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ESTIMATING SMALL SCALE MACROECONOMETRIC MODEL (SSMM) FOR
NIGERIA: A DYNAMIC STOCHASTIC GENERAL EQUILIBRIUM (DSGE)1
APPROACH
by
Adebiyi, Michael Adebayo, PhD
Central Bank of Nigeria,
Central Business District,
Garki, Abuja, Nigeria
E-mail [email protected] or [email protected]
Tel: 234-0946235913 or 234-8073356300
&
Charles N.O. Mordi
Central Bank of Nigeria,
Central Business District
Garki, Abuja, Nigeria
E-mail [email protected]
Tel: 234-0946235900 or 234-8037851373
1
The views expressed in this paper are those of the authors. No responsibility for them should be attributed to the
Central Bank of Nigeria
1
ESTIMATING SMALL SCALE MACROECONOMETRIC MODEL (SSMM) FOR
NIGERIA: A DYNAMIC STOCHASTIC GENERAL EQUILIBRIUM (DSGE)
APPROACH
ABSTRACT
This paper attempts to develop a small scale macroeconometric model for the Nigerian economy
using dynamic stochastic general equilibrium (DSGE) methodology. Particular attention is paid
to using impulse responses to explain the dynamic properties of the model. This model
incorporates expectation as an anchor in the forward-looking monetary policy objective of the
Central Bank of Nigeria (CBN). It captures most of the channels through which policymakers
believe monetary policy can influence a small open economy with a managed floating exchange
rate. The model was taken to the data by means of Bayesian estimation with the following major
findings. First, although inflation holds forward-looking component, the backward-looking one is
substantial. Also, income elasticity of the real demand for money in Nigeria is estimated to be
0.871, justifying the high volume of day-to-day transactions that use cash. Moreover, the paper
identifies the existence of exchange rate pass-through, confirming the import-dependent nature
of the Nigerian economy. Also, the paper estimates a sacrifice ratio of 1.306. Lastly, the paper
shows that the best Taylor-type policy rule for Nigeria is to focus on a monetary policy rule that
gives higher weight to inflation gap than output gap.
JEL Classifications: E32, C51
Keywords: DSGE Model, Bayesian Estimation, Nigeria, Structural Model
2
1
INTRODUCTION
Small Scale Macroeconometric Model (SSMM) emerged in the 1990s as a substitute for
comparing the results obtained from larger macroeconomic models. Since the model is implicitly
small, it is possible to carefully analyze key macroeconomic issues that are critical to the
economy rather than to concentrate on excessive details in the economy. SSMMs are
characterized by a compact system of equations that describe the behaviour of key
macroeconomic aggregates. The approach allows for a straightforward understanding of the
transmission mechanisms of policy actions to variables of interest, such as inflation, output or the
real exchange rate and their simplicity facilitates experiments with different assumptions
regarding agents or policy-makers preferences. This explains the popularity of SSMMs among
central bank modelling activities as evidenced from New Zealand, Canada, the United Kingdom,
Sweden, Finland, Australia, Spain, Brazil, Chile, and Venezuela (Arreaza et al. 2003;
Tanuwidjaja and Meng, 2005).
In recent times, SSMMs are estimated using dynamic stochastic general equilibrium (DSGE)
methodology. This technique is an improvement over the real business cycle (RBC) technique
because of its ability to combine explicit microeconomic foundations with nominal factors
(Christiano, Eichenbaum and Evans, 2005). With this discovery, there emerged new waves of
dynamic and stochastic models that integrate aggregate supply and demand responses based on
microeconomic theory. This is referred to as Dynamic Stochastic General Equilibrium Models
(DSGEMs) as developed by Nason and Cogley (1994); Schorfheide (2000); Kydland and
Prescott (1982); Smets and Wouters (2003); Bergeoing and Soto (2002) and Alege (2008, 2009).
Peiris and Saxegaard (2007) identify some features that make DSGE models unique. First,
DSGE model is structural and has the characteristics of a general equilibrium model. Apart from
the fact that the equations are interpreted based on economic theory, the main variables of
interest are also endogenous and depend on each other. In addition, the model is stochastic since
random shocks affect each endogenous variable. Thus, with the model, it is possible to measure
the impact of monetary policy transmission mechanism on output or price and determine
exchange rate pass-through to inflation, among others. The model can be employed to analyze
efficiency frontier and optimal monetary policy as well (Garcia, 2009). Lastly, DSGE
incorporates rational expectations, which assume that all economic agents utilize all available
3
information at their disposal to determine the expectation of some crucial macroeconomic
variables such as inflation rate and exchange rate.
While there is a large volume of literature on DSGE in Industrial, Latin America and Asian
economies2, few of these studies is based on the African economies and, in particular, Nigeria
(Alege, 2009). In Nigeria, with the pioneering work of Olekah and Oyaromade (2007) in this
area, other attempts were made by Olayeni (2009), Alege (2009) and Garcia (2009). Olekah and
Oyaromade (2007) estimate a small-scale DSGE model of the Nigerian economy with the aim of
guiding monetary policy decision makers in Nigeria. Using a VAR model, the results obtained
show that changes in prices are caused by volatility in real output while exchange rate and
inflation account for significant proportion of the variability in interest rate. The authors cited the
non- availability of relevant software at the time of the study as a major constraint of the study.
Alege (2009) develops a small business cycle model of the Nigerian economy using DSGE with
a view to identifying the source of a business cycle in Nigeria and draw policy analysis. Using
Bayesian econometrics, the paper concludes that Nigeria’s business cycles were driven by both
real and nominal variables.
Similarly, Olayeni (2009) develops a DSGE model of the Nigerian economy in order to analyse
how the Nigerian economy should be managed in the face of shocks such as the recent global
economic meltdown. Considering four monetary policy rules and estimating each of the resulting
models using DYNARE 4.0.2, the author finds that the Central Bank of Nigeria (CBN) places
little weight on the exchange rate behaviour in reacting to the shocks, resulting in overshooting
and persistence in the exchange rate. The CBN, however, react strongly to the behaviour of
inflation and, to a lesser degree to output or output gap following the shocks. The paper
concludes that it will be important for the CBN to pursue a guided exchange rate policy by
actively responding to exchange rate movements to avoid overshooting and persistence.
Garcia
(2009)
develops
a
Simple
Dynamic
General
Equilibrium
New
Keynesian
macroeconometric model for forecasting and policy analysis for the Nigerian economy. The
model, which focuses on the nominal interest rate and money supply, incorporates inflation
2
For examples Benhabib, Rogerson and Wright (1991) conduct the study for USA; Bergoeing and Soto (2002), for
Chile; Kose (1999) and Hofmaiser and Roldos (1997), for Asia; Maussner and Spatz (2005), for Germany and
Christodulakis, Dimeli and Kollintzas (1999), for the European countries.
4
expectation, thereby anchoring forward-looking monetary policy objective of the CBN.
Estimating with Nigerian quarterly data from 1995 to 2007, the results justify the current policy
actions of the CBN to control inflation.
In all these studies, little attention was paid to the interpretations of the posterior means and indepth analysis of the impulse response functions. Thus, attempt is made to contribute to the
literature in these areas. The objective of the paper, therefore, is to develop a macroeconometric
model for the Nigerian economy using dynamic stochastic general equilibrium (DSGE)
methodology. This is done in line with the studies by central banks in Canada, Chile, England,
Japan, Europe, New Zealand, Colombia, the United States and the International Monetary Fund
(IMF) that combine the New Keynesian approach with the real business cycle traditional
methods of dynamic stochastic general equilibrium (DSGE) with emphasis on modeling with
rational expectations. The study employs the Bayesian econometrics to address the following
objectives: first, establish the relationship between inflation and output gap (Phillips curve) and
determine sacrifice ratio; second, determine whether or not interest rate parity condition holds for
Nigeria and if it holds, to examine the impact of monetary policy shocks on exchange rate; third,
identify the link between exchange rate and prices and then estimate the coefficient of exchange
rate pass-through; and finally, determine the impact of the oil price shocks on money supply and
exchange rate.
Following the introduction, section 2 provides the theoretical framework, and describes the
SSMM of the Nigerian economy. The Bayesian Econometric techniques are discussed in
Sections 3, followed by a discussion of the data and estimation results in Section 4, paying
particular attention to the use of the impulse responses to a range of shocks to explain the
dynamic properties of the model. Section 5 concludes the paper, while areas for further work are
discussed in section 6.
5
2.0
THEORETICAL FRAMEWORK
2.1
Model Setup3
Most DSGE models available in the literature have a basic structure that incorporates elements of
the New Keynesian paradigm and the real business cycle approach. The benchmark DSGE model
is an opened or a closed economy fully micro-founded model with real and nominal rigidities
(see for instance Christiano, et al (2005) and Smets and Wouters (2003)). This model, which
fundamentally neoclassical in features in the long-run, has Keynesian short-term characteristics
as well. Thus, the model provides scope for monetary policy to influence the real sector of the
economy over the short- to medium-term (Roger, Restrepo and Garcia, 2009).
The basic structure of the model consists of perfectly competitive firms that produce a final
nontradable good which is consumed by a representative household and the fiscal authorities, in
addition to being used for investment. The inputs used in the production of the final good are
either produced domestically or imported by monopolistically competitive intermediate goods
firms (Peiris and Saxegaard 2007). The domestically produced goods, which are produced using
capital, labor and borrowing from a financial intermediary as inputs, are sold either in the
domestic market or exported overseas.
This model makes it possible to determine the steady state of the economy vis-a-vis the
permanent shocks that change the steady state. It focuses on the dynamics of the economy and
how the economy returns to the steady state following macroeconomic disturbances.
Households
Assuming a log-linearised system of equations, the model set up for households indicates that
d
domestically produced goods,
it

