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An Estimated Stochastic General Equilibrium Model with Partial Dollarization: A Bayesian Approach


In this paper, we develop and estimate a dynamic stochastic, general-equilibrium New Keynesian model with partial dollarization. Bayesian techniques and Peruvian data are used to evaluate two forms of dollarization: currency substitution (CS) and price dollarization (PD). The empirical results are as follow: first, it is noted that the two forms of partial dollarization are important in explaining the significance of the Peruvian data. Second, models with both forms of dollarization dominate models without dollarization. Third, a counterfactual exercise shows that by eliminating both forms of partial dollarization, the response of both output and consumption to a monetary policy shock doubles, making the interest rate channel of monetary policy more effective. Fourth, based on the variance decomposition of the preferred model (with CS and PD), it is found that demand type shocks explain almost all the fluctuation in CPI inflation, the monetary shock being the most important (39%). Remarkably, foreign disturbances account for 34% of the output fluctuations.

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  1. Castillo (2006a) and Felices and Tuesta (2007) build up small open economies with currency substitution showing the limitations of the central bank in stabilizing inflation and output gap. Batini et al. (2006) further analyzed the rational expectations determinacy under interest rate monetary policy rules in economies with CS. They found that conditions for determinacy of the rational expectation equilibrium are more difficult to meet when CS is present.

  2. We are not aware of any formal work comparing and estimating DSGE models with different forms of partial dollarization.

  3. Castillo et al. (2009) modeled financial dollarization along with currency substitution and price dollarization. They performed simulation exercises rather than estimation.

  4. Estimation of reduced-forms or partial equilibrium models might suffer from serious identification problems.

  5. Fernández-Villaverde and Rubio-Ramírez (2004) showed that, even in the case of misspecified models, Bayesian estimation and model comparison are consistent.

  6. In contrast, Galí and Monacelli (2005) modeled a small open economy by considering the world economy is determined by a continuum of small open economies.

  7. In the same vein, Castillo et al. (2009) extends this model including also financial dollarisation by introducing the financial accelerator mechanism in two currencies.

  8. Also, models with external habit formation have proven to be useful in accounting for asset prices empirical regularities. For instance Campbell and Cochrane (1999) showed that introducing a time-varying subsistence level to a basic isoelastic power utility function allowed to solve for a series of puzzles related to asset prices such as: the equity premium puzzle, countercyclical risk premium and forecastability of excess return of stocks.

  9. Since all households face the same consumption basket, we drop the superindex j in the definitions of the baskets.

  10. We follow Benigno (2009) who modeled incomplete markets in a two-country model. Schmitt-Grohe and Uribe (2003) developed a small open-economy models introducing the same cost to achieve stationarity.

  11. This assumption allows us to work with the aggregate economy as a representative agent model economy. Otherwise, we would have to keep track of the wealth position of each household in the economy.

  12. As Benigno (2009) pointed out, some restrictions on Ψ B (.) are necessary: Ψ B (0) = 1; assumes the value 1 only if \(B_{t}^{\ast }=0;\) differentiable; and decreasing in the neighborhood of zero.

  13. Notice that in this case, it is possible to obtain the unemployment rate as an equilibrium variable by comparing the level of hours employed when, λ wp  = 0 with those where real rigidities are present, thus we can define: \(ur_{t}=\frac{L_{\lambda _{wp},t}}{L_{t}}\), where ur t represents the unemployment rate, L t denotes equilibrium hours when Eq. 2.8 holds, finally, \(L_{\lambda _{wp},t}\) represents the level of hours when real rigidities at the labor market are present.

  14. It is worth noting that our terms of trade definition are the inverse to the traditional definitions in standard open economy literature.

  15. Following Benigno and Woodford (2005), the previous first-order condition can be written in a recursive way using two auxiliary variables, \(V_{t}^{D}\) and \(V_{t}^{N}\), defined as follows: \(\left[ \frac{\widetilde{P}_{t}^{H}(z)}{ P_{t+k}^{H}}\right] =\frac{V_{t}^{N}}{V_{t}^{D}},\) where \( V_{t}^{N}=MUP_{t}Y_{t}^{H}U_{C,t}MC_{t}^{H}+\theta ^{H}\beta E_{t}\left\{ \left[ \Pi _{t+1}^{H}\left( \Pi _{t}^{H}\right) ^{-\lambda _{\pi }}\right] ^{\varepsilon }V_{t+1}^{N}\right\} \)and \(V_{t}^{D}=Y_{t}^{H}U_{C,t}+\theta ^{H}\beta E_{t}\left\{ \left[ \Pi _{t+1}^{H}\left( \Pi _{t}^{H}\right) ^{-\lambda _{\pi }}\right] ^{\left( \varepsilon -1\right) }V_{t+1}^{D}\right\}.\)

  16. Similarly to the case of the Phillips Curve for the home-goods sector, it is possible to write first-order condition using two auxiliary variables, \( V_{t}^{N,M}\) and, \(V_{t}^{D,M}\), which in turn are defined as follows: \( V_{t}^{N,M}=MUP_{t}^{M}Y_{t}^{M}U_{C,t}LOP_{t}+\theta ^{M}\beta E_{t}\left[ \left( \Pi _{t+1}^{M}\right) ^{\epsilon }V_{t+1}^{N,M}\right],\) and \( V_{t}^{D,M}=Y_{t}^{M}U_{C,t}+\theta ^{M}\beta E_{t}\left[ \left( \Pi _{t+1}^{M}\right) ^{\epsilon -1}V_{t+1}^{D,M}\right] .\)

  17. As Woodford (2003) shows, interest rate smoothing might reflect an optimal behavior for the central bank when there exist transaction frictions.

