An update to the OBR's small economic model

The OBR uses a small model of the UK economy to assess the mechanisms through which shocks propagate in the UK economy. While the central macroeconomic forecast that we produce in our Economic and fiscal outlooks (EFOs) is compiled using our large-scale macroeconomic model, this smaller and stripped-down model is usually used to understand the impact of individual shocks.

We use the small model (and many other satellite models) both for informing elements of the central medium-term forecasts of the economy,1 and for calibrating some of the alternative economic scenarios that we produce alongside these forecasts. For example, we have used the small model, in conjunction with other evidence, to produce plausible estimates of the effects of government policy on Bank Rate. All OBR forecasts and scenarios ultimately remain a collective BRC judgement based on a wide range of evidence and tools.

This model was introduced in our Working paper No.4: A small model of the UK economy in 2012 and was calibrated on information available at the time.2 The model has evolved since 2012, and most recently underwent a re-estimation and re-calibration in 2024. Outturns since the original paper show a different picture of the economy now than in 2012, and the current sampling period now places more weight on the 21^st century following central bank independence. As part of our response to recommendations in our most recent external review, this article sets out the model in its current form.

The OBR’s small model has New Keynesian features in which prices do not adjust immediately in response to economic conditions. This induces a lag before shifts bring forth full price adjustment so that nominal variables affect real ones on transition paths. The OBR’s small model focuses on four variables: the output gap, rate of inflation, an interest rate, and the exchange rate. As such, it is a purely cyclical model: potential output is taken as exogenous and the determinants of long-run productive capacity lie outside its scope. Four equations describe the evolution of these variables:

• The IS relation, or investment-savings equation, shows how the output gap is influenced by interest and exchange rates. It models how changes in interest rates can affect decisions to invest or save, which in turn affect overall economic activity.
• The Phillips curve describes the relationship between inflation (the rate at which prices change), the output gap, and the exchange rate.3 It implies that inflation will be higher when the economy is operating above its potential and lower when there is unused capacity.
• The Taylor rule is a guideline for central banks on how to set interest rates based on the state of the economy. It implies the need to adjust interest rates when inflation is not at its target or when the economy is over- or under-utilising the resources available to it.
• Lastly, the uncovered interest parity (UIP) condition connects interest rates and exchange rates. Exchange rates fluctuate in response to changes in the differences between the real interest rate across countries, ensuring that investors cannot make risk-free profits borrowing in one country and investing in another one.

Since our original working paper in 2012, we have made various changes to better reflect the current economic environment. These include:

• Incorporating both backward- and forward-looking households into the IS relation to represent two types of households – unconstrained optimisers and liquidity-constrained consumers. Optimisers (around 30 per cent of households) look ahead and take decisions on borrowing or saving so as to smooth consumption, while liquidity-constrained consumers (around 70 per cent) spend what they earned in the previous period.4
• Updating the parameter controlling the persistence of inflation in the Phillips curve. Our 2012 calibration suggested persistence of 0.85, meaning that 85 per cent of the previous period’s inflation carries over to today’s inflation. In practice, we often vary this parameter based on our view of how, in the specific circumstances being considered, the level of inflation and source of inflation affects its persistence. Various evidence suggests inflation persistence has fallen since the introduction of central bank inflation targeting, meaning empirical estimates based on data prior to this regime change should be interpreted with caution.5 Therefore, we have lowered the baseline parameter to 0.75.
• Assuming a higher degree of forward-looking behaviour in the Taylor rule, increasing the lead horizon on inflation from three to four quarters and reducing the degree of interest rate smoothing. This improves model convergence over our five-year forecast period and is reflective of the Bank’s aim to target anchored inflation expectations and to favour smooth policy transitions.6
• Updating the UIP condition to include a purchasing power parity (PPP) convergence parameter, which delays convergence to the long run steady state. This reflects more observed persistence in deviations of the exchange rate from its long-run anchor rather than imposing immediate reversion following a shock to output, inflation or Bank Rate.7
• To better reflect the model’s practical uses, re-estimating the equations on a CPI basis (previously we had used the GDP price deflator at factor cost), on a GDP basis (previously non-North Sea GVA output gap definition as updated in Briefing paper No.8: Forecasting potential output), and aligning to the OBR’s macroeconomic model’s SONIA-based measure of short-term foreign interest rates (previously LIBOR-based).
• Re-estimating the model using a shadow rate to proxy the effective interest rate in the equations in the periods where quantitative easing was in effect.8 Between 2012 and 2019, Bank Rate was close to the effective lower bound so this better represents the overall stance of monetary policy during this period. When above the lower bound, Bank Rate was assumed to apply.
  

In the charts below we show impulse response functions for three simulations to illustrate the properties of our latest model (solid lines). Equivalent impulse response functions from the original 2012 specification are also shown for comparison (dashed lines). The model is operated using deviations from a given baseline and is used for scenario analysis around, and to assess shocks to, our central forecast.

Chart 1 shows the modelled impact of a four-quarter, 1 percentage point negative shock to the output gap. This generates increased spare capacity in the economy, which applies downward pressure to inflation via the Phillips curve. This results in lower interest rates through the Taylor rule, which lowers the exchange rate through the UIP condition. Lower interest rates reduce borrowing costs which, alongside the exchange rate depreciation, increases demand through the IS curve and returns the output gap to zero after five years.