, can be split into domestic consumption, ct , and goods
d
for exports,
3
xt
as shown in equation 14
This section benefits immensely from the work of Roger, Restrepo and Garcia (2009)
6

d
it 

c
d
ct 

d
x
d
i
Where


xd
and
i
x t ..................................................(1)
i
c
d
d
d
are the own share of consumption and exports in domestic output in the
i
steady state.

o
Decomposing consumption into Ricardian (inter-temporally optimized), c t , and non-Ricardian

r
(credit-constrained), ct , elements with a fraction of the domestic spending as  , we obtain:



c   ctr  (1   ) cto .................................................  2 
Ricardian consumption is optimized by maximizing household utility from consumption
Ct and leisure Lt such as:
(Ct   Ct 1 )1  1
U (Ct , Lt ) 
 Lvt ..............................(3)
1 
where

stands for the coefficient of relative risk aversion;

is the level of habit formation in consumption, which introduces an element of inertia into
consumption. This reflects the characteristics of typical New Keynesian models.
4
Variables with hats stand for the deviation of the variables from their steady state
7
Obtaining consumption from Euler equation, which is derived from the first order condition of
utility maximization5, we have:

o
t

 

 



 


1


o
o
1
1
c 
c 
c 
(r )  t 1  .........................................(4)
1   t 1
1   t 1
1     t

where:

rt stands for the current nominal interest rate

 t 1 is the expected inflation rate in period t+1

r
From equation 2, ct is based on current income and the share of 1   , which is described as
non-Ricardian consumption, is expected to be higher in many developing economies than in
more advanced economies due to weak development of financial system and/or
legal
weaknesses which inhibits collateralization of borrowing and limit access of some households to
borrowing and savings (Roger, Restrepo and Garcia, 2009; Gali, Lopez-Salido and Valles,
2007).

As indicated in Equation 1, domestically produced goods are classified into domestic, ct , and
 d
foreign, x t , demands. Export demand is determined by foreign real income and the real
exchange rate:
5
For more details on this, see Roger, Restrepo and Garcia, 2009, p 39.
8
d
x t   xd

 

x t 1  (1   xd )   et  yt*  ...............................................(5)


d
where:

yt* represents foreign real income

et
is the real exchange rate, which is the real cost of foreign currency
xd proxies the degree of persistence in domestically-produced exports

stands for the real exchange rate elasticity of foreign demand for domestically-produced
exports
Firms
In an given economy, two types of goods are commonly produced: composite good produced by
monopolistically competitive firms for both domestic consumption and also for export, and
natural endowment commodity basically for export. Constant elasticity of substitution (CES)
production technology is employed for domestically-produced composite good with inputs of
labor (L) and an imported input (M). This production technique incorporates a Cobb-Douglas
characteristics and a Leontief technology, thereby making it convenient to use. A CES
production technology is given as:

 1
 1  1


d

Yt  Kt  M t  (1   ) Lt   ......................................................  6


9
where:

is the elasticity of substitution in production
M t represents the imported intermediate input
Lt is the labor input
 stands for the share of the imported good in production, which reflects the openness of the
economy
Kt explains total factor productivity
In the production function identified in equation 6, the cost of production mirrors the costs of the
labor and the imported inputs with the real cost of imported inputs being explained by the real



exchange rate, et , and the real wage, ( wt  pt ) , which is determined by equating producers’

demand for labor, l t , with the supply of labor by households. Aggregate supply is positively
related to real wage and negatively to consumption (Roger, Restrepo and Garcia, 2009)6.

lt 

 



1   
 1


(
w

p
)


c

c
 t t
t
t

1
 1  
 .....................................(7)
1 
 1 

where:

l t is the number of work supplied

wt represents the nominal wage rate per unit of work supplied

p t is the price level
 stands for the coefficient of disutility of labor
6
See Roger, Restrepo and Garcia, 2009, for derivation of equation 7
10
From equation (7), the log deviation of the real marginal cost of production is obtained as follow:





c r  (1   ) ( wt  pt )   et  (1   )     kt ........................................................(8)
where:

kt is the deviation of
Kt
from its steady state value
From equation (8), it is obvious that the more open the economy (as indicated by higher value of

) the bigger the impact of exchange rate movements on production costs and inflation.
Similarly, the role of elasticity of substitution in production,
 , is visible in the sense that the
lesser the possibility of substituting domestic labor for imported inputs, the larger the impact of
exchange rate movement on costs.
In setting the price of the domestically-produced good, a group of firms, representing a fraction
of domestic sales, follows a simple, backward-looking approach to price setting (Smets and
Wouters, 2002) leading to an element of price indexation, which generates persistence in
inflation while all other firms take a forward-looking optimization approach but adjust their
prices on a random basis.
Against this background, therefore, the aggregate inflation rate in the economy, in the spirit of
the New Keynesian Phillips curve, is specified as:

          
t  
  t 1  
  t 1  
 c r ..................................(9)
1


1


1








where:

is the subjective rate of time preference
11

represents the proportion of price adjustment based on indexation to past prices
 proxies the average frequency of price adjustment by optimizing firms
From equation 9, three components in the Phillips curve are identified. The first component is an
expected inflation, which takes into consideration the assumption that firms adjust their prices
periodically rather than continuously, so that when prices are adjusted, firms take into account
the expected evolution of inflation. The second element is a lagged inflation component and it
reflects the indexation applied to a fraction

of prices, while the third term justifies the
inclusion of marginal costs in optimizing firms’ prices. Exchange rate movements feed into
inflation through their impact on marginal costs and the speed of pass-through into inflation
depends on both fraction of optimizing firms and on the average rate of price changes (Roger,
Restrepo and Garcia, 2009).

c
The second endowment type of good, x t , which is purely for exports, is exogenously


c
t
determined. Its value is influenced by the real exchange rate, et , and its price, p , at the
international markets. Thus, the value of production is given as:



xtc  et  ptc ....................................................(10)
Exchange rate determination
The determinants of real exchange rate are real uncovered interest parity condition, the lead
exchange rate, and a risk premium.




