bound on interest rates), and Latin hypercube sampling for efficient multi-variable sampling using lengthy time series.
- Nonlinearities generate visibly asymmetric forecast confidence intervals for key financial variables (short-term interest rate and BLT). The lower bound of the 95 percent confidence interval for the policy interest rate touches the zero axis almost throughout the 20-quarter forecast, and the BLT interval indicates more risk of unusual tightening than easing.
- The results imply a non-negligible risk that the Federal Reserve might be constrained by the zero bound over the medium term given severe credit market conditions.
- The output-gap confidence interval shows more downside than upside risk at the trough; the inflation interval is roughly symmetric, partly due to the forward-looking inflation expectations process anchored by the policy target.
- Decomposition suggests oil price shocks and U.S. domestic demand shocks have a stronger influence on confidence-interval widths than non-oil foreign shocks or U.S. supply shocks.

*Source: 2.1 Overview and 3.1 Construction (excerpt from provided IMF working paper section).*

### 2.1  Overview ................................................................................................4

### 2.1  Overview

### Model purpose and scope
- GPM (Global Projection Model) is a family of small quarterly macroeconomic models used for forecasting and monetary policy simulations, focusing on inflation, output and/or the unemployment rate, a short-term interest rate, and the exchange rate.
- Typical specification: 4 or 5 behavioral equations determining:
  - the inflation rate (an inflation expectations augmented Phillips curve);
  - the exchange rate; and
  - the interest rate (a monetary policy rule).
- Practical advantages: ease of updating forecasts and examining alternative assumptions; mainstream economic logic facilitating policy analysis and communication.

### Key nonlinearities incorporated
- zero bound to interest rates;
- amplified bank lending tightening effect beyond a specified threshold;
- asymmetric output gap effect on inflation—excess demand raises inflation by more than excess supply reduces it;
- adjustment mechanism to avoid implosive disinflation in the event of very large negative output gaps.
- These nonlinearities produce asymmetric model-derived confidence bands, especially far from long-run equilibrium.

### Estimation and simulation methods
- Bayesian estimation methods are used, blending theoretical priors and the data to estimate models with plausible dynamics and to estimate unobservables (e.g., potential output).
- Confidence intervals are calculated by simulating the model with multiple drawings from estimated distributions of disturbance terms.
- Computationally efficient Latin hypercube sampling is employed to draw from the large sample space.

### U.S.-specific features
- The U.S. model includes an empirically constructed bank lending tightening (BLT) index from the Federal Reserve Board’s Senior Loan Officer Survey; the index subtracts the percentage of “eased” responses from the percentage of “tightened.”
  - A BLT in excess of 50 percent means an exceptional degree of tightening.
  - In 2008 the index rose above 80 percent.
- For the United States, the model imposes a nonlinear BLT effect: the marginal impact on the output gap doubles when the index is above a 50 percent threshold, to capture financial accelerator effects when lenders cut off credit and borrowers lack alternatives.
- The model respects the zero lower bound to the interest rate.

### Variable representation and equilibrium concepts
- Variables are expressed as deviations from equilibrium (gap format); inflation is measured as the annualized quarterly change in the CPI (400 times the first difference of the log of the CPI).
- The model specifies underlying equilibrium or steady-state values and stochastic processes for:
  - the log of the real equilibrium price of oil in real U.S. dollar terms (,US t RPOIL);
  - the growth rate of real oil prices in U.S. dollar terms (,RPOIL g US t) with disturbances ,RPOIL g US t ε and adjustment coefficient 1−g US ρ;
  - the gap between the current real oil price and its equilibrium (rpoil us t) following a damped process with parameter rpoil ρ;
  - potential output (Y it) following a stochastic trend with growth rate Y it g, steady-state growth rate ss Y i g, and adjustment coefficient i τ;
  - NAIRU (U it) with level and growth-rate shocks and parameters UU g ε and U Ug α;
  - equilibrium real interest rate (it R) as a mean-reverting process around ss i R with coefficient 1−R it ρ;
  - equilibrium real exchange rate (it Z) following a random walk (Z it = Z it−1 + ε it).
- Output gap dynamics include leads and lags, the real interest rate gap, BLT (U.S. only), trading-partner output gaps, the real exchange rate gap, and a disturbance—plus a state-dependent amplification of negative demand shocks via the product yneg US t ε ⋅ yneg US t y.