  18. See Woodford (2003) chapter 2 for a brief discussion related to the consequences of nonseparable utility function in price determination.

  19. See Castillo (2006b) for a model in which price dollarization arises endogenously.

  20. Data on output, consumption and investment have been previously seasonally adjusted with the X-12 Arima method.

  21. See Castillo et al. (2011) for a discussion.

  22. For a detailed analysis of the stylized facts of the Peruvian economy for a broader period see Castillo et al. (2006).

  23. See the Appendices for some details on the estimation. See Fernández-Villaverde and Rubio-Ramírez (2004) for computational details. An and Schorfheide (2007) also provided useful details on the estimation procedure.

  24. Hence, we avoid the discussion on which de-trending method (linear, quadratic or HP-filter) to use.

  25. Previous studies for the Peruvian economy estimated this parameter in a range of 0.5--0.7.

  26. The elasticity of substitution between home and foreign goods is a source of controversy. Trade studies typically find values for the elasticity of import demand to respect to price (relative to the overall domestic consumption basket) in the neighborhood of 5 to 6 (see Trefler and Lai 1999). Most of the NOEM models consider values of 1 for this elasticity, which implies Cobb-Douglas preferences in aggregate consumption. Rabanal and Tuesta (2010), in an estimated two-country model found values for this elasticity–conditional on the asset market structure (complete and incomplete markets)—between 0 and 1.

  27. Selaive and Tuesta (2008) found values around 0.007 and 0.003 for OECD countries. This exogenous cost is only useful to make the net foreign debt position stationary.

  28. As is standard in the Markov Chain Monte Carlo methods, the initial 20% of draws were discarded, and the variance-covariance matrix of the perturbation term in the algorithm was adjusted such that the acceptance rate lies between 25 and 35%.

  29. To gauge the robustness of our baseline estimates, we allow for an even looser prior on δ CSand δ PD. We consider an alternative specification that is centered at 0.5 with standard deviation of 0.3. Estimations are available upon request.

  30. Rabanal and Tuesta (2007) found that non-traded goods explain 1/3 of the volatility of the real exchange rate.

  31. In constrast, Justiniano and Preston (2010a) found that foreign shocks (U.S. economy shocks) can account for at most 1% of the variation in Canadian output, inflation and interest rates.

  32. We calibrate the levels of the domestic and foreign productivity such that all relative prices are equal to one in steady state. That is: RER = TOT = T H,T = T M,T = 1.

  33. These types of shocks (PPP or UIP shocks) only helps to better fit individual equations. They do not appear anywhere else in the model, hence they do not imply any additional cross-equation restriction.

  34. This broad figure was calculated by comparing the volatility o non tradable productivity volatility with respect to the real exchange rate volatility at annual frequencies.


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We are especially grateful to an anonymous referee for helpful comments. We would like to thank Alberto Humala, Pau Rabanal, Marco Vega, David Vavra and participants at the Central Bank Workshop on Macroeconomic Modeling organized by the Central Bank of Chile and at the LACEA-LAMES 2006 meeting for useful suggestions and comments. The views expressed in this paper are those of the authors and should not be interpreted as reflecting the views of the Board of the Central Reserve Bank of Peru.

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Correspondence to Vicente Tuesta.


Appendix A: Solving the Steady-State

We know that non-stationary variables grow in the balanced growth path at rate g, we normalize this variables by the level of technology. We denote these variables with tilde, such that \(\widetilde{Z}_{t}=\frac{Z_{t}}{X_{t}}\) is stationary. Also, we denote variables without time subscprits as variables in the steady state.

Replace the functional form of the marginal utility in the Euler equation

$$ \frac{\xi _{t}}{\tilde{C}_{t}X_{t}-h\tilde{C}_{t-1}X_{t-1}}=\left( 1+i_{t}\right) \beta E_{t}\left\{ \frac{\xi _{t+1}}{\tilde{C}_{t+1}X_{t+1}-h \tilde{C}_{t}X_{t}}\right\}, $$

divide both sides in the numerator by X t and evaluated at the steady state, we obtain the interest rate in steady state

$$ \left( 1+i\right) =\frac{1+g}{\beta }, $$

Similarly, under the assumption that trading frictions in asset markets are zero in steady-state, thus, \(\phi \left( NFD\right) =1\), Eq. 2.2 implies, that DS = 1. From the Phillips Curve in the home produced goods sector, Eq. 2.3 we have

$$ MC^{H}=\frac{1}{\mu }=\frac{\varepsilon }{\varepsilon -1}, $$

From Eqs. 2.8 and 2.9 we get

$$ \left( 1+i\right) =R^{KH}=\frac{1+g}{\beta }, $$
$${R}^{H}=\frac{1+g}{\beta}-\left({1-\delta}\right) ={\delta}+\left(\frac{{1+g}-{\beta}}{\beta}\right),$$