*
t 1


et  et 1  (r   )  (rt   t 1 )   t ....................................................(11)
*
t
12
where:

et is the real exchange rate;

rt* stands for the foreign nominal interest rate;

 t*1 is the expected foreign inflation rate; and
 t mirrors the risk premium, which depends on debt, the exchange rate and output (Céspedes,
Chang, and Velasco, 2004).
Monetary policy
It is assumed that domestic vis-à-vis foreign central banks follow Taylor-type reaction functions since the
basic objective of the central bank is to stabilize both output and inflation. So to specify this reaction
function, it is expected that nominal interest rate is adjusted in response to deviations of inflation, a
measure of output and exchange rate depreciation from their targets. Typically, the central bank also
seeks to smooth the path of interest rates. Following Clarida, Gali and Gertler (2001), a simple
reaction function can be defined as:



 

rt  i rt 1  (1  i )   t   it   vt ....................................................(12)


Where:

rt is the deviation of policy target interest rate in period t, from its long-run steady-state value,



 t =(  tf   T ) is the deviation of the period t inflation forecast,  tf , from the inflation target,  T
13


it =(

it  i t ) is the deviation of real output , i t , from the estimated level of potential output,

i t , in period t;
vt
represents additional policy judgment and reflect the inaccuracy in policy implementation.
2.2
Model Structure and Description
The model is specified in gap and rate-of-change terms, so that all the variables are rendered
stationary. The model contains thirteen (13) equations as specified in equations 13 to 25 below:
ygt  1 ygt 1  11 ygt 1   2 (rir  rire)   3 (rert  rere)   4 ygf  1t ...............(13)
inft   1 inft 1  1   1  inft 1   2 ygt 1   3  ptf  (rert  rert 1 )    4 ( pot  pot 1 )   5 (mst  ms )   2t ........(14)
mst  1mst 1  (1  1 )mst 1   2 ygt   3rirt   3t .................................................................................(15)
rert  1rert 1  1  1  rert 1  2 (rirt  rsft )  3 (mst  mst 1 )  4 pft  5 ygt  6 ( pot  pot 1 )   4t .............(16)
rirt  1mprt 1  (1  1 )[riret  inft 1  2 (inft  inf et )  3 ygt ]   5t .................................................(17)
pf t   11 pf t 1   6t ......................................................................................................(18)
rsf t   21rsf t 1   7 t ....................................................................................................(19)
ygf t   31 ygf t 1   8t .......................................................................................................(20)
inf et   41 inf et 1   9t ................................................................................................(21)
riret   51riret 1  10t ..................................................................................................(22)
reret   61reret 1  11t ..................................................................................................(23)
pot   71 pot 1  12t .......................................................................................................(24)
mprt   81mprt 1  13t ....................................................................................................(25)
where:
14
ygt is the output gap in period t
ygft is the foreign output gap in period t
riret stands for potential or equilibrium real interest rate in period t
rert is the real exchange rate in period t defined as nominal exchange rate deflated by relative
prices
reret stands for the potential or equilibrium real exchange rate in period t
inft represents inflation rate in period t
infet stands for optimum or equilibrium inflation rate in period t
inft+1 is the expected inflation rate in period t+1
mst is the real money supply broadly supplied in period t
pft captures foreign price level in period t
rirt is the domestic short-term interest rate in period t
rsft is the foreign short-term interest rate in period t
pot represents crude oil price in period t
mprt is the monetary policy rate in period t
t-i represents the lagged of relevant variables
t+i stands for the lead of relevant variables
 ,  ,  ,  ,  , are all parameters to be estimated.
Equation 13 is the aggregate demand (IS) equation, which theoretically is linked to household
utility optimization. According to the theory, household maximizes discounted stream of utility
(consumption and labor supply) subjected to budget constraints (consumption expenditure and
wages). In calculating the present value of spending and wages, interest/ policy rate is
15
incorporated. Thus, a forward-looking proportion of the IS relationship (yg
t+1)
is obtained in
addition to the deviation of real interest rate (rir) from its equilibrium (rire). The lagged of
output gap (ygt-1) is included to give room for some degree of habit persistence in consumption
or adjustment costs of investment (Pongsaparn, 2008). Nigeria is a small open economy and
consequently, the deviation of real exchange rate (rer) from its equilibrium (rere) is included as a
variable that influences economic activities through the prices of imports and exports. Also,
foreign output gap (ygf) is added as a determinant of export demand. The influence of other
explanatory variables such as oil prices, fiscal policy and other demand shocks are captured by
the residual term.
Equation 14 is the inflation equation specified in the spirit of the Philips curve. The Equation
shows that inflation rate is influenced not only by past inflation but also by inflation
expectations, demand pressures, and external supply shocks captured by rert and pt f . From this
equation, inflation depends on its expected future value and its own lagged value. The inclusion
of this lagged term shows the existence of a short-run trade-off between output and inflation. In
the specification of inflation equation, exchange rate effect on domestic prices is considered. The
inclusion of the real exchange rate attempts to capture the exchange rate pass-through to
domestic prices due to the openness of the economy. Domestic sources of inflation are captured
by the inclusion of money supply, mst and output gap, ygt, while the inclusion of oil price and
foreign price reflects the peculiarities of the Nigerian economy as an oil exporting and importdependent economy.
Equation 15 is the demand for real money balances. Conventionally, the equation shows that the
demand for real money balances is explained by the past and future money supply, real interest
rate, reflecting the opportunity cost of holding money and real output (in gap form). Velocity
shocks are explained by the innovations in  3t
Equation 16 is the uncovered interest parity (UIP) equation for an open economy, like Nigeria.
rir and rsft are the real domestic and real foreign short-term interest rates, respectively. In the
literature, many models that assume interest parity condition do not provide enough persistence
16
to generate a hump-shaped response of the real exchange rate after a shock to monetary policy,
