Forecasting output growth using a DSGE-based decomposition of the South African yield curve

Abstract

Evidence in favour of the ability of the term spread to forecast economic growth of the South African economy is non-existent. This could be due to the term spread aggregating information contained in the expected spread and the term premium. To decompose the term spread into its subcomponents, we develop an estimable small open economy new Keynesian dynamic stochastic general equilibrium (SOENKDSGE) model of the inflation targeting South African economy. The SOENKDSGE model is estimated with Bayesian methods over the quarterly period of 2000:01–2014:04. We then use a linear predictive regression framework to analyse the out-of-sample forecasting ability of the aggregate term spread, as well as the expected spread and term premium. Our forecasting results fail to detect forecasting gains from the aggregate term spread and also the term premium, but the expected spread is found to contain important information in forecasting output growth over short- to medium-run horizons, over the period of 2004:01–2014:04, using an in-sample period of 2000:01–2003:04. The results therefore highlight the importance of the forward-looking component of the term spread—the expected spread—in forecasting the output growth of South Africa.

A The linearized model

1.1 Households

Optimal L-period bond holdings, $\hat{b}_{L,t}$

$$\begin{aligned} \hat{R}_{L,t}&= \frac{1}{L}\left[ \left( \frac{R_L}{R}\right) ^L+\beta \phi _L y\right] ^{-1}\left\{ \beta \phi _L y E_t\left[ L\hat{R}_{L,t+1} - \hat{\psi }_{t+1}^z - 3\left( \hat{b}_{L,t+1}-\hat{b}_{L,t}\right) -y_{t+1}\right] \right. \nonumber \\&\quad +\, \left( 1 + \frac{3}{2}\phi _L y \right) \hat{\psi }_{t}^z + \frac{3}{2}\phi _L y\left[ 2\left( \hat{b}_{L,t}-\hat{b}_{L,t-1}\right) + y_t\right] - \frac{\nu _L}{\kappa _L}\left( R_L\right) ^L y \left[ \hat{m}_{t}-\hat{b}_{L,t}\right] \nonumber \\&\quad \left. +\,\left( \frac{R_L}{R}\right) ^L E_t\left[ \sum _{k=1}^{L}\hat{\mu }_{t+k}^z + \sum _{k=1}^{L}\hat{\pi }_{t+k}^d - \hat{\psi _{t+L}^z}\right] \right\} . \end{aligned}$$

(A.1)

Money holdings, $\hat{m}_t$

$$\begin{aligned} \hat{m}_t = \left[ \frac{A_m\sigma ^m m ^{-\sigma _m}}{\psi ^z} + \nu _L y\right] ^{-1}\left\{ \frac{1}{R}E_t\left[ \hat{\psi }_{t+1}^z - \hat{\pi }_{t+1}^d - \hat{\mu }_{t+1}^z\right] - \hat{\psi }_t^z + \nu _L y \hat{b}_{L,t}\right\} . \end{aligned}$$

(A.2)

Wage setting (real) $\hat{w}_{t}$

$$\begin{aligned} \hat{w}_{t} = -\frac{1}{\eta _1}\left[ \begin{array}{c} \eta _0\hat{w}_{t-1} + \eta _2 E_t\hat{w}_{t+1} + \eta _3\left( \hat{\pi }_{t}^d-\hat{\bar{\pi }}_t^c\right) + \eta _4\left( E_t\hat{\pi }_{t+1}^d-\rho _{\pi }\hat{\bar{\pi }}_t^c\right) \\ + \eta _5\left( \hat{\pi }_{t-1}^c-\hat{\bar{\pi }}_t^c\right) + \eta _6\left( \hat{\pi }_{t}^c-\rho _{\pi }\hat{\bar{\pi }}_t^c\right) +\eta _7\hat{\psi }_t^z + \eta _8 \hat{H}_t + \eta _9\hat{\xi }_t^h\end{array}\right] . \end{aligned}$$

(A.3)

