Monetary and Exchange Rate Policy Under Remittance Fluctuations
Technical Appendix and Additional Results
Federico Mandelman1
February 2011
In this appendix, I provide technical details on the Bayesian estimation. I include: (a) a brief description
of the estimation methodology (b) The prior and posterior densities of the coe¢ cients of the benchmark
model. (c) Median impulse responses to all model shocks, including the 10 and 90 percent posterior intervals.
(d) Markov Chain Monte Carlo (MCMC) multivariate convergence diagnostics.
A
A.1
The Bayesian Estimation
Data Sources
Real output is from the National Economic and Development Authority (1985 pesos, seasonally adjusted
by Haver Analytics). The consumer price index is from the National Statistic O¢ ce. Data on remittances
is provided by the Central Bank of the Philippines. Original data is converted in Philippine Pesos and
seasonally adjusted with X12-ARIMA method from the US Bureau. The foreign interest rate of reference is
the US T-Bill rate (90 days) + EMBI Global Spread for the Philippines. Bank System’s domestic currency
interest rate data is from the Central Bank of the Philippines. I use the sample of 10 commercial banks’
actual interest expenses on peso-savings deposits to the total outstanding level of these deposits.
A.2
Estimation Methodology
In this section I brie‡y explain the estimation approach used in this paper. A more detailed description of
the method can be found in An and Schorfheide (2007), Fernández-Villaverde and Rubio-Ramírez (2004)
among others. Let’s de…ne
T
as the parameter space of the DSGE model, and z T = fzt gt=1 as the data
series used in the estimation. From their joint probability distribution P (z T ; ), I can derive a relationship
between the marginal P ( ) and the conditional distribution P (z T j ); which is known as the Bayes theorem:
P ( jz T ) / P (z T j )P ( ): The method updates the a priori distribution using the likelihood contained in
the data to obtain the conditional posterior distribution of the structural parameters. The resulting posterior
1 Beyond the usual disclaimer, I must note that any views expressed herein are those of the author and not
necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System.
1
density P ( jz T ) is used to draw statistical inference on the parameter space
. The likelihood function is
obtained combining the state-form representation implied by the solution for the linear rational expectation
model and the Kalman …lter. The likelihood and the prior permit a computation of the posterior that can
be used as the starting value of the random walk version of the Metropolis-Hastings (MH) algorithm, which
is a Monte Carlo method used to generate draws from the posterior distribution of the parameters. In this
case, the results reported are based on 500,000 draws following this algorithm. I choose a normal jump
distribution with covariance matrix equal to the Hessian of the posterior density evaluated at the maximum.
The scale factor is chosen in order to deliver an acceptance rate between 20 and 45 percent depending on
the run of the algorithm. Measures of uncertainty follow from the percentiles of the draws.
A.3
Empirical Performance
De…ne the marginal likelihood of a model A as follows: MA =
R
P ( jA)P (Z T j ; A)d : Where P ( jA) is
the prior density for model A, and P (Z T j ; A) is the likelihood function of the observable data, conditional
on the parameter space
and the model A. The Bayes factor between two models A and B is the de…ned
as: FAB = MA =MB . The marginal likelihood of a model (or the Bayes factor) is directly related to the
T +m
R
predicted density of the model given by: p^TT +m
P ( jZ T ; A)
P (zt jZ T ; ; A)d : Where p^T0 = MT :
+1 =
t=T +1
Therefore the marginal likelihood of a model also re‡ects its prediction performance.
B
Additional Results
Figure A1 shows the prior (grey line) and posterior density (black line) for the benchmark model. Figure A2
reports impulse responses to all shocks: remittance, foreign rate (country risk premium), technology, credit
(…nancial), terms of trade. I depict the median response (solid lines) to a one standard deviation of the
shocks, along with the 10 and 90 percent posterior intervals (dashed lines).
C
Convergence Diagnostics
I monitor the convergence of iterative simulations with the multivariate diagnostic methods described in
Brooks and Gelman (1998). The empirical 80 percent interval for any given parameter, %, is taken from
each individual chain …rst. The interval is described by the 10 and 90 percent of the n simulated draws.