which is commonly found in estimated VARs (Eichenbaum and Evans, 1995; Faust and Rogers,
2003). Given the degree of openness of the Nigerian economy, it is plausible to assume that
interest parity condition holds in Nigeria. Thus, real exchange rate depends on its own lag and
lead values. Oil price and output gap are included in the equation to measure the effect of the
country risk premium on the exchange rate. These variables measure the foreign investors’
perception about the Nigerian economy (Garcia, 2009).
Equation 17 is the modified Taylor rule, which explains the interest rate path for the monetary
authority. Apart from inflation and output gaps, monetary authorities react immediately to the
changes in the nominal exchange rate and money supply by altering its monetary policy rate to
stabilize both the nominal and real exchange rates. The exchange rate plays an important role in
aggregate demand through its effects on net export and also on inflation through the pass-through
effect. The UIP shows the links between exchange rate and interest rates. In reaction to a
depreciation of the exchange rate, for example, the monetary authority is expected to raise
interest rates subsequently. Equations 18, 19, 20, 21, 22, 23 24 and 25 are foreign price (US
CPI), foreign interest rate (US short term deposit rate), foreign output gap, potential inflation,
potential real interest rate, potential real exchange rate, oil price and monetary policy rate,
respectively. These variables are exogenously determined but are assumed to be generated by an
autoregressive of order 1 (AR,1).
Figure 1 gives the overall mechanism of the model. While inflation is determined by output, oil
price, money supply, foreign inflation and exchange rate, output itself is influenced by the real
interest rate, exchange rate and foreign output. Both inflation and output consequently
determines interest rate through the monetary policy rule, while the interest rate feeds back into
output directly and indirectly through the exchange rate.
17
Figure 1: The Model Flowcart
Foreign
inflation
Oil Price
Monetary
Policy
rate
Foreign
interest
rate
Inflation
Interest
rate
Expected
exchange
rate
Exchange
rate
Money
Supply
Output
gap
Foreign
output gap
18
3.0
BAYESIAN ESTIMATION TECHNIQUES7
In the literature, two main methods for evaluating DSGE models are calibration and econometric
estimation. While calibration methods were very popular a few years ago, their popularity has
diminished and has given way to the development of econometric estimation, which is the focus
of this paper. Different econometric techniques for estimating DSGE models have been
identified in the literature, namely method of moments (GMM), minimum distance estimation
based on the discrepancy between VAR and DSGE, impulse response functions, maximum
likelihood (Canova, 2007; An and Shorfheide, 2007; Ruge-Murcia, 2007; and Favero, 2001). An
important difference that arises among these different approaches has to do with the amount of
information that each method is able to handle.
However, because of the weaknesses of these methods, Bayesian methods for estimating DSGE
models have developed. These methods allow construction of posterior distributions from which
to draw inferences about the parameters. Bayesian methods are considered to be able to handle
the weaknesses of pure maximum likelihood methods for several reasons. First, the prior
reweighs the likelihood, so that if the likelihood reaches a peak at a point that is at odds with the
prior then the marginal density of the DSGE model will be low (Tover, 2008). Equally
important, posterior inference does not depend on the model being the correct data generating
process and Bayesian methods are able to incorporate the policy makers’ views about what has
been learned in the past about the economy.
Technically speaking, Bayesian estimation is a mix between calibration and maximum
likelihood, which are connected by Bayes’ rule. The calibration part is the specification of priors.
And the maximum likelihood approach enters through standard econometrics based on adjusting
the model with data.
7
This section benefits immensely from the work of Griffoli (2007)
19
The description of the priors is given by a density function
p  B B 
Where B indicates a specific model, B represents the parameters of model B, p   is the
probability density function (pdf) such as a normal gamma, shifted gamma, inverse gamma, beta,
generalized beta, or uniform function. Also, the density of the observed data is described by the
likelihood function given the model and its parameters:
L  B KT B   p  KT B , B 
Where K T stands for the observations until period T, and where in our case the likelihood is
recursive and can be written as:
T
p  KT B , B   p  y0 B , B   p  yt K t 1 , B , B 
t 1
Taking a step back, we have a prior density p    on the one hand, and a likelihood of p  KT  
on the other hand. Since our interest is on the posterior density, p   KT  , we obtain this density
of parameters knowing the data by using the Bayes theorem and we have:
p   KT  
p   ; KT 
p  KT 
20
Given that
p  KT   
p   ; KT 
p  KT 
 p   ; KT   p  KT    p   
by using these identities, we combine the prior density and the likelihood function to get the
posterior density:
p  B KT , B  
p  KT B , B  p  B B 
p  KT B 
where p  KT B  is the marginal density of the data condition on the model:
p  KT B  
 p 
B
; KT B  d B
B
Lastly, the posterior kernel corresponds to the numerator of the posterior density:
p  B KT , B   p  KT B , B  p  B B     B KT , B 
21
This equation permits us to rebuild all posterior moments of interest by estimating the likelihood
function with the help of the Kalman filter and then simulating the posterior kernel using Monte
Carlo method such as Metropolis-Hastings.
The models’ parameter values are estimated from the data using a Bayesian estimation
technique. The Bayesian approach starts with prior distributions for the model parameters that
are then combined with the data using the likelihood function to estimate the posterior
distributions for the parameters. This approach has two important strengths. First, starting with
prior distributions for the parameters allows other empirical evidence from a range of sources to
enter into the estimation. Secondly, use of prior distributions makes the highly nonlinear
optimization algorithm considerably more stable, making it feasible to apply the technique when
sample periods are short. In addition, the estimation procedure also allows for measurement
errors in the data (Honjo and Hunt, 2006).
Although Generalized Methods of Moments (GMM) has fully developed econometric
framework with inference methods for parameters and specification tests which is easy to
implement without requiring equilibrium solution, it only consider the model in parts not
‘general equilibrium’ and ignore cross-coefficients restrictions (Pongsaparn, 2008). Thus,
identification problems tend to be more pronounced. In DSGE, the state of ‘general equilibrium’
is ensured and the whole system of the model is consistent, particularly at the steady state.