Consumption Euler equation, $\hat{c}_{t}$

$$\begin{aligned} \hat{c}_{t}= & {} \frac{\mu ^z\, b}{(\mu ^z)^{2}+\beta {b^{2}}}\, \hat{c}_{t-1}+\frac{\beta \, \mu ^z\, b}{(\mu ^z)^{2}+\beta {b^{2}}}\, E_t \hat{c}_{t+1}-\frac{\mu ^z\, b}{(\mu ^z)^{2}+\beta {b^{2}}}\, \left( \hat{\mu }^z_{t}-\beta \, E_t \hat{\mu }^z_{t+1}\right) \nonumber \\&-\frac{\left( \mu ^z-b\right) \, \left( \mu ^z-\beta \, b\right) }{(\mu ^z)^{2}+\beta {b^{2}}}\, \left( \hat{\psi }^z_{t}+\hat{\gamma }^{c,d}_{t}\right) +\frac{\mu ^z - b}{(\mu ^z)^{2}+\beta {b^{2}}}\left( \mu ^z\hat{\xi }^{c}_{t} - \beta {b}E_t \hat{\xi }^{c}_{t+1}\right) .\nonumber \\ \end{aligned}$$

(A.4)

Investment Euler equation, $\hat{i}_{t}$

$$\begin{aligned} \hat{i}_{t}=\frac{1}{1+\beta }\left[ \beta E_t \hat{i}_{t+1} + \hat{i}_{t-1} + \beta E_t \hat{\mu }_{t+1}^z - \mu _t^z\right] + \frac{1}{(\mu ^z)^2\phi _i(1+\beta )}\left( \hat{P}^k_{t}-\hat{\gamma }^{i,d}_{t}+\hat{\xi }^{i}_{t}\right) .\nonumber \\ \end{aligned}$$

(A.5)

Price of installed capital, $\hat{P}^k_{t}$

$$\begin{aligned} \hat{P}^k_{t}=E_t \left[ \frac{\left( 1-\delta \right) \beta }{\mu ^z}\, \hat{P}^k_{t+1}+\hat{\psi }^z_{t+1}-\hat{\psi }^z_{t}-\hat{\mu }^z_{t+1}+\frac{\mu ^z-\left( 1-\delta \right) \beta }{\mu ^z}\, \hat{r}^k_{t+1}\right] . \end{aligned}$$

(A.6)

Capital’s law of motion, $\hat{k}_{t}$

$$\begin{aligned} \hat{k}_{t+1}= \frac{1-\delta }{\mu ^z}\left( \hat{k}_{t}-\hat{\mu }^z_{t}\right) + \left( 1-\frac{1-\delta }{\mu ^z}\right) \left( \hat{i}_{t} + \hat{\xi }^{i}_{t}\right) . \end{aligned}$$

(A.7)

Capital utilization, $\hat{u}_{t}$

$$\begin{aligned} \hat{u}_{t}=\frac{1}{\sigma _a}\hat{r}^k_{t}. \end{aligned}$$

(A.8)

Capital services, $\hat{k}^s_{t}$

$$\begin{aligned} \hat{k}^s_{t} = \hat{k}_{t} + \hat{u}_{t}. \end{aligned}$$

(A.9)

Optimal 1-period asset holdings, $\hat{\psi }^z_{t}$

$$\begin{aligned} \hat{\psi }^z_{t}=E_t\left( \hat{\psi }^z_{t+1} - \hat{\mu }^z_{t+1}\right) + \left( \hat{R}_{t}-E_t \hat{\pi }^{d}_{t+1}\right) . \end{aligned}$$

(A.10)

Exchange rate’s modified UIP condition, $\hat{S}_{t}$

$$\begin{aligned} \hat{R}_t^{}-\hat{R}_t^* = (1- \tilde{\phi _s})E_t \Delta \hat{S}_{t+1}-\tilde{\phi }_s\Delta \hat{S}_t-\tilde{\phi }_a \hat{a_t}+\hat{\tilde{\phi }}_t. \end{aligned}$$

(A.11)