(i)
In this multivariate approach, I de…ne % as a vector parameter based upon observations %jt denoting the
ith element of the parameter vector in chain j at time t: The direct analogue of the univariate approach in
higher dimensions is to estimate the posterior variance-covariance matrix as: V^ =
2
n 1
n W
+ (1 +
1
m )B=n;
where W =
1
m(n 1)
Pm Pn
j=1
t=1 (%jt
%j :)(%jt
%j :)0 and B=n =
1
m 1
Pm
j=1 (%j:
%:: )(%j:
%:: )0 : It is possible
to summarize the distance between V^ and W with a scalar measure that should approach 1 (from above)
as convergence is achieved, given suitably overdispersed starting points. I can monitor both V^ and W;
determining convergence when any rotationally invariant distance measure between the two matrices indicates
that they are su¢ ciently close. Figure A3 reports measures of this aggregate.2 Convergence is achieved before
100,000 iterations.3 General univariate diagnostics are available are not displayed but are available upon
request.
2 Note that, for instance, the interval-based diagnostic in the univariate case becomes now a comparison of volumes of total
and within-chain convex hulls. Brooks and Gelman (1998) propose to calculate for each chain the volume within 80%, say, of
the points in the sample and compare the mean of these with the volume from 80% percent of the observations from all samples
together.
3 Standard general univariate diagnostics are not displayed but are available upon request .
3
References
[1] An, S., Schorfheide, F., 2007. Bayesian Analysis of DSGE Models. Econometric Reviews 26, 113–172.
[2] Brooks, S., Gelman, A., 1998. General Methods for Monitoring Convergence of Iterative Simulations.
Journal of Computational and Graphical Statistics 7(4), 434–455.
[3] Fernández-Villaverde, J., Rubio-Ramírez, J., 2004. Comparing Dynamic Equilibrium Models to Data: A
Bayesian Approach. Journal of Econometrics 123, 153–187.
4
Figure A1. Prior and posterior distributions
St dev Technology
St dev Foreign rate
150
St dev Remit Shock
150
2000
1500
100
100
1000
50
50
500
0
0.01
0.02
0.03
0.04
0
0.05
0.01
0.02
St dev Credit Shock
0.03
0.04
0
0.05
0.05
St dev TOT shock
150
150
100
100
50
50
0.1
0.15
0.2
0.25
Taylor Rule Inflation Coefficient
2
1.5
1
0
0
0.02
0.04
0.06
0.08
0.5
0
-0.5
0.1
0
Taylor Rule Output Coefficient
0.5
1
1.5
2
2.5
0
0.5
1
1.5
Taylor Rule Exch Rate Coefficient
2.5
3
Inertia interest rate
5
1.5
2
4
4
3
3
1
2
2
0.5
1
1
0
-0.5
0
0.5
1
1.5
2
2.5
3
0
0
1
elast price of capital
2
0
3
-0.2
0
0.2
Prob price not adj
1.5
15
1
10
0.5
5
0.4
0.6
0.8
Share rule of thumb
4
3
2
0
0
0.5
1
1.5
2
2.5
3
3.5
1
0
0.2
0.4
Inv Intertemporal elast
0.6
0.8
1
0
0.2
Export elasticity
0.4
0.3
0.4
0.6
0.8
1
Elasticity Remittances
2.5
0.4
2
0.3
1.5
0.2
0.2
1
0.1
0
0
0.1
0.5
0
2
4
6
8
10
0
0.5
1
Export inertia
2
3
1.5
2
1
1
0.5
0
0.2
0.4
0
2
-8
-6
-4
Depreciation elasticity
4
0
-0.2
1.5
0.6
0.8
0
-2
0
2
4
6
Persistence technology shock
4
3
2
1
0
0.5
Persistence foreign rate shock
1
1.5
0
0.4
2
Persistence TOT shock
0.5
0.6
0.7
0.8
0.9
1
1.1
rPersistence Remittances Shock
8
6
6
6
4
4
4
2
2
0
0
0.5
0.6
0.7
0.8
0.9
1
2
0.5
0.6
0.7
0.8
0.9
1
0
0.4
0.5
0.6
0.7
0.8
0.9
Persistence credit market shock
8
6
4
2
0
0.4
0.6
0.8
1
Note: Benchmark Model. Results based on 500,000 draws of the Metropolis algorithm. Gray line: prior. Black line:
posterior.
