G-Cubed Monetary policy in G-Cubed
Table of contents
Central banks in G-Cubed set short-term nominal interest rates using a Henderson–McKibbin–Taylor (HMT) monetary policy rule. The rule responds systematically to deviations of inflation and economic activity from their targets. It can also respond to an exchange-rate target, which allows different regions to represent different monetary regimes.
The reaction function can provide a nominal anchor without requiring the central bank to set the long-run nominal interest rate independently of the rest of the model. In an inflation-targeting parameterisation, the variable being anchored is the long-run inflation rate and the value it is anchored to is the exogenous inflation target. The nominal policy rate is the instrument used to achieve that outcome; its long-run level is not itself the nominal anchor. Asset-market equilibrium and the Fisher relation determine the nominal-rate level consistent with the equilibrium real interest rate and the anchored inflation rate.
Money is required for transactions in G-Cubed. The central bank supplies the money demanded at the short-term nominal interest rate, so the policy instrument is the interest rate rather than an independently fixed quantity of money. Money demand then responds to the policy rate, prices, and economic activity.
What is the nominal anchor?
A nominal anchor is the nominal variable, or path for a nominal variable, that provides the long-run reference point for prices expressed in money. It stops the model’s nominal variables from all drifting without reference to a stated policy objective. The policy instrument and the nominal anchor are not the same thing:
INTN, the short-term nominal interest rate, is the central bank’s policy instrument.- The target selected by the active reaction-function terms supplies the nominal anchor.
For the commonly used inflation-targeting parameterisation, positive feedback on INFL - INFX anchors long-run consumer price inflation, INFL, to the inflation target, INFX, once the other active policy gaps have closed. It does not anchor the consumer price level to a fixed number. The price level can retain the effect of past inflation surprises and subsequently grows at the anchored inflation rate.
This is not the nominal anchor under every G-Cubed monetary regime. The active target-gap terms and their coefficients determine what is anchored and what it is anchored to. A model build can instead represent a price-level target, an exchange-rate regime, a nominal-income target, or a combination of objectives.
The Henderson–McKibbin and Taylor rules
Henderson and McKibbin (1993) compared monetary regimes in which the central bank changes its interest-rate instrument to eliminate, or progressively reduce, deviations of intermediate targets from their desired values. One of the regimes responds jointly to inflation and output relative to potential. Taylor (1993) proposed a closely related rule with different response weights. The combined family is now commonly described as Henderson–McKibbin–Taylor rules.
A level form of this family can be written as:
\[i_t = r_t^{\ast} + \pi_t + \alpha(\pi_t-\pi_t^{\ast}) + \beta(y_t-y_t^{\ast}),\]where $i_t$ is the nominal policy interest rate, $r_t^{\ast}$ is the equilibrium real interest rate, $\pi_t^{\ast}$ is the inflation target, and $y_t-y_t^{\ast}$ is the output gap. Henderson and McKibbin placed more weight on inflation and output gaps than the calibration proposed by Taylor.
The superscript $\ast$ marks the relevant benchmark or target for each variable. It denotes the equilibrium real rate for $r_t^{\ast}$, the inflation target for $\pi_t^{\ast}$, and potential output for $y_t^{\ast}$. In the G-Cubed adaptation below, it likewise marks the policy-target series for inflation, potential-output growth, and exchange-rate change.
The term $r_t^{\ast}+\pi_t$ compensates for the current inflation rate. A positive $\alpha$ then makes the nominal rate rise by more than one-for-one with a sustained increase in inflation. The real policy rate consequently rises, restraining demand and inflation.
At a long-run equilibrium with output at potential, the Fisher relation gives $i^{\ast}=r^{\ast}+\pi$. Substitution into the rule then requires $\pi=\pi^{\ast}$. The rule anchors the rate of inflation; it does not require the price level to return to the path it followed before an inflation surprise.
Adaptation to G-Cubed
Cagliarini and McKibbin (2009) describe the G-Cubed form of the HMT rule as:
\[i_t = i_{t-1} + \beta_1(\pi_t-\pi_t^{\ast}) + \beta_2(\Delta y_t-\Delta y_t^{\ast}) + \beta_3(\Delta e_t-\Delta e_t^{\ast}).\]The central bank therefore changes its policy rate in response to three possible target gaps:
| Term | Interpretation |
|---|---|
| $\pi_t-\pi_t^{\ast}$ | Inflation relative to the inflation target |
| $\Delta y_t-\Delta y_t^{\ast}$ | Output growth relative to potential-output growth |
| $\Delta e_t-\Delta e_t^{\ast}$ | Exchange-rate change relative to its target |
The response coefficients can differ between regions. For example, a central bank operating an inflation-targeting regime can place positive weights on inflation and output growth and no weight on the exchange rate. A region that manages its exchange rate can place greater weight on the exchange-rate term. In the 2009 application, most advanced economies had weights of 0.5 on both inflation and output growth and zero on the exchange rate. Several developing regions instead had non-zero exchange-rate weights. Those values describe the regimes used in that application rather than universal G-Cubed defaults.