Similarly using Eqs. 2.24 and 2.25, we obtain from the law of motion of capital:

$$ \widetilde{INV}^{H}=\frac{\delta +g}{1+g}\widetilde{K}^{H}, $$

From the Eq. 2, the marginal productivity of capital, we obtain the capital—output ratio

$$ \frac{\widetilde{K}^{H}}{\widetilde{Y}^{H\ }}=\alpha ^{H}\left( 1+g\right) T^{H}\frac{MC^{H}}{R^{H}}, $$

after replacing out the steady state values of T H, MC H and R H, the above equation can be re-written as:Footnote 32

$$ \frac{\widetilde{K}^{H}}{\widetilde{Y}^{H\ }}=\frac{\alpha ^{H}\left( 1+g\right) }{\mu \left[ \delta +\left( \frac{1+g-\beta }{\beta }\right) \right] }, $$

Plugging Eq. A.6 in Eq. A.5, we obtain the investment-output ration

$$ \frac{\widetilde{INV}^{H}}{\widetilde{Y}^{H}}=\frac{\left( \delta +g\right) \alpha ^{H}}{\mu \left[ \frac{1+g}{\beta }-\left( 1-\delta \right) \right] }, $$

We take as given net foreign asset-output ratio

$$ \frac{\widetilde{NFD}}{\widetilde{Y}^{H}}=\gamma _{B}, $$

From the agregate resource constraint, the net exports-output ratio is

$$ \frac{\widetilde{NX}}{\widetilde{Y}^{H}}=-\gamma _{B}\frac{1-\beta }{1+g}, $$

the absortion-output ratio is

$$ \frac{\widetilde{ABS}}{\widetilde{Y}^{H}}=1-\frac{\widetilde{NX}}{\widetilde{Y}^{H}}, $$

and the consumption-output ratio is

$$ \frac{\widetilde{C}}{\widetilde{Y}^{H}}=\frac{\widetilde{ABS}}{\widetilde{Y} ^{H}}-\frac{\widetilde{INV}^{H}}{\widetilde{Y}^{H}}, $$

The steady state of the rest of the variables are a function of these ratios (Table 7).

Table 7 Implied steady state relationships

Appendix B: Log-linear system, benchmark model without dollarization

This section summarizes the log-linear equations of the benchmark economy. We take a log-linear approximation of the equations of the model around a deterministic steady-state with zero inflation defined in the previous section. Variables in log linear deviations from the steady-state are denoted by lower case letters, \(z=\log (\frac{Z_{t}}{Z}).\) We normalize all real variables by the level of technology to make them stationary. Normalized variables are denoted with tilde, i.e, \(\widetilde{Z}_{t}=\frac{ Z_{t}}{X_{t}}\).

The benchmark economy contains 46 equations for 48 endogenous variables and 8 exogenous shocks. We include 4 more equations when we add both CS and PD to the benchmark model.

We divide the system of equations as follows: a) households and firms optimal allocation decisions, b) monetary policy rule, c) market clearing conditions, d) the equilibrium for the foreign economy, e) the exogenous process of shocks and, f) observable variables.

B.1  Households

B.1.1  First Order Conditions

The euler equation for the representative consumer is

$$ \widetilde{u_{ct}}=\left( i_{t}-E_{t}\pi _{t+1}\right) +E_{t}\widetilde{ u_{ct+1}}, $$

The uncovered interest rate parity condition

$$ \left( i_{i}-E_{t}\pi _{t+1}\right) -\left( i_{t}^{\ast }-E_{t}\pi _{t}^{M^{\ast }}\right) =E_{t}\Delta rer_{t+1}-\psi _{B}nfd_{t}+uip_{t}, $$

where uip t denotes the shock to the uncovered interest rate parity condition, \(nfd_{t}=\left( NFD_{t}-\widetilde{NFD}\right) /\widetilde{Y^{H}}\) and \(NFD_{t}=\frac{B_{t}^{\ast }S_{t}}{P_{t}}\).

The marginal rate of substitution is equal to

$$ \widetilde{mrs}_{t}=v_{lt}-u_{ct}, $$

where v lt is the marginal desutility of labor

$$ v_{lt}=\eta l_{t}, $$

and u ct is the marginal utility of consumption

$$ u_{ct}=-\left( \left[ \frac{1+g}{1+g-h}\right] \widetilde{c}_{t}-\frac{h}{ \left( 1+g-h\right) }\widetilde{c}_{t-1}-\frac{h}{\left( 1+g-h\right) }\mu _{t}^{x}\right) +\xi _{t}, $$

where \(\mu _{t}^{x}=\Delta x_{t}\) is the unit root shock.