Therefore, GMM estimation does not appear to fit in well. Similarly, ordinary least square (OLS)
method of estimation requires certain conditions be satisfied to ensure unbiasness and efficiency.
Using OLS technique, estimated coefficients may not conform to economic theory. Thus,
Bayesian estimation is chosen over other alternatives.
4.0
DATA AND THE EMPIRICAL RESULTS
4.1
Data
The observable variables used to estimate the model for Nigeria are plotted in Figure 2. The plot
shows the trend and de-trended measures of logarithm of consumer price index (inf), short term
real interest rate (rir), monetary policy rate (mpr), logarithm of real gross domestic output gap
22
(gy), logarithm of US CPI (Pf), inflation potential inflation (phie), US three-month deposit rate
(rsf), logarithm of real broad money (ms), foreign output gap (ygf), and oil price (po).
Equilibrium values of these variables are exogenous and are derived using a variant of the
Hodrick Prescott (1997) filter that allows for additional constraints to be added to the
minimization problem to prevent the resulting equilibrium value from converging to the actual
observed data at the end of the sample period (Honjo and Hunt (2006). The sample period for the
estimation is 1985Q1 -2008Q4
Figure 2: Plot of Variables Used in the Model
RIR (gap)
RIR
40
Potential
50
40
20
30
20
0
10
-20
0
-10
-40
-20
-60
-30
-40
-80
86
88
90
92
94
Lcpi (inf)
9
96
98
00
02
04
06
86
08
88
90
92
94
96
98
00
02
04
06
08
inf (gap)
Potential
40
8
0
7
-40
6
-80
5
-120
4
3
-160
86
88
90
92
94
96
98
00
02
04
06
08
86
23
88
90
92
94
96
98
00
02
04
06
08
RER (gap)
50
RER
160
Potential
40
30
120
20
80
10
40
0
-10
0
-20
-40
-30
-40
-80
86
88
90
92
94
96
98
00
02
04
06
86
08
88
90
92
94
96
98
00
02
04
06
04
06
08
Y (gap)
Log of Y
potential
.08
12.2
.06
12.0
.04
11.8
.02
11.6
11.4
.00
11.2
-.02
11.0
-.04
10.8
-.06
-.08
10.6
86
88
90
92
94
96
98
00
02
04
06
86
08
24
88
90
92
94
96
98
00
02
08
PF (gap)
Log of Pf
4.9
Potential
.015
4.8
.010
4.7
.005
4.6
.000
4.5
-.005
4.4
-.010
4.3
-.015
4.2
-.020
4.1
86
88
90
92
94
96
98
00
02
04
06
86
08
88
90
92
94
96
98
00
02
04
06
08
RSF (gap)
RSF
10
Potential
2
8
1
6
0
4
-1
2
-2
-3
0
86
88
90
92
94
96
98
00
02
04
06
86
08
25
88
90
92
94
96
98
00
02
04
06
08
Log of MS
8.0
Ms (gap)
Potential
.4
.3
7.6
.2
7.2
.1
6.8
.0
-.1
6.4
-.2
6.0
-.3
5.6
-.4
86
88
90
92
94
96
98
00
02
04
06
08
86
88
90
92
94
96
98
00
02
04
06
02
04
08
PO (gap)
PO
50
Potential
140
40
120
30
20
100
10
80
0
60
-10
40
-20
20
-30
-40
0
86
88
90
92
94
96
98
00
02
04
06
86
08
26
88
90
92
94
96
98
00
06
08
Potential
ygf (gap)
LYF
9.8
.03
9.6
.02
9.4
.01
9.2
9.0
.00
8.8
-.01
8.6
-.02
8.4
8.2
-.03
86
88
90
92
94
96
98
00
02
04
06
08
86
88
90
92
94
96
98
00
02
04
06
08
MPR (gap)
MPR
4
Potential
28
2
24
0
20
-2
16
-4
12
-6
8
-8
-10
4
86
4.2
88
90
92
94
96
98
00
02
04
06
86
08
88
90
92
94
96
98
00
02
04
06
08
Empirical Results
Tables 1-5 and Appendix 1 contain the prior and the posterior mean values. Table 1 shows the
parameters estimated for the aggregate demand (IS) curve. The general guideline on the posterior
mean are satisfied. For instance, lags in monetary policy transmission mechanism imply
relatively higher inertia than forward looking components in aggregate demand equation;
therefore, β11 is greater than β1 while the sum of β2 and β3 is less than β11. In other words, the
output gap in the Nigerian economy is not influenced by expectations of future trajectories of the
interest and exchange rates. This confirms the findings that the parameter on the lagged output
27
gap would typically range between 0.50 and 0.90 and a small coefficient on the lead of the
output gap ranging between 0.05 and 0.15 would be expected (Berg, Karam and Laxton, 2006).
The coefficients of real interest rate and real exchange rate are 0.011 and 0.014, respectively.
This confirms the degree of openness of the Nigerian economy. The parameter of foreign output
gap (  4 ) is 0.014, which implies that a one percent increase in foreign output gap would lead to
0.014 percent increase in the domestic output gap the following period.
Table1: Model Parameters Estimation Results of the Aggregate Demand (IS) Curve
Parameter
Prior Mean
Posterior Mean Distribution
1 (coefficient of lead of output gap in gy)
0.100
0.132
Beta
1 1 (coefficient of lag of output gap in gy)
0.700
0.717
Beta
 2 (coefficient of the deviation of real 0.100
0.011
Gamma
0.014
Gamma
0.124
Beta
interest rate from it potential in gy)
 3 (coefficient of deviation of real exchange 0.100
rate from its potential in gy)
 4 coefficient of foreign output gap in gy)
0.100
Table 2 shows the estimation of the Phillips curve. The prior mean for the estimation are
consistent with estimates obtained in other countries (Pongsaparn, 2008). The behavior of the
economy depends critically on the value of  1 . If inflation expectations are entirely forward
looking (  1 =1), then inflation is equal to the sum of all future output and exchange rate gaps,
which is referred to as “speedboat” economy (Berg, Karam and Laxton, 2006). In this type of
economy, small re-calibrations of the monetary-policy wheel, if perceived to be persistent, will
cause large jumps in inflation through forward-looking inflation expectations. From the
empirical results, inflation expectation in Nigeria is more backward looking ([1-  1 ] = 0.616)
than forward looking (  1 = 0.384), which is closer to the figure of 0.57 obtained by Garcia (2009)
28
and consistent with the assertion that for most countries, the values of forward looking inflation
expectation significantly below 0.50 produced results that were usually considered to be more
consistent with data (Berg, Karam and Laxton, 2006). It seems that economic agents in Nigeria
tend to be more backward rather than forward-looking in forming their expectations of inflation.
The high persistence of inflation means that the current rate of inflation is strongly influenced by
the previous period’s inflation rate and not so much by the expected future rate of inflation.
Monetary policy influences inflation through its effects on output and the exchange rate. The
estimated posterior value of the output gap (  2 =0.306) is very close to the prior (0.300) and this
gives the CBN an important tool to control inflation through output gap. Also, the impact of the
exchange rate on prices (  3 = 0.014) is far from the prior (0.100), indicating that the exchange
rate pass-through into consumer prices is 1.4%, which is very low.
For monetary policy to have an impact on inflation, the coefficients on the output and exchange
rate gaps must be greater than zero. This assertion is established with the coefficients of output
(  2 = 0.306) and exchange rate (  3 = 0.014). Oil price and money supply have significant impact
on inflation with a posterior value of  4 =0.126 and  5 =0.240, respectively, which is less than the
prior values of 0.200 and 0.300. They imply that a one percent increase in oil price and money
supply would lead to 0.126 and 0.240 percent rise in inflation rate, respectively, the following
period.
