Behavioural equations

There are four behavioural equations at the core of each country model. The first behavioural equation is an aggregate demand, or IS, curve that relates the level of real activity to expected future and past real activity, the real interest rate, the real exchange rate, and the level of real activity in each country’s trading partners. All variables are expressed as deviations from their equilibrium values so that the aggregate demand equation relates the output gap to its determinants as follows:

y_i,t = β_i,1y_i,t−1+ β_i,2y_i,t+1− β_i,3(R_i,t−1− R_i,t−1) (1) + β_i,4X

j

w_i,j,4(Z_i,j,t−1− Z_i,j,t−1) + β_i,5X

j

w_i,j,5y_j,t−1+ ε^y_i,t,

where y is the output gap, R is the real interest rate, R is the equilibrium real interest rate, Zi,j is the bilateral real exchange rate of country i relative to that of country j (such that an increase in Zi,j represents a real depreciation of the currency of country i relative to the currency of country j ), Z_i,j is the equilibrium bilateral real exchange rate, and ε^y is a disturbance term.⁶ The foreign output-gap term is defined as a weighted average of the foreign output gaps where the weights (w_i,j,5) used are the ratios of the exports of country i to country j to its total exports to all the countries in the model.

The lead term captures the forward-looking elements in aggregate demand that arise in a framework where forward-looking households optimize their consumption. Thus, expecta-tions of the future performance of the economy are assumed to influence current aggregate demand because of the forward-looking nature of decisions made by individual households and firms. The own-lag term allows for inertia in the system and permits shocks to have persistent effects. As discussed in Clarida, Gali, and Gertler (1999), the primary justification for allowing some form of lagged dependence in the IS curve is empirical, although it may be possible to explain its presence by appealing to some form of adjustment costs. The real interest rate term provides the crucial link between monetary policy actions and the real economy. Given the sluggish adjustment of prices, by varying the nominal interest rate the central bank is able to influence the real interest rate gap, and hence aggregate demand. And the foreign activity variable and the real exchange rate term allow for critical links between the domestic economy and the other economies in the model.

6The weights used to construct the effective real exchange rate term (wi,j,4) are the ratios of the sum of the exports and imports of country i with country j to the sum of its exports and imports with all the countries in the model.

The second equation is an inflation equation, or Phillips curve, which links inflation to its past and future values, the lagged output gap, the effective exchange rate gap, and oil-price inflation. The inflation equation is thus assumed to take the following form:

πi,t = λi,1π4i,t+4+ (1 − λi,1)π4i,t−1+ λi,2yi,t−1+ λi,3

X

j

wi,j,3∆Zi,j,t (2) + ν_i,1π_i,t^{RP OIL}+ ν_i,2π_i,t−1^{RP OIL}− ε^π_i,t,

where π is the quarterly rate of inflation (at annual rates), π4 is the average annual inflation rate, y is the output gap, Z_i,j is the bilateral real exchange rate of country i relative to that of country j, Zi,j is the equilibrium bilateral real exchange rate, π^{RP OIL} is the rate of inflation of real oil prices (denominated in the domestic currency), and ε^π is a disturbance term. The weights (w_i,j,3) used to construct the effective exchange rate term are the ratios of the imports of country i from country j to its total imports from all the countries in the model.

This inflation equation is in the spirit of the New Keynesian Phillips curve (NKPC), which evolves from the optimal price-setting behaviour of forward-looking firms in an environment of imperfect competition and price stickiness (i.e., it is assumed that firms set prices on a staggered basis). Equation (2) thus emphasizes the forward-looking process for inflation.

And, as discussed in Gali and Gertler (1999), lagged inflation can also influence inflation dynamics in the NKPC framework if it is assumed that a fraction of firms set prices using a backward-looking rule of thumb. Inflation is also a function of the output gap, which in the NKPC set-up would be a proxy for marginal cost.⁷

The third equation is the monetary policy reaction function, where the short-term nominal interest rate is determined as a function of its own lag and of the central bank’s policy responses to movements of the output gap and deviations of the expected inflation rate from its target. This Taylor-type rule takes the following form:

I_i,t = (1 − γ_i,1)[R_i,t + π4_i,t+3+ γ_i,2(π4_i,t+3− π_i^tar) (3) + γ_i,4y_i,t] + γ_i,1I_i,t−1+ ε^I_i,t,

where I is the short-term nominal interest rate, R is the equilibrium real interest rate, π4 is

7Since it is assumed in the NKPC framework that firms adjust their prices in response to expected movements in marginal cost.

the average annual inflation rate, π^tar is the inflation target, y is the output gap, and ε^I is a disturbance term.

More precisely, this reaction function assumes that the central bank aims at achieving a measure of the equilibrium nominal interest rate over the longer run, but adjusts its rate in response to deviations of the expected inflation rate from target, and to the current output gap. This specification for the monetary policy rule assumes that the central bank smooths interest rates, adjusting them gradually to the desired value. This behaviour is widely observed in practice and has been shown by Woodford (2003) to be a desirable outcome in a model of optimizing private sector behaviour, because it can help to steer private sector expectations of future policy. For the Chinese economy, there are important departures from the structure of the monetary policy reaction function outlined here for the G-3 economies;

changes to this equation that address the Chinese economy are provided in Appendix A.

In the fourth equation, the bilateral real exchange rate (versus the U.S. dollar) is modelled using the following version of uncovered interest rate parity (UIP):

4(Z_i,t+1^e − Z_i,t) = (R_i,t− R_us,t) − (R_i,t − R_us,t) + ε^Z−Z_i,t ^e, (4)

where the expected real exchange rate is assumed to evolve as follows:

Z_i,t+1^e = φ_iZ_i,t+1+ (1 − φ_i)Z_i,t−1. (5)

Equation (4) therefore states that the difference between the real exchange rate and its expected value is a function of the real interest rate differential and the equilibrium real interest rate differential between the two countries. Thus, any deviation in the real interest rates across the two countries should result in either an expected change in the exchange rate or a deviation in the equilibrium real interest rates in the two countries. Any other movement in the exchange rate is captured by the residual in the equation (which can be thought of as a temporary shock to the risk premium). As shown in Equation (5), in this “hybrid” version of UIP, the expected real exchange rate is not fully model-consistent, but also depends, in part, on past values of the real exchange rate.

In the Bank of Canada’s version of the GPM, the real bilateral exchange rates are related to the U.S. real effective exchange rate, which is anchored in MUSE by a long-run assumption on the stock of U.S. net foreign assets. Because the U.S. real effective exchange rate is

determined in MUSE exogenously to the GPM, the RoW real bilateral exchange rate is determined as a residual, so that all exchange rates are consistent.