Real wages evolve according to the following equation:

$$ \widetilde{wp}_{t}=\lambda _{wp}\left( \widetilde{wp}_{t-1}-\mu _{t}^{x}\right) +\left( 1-\lambda _{wp}\right) \widetilde{mrs}_{t} $$

B.1.2  Consumption Demands

The domestic demands for home produced and imported goods are

$$ c_{t}^{H} =-\varepsilon _{H}t_{t}^{H}+\widetilde{c}_{t}, $$
$$ c_{t}^{M} =-\varepsilon _{H}t_{t}^{M}+\widetilde{c}_{t}, $$

and the foreign demand for domestic goods is

$$ c_{t}^{H^{\ast }}=-\varepsilon _{H}\left( t_{t}^{H}-rer_{t}\right) +y_{t}^{\ast } $$

B.1.3  Price Indexes and Relative Prices

The price indexes are defined from the optimal allocation of consumption across goods. Total inflation is given by:

$$ \pi _{t}=\gamma _{H}\pi _{t}^{H}+(1-\gamma _{H})\pi _{t}^{M}, $$

We have defined the terms of trade (TOT) as follows: \(TOT=\frac{ S_{t}P_{t}^{H^{\ast }}}{P_{t}^{M}}=\frac{P_{t}^{H}}{P_{t}^{M}}\). Hence, relative prices of domestic goods/tradable goods and imported goods/tradable goods are

$$ t_{t}^{H}=\left( 1-\gamma _{H}\right) tot_{t}, $$
$$ \gamma _{H}t_{t}^{H}+(1-\gamma _{H})t_{t}^{M,T}=0, $$

The evolution of the real exchange rate, the terms of trade and deviations to the law of one price are given by:

$$ rer_{t}=-\gamma _{H}tot_{t}+lop_{t}, $$
$$ tot_{t}=tot_{t-1}+\pi _{t}^{H}-ds_{t}-\pi _{t}^{M^{\ast }}, $$
$$ lop_{t}=lop_{t-1}+\pi _{t}^{M}-ds_{t}-\pi _{t}^{M^{\ast }} $$

B.2  Firms

B.2.1  Intermediate Goods Firms

The production function for the intermediate goods firm

$$ \widetilde{y}_{t}^{H}=\left( 1-\alpha _{H}\right) l_{t}^{H}+\alpha _{H}\left( \widetilde{k}_{t-1}^{H}-\mu _{t}^{x}\right) +z_{t}^{H}, $$

The first order conditions for the firm equalize the marginal productivities to the rental price of labor and capital

$$ \widetilde{wp}_{t} =mc_{t}^{H}+t_{t}^{H}+\widetilde{y}_{t}^{H}-l_{t}^{H}, $$
$$ r_{t}^{H} =mc_{t}^{H}+t_{t}^{H}+\widetilde{y}_{t}^{H}-\widetilde{k} _{t-1}^{H}+\mu _{t}^{x} $$

B.2.2  Capital Goods firms

New capital is produced using the following technology

$$ \widetilde{k}_{t}^{H}=\frac{\left( 1-\delta \right) }{1+g}\widetilde{k} _{t-1}^{H}+\left( 1-\frac{\left( 1-\delta \right) }{1+g}\right) \widetilde{ inv}_{t}^{H}-\frac{\left( 1-\delta \right) }{1+g}\mu _{t}^{x}, $$

Optimal investment made by the firms that produce unfinished capital goods satisfies the following optimality condition

$$ q_{t}^{H}=\psi _{K}\left( \widetilde{inv}_{t}^{H}-\widetilde{k} _{t-1}^{H}+\mu _{t}^{x}\right), $$

where \(\psi _{K}=\frac{\Psi _{K}^{\prime \prime }\left( INV^{H}/K^{H}\right) }{\Psi _{K}^{\prime }\left( INV^{H}/K^{H}\right) }\frac{INV^{H}}{K^{H}}\) is the adjustment costs elasticity and \(q_{t}^{H}\) is the relative price of capital goods with respect to final goods.

B.2.3  Investors

The optimal conditions that determines the level of new capital goods are given by:

$$ \begin{array}{rll} 0 &=&-E_{t}r_{t+1}^{KH}+E_{t}\widetilde{u_{ct+1}}-\widetilde{u_{ct}}, \\ \left( q_{t}^{H}+r_{t+1}^{KH}\right) &=&\left[ \left( 1-\left( 1-\delta \right) \beta \right) r_{t+1}^{H}+\left( 1-\delta \right) \beta q_{t+1}^{H}-\beta \psi _{K}\left( \widetilde{inv}_{t+1}^{H}-\widetilde{k} _{t}^{H}\right) \right], \end{array} $$

plugging the above equations into the euler equation we get

$$\begin{array}{rll} &&E_{t}\left[ \left( 1-\left( 1-\delta \right) \beta \right) r_{t+1}^{H}+\left( 1-\delta \right) \beta q_{t+1}^{H}-\beta \psi _{K}\left( \widetilde{inv}_{t+1}^{H}-k_{t}^{H}\right) \right] -q_{t}^{H}\\ &&i_{t}-E_{t}\pi _{t+1} \end{array}$$

B.2.4  Final Goods Producers (Retailers)

Phillips curve for the home produced goods

$$ \pi _{t}^{H}-\lambda _{H}\pi _{t-1}^{H}=\kappa _{H}\left( mc_{t}^{H}+mup_{t}\right) +\beta E_{t}\left( \pi _{t+1}^{H}-\lambda _{H}\pi _{t}^{H}\right), $$

where \(\kappa _{H}\equiv \frac{\left( 1-\theta _{H}\right) }{\theta _{H}} \left( 1-\theta _{H}\beta \right) \).