The lower persistence of inflation in this model brings more rapid convergence to target than in the original 2012 model specification, and is more consistent with our convention that the output gap normally closes by our forecast horizon. The initial fall in Bank Rate of around 80 basis points is less than the 100 basis points implied by the 2012 model. This is due to the more moderate inflation response, increased lead length on inflation in the Taylor rule and reduced weight on interest rate smoothing.

Chart 2 shows the modelled impact of a four-quarter, 1 percentage point exogenous positive shock to Bank Rate. This shock raises borrowing costs and increases incentives to save, leading to a fall in aggregate demand through the IS curve. This generates more economic slack, reducing pressure on prices and lowering inflation. This reduction in output and inflation feeds into the Taylor rule to subsequently lower Bank Rate, and return output back in line with potential, and inflation to target. The larger fall in the output gap than in the 2012 specification is due to the introduction of forward-looking optimising households.

Finally, Chart 3 shows a four-quarter, 1 percentage point positive shock to the inflation rate. This generates a rise in Bank Rate through the forward-looking Taylor rule. In response to higher interest rates, households and businesses save more and borrow less, which causes household and business spending to fall, and the output gap to widen. This brings inflation back to target, allowing Bank Rate to fall back through the Taylor rule and closing the output gap again. The inflation shock is shorter-lived in our updated model compared to the 2012 specification, due to the lower assumed persistence of inflation. But the increased lead time and lesser weight on interest rate smoothing implied by this model means the interest rate rises more rapidly and inflation falls further and faster to converge around its target after around three years.

In practice, we use the small model for a range of purposes and may vary the parameters from those described here to achieve a particular effect. For example, during scenario analysis, we may alter our view of the persistence of inflation or the weight that the Bank of England’s Monetary Policy Committee might place on smooth policy transitions, due to specific prevailing economic or policy conditions. We may also consider imposing convergence on the model if we do not expect the effects of a shock to persist beyond five years, or to prevent unlikely overcorrections.

IS relation

Description: The investment-savings equation, which shows how the output gap is influenced by lagged real interest and exchange rates.

Equation:

yₜ = βᵧyₜ_-1 + (1 − βᵧ)yₜ₊₁ + βᵣ(Rₜ_-1 − πₜ_-1) +βₑΔqₜ_{-1 +} μ_t

Where:

• yₜ is the output gap in period t;
• Rₜ is the nominal effective interest rate;
• πₜ is inflation;
• Δqₜ_-1 is the lagged change in the exchange rate;
• βᵧ, βᵣ, βₑ are coefficients on persistence, the real interest rate, and the exchange rate, respectively; and
• μ_t is a residual.
  

Phillips curve

Description: The relationship between inflation, the output gap, and exchange rates.

Equation:

πₜ = ςπₜ_-1 + λᵧyₜ₋₂ + λₑΔeₜ₋₁ + ε_t

Where:

• πₜ is inflation in period t;
• yₜ₋₂ is the output gap lagged by two periods;
• Δeₜ₋₁ is the lagged change in the exchange rate;
• ς is the coefficient on inflation persistence;
• λᵧ, λₑ are coefficients on the output gap and the change in the exchange rate, respectively; and
• ε_t is a residual.
  

Taylor rule

Description: A monetary policy reaction function, where the policy effective nominal interest rate reacts to exchange rate changes, future inflation and the future output gap, with interest rate smoothing.

Equation:

Rₜ = ΨRₜ_-1 + (1 − Ψ) (γᵧyₜ₊₁ + γ_ππₜ₊₄)

Where:

• Rₜ is the effective nominal interest rate;
• yₜ₊₁ is the one-period-ahead output gap;
• πₜ₊₄ is inflation four quarters ahead;
• Ψ is the interest rate smoothing parameter; and
• γᵧ and γ_π, are coefficients on sensitivity to the output gap and inflation.
  

Uncovered interest parity condition

Description: A condition relating the exchange rate to changes in domestic and foreign real interest rates, as well as the past exchange rate.

Equation:

qₜ = qₜ₊₁ + (Rₜ − πₜ₊₁) − (ifₜ − pfₜ₊₁) − ρqₜ_-1

Where:

• qₜ is the exchange rate in period t;
• Rₜ − πₜ₊₁ is the domestic real effective interest rate;
• ifₜ − pfₜ₊₁ is the foreign real effective interest rate, where foreign interest rates are defined via an autoregressive process of the form pf_t = δ_pf pf_t-1, and foreign inflation is defined as if_t = δ_if if_t-1 , where δ_pf and δ_if are persistence parameters; and
• ρ is a parameter governing the speed at which convergence to long-run purchasing power parity occurs as described in Murray, 2013.
  

Effective interest rate

Description: The model uses an effective interest rate which is defined as Bank Rate plus credit spreads. To estimate the equations, a shadow rate was used to account for the unconventional monetary policy in use over the sample period.

Equation:

Rₜ = rₜ + cs

Where:

• Rₜ is the effective nominal interest rate;
• rₜ is Bank Rate; and
• cs is the credit spread, defined via an autoregressive process of the form csₜ = θ_cscsₜ_-1 where θ_cs is a persistence parameter

Table 1: Changes to calibrated small model parameters

Acknowledgements

The author would like to thank Scott Bowman, James Watson, Rosanna Colthorpe, Charlotte Bunney, and various other OBR staff past and present for their valuable contributions to this article.

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