Targeting Rules for an Open Economy

Abstract

This study extends the formal analysis of inflation targeting monetary policy using the standard New Keynesian framework to a small open economy by adding inflation and output persistence as well as a direct exchange rate channel to domestic inflation. We find that output variability is lower under CPI inflation targeting than under domestic inflation targeting. However, CPI inflation results in higher variability of the real exchange rate than domestic inflation targeting. Output and the nominal interest rate are less volatile under flexible inflation targeting than under almost-strict inflation targeting. We also find that almost-strict domestic inflation targeting cannot completely insulate domestic inflation from foreign shocks due to a direct exchange rate channel. The model is calibrated to Canadian data.

Appendices

Appendix

Appendix A: State-Space Form of the Model

We can write the model in a convenient state-space form as

$$ H_{0} \begin{bmatrix} x_{1t+1}\\[4pt] x_{2t+1} \end{bmatrix} =A_{0} \begin{bmatrix} x_{1t}\\[4pt] x_{2t} \end{bmatrix} +B_{0}u_{t}+C_{0}W_{t+1},\label{A.1} $$

(42)

where x _1t is a (column) vector of predetermined (backward- looking) variables with the initial values given, and x _2t is a vector of non-predetermined or jump (forward looking) variables defined in Section 4.1. u _t is a vector of policy instruments, the nominal interest rate:

$$ u_{t}= [r_{t}]. $$

W _t + 1 is a 15×1 vector of NID(0,1) variables, and

$$ H_{0}= \begin{bmatrix} e_{1}\\ e_{2}\\ e_{3}\\ H_{4}\\ -\bar{y}_{0}e_{3}+e_{5}\\ H_{6}\\ e_{7}\\ e_{8}\\ e_{9}\\ e_{10}\\ e_{11}\\ \frac{1}{2}e_{12}-\frac{\alpha}{1-\alpha}e_{15}\\[2pt] \frac{1}{[\sigma+\delta(\sigma-1)](1-\alpha)}e_{12}+\frac{\sigma}{\sigma+\delta(\sigma-1)}e_{13}\\[2pt] e_{12}+e_{15}\\[2pt] \frac{1}{[\sigma+\delta(\sigma-1)](1-\alpha)}e_{12}+\frac{\sigma}{\sigma+\delta(\sigma-1)}e_{14} \end{bmatrix}, A_{0}= \begin{bmatrix} \rho_{\zeta}e_{1}\\ \rho_{\pi^{\ast}}e_{2}\\ \rho_{y^{\ast}}e_{3}\\ e_{0}\\[2pt] \bar{y}_{2}e_{3}+\bar{y}_{1}e_{5}\\ rr_{3}e_{3}+rr_{1}e_{5}\\ e_{3}\\ e_{12}\\ e_{13}\\ e_{14}\\[2pt] e_{12}-\frac{\alpha}{1-\alpha}e_{15}\\[2pt] A_{12}\\ A_{13}\\ \rho_{\pi^{\ast}}e_{2}-e_{4}\\ A_{15} \end{bmatrix},$$

$$B_{0}= \begin{bmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\[2pt] \frac{1}{[\sigma+\delta(\sigma-1)](1-\alpha)}\\[2pt] 1\\[2pt] \frac{1}{[\sigma+\delta(\sigma-1)](1-\alpha)} \end{bmatrix}, C0= \begin{bmatrix} \sigma_{\zeta}e_{1}\\ \sigma_{\pi^{\ast}}e_{2}\\ \sigma_{y^{\ast}}e_{3}\\ \sigma_{r^{\ast}}e_{4}\\ e_{0}\\ e_{0}\\ e_{0}\\ e_{0}\\ e_{0}\\ e_{0}\\ e_{0}\\ e_{0}\\ e_{0}\\ e_{0}\\ e_{0} \end{bmatrix},$$

$$\begin{array}{lll}H_{4}&=&-f_{\pi^{\ast}}e_{2}-f_{y^{\ast}}e_{3}+e_{4},\notag\\ H_{6}&=&-rr_{2}e_{3}-rr_{0}e_{5}+e_{6},\notag\\ A_{12}&=&-e_{1}-\frac{1}{2}e_{8}-\lambda e_{9}+e_{12}-\lambda e_{13}-\frac{\alpha}{1-\alpha}e_{15},\notag\\ A_{13}&=&\frac{1}{[\sigma+\delta(\sigma-1)](1-\alpha)}e_{6}-\frac{\sigma(\delta-1)}{\sigma+\delta(\sigma-1)}e_{9}+e_{13},\notag\\ A_{15}&=&-y_{0}e_{3}-y_{1}e_{7}-\frac{\delta}{1+\delta}e_{10}+e_{14},\notag \end{array}$$