B.2.5  Distributors of Imported Goods

Similarly, aggregating the optimal price setting in the importing sector

$$ \pi _{t}^{M}=\kappa _{M}\left( lop_{t}+mup_{t}^{M}\right) +\beta E_{t}\pi _{t+1}^{M}, $$

where \(\kappa _{M}\equiv \frac{\left( 1-\theta _{M}\right) }{\theta _{M}} \left( 1-\theta _{M}\beta \right) .\)

B.3  Policy Rule

The policy rule followed by the monetary authority is

$$ i_{t}=\varphi _{i}i_{t-1}+\left( 1-\varphi _{i}\right) \left[ \varphi _{\pi }E_{t}\pi _{t}+\varphi _{y}\bigtriangleup y_{t}+\varphi _{s}ds_{t}\right] +mon_{t} $$

B.4  Market Clearing

Absortion is defined by the sum of consumption and investment

$$ abs_{t}=\frac{\widetilde{C}}{\widetilde{ABS}}\widetilde{c}_{t}+\frac{ \widetilde{INV}}{\widetilde{ABS}}\widetilde{inv}_{t}, $$

Aggregating the resources constraint of the economy, we obtain an equation for the net foreign asset accumulation

$$ \left( 1\!+\!\gamma _{B}\psi _{B}\right) \beta nfd_{t}\!=\!\gamma _{B}\left( \beta i_{t}^{\ast }\!-\!\pi _{t}\!+\!ds_{t}\right) \!+\!\frac{1}{1+g}nfd_{t-1}\!+\!\widetilde{y} _{t}^{H}\!+\!t_{t}^{H}\!-\!\frac{\widetilde{ABS}}{\widetilde{Y}^{H}}abs_{t}, $$

The demand for domestic produced good is given by:

$$ \widetilde{y}_{t}^{H}=\gamma _{H}\frac{\widetilde{C}}{\widetilde{Y}^{H}} c_{t}^{H}+\left( 1-\gamma _{H}\right) \frac{\widetilde{C}}{\widetilde{Y}^{H}} c_{t}^{H^{\ast }}+\gamma _{H}\frac{\widetilde{INV}^{H}}{\widetilde{Y}^{H}} inv_{t}^{H,d}, $$

The demand for investment in home produced and imported goods

$$ inv_{t}^{H,d} =-\varepsilon _{H}t_{t}^{H}+\widetilde{inv}_{t}^{H}, $$
$$ inv_{t}^{M,d} =-\varepsilon _{H}t_{t}^{M}+\widetilde{inv}_{t}^{H}, $$

Net exports are defined as:

$$ \frac{NX}{Y^{H}}nx_{t}=y_{t}+t_{t}^{H}-\frac{ABS}{Y^{H}}abs_{t} $$

B.5  Foreign Economy

The aggregate demand, the Phillips Curve and the monetary policy rule for the foreign economy are the following:

$$ E_{t}\left( y_{t+1}^{\ast }-y_{t}^{\ast }\right) =\left( i_{t}^{^{\ast }}-E_{t}\pi _{t+1}^{M^{\ast }}\right), $$
$$ \pi _{t}^{M^{\ast }} =E_{t}\pi _{t+1}^{M^{\ast }}+\kappa ^{\ast }y_{t}^{\ast }, $$
$$ i_{t}^{^{\ast }} =\varphi _{\pi ^{\ast }}\pi _{t}^{M^{\ast }}+\varphi _{y^{\ast }}y_{t}^{\ast }+mon_{t}^{\ast }, $$

where \(\kappa ^{\ast }=\frac{\left( 1-\theta ^{\ast }\right) }{\theta ^{\ast }}\left( 1-\theta ^{\ast }\beta \right) .\)

B.6  Exogenous Shocks

Preferences shock

$$ \xi _{t}=\rho _{\xi }\xi _{t-1}+\mu _{t}^{\xi }, $$

domestic productivity shock

$$ z_{t}^{^{H}}=\rho _{Z^{H}}Z_{t-1}^{^{H}}+\mu _{t}^{Z^{H}}, $$

domestic interest rate shock

$$ mon_{t}=\rho _{MON}mon_{t-1}+\mu _{t}^{MON}, $$

mark-up shock

$$ mup_{t}=\rho _{MUP}mup_{t-1}+\mu _{t}^{MUP}, $$

imported sector mark-up shock

$$ mup_{t}^{M}=\rho _{MUP^{M}}mup_{t-1}^{M}+\mu _{t}^{MUP^{M}}, $$

uncovered interest rate parity shock

$$ uip_{t}=\rho _{uip}uip_{t-1}+\mu _{t}^{uip}, $$

foreign interest rate shock

$$ mon_{t}^{\ast }=\rho _{MON^{\ast }}mon_{t-1}^{\ast }+\mu _{t}^{MON^{\ast }} $$