1.2 Firms

1.2.1 Domestic goods

Production, $\hat{y}_{t}$

$$\begin{aligned} \hat{y}_{t}=\lambda ^d\, \left( \hat{\varepsilon }^{}_{t}+\alpha \, \left( \hat{k}^s_{t}-\hat{\mu }^z_{t}\right) +\left( 1-\alpha \right) \, \hat{H}_{t}\right) . \end{aligned}$$

(A.12)

Rental rate of capital, $\hat{r}^k_{t}$

$$\begin{aligned} \hat{r}^k_{t}=\hat{w}_{t}+\hat{\mu }^z_{t}-\hat{k}^s_{t}+\hat{H}_{t}. \end{aligned}$$

(A.13)

Real marginal cost, $\hat{mc}^{d}_{t}$

$$\begin{aligned} \hat{mc}^{d}_{t}=\alpha \, \hat{r}^k_{t}+\left( 1-\alpha \right) \, \left( \hat{w}_{t}\right) -\hat{\varepsilon }^{}_{t}. \end{aligned}$$

(A.14)

Price setting, $\hat{\pi }^{d}_{t}$

$$\begin{aligned} \hat{\pi }^{d}_{t}-\hat{\bar{\pi }}^c_{t}= & {} \frac{\beta }{1+\beta \, \kappa _d}\, \left( E_t\hat{\pi }^{d}_{t+1}-\rho _{\pi }\hat{\bar{\pi }}^c_{t}\, \right) +\frac{\kappa _d}{1+\beta \, \kappa _d}\, \left( \hat{\pi }^{d}_{t-1}-\hat{\bar{\pi }}^c_{t}\right) \nonumber \\&- \frac{\beta \, \kappa _d\, \left( 1-\rho _{\pi }\right) }{1+\beta \, \kappa _d}\hat{\bar{\pi }}^c_{t}+\frac{\left( 1-\theta _d\right) \, \left( 1-\beta \, \theta _d\right) }{\left( 1+\beta \, \kappa _d\right) \, \theta _d}\, \left( \hat{mc}^{d}_{t}+\hat{\lambda }^d_{t}\right) . \end{aligned}$$

(A.15)

1.2.2 Imported goods

Price setting: imported consumption goods, $\hat{\pi }^{m,c}_{t}$

$$\begin{aligned} \hat{\pi }^{m,c}_{t}-\hat{\bar{\pi }}^c_{t}= & {} \frac{\beta }{1+\beta \kappa _{m,c}} \left( E_t \hat{\pi }^{m,c}_{t+1}-\rho _{\pi }\hat{\bar{\pi }}^c_{t}\right) +\frac{\kappa _{m,c}}{1+\beta \kappa _{m,c}} \left( \hat{\pi }^{m,c}_{t-1}-\hat{\bar{\pi }}^c_{t}\right) \nonumber \\&- \frac{ \kappa _{m,c}\beta \left( 1-\rho _{\pi }\right) }{1+\beta \kappa _{m,c}}\hat{\bar{\pi }}^c_{t}+\frac{\left( 1-\theta _{m,c}\right) \left( 1-\beta \theta _{m,c}\right) }{\left( 1+\beta \kappa _{m,c}\right) \theta _{m,c}} \left( \hat{mc}^{m,c}_{t}+\hat{\lambda }^{m,c}_{t}\right) .\nonumber \\ \end{aligned}$$

(A.16)

Marginal cost: imported consumption goods, $\hat{mc}^{m,c}_{t}$

$$\begin{aligned} \hat{mc}^{m,c}_{t}=-\hat{\gamma }^{f}_{t}-\hat{\gamma }^{mc,d}_{t}. \end{aligned}$$

(A.17)