and

$$\begin{array}{lll} \bar{y}_{0}&=&\kappa^{-1}\alpha(1-\sigma), \bar{y}_{1}=\kappa^{-1}(\sigma-)\delta(1-\alpha),\bar{y}_{2}=\kappa^{-1}(\sigma-1)\delta\alpha,\\ rr_{0}&=&\sigma(1-\alpha)\left(\frac{\bar{y}_{1}\sigma}{\sigma+\delta(\sigma-1)}-1\right),rr_{1}=\sigma(1-\alpha)\frac{\sigma(\delta-1)}{\sigma+\delta(\sigma-1)},\\ rr_{2}&=&\frac{\sigma(1-\alpha)}{\sigma+\delta(\sigma-1)}\left(\bar{y}_{0}\rho_{y^{\ast}}+\bar{y}_{2}\right)+\frac{\alpha\sigma}{\sigma+\delta(\sigma-1)}\rho_{y^{\ast}}-\alpha(\sigma-1), \\ y_{0}&=&\frac{\alpha}{1-\alpha}\frac{\sigma}{\sigma+\delta(\sigma-1)}\rho_{y^{\ast}}-\frac{\alpha}{1-\alpha}, y_{1}=\frac{\alpha}{1-\alpha}\frac{\delta(\sigma-1)}{\sigma+\delta(1-\sigma)},\end{array}$$

where e _j , j = 0,....,n, denotes a 1×n row vector, the j-th element of which equals 1 and all other elements are zero, and $\sigma_{\zeta}, \sigma_{\pi^{\ast}}, \sigma_{y^{\ast}}$, and $\sigma_{r^{\ast}}$ are the standard deviations of cost-push shocks, foreign inflation and income, and foreign nominal interest rate shocks, respectively.

Then, we can express Eq. 42 as

$$\begin{bmatrix} x_{1t+1}\\ x_{2t+1} \end{bmatrix} =A \begin{bmatrix} x_{1t}\\ x_{2t} \end{bmatrix} +Bu_{t}+CW_{t+1},\label{A.2} $$

(43)

or

$$ X_{t+1}=AX_{t}+Bu_{t}+CW_{t+1},\label{A.3} $$

(44)

where

$$ X_{t}= \begin{bmatrix} x_{1t} & x_{2t} \end{bmatrix}, $$

and $A=H^{-1}_{0}A_{0}, B=H^{-1}_{0}B_{0}$, $C=H^{-1}_{0}C_{0}$.

Appendix B: The Period Loss Function

We express the period loss function in Eq. 41 for each targeting case in the state-space form. First, we consider the CPI inflation targeting case. The period loss function for CPI inflation targeting is

$$ L_{t}=\omega x^{2}_{t}+\pi^{2}_{t}=X'_{t}Q_{CPI}X_{t}+2X'_{t}Uu_{t}+u'_{t}Ru_{t},\label{A.4} $$

(45)

where

$$ Q_{CPI}=\left[e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{12},\omega e_{13}, e_{0}, e_{0}\right]'. $$

Then the period loss function for domestic inflation targeting will be

$$ L_{t}=\omega x^{2}_{t}+\pi^{2}_{H,t}.\label{A.5} $$

(46)

Making use of Eqs. 15 and 17, the period loss function (Eq. 46) can be expressed in terms of CPI inflation as

$$\begin{array}{rll} L_{t}&=&\omega x^{2}_{t}+\pi^{2}_{t}-\frac{2\alpha}{1-\alpha}\pi_{t} \Delta q_{t}+\left(\frac{\alpha}{1-\alpha}\right)^{2}\Delta q_{t}^{2}\\ &\approx&\omega x^{2}_{t}+\pi^{2}_{t}+\left(\frac{\alpha}{1-\alpha}\right)^{2}\Delta q_{t}^{2}\\ &=&X'_{t}Q_{D}X_{t}+2X'_{t}UU_{t}+U'_{t}RU_{t,}\label{A.6} \end{array}$$

(47)

where

$$ Q_{D}=\left[e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{0},e_{12},\omega e_{13},e_{0},\left(\frac{\alpha}{1-\alpha}\right)^{2}e_{15}\right]'\notag. $$