B.7  Observable Variables

We use the following eight observable variables for the estimation:

$$ \left\{ \bigtriangleup c_{t},\bigtriangleup rer_{t},\bigtriangleup y_{t},\bigtriangleup inv_{t},\Delta wp_{t},\bigtriangleup tot_{t},i_{t,}\pi _{t}\right\} $$

The corresponding transformed equations that describe the dynamics of the previous variables are given by:

$$ \widetilde{c}_{t}-\widetilde{c}_{t-1} =\bigtriangleup c_{t}-\mu _{t}^{x}, $$
$$ \widetilde{inv}_{t}-\widetilde{inv}_{t-1} =\bigtriangleup inv_{t}-\mu _{t}^{x}, $$
$$ \widetilde{y}_{t}-\widetilde{y}_{t-1} =\bigtriangleup y_{t}-\mu _{t}^{x}, $$
$$ \widetilde{wp}_{t}-\widetilde{wp}_{t-1} =\bigtriangleup wp_{t}-\mu _{t}^{x}, $$
$$ \bigtriangleup rer_{t} =rer_{t}-rer_{t-1}, $$
$$ \bigtriangleup tot_{t} =tot_{t}-tot_{t-1} $$

B.8  Dollarization

B.8.1  Currency Substiturion

Under this new formulation the marginal utility of consumption can be expressed in terms not only of consumption but also of both foreign and domestic interest rates and their relative weights are sensitive to the ratio of foreign currency in the total money aggregates. The marginal utility of consumption with CS adopts the following log-linear form:

$$ u_{ct}^{CS}=u_{ct}+\Lambda \left[ \left( 1-\delta ^{cs}\right) i_{t}+\delta ^{cs}i_{t}^{\ast }\right], $$

where Λ ≡ β(ω − 1)(1 − b). where u ct corresponds to the marginal utility of consumption of the benchmark economy.

B.8.2  Price Dollarizartion

We introduce price dollarization (PD) by exogenously assuming that a subset of firms that produce home goods set their prices in foreign currency. We further assume that prices in foreign currency are also sticky. The derivation of this new Phillips curve follows exactly the same steps as the once described in Section 2.6.2 of the paper. We present the log-linear expressions that characterize the model with PD. The log-linear expressions read as follows:

$$ \pi _{Ht}=\left( 1-\delta ^{pd}\right) \pi _{s,t}+\delta ^{pd}\left( \pi _{d,t}+ds_{t}\right), $$
$$ \pi _{s,t}-\lambda _{\pi _{s}}\pi _{s,t-1}=\beta \left( E_{t}\pi _{s,t+1}-\lambda _{\pi _{s}}\pi _{s,t}\right) +\kappa _{H}mc_{t}+\kappa _{H}\delta ^{pd}rpd_{t}, $$
$$ \pi _{d,t}\!-\!\lambda _{\pi _{d}}\pi _{d,t-1}\!=\!\beta \left( E_{t}\pi _{d,t+1}\!-\!\lambda _{\pi _{d}}\pi _{d,t}\right) \!+\!\kappa _{PD}mc_{t}\!-\!\kappa _{PD}\left( 1\!-\!\delta ^{pd}\right) rpd_{t}, $$
$$ \Delta rpd_{t}=ds_{t}+\pi _{d,t}-\pi _{s,t}, $$

The dynamics of domestic inflation is determined by four endogenous variables: the inflation of goods that arises from firms that set prices in domestic currency (soles), π s,t , the inflation of goods that comes from firms that set prices in foreign currency (dollars), π d,t , the marginal cost, mc t , and the relative price between soles and dollars denoted by rpd t  ≡ p s,t  − s t  − p d,t , where \(\lambda _{\pi _{s}}\) and \(\lambda _{\pi _{d}}\) indicates the degrees of price indexation for each type of firms, \(\kappa _{PD}=\frac{\left( 1-\theta _{PD}\right) \left( 1-\beta \theta _{PD}\right) }{\theta _{PD}}\) is the slope of the Phillips Curve with respect to marginal costs for the case in which firms set prices in foreign currency. To the extent that firms setting price in dollars face nominal rigidities, nominal prices in domestic currency differs from those set in foreign currency, hence p s,t  ≠ s t p d,t . Variable rpd t could be interpreted as a form of deviations from the law of one price within the country.

Overall, the main implications of PD are that, on the one hand, it increases the sensitivity of domestic inflation, π Ht , to the depreciation of the nominal exchange rate, and on the other hand, it adds endogenous persistence to both inflation and international relative prices.

Appendix C: The Likelihood Function and The Metropolis-Hastings Algorithm

C.1  The Law of Motion and the Likelihood Function

Let Ψ denote the vector of parameters that describe preferences, technology, the monetary policy rules, and the shocks in the small open economy model, d t be the vector of all endogenous variables (state and forward looking), z t the vector of exogenous variables (i.e. shocks), and ε t the vector of innovations. x t is the vector of the nine observable variables that will enter the likelihood function. The system of equilibrium conditions and the process for the exogenous shocks can be written as a second-order difference equation

$$ \begin{array}{rll} &&A({\Psi })E_{t}d_{t+1}=B({\Psi })d_{t}+C({\Psi })d_{t-1}+D({\Psi })z_{t}, \\ &&\begin{array}{lr} z_{t}=N({\Psi })z_{t-1}+\epsilon _{t}, &\quad\quad\quad \ \ E(\epsilon _{t}\epsilon _{t}^{^{\prime }})=\Sigma ({\Psi }). \end{array} \end{array} $$