Price setting: imported investment goods, $\hat{\pi }^{m,i}_{t}$

$$\begin{aligned} \hat{\pi }^{m,i}_{t}-\hat{\bar{\pi }}^c_{t}= & {} +\frac{\beta }{1+\beta \kappa _{m,i}} \left( E_t \hat{\pi }^{m,i}_{t+1}-\rho _{\pi }\hat{\bar{\pi }}^c_{t} \right) +\frac{\kappa _{m,i}}{1+\beta \kappa _{m,i}} \left( \hat{\pi }^{m,i}_{t-1}-\hat{\bar{\pi }}^c_{t}\right) \nonumber \\&- \frac{\kappa _{m,i}\beta \left( 1-\rho _{\pi }\right) }{1+\beta \kappa _{m,i}}\hat{\bar{\pi }}^c_{t}+\frac{\left( 1-\theta _{m,i}\right) \left( 1-\beta \theta _{m,i}\right) }{\left( 1+\beta \kappa _{m,i}\right) \theta _{m,i}}\left( \hat{mc}^{m,i}_{t}+\hat{\lambda }^{m,i}_{t}\right) .\nonumber \\ \end{aligned}$$

(A.18)

Marginal cost: imported investment goods, $\hat{mc}^{m,i}_{t}$

$$\begin{aligned} \hat{mc}^{m,i}_{t}=-\hat{\gamma }^{f}_{t}-\hat{\gamma }^{mi,d}_{t}. \end{aligned}$$

(A.19)

1.2.3 Exported goods

Price setting: exported goods, $\hat{\pi }^{x}_{t}$

$$\begin{aligned} \hat{\pi }^{x}_{t}-\hat{\bar{\pi }}^c_{t}= & {} \frac{\beta }{1+\beta \kappa _x} \left( E_t \hat{\pi }^{x}_{t+1}-\rho _{\pi }\hat{\bar{\pi }}^c_{t}\right) +\frac{\kappa _x}{1+\beta \kappa _x} \left( \hat{\pi }^{x}_{t-1}-\hat{\bar{\pi }}^c_{t}\right) -\frac{\kappa _x\beta \left( 1-\rho _{\pi }\right) }{1+\beta \kappa _x} \hat{\bar{\pi }}^c_{t}\nonumber \\&+ \frac{\left( 1-\theta _x\right) \left( 1-\beta \theta _x\right) }{\left( 1+\beta \kappa _x\right) \theta _x}\left( \hat{mc}^{x}_{t}+\hat{\lambda }^{x}_{t}\right) . \end{aligned}$$

(A.20)

Marginal cost: exported goods, $\hat{mc}^{x}_{t}$

$$\begin{aligned} \hat{mc}^{x}_{t}=\hat{mc}^{x}_{t-1}+\hat{\pi }^{d}_{t}-\hat{\pi }^{x}_{t} -\Delta \hat{S}_{t}. \end{aligned}$$

(A.21)

1.3 Government

Liabilities, $\hat{\ell }_{t}$

$$\begin{aligned} \ell \hat{\ell }_{t}= & {} \frac{1}{\mu ^z\pi } \Bigg (b\hat{b}_{t-1} + m \hat{m}_{t-1} -(b+m)\left[ \hat{\mu }^z_{t} + \hat{\pi }^{d}_{t}\right] \Bigg )\nonumber \\&+ \frac{b_L}{\left( \mu ^z\, \pi \right) ^{L}} \Bigg (\hat{b}_{L,t-L} - \sum _{k=0}^{L-1}\hat{\pi }^{d}_{t-k} -\sum _{k=0}^{L-1}\hat{\mu }^z_{t-k}\Bigg ). \end{aligned}$$

(A.22)

Expenditure’s budget constraint, $\hat{g}_t$

$$\begin{aligned} g\hat{g}_t + \ell \hat{\ell }_t = \tau \hat{\tau }_t + m\hat{m}_t + \frac{b}{R}\left( \hat{b}_t - \hat{R}_t\right) +\frac{b_L}{(R_L)^L}\left( \hat{b}_{L,t} - L\hat{R}_{L,t}\right) . \end{aligned}$$

(A.23)

Taxation, $\hat{\tau }_{t}$

$$\begin{aligned} \hat{\tau }_{t}=\psi _1 \frac{\ell }{\tau }\hat{\ell }_{t}. \end{aligned}$$