### Inflation and monetary policy treatment
- Inflation is modeled via an augmented Phillips curve where inflation depends on:
  - inflation expectations (weighted average of past and model-consistent future rates);
  - the lagged output gap;
  - the change in the real exchange rate;
  - current and lagged increases in the real price of oil;
  - a disturbance term.
- The Phillips curve contains a mechanism that sets the marginal negative effect on inflation to zero for output gaps wider than -3 percent of potential output to avoid implosive disinflation in large contractions.
- Monetary policy is represented by a Taylor-type, forward-looking policy rule for the short-term interest rate (Federal Funds rate for the United States), including smoothing (own lag) and responding to expected inflation deviations and output gaps.
- The model assumes a Fed target of 2.5 percent for current purposes.
- The long-run FOMC forecasts for the consumption deflator (PCE) in February 2009 were 1.7-2.0 percent; in terms of the CPI, the FOMC range would imply a long-run target of about 2 ¼ to 2 ½ percent, given historical differences between PCE and CPI.

### Cross-equation correlations in disturbances
- Three cross-equation correlations are included:
  - potential output and inflation disturbances: negative covariance (supply shocks; e.g., a positive shock to potential output reduces current inflation);
  - potential output growth and output gap disturbances: positive covariance (anticipated income effects producing short-run excess demand);
  - potential output growth and BLT disturbances in the United States: negative covariance (higher potential growth eases bank credit; anticipatory spending can produce a positive output gap).

*Source: 2.1 Overview (excerpt from provided IMF working paper section).*

### 3.1 Construction

### 3.1 Construction

### Sampling challenge and solution
- Routine Monte-Carlo methods are infeasible due to the number of interacting variables in GPM and the dimensionality of the underlying joint distribution. The dimensionality is represented as MN⋅, where N is the number of periods, and M is the minimum of state variables and exogenous shocks. In our exercise MN⋅ is equal to 7203620=×.
- To reduce required simulations and ensure representative coverage of the random terms, Latin hypercube sampling is employed (McKay, Beckman and Conover, 1979).
- The LHSA algorithm (Fang, Li and Sudjianto, 2006) is used to generate Latin Hypercube samples that evenly spread a given number of points in a D-dimensional unit hypercube.

### LHSA algorithm mechanics (as described)
- Let D = MND⋅ be the dimension, and K be the number of samples. For the experiment, we set 4800=K.
- The hypercube is conceptually split into D K D-dimensional hypercubes by splitting each coordinate into K sections.
- Latin hypercube sampling selects K cubes such that in each coordinate hyperplane there is one and only one selection.
- LHSA achieves this by generating D random independent permutations D ππ,,1  of integers K,,1. Each selection is defined as )](,),(,[ 1 kk D ππ  for Kk,,1=.
- The Latin Hypercube samples are obtained as centers of the selected cubes perturbed by uniform D-dimensional distributions not exceeding the boundaries of the selected cubes.
- When samples are scaled by the length of edges of the hypercube, we obtain K draws from a D-dimensional uniform distribution.
- Illustration: for 2D= and 6=K the square is divided into 66× squares, two random permutations of integers 61 select 6 squares (one per column and row), and each selected center is perturbed by a uniform distribution to obtain 6 draws )](2),(1[iXiX for .61=i

### Converting LHSA points into model shock draws
- LHSA produces K points in the MN⋅ = MND⋅ dimensional unit hypercube.
- To obtain K series of N independent draws from ),0(VN (V is the variance-covariance matrix of the M exogenous shocks):
  - For each of the K points with MN⋅ coordinates, split the coordinates into N sections containing M numbers to obtain K series of N independent draws from M-dimensional uniform distributions.
  - Convert each uniform M-dimensional draw to represent a draw from ),0( M IN (M-dimensional Gaussian distribution).
  - Multiply each draw by a matrix A, where VAA=⋅ * , to obtain Gaussian shocks with the given variance-covariance matrix V.