We use standard solution methods for linear models with rational expectations to write the law of motion in state-space form. The transition and measurement equations are

$$ \begin{array}{rll} &{d}_{t}=F\left({\Psi}\right){d}_{t-1}+G\left({\Psi}\right){z}_{t},& \\ &\begin{array}{lr}{z}_{t}=N\left({\Psi}\right){z}_{t-1}+\epsilon_{t},& \quad\quad\quad \ \ E\left(\epsilon_{t}\epsilon_{t}^{^{\prime}}\right)=\Sigma\left({\Psi}\right), \end{array}& \end{array} $$


$$ x_{t}=Hd_{t} $$

Let \(y_{t}=[d_{t}^{^{\prime }},z_{t}^{^{\prime }}]^{\prime }\) be the vector of all variables, endogenous and exogenous. The evolution of the system can be rewritten as

$$ y_{t}=\widetilde{A}y_{t-1}+\widetilde{B}\xi _{t}\ {\rm where}\ E\left( \xi _{t}\xi _{t}^{^{\prime }}\right) =I,\widetilde{B}=\widetilde{C}\Sigma ^{1/2}, \ {\rm and}\ \epsilon _{t}=\Sigma ^{1/2}\xi _{t} $$


$$ x_{t}=\widetilde{D}y_{t}, $$

The \(\widetilde{A},\widetilde{B},\widetilde{C}\) and \(\widetilde{D}\) matrices are functions of F, G, N, and Σ. The matrix \(\widetilde{D}\) contains zeros everywhere, and a one in each row to select the variable of interest from the vector of all variables y t . We can evaluate the likelihood function of the observable data conditional on the parameters \( L(\{x_{t}\}_{t=1}^{T}\mid {\Psi })\), by applying the Kalman filter recursively as follows

Define the prediction error as

$$ v_{t}=x_{t}-x_{t\mid t-1}=x_{t}-\widetilde{D}y_{t\mid t-1} $$

whose mean squared error is

$$ K_{t}=\widetilde{D}P_{t\mid t-1}\widetilde{D}^{\prime } $$

where x t|t − 1 is the conditional expectation of the vector of observed variables using information up to t − 1, and

$$ P_{t\mid t-1}=E\left[(y_{t}-y_{t\mid t-1})(y_{t}-y_{t\mid t-1})^{\prime }\right] $$

The updating equations are:

$$ y_{t}=y_{t\mid t-1}+P_{t\mid t-1}\widetilde{D}^{\prime }K_{t}^{-1}v_{t}\text{ and }P_{t}=P_{t\mid t-1}-P_{t\mid t-1}\widetilde{D}^{\prime }K_{t}^{-1} \widetilde{D}P_{t\mid t-1} $$

And the prediction equations are:

$$ y_{t+1\mid t}=\widetilde{A}y_{t},\text{ and }P_{t+1\mid t}=\widetilde{A}P_{t} \widetilde{A}^{\prime }+\widetilde{C}\Sigma \widetilde{C}^{\prime } $$

Then, the log-likelihood function is equal to

$$ L_{t}=-\frac{1}{2}\sum\limits_{t=1}^{T}\left\{n\log (2\pi)+\log [\det (K_{t})]+v_{t}^{^{\prime }}K_{t}^{-1}v_{t}\right\}. $$

where n is the size of the vector of observable variables x. Note that the log-likelihood function has to be computed recursively. To initialize the filter, we set y 0 = x 0 = 0, and we set P 0 as the solution to the nonlinear system of equations.

C.2  Drawing from the Posterior

To obtain a random draw of size N from the posterior distribution, a random walk Markov Chain using the Metropolis-Hastings algorithm is generated. The algorithm is implemented as follows:

  1. 1.

    Start with an initial value (Ψ0). From that value, evaluate the product \(L(\{x_{t}\}_{t=1}^{T}\mid {\Psi }^{0})\Pi ({ \Psi }^{0})\)

  2. 2.

    For each i:

    $$ \left\{ \begin{array}{c} {\Psi }^{i}={\Psi }^{i-1}\text{with probability }1-R \\[4pt] {\Psi }^{i}={\Psi }^{i,\ast }\text{ with probability }R \end{array} \right. $$

    where Ψi, ∗  = Ψi − 1 + v i, v i follows a multivariate Normal distribution, and

    $$ R=\min \left\{1,\frac{L(\{x_{t}\}_{t=1}^{T}\mid {\Psi })\Pi ({\Psi }^{i,\ast })}{ L(\{x_{t}\}_{t=1}^{T}\mid {\Psi }^{i-1})\Pi ({\Psi }^{i-1})}\right\} $$

The idea for this algorithm is that, regardless of the starting value, more draws will be accepted from the regions of the parameter space where the posterior density is high. At the same time, areas of the posterior support with low density (the tails of the distribution) are less represented, but will eventually be visited. The variance-covariance matrix of v i is proportional to the inverse Hessian of the posterior mode and the constant of proportionality is specified such that the random draw has some desirable time series properties.