G-Cubed uses output growth relative to potential growth rather than the level of output relative to potential. McKibbin and Panton (2018) explain that average trend output growth is easier to measure than the level of potential output at every future date. This specification allows the central bank to stabilise the transition without requiring it to reverse every past loss or gain in the level of output.
Model variables
The principal G-Cubed variables corresponding to the 2009 rule are:
| Model variable | Role in the rule |
|---|---|
INTN | Current nominal policy interest rate, $i_t$ |
INTL | Lagged nominal policy interest rate, $i_{t-1}$ |
INFL | Consumer price inflation, $\pi_t$ |
INFX | Inflation target, $\pi_t^{\ast}$ |
OUTP and OUTL | Current and lagged output used to calculate output growth |
ROGY | Central bank’s estimate of potential-output-growth, $\Delta y_t^{\ast}$ |
EXCH, EXCL, and EXCX | Exchange-rate change and its target |
INTX | Exogenous adjustment to the policy rate |
The mrule_* parameters determine which target gaps are active and the weight placed on each one. When the coefficient on the lagged interest rate, mrule_r, equals one, INTL and INTN implement the first-difference form shown above. Other monetary-policy modules or parameter settings can represent partial adjustment and alternative regimes.
Some model builds first calculate a preferred HMT policy rate, INPN, and then set INTN equal to that rate or allow INTN to adjust towards it. Other builds place the lagged rate and target-gap terms directly in the equation for INTN. These are alternative implementations of the same policy architecture; the active SYM module and its parameter values define the regime for a particular model build.
INTX allows a scenario to impose a monetary-policy adjustment that is not generated by the systematic rule. A permanent non-zero INTX would alter the long-run condition described below, so its path is part of the definition of a scenario’s monetary regime.
How parameterisation changes the nominal anchor
The identity of the nominal anchor depends on which reaction-function coefficients are non-zero:
| Active feedback | What is anchored | What it is anchored to |
|---|---|---|
Consumer price inflation, such as mrule_1 > 0 | Long-run consumer price inflation (INFL) | The inflation target (INFX), provided the other active gaps close |
| Producer or core inflation | The selected inflation measure | Its specified inflation target |
| A price-level term | The selected price-index path | The specified price-level target path, such as PTAR |
| Exchange-rate change | Nominal exchange-rate depreciation or appreciation | The exchange-rate-change target, such as EXCX, and therefore the monetary conditions of the anchor currency |
| Nominal-income or nominal-output feedback | The selected nominal aggregate or its growth rate | Its specified target path; the division between real growth and inflation is determined with the rest of the model |
| Several target gaps | Their coefficient-weighted combination | The corresponding coefficient-weighted combination of targets |
The magnitude of a positive coefficient affects the strength and speed of the policy response. Setting a coefficient to zero removes that target from the reaction function and can change the identity of the nominal anchor. In particular:
- If inflation and exchange-rate terms are both active, the rule directly restricts their weighted combination. It does not independently force both variables to their targets unless the wider model also closes the other gap.
- If the inflation coefficient is zero and the exchange-rate coefficient dominates, the exchange-rate target rather than
INFXsupplies the nominal anchor. - If the rule responds only to real activity, with no nominal target and no separately specified nominal-rate target, the reaction function does not supply a nominal anchor.
- A lagged-interest-rate term changes interest-rate persistence. It does not, by itself, say what any nominal variable should converge to.
The 2009 G-Cubed application illustrates this flexibility. Most advanced regions used inflation and output-growth feedback and no exchange-rate feedback. Other regions can also respond to the exchange rate. For example, in some model versions, China’s parameterisation placed the policy emphasis on its exchange-rate target.
How an inflation-targeting parameterisation anchors inflation
The first-difference rule does not contain an explicit value for the equilibrium real interest rate. When its parameterisation selects inflation as the nominal anchor, its anchoring mechanism instead has three parts.
1. Interest-rate changes accumulate
If inflation remains above target, the inflation term remains positive and the central bank raises the nominal policy rate again in each period. The response is cumulative because each period starts from the previous period’s interest rate. A permanent positive inflation gap therefore cannot coexist with a finite, unchanged policy rate unless another active target gap offsets it.
Higher nominal rates raise real rates when expected inflation does not rise by the same amount. With nominal rigidities, the higher real rates reduce interest-sensitive consumption and investment, affect asset prices and exchange rates, and reduce the demand and wage pressures supporting the inflation gap.