From the inflation equation, the lead term in the equation (  1 ) is 0.384. This implies a weight on
the lagged inflation term (1-  1 ) of 0.616, which indicates a significant intrinsic inertia in the
inflation process. These parameters, combined with the coefficient on the output gap (  2 =0.306),
are the principal determinants of the output costs of disinflation. With the posterior mean of the
output gaps of 0.306, the sacrifice ratio in Nigeria is 1.3068.
8
Sacrifice ratio is defined as the cumulative output losses associated with a permanent one percentage point decline
in inflation.
29
Table 2: Model Parameters Estimation Results of the Phillip Curve
Parameter
Prior Mean
 1 (coefficient of the lead of inflation rate 0.200
Posterior Mean Distribution
0.384
Gamma
0.300
0.306
Gamma
 3 (coefficient of real exchange rate in inf 0.100
0.014
Gamma
0.200
0.126
Gamma
 5 (coefficient of real money supply in inf 0.300
0.240
Beta
inf equation)
 2 (coefficient of output gap in inf equation)
equation
 4 (coefficient of oil price in inf equation)
equation)
Table 3 shows the estimated parameters of the interest rate parity condition. In the literature, it is
argued that more robust policies should assume a much smaller value of  1 (below 0.5),
because it might be imprudent to rely so heavily on these forward looking linkages in the face of
uncertainty (Isard and Laxton, 1998; Berg, Karam and Laxton, 2006). The estimated coefficient
in Table 3 (  1= 0.284) satisfies this requirement. The condition for uncovered interest rate
parity is held, the posterior value for  2 indicates that a one percentage point increase in interest
rate differential between the domestic and foreign interest rates (rir-rsf) would lead to an
appreciation of naira by 0.456 percent. The posterior values of output gap (  5) and oil price
(  6) are 0.345 and 0.104, respectively. This indicates that a one percent rise in output gap and
oil price would lead to 0.345 percent increase and 0.104 percent reduction in real exchange rate,
respectively.
30
Table 3: Model Parameters Estimation Results of the Uncovered Interest Parity (UIP)
Parameter
Prior Mean
Posterior mean
Distribution
0.284
Gamma
0.456
Gamma
0.527
Beta
0.450
Beta
 5 (coefficient of output gap in rer equation) 0.400
0.345
Beta
 6 (coefficient of oil price in rer equation)
0.104
Beta
 1(coefficient of the lead of real exchange 0.300
rate in rer equation)
 2 (coefficient of interest rate differential in 0.500
rer equation)
 3 (coefficient of real money supply in rer 0.500
equation)
 4 (coefficient of foreign interest rate in rer 0.450
equation)
0.250
In the literature, it is shown that a stable inflation rate requires a positive  2, and how monetary
authorities should react is dependent on the other features of the economy (Berg, Karam and
Laxton, 2006). Monetary policy rule in Table 4 shows that the posterior value (  2 = 1.559) is in
line with the literature. However, the original work of Taylor rule imposed a zero weight on
interest rate smoothing (  1 = 0) and implying  2 of 0.5 and a value of  3 of 0.5, which is very
close to the posterior value estimated in Table 4 (  3 =0.433). The posterior value (  1 = 0.624)
explains the possibility that the central bank can moderate interest rates and adjust them fairly
rapidly to the desired value based on deviation of the inflation and output from equilibrium. This
implies that the best Taylor-type policy rule for Nigeria is a monetary policy rule with a higher
weight attached to inflation than output gap.
31
Table 4: Model Parameters Estimation Results of the Monetary Policy Rules
Parameter
Prior mean
 1 (coefficient of lag of monetary policy rate in 0.500
Posterior Mean
Distribution
0.624
Beta
1.559
Gamma
0.433
Gamma
mpr equation)
 2 (coefficient of the deviation of inflation rate 1.500
from its equilibrium in mpr equation)
 3 (coefficient of output gap in mpr equation)
0.500
Table 5 shows the determinants of money supply. In the money supply equation, the income
elasticity of the real demand for money is estimated to be 0.871. This estimate, which is close to
unity, seems to be plausible for a developing economy such as Nigeria in view of the high
volume of day-to-day transactions that use cash. The real interest rate effect on the demand for
money is 0.334, which implies that when the interest rate is raised by one percentage point, the
real demand for money would reduce by 0.334 percent. Such a relatively small coefficient is
expected of the Nigerian economy, which is an evidence of underdeveloped nature of the
financial market in Nigeria and the interest rate inelasticity of the demand for real money
balances.
Table 5: Model Parameters Estimation Results of the Money Supply
Parameter
Prior mean
 1 (coefficient of the lead of money 0.300
Posterior Mean
Distribution
0.039
Gamma
0.871
Gamma
0.334
Gamma
supply in money supply equation)
 2 (coefficient of output gap in money 0.900
supply equation)
 3 (coefficient of real interest rate in 1.300
money supply equation)
32
4.3
Model Applications
A small model of this nature can facilitate an understanding of how the economy works and how
shocks are transmitted throughout the macroeconomy. In this part we examine how shocks to
inflation and output propagate. The shocks we refer to here are shocks relative to the steady
states.
Impulse Response Functions
The impulse response functions in Figures 1-10 (Appendix 2) illustrate how the economy
adjusts, i.e. how the disturbance causes each variable to move away from its steady state (zero
line) and how it reverts back to it. The figures illustrate the response of the system to a structural
shock. Figure 1 is the monetary policy (interest rate) shock and it is consistent with the
conclusion reached by Garcia (2009). As shown in the Figure, a real interest rate shock leads to
an instantaneous decrease in the inflation rate and the output gap. The results show a peak for the
nominal interest rate at the beginning of the adjustment period, which implies that the Central
Bank of Nigeria needs to react strongly to a real interest rate shock to bring the inflation rate to
its optimal path. In order to bring about the observed interest rate response, the Central Bank of
Nigeria must engineer a reduction in money supply. This shock also produces an appreciation of
the Naira due to the partial existence of uncovered interest rate parity, evidenced in the fact that
higher interest rate generates capital inflows that appreciate the Naira as indicated in equation 4.
However, all variables tend towards a steady state (equilibrium) in the long run. Thus, an
increase in the monetary policy rate by 10 percentage points will reduce inflation rate
immediately by 4 percentage points, output is reduced by 0.3 percent and nominal exchange rate
appreciates by 10 percent.
33
Figure 1: Impulse Response Function to a Transitory Monetary Policy (Interest Rate)
Shocks
y
phi
0
rir
0
10
-0.2
-2
5
-0.4
-4
0
5
10
15
20
5
rer
10
15
20
-2
-3
-10
10
15
20
15
20
15
20
-2.5
-3
-3.5
-4
-4.5
-1
-5
10
ms
0
0
5
5
mpr
5
10
15
20
5
10
Figure 2 explains the dynamic impact of oil price shocks on the economy. An oil price shock