In all cases, the acceptance rates were between 25 and 35%, and the autocorrelation functions of the parameters decay fairly fast. First, we find the posterior mode using standard optimization algorithms to be used as initial value. Then, we generate a chain of 250,000 draws.

Appendix D: Robustness Analysis

D.1  PPP Shocks

Even so we find evidence that favors the model with both type of dollarization, the model (CS and PD) does not perform well in matching some second moments like the volatility of real variables and the co-movement between international relative prices and output. As Lubik and Schorfheide (2006) and Rabanal and Tuesta (2010) pointed out, estimated open economies model predict a tension between the fit of domestic aggregate variables and international relative prices. Some authors have suggested some form of PPP shocks as a resolution.Footnote 33

In our economies we have not considered the role of non tradable goods. In the Peruvian economy nontradable goods accounts for about 2/3 of the real exchange rate volatility and they represent around 50% of the aggregate consumption bundle.Footnote 34 In order to capture this potential model misspecification we append a shock to the traditional equation of the real exchange rate dynamics

$$ rer_{t}=-\gamma _{H}tot_{t}+lop_{t}+ppp_{t} $$

where ppp t denotes the PPP shock. This shock might be capturing an additional channel through which we can induce deviations from PPP. In order to factor the importance of this shock we re-estimate the preferred model assuming the same prior for this shock as that considered for the rest of the shocks \(\left( ppp_{t}=\rho _{ppp}ppp_{t-1}+\mu _{t}^{ppp}\right) .\)

Results of the estimated parameters are reported in Table 8. Remarkably, this shock significantly improves the fit of the preferred model. Parameters estimates change in important dimensions. First the autoregressive coefficient of the monetary policy shock increases from 0.01 to 0.65. Likewise, the smoothness parameter in the policy reaction function of the bank increases from 0.03 to 0.22. Changes in those parameters might induce a larger effect of monetary policy over aggregate variables. The parameter that captures PD decreases from 0.47 to 0.39 whereas the one that measures the degree of CS increases from 0.40 to 0.56. Finally, prices in the domestic good sector become significant more flexible (θ H decreases form 0.52 to 0.27) where as prices in the imported distributor sector become stickier (θ H goes up from 0.45 to 0.82).

Table 8 Posterior distributions—robustness

More interestingly the PPP shock has a relative large standard deviation (around 6.04%) that compensates other shocks volatility. Indeed, we observe an important reduction in the estimated standard deviations of preference shocks with respect to the preferred model without PPP shock. Finally, the overall fit of the model increases as evidence of the reduction of the marginal likelihood.

D.1.1  Second Moments

Table 9 reports some selected moments comparing the preferred model with and without PPP shock. It shows that, in terms of both standard deviations and autocorrelations, the PPP model performs better than the one without this shock. For instance, the volatility of almost all variables moves closer to the data. Furthermore, the autocorrelation coefficient of the nominal interest rate increases from 0.09 in a model without PPP shock to 0.46 in a model with PPP, getting much closed to the value of 0.5 observed in the data.

Table 9 Selected second moments of the models—robustness

D.1.2  Variance Decomposition

The variance decomposition of the preferred model with PPP shock is reported in Table 10. Compared to the model without a PPP shock the variance decomposition changes in many grounds. The first observation is that PPP shocks mitigate all the effect of foreign interest rate shock over domestic business cycles. Indeed, the effect of foreign interest rate over aggregate variables and international relative prices become almost negligible.

Table 10 Contributions of the shocks to the variance, robustness (model with currency substitution, price dollarization and PPP shocks)

Monetary policy shocks still explain most of CPI inflation (58%). Finally, unlike the model without PPP shock, most of the variances of both the RER and the TOT are explained by the PPP shock.

D.2  Shorter Sample Period (1995–2006)

A simple inspection of Fig. 1 in the main paper shows that volatilities of real variables during the two first years of the sample were relatively high as a consequence of the post stabilization period. In this subsection we briefly discuss the results of reducing our sample period to starting in 1995:01, disregarding the period 1992–1994.

The main differences with respect to the preferred model are. First, the parameter on the reaction of the Taylor rule to both CPI inflation and output growth are estimated to be larger (from 1.94 to 2.20 and from 0.09 to 0.22, respectively). Second, the autoregressive coefficient of monetary policy shock increases significantly (from 0.01 to 0.31) although not that much as in the case of the preferred model with a PPP shock. Third, both the elasticity of substitution between tradable goods, ε H, and the inverse of the elasticity of labor supply, η, increase. In both cases the values almost double in the shorter sample. The large value of η is consistent with the high relative volatility of real wages with respect to the low volatility of output. The remaining parameters of the model do no change significantly.

In terms of second moments the implied standard deviations of the observable variables decrease with respect to the preferred model, getting closer to the one observed in the data (see Table 9).

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Castillo, P., Montoro, C. & Tuesta, V. An Estimated Stochastic General Equilibrium Model with Partial Dollarization: A Bayesian Approach. Open Econ Rev 24, 217–265 (2013).

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  • Bayesian estimation
  • DSGE
  • Partial dollarization

JEL Classification

  • F31
  • F32
  • F41
  • C11