2. A stationary policy rate restricts the target gaps
In a long-run equilibrium, $i_t=i_{t-1}$. The HMT equation therefore requires:
\[0 = \beta_1(\pi-\pi^{\ast}) + \beta_2(\Delta y-\Delta y^{\ast}) + \beta_3(\Delta e-\Delta e^{\ast}) + INTX.\]If output growth has converged to potential growth, the exchange-rate change has converged to its target, INTX is zero, and $\beta_1>0$, the remaining condition is:
The qualification about the other target gaps is important. The rule directly sets the weighted sum of its target gaps to zero. The wider model supplies the long-run conditions under which potential-output growth and any exchange-rate target are also attained. If one of those gaps is deliberately maintained, the monetary authority is trading off its objectives rather than operating as a pure inflation targeter.
3. The wider model determines the interest-rate level
Once the target gaps have closed, the first-difference rule becomes $i_t=i_{t-1}$. This does not mean that any nominal interest rate is a long-run equilibrium of the complete G-Cubed model. It means that the monetary rule is no longer changing a rate whose equilibrium level is determined elsewhere.
G-Cubed distinguishes between three short-term rates:
| Variable | Meaning |
|---|---|
INTN | Nominal policy interest rate set by the central bank |
INTF | Risk-free real short-term interest rate |
INTR | Risk-adjusted real interest rate used in intertemporal decisions |
The Fisher relation in the model is:
\[INTF_t = INTN_t - E_t\pi_{t+1},\]where expected inflation is the expected change in the relevant price index. The risk-adjusted rate is:
\[INTR_t = INTF_t + RISR_t,\]where RISR is an exogenous risk premium. Equivalently:
The equilibrium real interest rate is an outcome of the model’s real and financial equilibrium, not an arbitrary nominal-policy choice. Household saving and consumption choices depend on real returns and time preference. Firms compare the cost of funds with the expected return on installed capital when determining investment. Government debt and international asset positions must satisfy their accumulation equations and intertemporal budget constraints. Purchases and sales of financial assets must also clear across regions.
International interest parity links differences in regional real interest rates to expected real-exchange-rate changes and exogenous risk premia. In a long-run equilibrium with stable real exchange rates, these conditions make regional real rates mutually consistent, allowing for any maintained risk premia. Collectively, these saving, investment, portfolio, capital-return, and market-clearing conditions determine the equilibrium path and long-run level of INTF and INTR.
In summary:
- Asset-market equilibrium and intertemporal behaviour determine the equilibrium real interest rate.
- The cumulative HMT response makes a sustained inflation gap incompatible with an unchanged policy rate once the other policy gaps have closed.
- The Fisher relation converts the equilibrium real rate and expected inflation into the corresponding nominal policy rate.
In a stationary equilibrium with expected inflation equal to the inflation target:
\[INTN^{\ast} = INTF^{\ast} + \pi^{\ast} = INTR^{\ast} - RISR^{\ast} + \pi^{\ast}.\]This is how a first-difference G-Cubed rule parameterised to target inflation can anchor INFL to INFX without placing an explicit $r^{\ast}$ term in the monetary-policy equation. The policy rule determines the inflation outcome through feedback; the complete model determines the compatible real and nominal interest-rate levels. A reaction function parameterised around a different nominal target uses the same general architecture but does not make INFX the independent long-run anchor.
Policy transmission and long bond rates
INTR enters the intertemporal decisions of households and firms, including wealth valuation and the cost of financing capital. It also affects real exchange rates through international interest parity. These channels transmit changes in INTN to aggregate demand, investment, trade, and inflation.
G-Cubed also reports real and nominal rates for bonds of several maturities. These rates are derived from the expected path of one-year rates rather than being separate policy instruments. See Long bond rates for details.
References
Cagliarini, A. & McKibbin, W. J. (2009), “Global Relative Price Shocks: The Role of Macroeconomic Policies”, Reserve Bank of Australia Research Discussion Paper 2009-10.
Henderson, D. W. & McKibbin, W. J. (1993), “A Comparison of Some Basic Monetary Policy Regimes for Open Economies: Implications of Different Degrees of Instrument Adjustment and Wage Persistence”, Carnegie-Rochester Conference Series on Public Policy, 39, 221–318.
McKibbin, W. J. & Panton, A. J. (2018), “Twenty-five Years of Inflation Targeting in Australia: Are There Better Alternatives for the Next Twenty-five Years?”, Reserve Bank of Australia Conference Volume.
Taylor, J. B. (1993), “Discretion versus Policy Rules in Practice”, Carnegie-Rochester Conference Series on Public Policy, 39, 195–214.