(A.24)

1.4 The Central Bank

Interest rate policy rule, $\hat{R}_{t}^{}$

$$\begin{aligned} \hat{R}_{t}^{}=\rho _R\hat{R}_{t-1}^{} + \left( 1-\rho _R\right) \left[ \hat{\bar{\pi }}_t^c + \phi _{\pi }\left( \hat{\pi }_{t+1}^{c,4}-\bar{\hat{\pi }}_{t}^{c}\right) + \phi _{\Delta \pi }\hat{\pi }_{t}^{c} + \phi _y\hat{y}_{t}^{} + \phi _{\Delta y}\Delta \hat{y}_{t}^{}\right] + \varepsilon ^R_t, \end{aligned}$$

(A.25)

where CPI inflation is given by

$$\begin{aligned} \hat{\pi }^c_{t}=\left( 1-\vartheta _c\right) \left( \frac{1}{\gamma ^{c,d}} \right) ^{1-\eta _c}\hat{\pi }^{d}_{t}+\vartheta _c\left( \gamma ^{mc,c} \right) ^{1-\eta _c}\hat{\pi }^{m,c}_{t}. \end{aligned}$$

(A.26)

1.5 Relative prices, $\hat{\gamma }_{t}^{i,j}$

Consumption and investment goods

$$\begin{aligned} \hat{\gamma }_{t}^{c,d}= & {} \hat{\gamma }^{i,d}_{t-1}+\hat{\pi }_{t}^{c}-\hat{\pi }^{d}_{t}, \end{aligned}$$

(A.27)

$$\begin{aligned} \hat{\gamma }_{t}^{i,d}= & {} \hat{\gamma }^{i,d}_{t-1}+\hat{\pi }_{t}^{i}-\hat{\pi }^{d}_{t}. \end{aligned}$$

(A.28)

Imported consumption and investment goods

$$\begin{aligned} \hat{\gamma }_{t}^{mc,d}= & {} \hat{\gamma }^{mc,d}_{t-1}+\hat{\pi }_{t}^{m,c}-\hat{\pi }^{d}_{t}, \end{aligned}$$

(A.29)

$$\begin{aligned} \hat{\gamma }_{t}^{mi,d}= & {} \hat{\gamma }^{mi,d}_{t-1}+\hat{\pi }_{t}^{m,i}-\hat{\pi }^{d}_{t}. \end{aligned}$$

(A.30)

Export goods

$$\begin{aligned} \hat{\gamma }^{x,*}_{t} = \hat{\gamma }^{x,*}_{t-1}+\hat{\pi }_{t}^{x}-\hat{\pi }^{*}_{t}. \end{aligned}$$

(A.31)

Domestic–foreign goods relative price

$$\begin{aligned} \hat{\gamma }_t^f = \hat{mc}^{x}_{t}+\hat{\gamma }^{x,*}_{t}. \end{aligned}$$

(A.32)

Real exchange rate

$$\begin{aligned} \hat{\gamma }_t^s = -\vartheta _c \left( \frac{1}{\gamma ^{mc,c}}\right) ^{\eta _c-1}\hat{\gamma }_t^{mc,d} -\hat{\gamma }^{x,*}_{t} - \hat{mc}^{x}_{t}. \end{aligned}$$

(A.33)

1.6 Market clearing

Domestic goods market, $\hat{y}_{t}$

$$\begin{aligned} \hat{y}_{t}= & {} \frac{1}{1-\dfrac{\phi _L}{2}}\bigg [\left( 1-\vartheta _c\right) \left( \gamma ^{c,d}\right) ^{\eta _c}\frac{c}{y}\left( \hat{c}_{t}+\eta _c \hat{\gamma }^{c,d}_{t}\right) +\left( 1-\vartheta _i\right) \left( \gamma ^{i,d}\right) ^{\eta _i}\frac{i}{y} \left( \hat{i}_{t}+\eta _i\hat{\gamma }^{i,d}_{t}\right) \nonumber \\&+\, g_y\hat{g}_{t}+\frac{y^*}{y}\left( \hat{y}^{*}_{t}-\eta _f \hat{\gamma }^{x,*}_{t}+\hat{\tilde{z}}^{*}_{t}\right) +\frac{r^k}{\mu ^z}\frac{k}{y}\left( \hat{k}^s_{t}-\hat{k}_{t}\right) + \phi _L\left( \hat{b}_{L,t}-\hat{b}_{L,t-1}\right) \bigg ].\nonumber \\ \end{aligned}$$