1
1.1
Figure A2. Impulse response functions to the model’s shock.
Remittance shock
rem ittance shock
0.02
0.04
0.01
0.015
0.06
0.01
0.04
0.005
0.02
0.02
0
-0.01
0
5
10
Output
15
20
0
0
5
10
15
Aggregate Consumption
20
0
-3
4
0
5
10
15
"Ricardian" Consumption
20
0
-3
x 10
x 10
5
2
0.01
0
0
0
0
-5
-1
0
5
10
15
Aggregate Employment
20
-0.01
0
5
10
15
"Ricardian" Employment
20
-10
0
5
10
15
"Rule-of-Thumb" Consumption
20
-3
0.02
-2
0
1
5
10
15
"Rule-of-Thumb" Employment
20
-2
x 10
0
5
10
Investment
15
20
0
5
10
15
Real Interest Rate
20
0
5
20
-3
0
0.015
0.04
2
0.01
0.02
1
0.005
0
0
x 10
-0.005
-0.01
0
5
10
Exports
15
20
0
0
5
10
Imports
15
20
-0.02
0
5
10
15
Consumption Domestic Goods
20
-1
-4
10
x 10
0
5
0.1
0.06
0.05
0.04
0
0.02
-0.005
0
-5
0
5
10
CPI inflation
15
20
-0.01
0
5
10
15
Real Exchange Rate
20
-0.05
0
5
10
Real Wages
15
20
0
10
15
Remittances
foreign (country ris k ) rate
Foreign Rate (Country risk premium) shock
-3
5
-3
x 10
5
-3
x 10
0.02
5
0
0
0
0
-5
-5
-0.02
-5
-10
0
5
10
Output
15
20
-10
-3
1
0
5
10
15
Aggregate Consumption
20
-0.04
-3
x 10
2
0
5
10
15
"Ricardian" Consumption
20
-10
-3
x 10
2
0
x 10
0
5
10
15
"Rule-of-Thumb" Consumption
20
-3
x 10
0
0
-2
-2
-4
x 10
0
-1
-2
0
5
10
15
Aggregate Employment
20
-2
-3
4
0
5
10
15
"Ricardian" Employment
20
-4
-3
x 10
5
0
5
10
15
"Rule-of-Thumb" Employment
20
-6
-3
x 10
5
0
5
10
Investment
15
20
0
5
10
15
Real Interest Rate
20
0
5
20
-3
x 10
2
x 10
2
0
0
1
0
-2
0
5
10
Exports
15
20
-5
-4
2
5
10
Imports
15
20
-5
0
5
10
15
Consumption Domestic Goods
20
0
-3
x 10
5
0
0
-2
-5
0
0
5
10
CPI inflation
15
20
x 10
0
5
10
15
Real Exchange Rate
20
0.01
0.02
0
0.01
-0.01
0
-0.02
-0.01
0
5
10
Real Wages
15
20
10
15
Remittances
Figure A2 (cont.). Impulse response functions to the model’s shock.