### Decomposition of confidence intervals (counterfactual experiments)
- Confidence intervals can be decomposed by stepwise elimination of random components and recalculation of confidence intervals. Eliminating components generally results in narrower intervals (but not always, due to negative covariances).
- The sequence followed in the experiment (cumulatively removed) was:
  - first, foreign shocks other than the oil price
  - second, domestic supply shocks
  - third, oil-price shocks — leaving U.S. demand shocks as the only contributor to the confidence interval

### Key quantified model outcomes and contributions to uncertainty
- Oil price fan: baseline oil-price path assumed in March 2009 baseline rises from about $40 per barrel to almost $60 over 20 quarters. Calculated confidence intervals for the oil price: $30 to $70 at the 4-quarter horizon, $30 to $90 at 8-quarters, and $25 to $115 at the 12-quarter horizon.
- U.S. CPI inflation:
  - Baseline inflation falls from an initial rate near zero to a trough of -3 percent in quarter 3; inflation becomes positive again at about quarter 8; after 20 quarters it remains short of 2.5 percent.
  - At 4 quarters, the All Shocks 95 percent confidence interval for the annual inflation rate is about -2.0 to +1.0 percent.
  - At 8 quarters, the interval is just inside the ± 2 percent range.
  - Decomposition effects:
    - Eliminating foreign (non-oil) shocks does not materially change the U.S. inflation confidence band.
    - Eliminating domestic supply shocks slices about 1 percent from each side of the All Shocks band at horizons of 4 quarters and beyond.
    - Eliminating oil-price shocks further narrows the band; domestic supply shocks and oil-price shocks account for almost all risk to the inflation forecast, leaving U.S. demand shocks with only a small contribution.
- U.S. interest rate:
  - Baseline policy-determined rate stays near zero for 8 quarters and rises slowly thereafter, remaining below equilibrium at 20 quarters.
  - All Shocks 95 percent confidence interval at 20 quarters goes from near zero to almost 7 percent.
  - Removal of foreign non-oil shocks significantly narrows the confidence interval for the interest rate—by at least ½ percentage point after 4 quarters.
  - Eliminating U.S. supply shocks has little effect on interest-rate variability; removing oil-price shocks produces major narrowing.
  - Domestic demand shocks account for a substantial part of the interest-rate confidence interval at all horizons.
- U.S. output gap:
  - No Shocks baseline: a 5-quarter decline with a trough near -6 percent; baseline output not back to potential until the very end of the 20-quarter horizon.
  - Asymmetry: at 8 quarters, the 95 percent confidence interval extends from about 4 percent below to 2 percent above the 5 percent midpoint (greater downside risk).
  - Eliminating foreign shocks has no material effect on output-gap uncertainty; removing U.S. supply shocks has virtually no effect.
  - Removal of oil price shocks halves the interval. Oil price and domestic demand shocks together account for almost the entire confidence interval around the forecast output gap.
- Bank lending tightening (BLT):
  - No Shocks forecast reverts from an extremely high starting point to near zero within 5 quarters; zero is high under current circumstances, and return to a normal level would require extended easing (negative BLT values).
  - Uncertainties are considerable: at the 8-quarter horizon the width of the All Shocks band is about 80 percentage points, and the interval is asymmetric with much more risk of unexpected tightening than easing.
  - Decomposition: oil price shocks dominate the confidence interval for BLT; domestic demand shocks (including BLT shocks) have a surprisingly small independent influence.

### Methodological and substantive conclusions (summary)
- The confidence-interval construction includes innovations: Bayesian estimation, nonlinearities (notably the zero bound on interest rates), and Latin hypercube sampling for efficient multi-variable sampling using lengthy time series.
- Nonlinearities generate visibly asymmetric forecast confidence intervals for key financial variables (short-term interest rate and BLT). The lower bound of the 95 percent confidence interval for the policy interest rate touches the zero axis almost throughout the 20-quarter forecast, and the BLT interval indicates more risk of unusual tightening than easing.
- The results imply a non-negligible risk that the Federal Reserve might be constrained by the zero bound over the medium term given severe credit market conditions.
- The output-gap confidence interval shows more downside than upside risk at the trough; the inflation interval is roughly symmetric, partly due to the forward-looking inflation expectations process anchored by the policy target.
- Decomposition suggests oil price shocks and U.S. domestic demand shocks have a stronger influence on confidence-interval widths than non-oil foreign shocks or U.S. supply shocks.

*Source: _wp09214 - 3.1 Construction (IMF PDF).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp09214.pdf_