(A.34)

Net foreign assets, $\hat{a}_{t}$

$$\begin{aligned} \hat{a}_{t}= & {} -y^*\hat{mc}^{x}_{t}-\eta _f y^* \hat{\gamma }^{x,*}_{t}+ y^*\hat{y}^{*}_{t}+ y^*\hat{\tilde{z}}^{*}_{t}+\left( c^m+i^m\right) \hat{\gamma }^{f}_{t}\nonumber \\&-\,\left[ c^m\left( -\eta _c\left( 1-\vartheta _c\right) \left( \gamma ^{c,d} \right) ^{\eta _c-1}\right) \hat{\gamma }^{mc,d}_{t}+\hat{c}_{t}\right] \nonumber \\&-\,\left[ i^m\left( -\eta _i\left( 1-\vartheta _i\right) \left( \gamma ^{i,d} \right) ^{\eta _i-1}\right) \hat{\gamma }^{mi,d}_{t}+\hat{i}_{t}\right] \nonumber \\&+\frac{\pi ^*}{\pi }\, \frac{1}{\beta }\, \hat{a}_{t-1}. \end{aligned}$$

(A.35)

1.7 AR(1) shock processes

$$\begin{aligned} \Xi _t = \rho \Xi _{t-1} + \Gamma _t, \end{aligned}$$

(A.36)

where

$$\begin{aligned} \Xi _t= & {} [\hat{\xi }^{c}_{t}\;\;\hat{\xi }^{i}_{t}\;\;\hat{\tilde{\phi }}^{}_{t} \;\;\hat{\varepsilon }^{}_{t}\;\;\hat{\xi }^{H}_{t}\;\;\hat{\lambda }^{x}_{t}\;\; \hat{\lambda }^d_{t}\;\;\hat{\lambda }^{m,c}_{t}\;\;\hat{\lambda }^{m,i}_{t}\;\; \hat{\tilde{z}}^{*}_{t}\;\;\hat{\mu }^z_{t}\;\;\hat{g}_{t}\;\;\hat{\bar{\pi }}_t^c\;\; \hat{b}_{L,t}]'\\ \rho= & {} [\rho _{c}\;\;\rho _{i}\;\;\rho _{\tilde{\phi }}\;\;\rho _{\varepsilon } \;\;\rho _{H}\;\;\rho _{\lambda ^{x}}\;\;\rho _d\;\;\rho _{\lambda ^{m,c}}\;\; \rho _{\lambda ^{m,i}}\;\;\rho _{\tilde{z}^*}\;\;\rho _{\mu ^z}\;\;\rho _{g}\;\; \rho _{\bar{\pi }^c}\;\;\rho _L]'\\ \Gamma _t= & {} [\varepsilon ^{c}_{t}\;\;\varepsilon ^{i}_{t}\;\; \varepsilon _t^{\tilde{\phi }}\;\;\varepsilon ^{\varepsilon }_{t}\;\; \varepsilon ^{H}_{t}\;\;\varepsilon ^{x}_{t}\;\;\varepsilon ^{d}_{t}\;\; \varepsilon ^{m,c}_{t}\;\;\varepsilon ^{m,i}_{t}\;\;\varepsilon ^{\tilde{z}^*}_{t}\;\; \varepsilon ^{\mu ^z}_{t}\;\;\varepsilon ^{g}_{t}\;\;\varepsilon ^{\bar{\pi }^c}_t\;\; \varepsilon ^{L}_t]'. \end{aligned}$$