Technology Shock
technology
0.02
0.02
0.01
0.02
0
0
0
-0.02
-0.01
-0.02
0
-0.02
0
5
10
Output
15
20
-0.04
0
5
10
15
Aggregate Consumption
20
-0.02
0
5
10
15
"Ricardian" Consumption
20
-0.04
0
5
10
15
"Rule-of-Thumb" Consumption
20
-3
0.01
0.02
0.01
4
0
0
0
2
-0.01
-0.02
-0.01
0
-0.02
0
5
10
15
Aggregate Employment
20
-0.04
0
5
10
15
"Ricardian" Employment
20
-0.02
0
5
10
15
"Rule-of-Thumb" Employment
20
-2
x 10
0
5
10
Investment
15
20
0
5
10
15
Real Interest Rate
20
0
5
20
-3
0.015
0
0.01
0.01
1
0
0
-0.01
-1
x 10
-0.005
0.005
0
0
5
10
Exports
15
20
-0.01
-3
2
x 10
10
5
-2
0
0
5
10
Imports
15
20
-0.02
0
5
10
15
Consumption Domestic Goods
20
-2
-3
0
-4
0
5
10
CPI inflation
15
20
-5
x 10
0
5
10
15
Real Exchange Rate
20
0.1
0.1
0
0
-0.1
0
5
10
Real Wages
15
20
-0.1
10
Remittances
15
c redit s hoc k
Credit Shock
0.02
0.01
0.02
0.01
0
0
0
0
-0.02
-0.01
-0.02
-0.01
-0.04
0
5
10
Output
15
20
-0.02
-3
5
0
5
10
15
Aggregate Consumption
20
-0.04
-3
x 10
5
0
5
10
15
"Ricardian" Consumption
20
-0.02
x 10
5
x 10
0
0
0
-5
-5
-5
-0.01
0
5
10
15
Aggregate Employment
20
-10
0
5
10
15
"Rule-of-Thumb" Consumption
20
0.01
0
-10
0
-3
5
10
15
"Ricardian" Employment
20
-10
0
5
10
15
"Rule-of-Thumb" Employment
20
-0.02
0
5
10
Investment
15
20
0
5
10
15
Real Interest Rate
20
0
5
20
-3
0
0.05
-0.01
0
-0.02
0
5
10
Exports
15
20
-0.05
0
5
10
Imports
15
20
0.01
4
0
2
-0.01
0
-0.02
0
5
10
15
Consumption Domestic Goods
20
-2
x 10
-3
1
x 10
0.02
0
0.05
0.04
0
0.02
0
-1
-2
-0.02
0
5
10
CPI inflation
15
20
-0.04
0
0
5
10
15
Real Exchange Rate
20
-0.05
0
5
10
Real Wages
15
20
-0.02
10
15
Remittances
Figure A2 (cont.). Impulse response functions to the model’s shock.
Terms of Trade Shock
-3
TOT shock
5
x 10
0.01
0
0
-5
-0.01
0.02
0
-0.005
0
-10
0
5
10
Output
15
20
-0.02
-0.01
0
5
10
15
Aggregate Consumption
-0.02
20
-3
1
0
5
10
15
"Ricardian" Consumption
20
x 10
0
2
0
x 10
0
5
10
15
"Rule-of-Thumb" Consumption
20
0
1
-0.005
-0.005
-1
-2
-0.015
-3
0
0
5
10
15
Aggregate Employment
20
-0.01
0
5
10
15
"Ricardian" Employment
-1
20
0
5
10
15
"Rule-of-Thumb" Employment
20
-0.01
0
5
10
Investment
15
20
0
5
10
15
Real Interest Rate
20
0
5
20
-3
0
0
0.015
-0.005
-0.01
0.01
-0.01
-0.02
0.005
5
x 10
0
-0.015
0
5
10
Exports
15
20
-0.03
0
5
10
Imports
15
0
20
0
5
10
15
Consumption Domestic Goods
20
-5
-3
10
x 10
0
0.01
5
-0.01
0
0
-0.02
-0.01
0.01
0
-5
0
5
10
CPI inflation
15
20
-0.03
0
5
10
15
Real Exchange Rate
-0.02
20
0
5
10
Real Wages
15
20
-0.01
10
Remittances
15
Note: The Solid line is the median impulse response to one standard deviation of the shocks; the dotted lines are the 10
and 90 percent posterior intervals.
Figure A3. MCMC multivariate convergence diagnostics
Interval
10
8
6
4
1
2
3
4
5
6
7
8
9
10
4
x 10
m2
15
10
5
1
2
3
4
5
6
7
8
9
10
4
x 10
m3
80
60
40
20
1
2
3
4
5
6
7
8
9
10
4
x 10
Note: Multivariate convergence diagnostics (Brooks and Gelman, 1998). The eighty percent interval, second and third
moments are depicted respectively.