motivates the monetary authority to raise interest rate in order to mitigate the effect of inflation.
This leads to an appreciation of the Naira arising from capital inflow and decline in output.
However, all variables return to their equilibrium values when oil price shock disappears. The
main important effect of the oil price shock is an appreciation of the real exchange rate, which
tends to push down the inflation rate. However, inflation equally increases on account of the
increase in money supply related to the oil price shocks as well. But the increase in inflation
arising from money supply due to the oil price shocks exceeds its reduction due to the
appreciation of the naira.
34
Figure 2: Impulse Response Function to a Transitory Oil Price Shocks
y
phi
po
1
0.1
10
8
6
4
2
0.5
0
0
-0.1
5
10
15
20
-0.5
5
rir
10
15
20
0.5
15
20
15
20
15
20
-0.5
-2
10
20
0
-1
-0.5
15
0.5
0
0
10
mpr
1
5
5
rer
5
10
15
20
5
10
ms
0.2
0
-0.2
-0.4
5
10
Figure 3 is the aggregate demand shocks. The responses of the variables to shocks are in line
with economic theory. For example, excess demand puts upward pressure on inflation. In
response to the positive output gap and higher inflation, the central bank raises real interest rates,
which leads to a depreciation of the naira. The depreciation tends to heighten the rise in inflation
directly via pass-through effects, as well as indirectly, by adding to the demand stimulus through
greater competitiveness (see Figure 3). However, these shocks die off in the long run.
35
Figure 3: Impulse Response Function to a Transitory Aggregate Demand Shocks
phi
y
2
2
1
1
0
0
5
10
15
20
rir
3
2
1
0
-1
5
rer
10
15
20
0
15
20
15
20
15
20
2
1.5
1
0.5
2
10
10
ms
4
0
-2
-4
-6
-8
5
5
mpr
5
10
15
20
5
10
Figure 4 is the aggregate supply shocks, which explains the dynamic impact of inflation rate on
the economy. From the Figure, a positive shock to price leads to an increase in inflation rate due
to the dynamics on inflation arising from both backward and forward-looking components. An
increase in inflation leads to a rise in the real interest rate as well as appreciation of naira. As a
result, higher interest rate and real exchange rate appreciation reduce aggregate demand as well
as a fall in output gap and inflation.
36
Figure 4: Impulse Response Function to a Transitory Supply Shocks
y
phi
-0.2
-0.4
-0.6
-0.8
-1
5
10
15
20
4
2
0
-2
5
rer
10
15
20
-5
-10
10
15
20
10
15
20
15
20
ms
4
2
0
-2
5
5
mpr
0
-15
rir
8
6
4
2
0
-2
-2
-4
5
10
15
20
-6
5
10
A positive shock on the money stock leads to an increase in inflation, which induces a monetary
policy tightening (Figure 5). The inflation rate reaches its equilibrium level while the real interest
rate adjusts with a peak in the second quarter. It is important to note that the interest rate reaches
a peak and its reaction is as strong as that of inflation rate at the beginning of the adjustment
period. This implies that the Central Bank of Nigeria should react strongly to a money demand
shock in order to stabilize the inflation rate.
37
Figure 5: Impulse Response Function to a Transitory Money Supply Shocks
phi
y
0.08
0.06
0.04
0.02
0
-0.02
-0.04
1.5
1
0.5
0
2
1
0
5
10
15
5
20
10
15
5
20
10
15
20
15
20
ms
mpr
rer
3
2
1
0
-1
rir
2
6
1
5
0
5
10
15
20
5
10
15
20
5
10
Figure 6 shows that a positive shock to the real exchange rate leads to an immediate real interest
rate tightening due to a surge in inflation, thus, confirming the existence of exchange rate passthrough in the Nigerian economy. The model shows a peak for the real interest rate, which
implies that the Central Bank of Nigeria should react strongly to a real exchange rate shock at the
very beginning of the adjustment period.
38
Figure 6: Impulse Response Function to a Transitory Exchange Rate shocks
y
phi
3
rir
15
20
2
10
10
1
0
5
0
5
10
15
20
5
rer
10
15
20
5
0
15
20
0
15
20
15
20
-2
-4
-6
-8
-10
10
20
10
ms
15
40
10
5
mpr
60
5
0
5
10
15
20
5
10
As Figure 7 shows, an external (or foreign price) shock leads to an increase in domestic price and
the real interest rate, which induces a depreciation of the naira. A depreciation of the domestic
currency raises output and money supply. The peak of real interest rate, which occurs at about
the 8 period after the shock, indicates that the Central Bank of Nigeria needs to react strongly to
an external (foreign price) shock in order to eliminate the output gap.
39
Figure 7: Impulse Response Function to a Transitory External (Foreign Price) Shocks
y
phi
rir
0.08
0.15
0.1
0.05
0.05
0.06
0.04
0
0.02
0
-0.05
5
10
15
20
5
rer
10
15
20
pf
1
0.6
0.5
0.4
15
20
15
20
15
20
15
20
0.15
0.1
0.05
0
0.2
10
10
mpr
0
5
5
5
10
15
20
5
10
ms
0.2
0.15
0.1
0.05
5
10
Figure 8 explains the dynamic effect of monetary policy rate (MPR) shocks. Shocks to MPR
raise interest rate and reduces money supply. The increase in interest rate leads to an appreciation
of the naira arising from capital inflow and decline in money stock and output. However, all
variables return their equilibrium values when the shock disappears. The main effects of the
MPR are an increase in the interest rate and an appreciation of the naira. It is important to note
that the interest rate reaches a peak and its reaction is as strong as that of inflation rate at the
beginning of the adjustment period. This is an indication that the Central Bank of Nigeria should
rely on the use of MPR to stabilize prices.
40
Figure 8: Impulse Response Function to a Transitory Monetary Policy Rate Shocks
y
phi
rir
0
0
20
-0.5
-5
10
-1
-10
0
-15
-1.5
5
10
15
20
5
rer
10
15
20
mpr
0
20
-10
10
10
15
20
10
15
20
15
20
ms
-2
-4
-6
-8
0
-20
5
5
5
10
15
20
5
10
Figure 9 shows the dynamic impact of the external (foreign interest rate) shocks. A shock on
foreign interest rate, relative to domestic interest rate, leads to a depreciation of the naira arising
from capital outflow. This consequently raises output. However, all the variables are restored
back to steady state (equilibrium) in the long. It is also important to note that interest rate peaks
in the short run, indicating the need for the monetary authority to react to external shock to
stabilize inflation and stimulate output.
41
Figure 9: Impulse Response Function to a Transitory External (Foreign Interest Rate)
Shocks
phi
y
rir
1
0.15
1
0.1
0.5
0.5
0.05
0
5
10
15
20
5
rer
10
15
20
1
0
15
20
15
20
10
15
20
15
20
mpr
1.4
1.2
1
0.8
0.6
0.4
0.2
2.5
2
1.5
1
0.5
10
5
rsf
2
5
0
5
10
15
20
5
10
ms
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
5
10
Figure 10 is the foreign output gap shocks. The result indicates that a shock to foreign output gap
increases domestic output gap, inflation and interest rates. However, the impact on exchange rate
and money supply is not noticeable in the short run.
42
Figure 10: Impulse Response Function to a Transitory Foreign Output Gap Shocks
y
yf
0.15
0.1
0.05
0
-0.05
phi
0.15
0.1
0.05
0
-0.05
0.4
0.2
5
10
15
20
5
rir
10
15
20
rer
0.3
0.2
0.1
0
-0.1
-1
15
20
15
20
15
20
15
20
0.3
0.2
0.1
0
-0.1
-0.5
10
10
mpr
0
5
5
5
10
15
20
5
10
ms
0.3
0.2
0.1
5
5
10
POLICY IMPLICATIONS AND CONCLUSION
This paper attempts to develop a small scale macroeconometric model for the Nigerian economy
using dynamic stochastic general equilibrium (DSGE). Particular attention is paid to using
impulse responses to explain the dynamic properties of the model. This model incorporates
expectation as an anchor in the forward-looking monetary policy objective of the CBN. It
captures most of the channels through which policymakers believe monetary policy can
influence a small open economy with a managed floating exchange rate. The model was
estimated with Nigerian quarterly data from 1985 to 2008, to replicate some facts of this
economy.
The following policy implications can be drawn from the study. First, the small scale
macroeconomic model developed is capable of capturing the short-to-medium-term economic
43
dynamics in Nigeria. The finding suggests that the implementation of a small scale model allows
consideration in policy-making and can also serve as a guide to building large scale
macroeconometric model, as currently embarked on by the Central Bank of Nigeria.
Also, this model explicitly considers the role of expectations in the form of model-consistent
expectations. This is important because the lack of forward-looking element significantly reduces
the ability of the model to provide a credible description of the economy which is valuable to
policymakers. The finding shows that inflation expectation in Nigeria is more backward than
forward-looking inflation expectation. This implies that economic agents in Nigeria tend to be
more backward rather than forward-looking in forming their expectations of inflation. The high
persistence of inflation in Nigeria means that the current rate of inflation is strongly influenced
by the previous period’s inflation rate and not so much by the expected future inflation rate.
Income elasticity of the real demand for money in Nigeria is estimated to be 0.871, which
justifies the high volume of day-to-day transactions that use cash. Also, the impact of the
exchange rate on prices is 0.014, which supports the existence of exchange rate pass-through to
inflation and the import-dependent nature of the Nigerian economy. It also supports the
uncovered interest rate parity. Moreover, the posterior value (  1 = 0.624) explains the
possibility that the central bank can moderate interest rates and adjust them fairly rapidly to the
desired value based on deviation of the inflation and output from equilibrium. This implies that
the best Taylor-type policy rule for Nigeria is to focus on a monetary policy rule with a higher
weight given to inflation gap than output gap.
From the inflation equation, the lead terms in the equation (  1 ) is 0.384. This implies weights on
the lagged inflation terms (1-  1 ) of 0.616, which indicates significant intrinsic inertia in the
inflation process. These parameters, combined with the coefficient on the output gap (  2 =0.306),
give a sacrifice ratio of 1.306.
44
6.0
FURTHER WORK
To make the model versatile and suitable for forecasting, further work is required in the model
by incorporating assumptions and judgments into the forecasts. With this, fan charts can be
constructed by incorporating forecast uncertainties both from the model and the joint distribution
of assumptions.
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49
Appendix 1: Priors and Posteriors
SE_e1
SE_e2
SE_e3
2
2
1
1
1
0
5
10
15
0
0.5
20 40 60 80 100120
SE_e4
0
2 4
SE_e5
6 8 10 12
SE_e6
0.5
10
0.5
0
100 200 300
0
SE_e7
10
20
30
0
SE_e9
2
20
1
1
10
0.5
0
2
4
6
8 10 12
0.5 1 1.5 2 2.5
SE_e8
0
2
4
50
0
10
20
30
SE_e10
SE_e11
SE_e12
1
5
1
0.5
0
20
40
60
0.5
0
100
SE_e13
200
300
0
20 40 60 80 100
beta1
beta11
40
0.5
50
20
0
20 40 60 80 100
0
0.1
beta2
0.2
0.3
0
0.6
beta6
0.7
0.8
beta8
100
50
0
20
50
0
0.1
0.2
0.3
0
0
0.1
beta3
0.2
0.3
10
0.2
0.4
100
20
50
0
0.2
0.3
0
0.4
0.1
0.2
0.3
0.2
beta9
0
rho2
0.3
0.3
0.4
rho1
50
50
50
0.2
beta5
40
beta7
0
0.1
beta4
20
0
0
0.2
0.3
0
0.4
0
0.2 0.4 0.6 0.8
rho3
tau1
50
200
20
100
0
0.88 0.9 0.92
0
1
2
51
0
0.2
0.3
0.4
tau2
rho4
rho5
50
50
20
0
0.4
0.5
0
0.6
0.4
rho6
0.5
0.6
0
0.4
rho7
0.5
0.6
alpha1
50
40
50
0
0.2
0.4
0
20
0.3
alpha2
0.4
0.5
0
0.4
gamma1
0.5
0.6
tau6
40
20
0
20
20
1.4
1.5
1.6
0
0.4
tau7
0.5
0.6
0
0.4
tau8
0.5
0.6
mu1
50
40
100
20
0
0.4
0.6
0
0.6
mu2
0.7
0.8
mu3
0.6
0.8
40
50
200
20
0.5 0.6 0.7 0.8 0.9
0
0.6
0.7
0.8
mu5
50
0
0.4
mu4
400
0
0
0.6
0.8
52
0
0.6
0.8
Figure 2: MCMC Univariate Diagnostic
0.02
0.05
0.01
0.5
0
SE_e1 (m3)
SE_e1 (m2)
SE_e1 (Interval)
1
5000 10000 15000
0
5000 10000 15000
0
SE_e2 (m3)
SE_e2 (m2)
SE_e2 (Interval)
2
0.4
0.2
1
0.2
0.1
0
5000 10000 15000
0
0.2
0.5
0.1
5000 10000 15000
0
0
5000 10000 15000
SE_e3 (m3)
SE_e3 (m2)
SE_e3 (Interval)
1
0
5000 10000 15000
5000 10000 15000
0.05
5000 10000 15000
53
0
5000 10000 15000
tau2 (Interval)
0.04
4
0.02
2
0
5000 10000 15000
0
5000 10000 15000
5
x 10rho4 (m2)
0.01
0
5
0
-5
rho4 (Interval)
0.02
-4
x 10tau2 (m2)
4
-6
x 10tau2 (m3)
5000 10000 15000
-7
x 10rho4 (m3)
2
5000 10000 15000
0
0
-4
rho5 (Interval)
0.05
5000 10000 15000
4
x 10rho5 (m2)
5
5000 10000 15000
-6
x 10rho5 (m3)
2
0
5000 10000 15000
0
rho6 (Interval)
0.04
2
0.02
1
0
5000 10000 15000
0
0.04
1
0.02
0.5
5000 10000 15000
0
4
5000 10000 15000
0
5000 10000 15000
-6
x 10rho6 (m3)
1
5000 10000 15000
x 10rho7 (m2)
0
2
5000 10000 15000
-6
x 10rho7 (m3)
1
5000 10000 15000
x 10alpha1 (m2)
2
0
2
0
-4
alpha1 (Interval)
0.05
-4
x 10rho6 (m2)
0
-4
rho7 (Interval)
0
5000 10000 15000
1
5000 10000 15000
-5
x 10alpha1 (m3)
0.5
5000 10000 15000
54
0
5000 10000 15000
alpha2 (Interval)
0.04
2
0.02
1
0
5000 10000 15000
0
0.04
1
0.02
0.5
5000 10000 15000
0
2
-6
x 10alpha2 (m3)
1
5000 10000 15000
0
gamma1 (m2)
x 10
2
5000 10000 15000
-6
gamma1
(m3)
x 10
1
5000 10000 15000
0
-4
tau6 (Interval)
0.05
2
-4
gamma1 (Interval)
0
-4
x 10alpha2 (m2)
x 10tau6 (m2)
5
5000 10000 15000
-6
x 10tau6 (m3)
1
0
5000 10000 15000
0
tau7 (Interval)
0.04
1
0.02
0.5
0
5000 10000 15000
0
0.04
2
0.02
1
5000 10000 15000
0
0.02
4
0.01
2
5000 10000 15000
2
0
5000 10000 15000
-6
x 10tau7 (m3)
1
5000 10000 15000
0
x 10tau8 (m2)
4
5000 10000 15000
-6
x 10tau8 (m3)
2
5000 10000 15000
0
-5
mu1 (Interval)
0
-4
x 10tau7 (m2)
0
-4
tau8 (Interval)
0
5000 10000 15000
x 10mu1 (m2)
4
5000 10000 15000
-7
x 10mu1 (m3)
2
5000 10000 15000
55
0
5000 10000 15000
mu2 (Interval)
0.02
4
0.01
2
0
5000 10000 15000
4
5000 10000 15000
0
-5
5
-7
x 10mu2 (m3)
2
0
mu3 (Interval)
0.02
-5
x 10mu2 (m2)
x 10mu3 (m2)
5
5000 10000 15000
-7
x 10mu3 (m3)
0.01
0
5000 10000 15000
0
0.04
4
0.02
2
5000 10000 15000
0
-4
mu4 (Interval)
0
5000 10000 15000
x 10mu4 (m2)
0
4
5000 10000 15000
-6
x 10mu4 (m3)
2
5000 10000 15000
0
5000 10000 15000
mu5 (Interval)
0.04
0.02
0
2000
4000
6000
-4
1
8000
10000
12000
14000
10000
12000
14000
10000
12000
14000
mu5 (m2)
x 10
0.5
0
2000
4000
6000
-6
2
8000
mu5 (m3)
x 10
1
0
2000
4000
6000
8000
56
SE_e4 (Interval)
SE_e4 (m2)
SE_e4 (m3)
4
1
1
2
0.5
0.5
0
5000 10000 15000
0
SE_e5 (Interval)
5000 10000 15000
0
SE_e5 (m2)
SE_e5 (m3)
4
1
2
2
0.5
1
0
5000 10000 15000
0
0.2
4
0.1
2
5000 10000 15000
0
-3
SE_e6 (Interval)
0
5000 10000 15000
0
SE_e7 (Interval)
SE_e6 (m2)
x 10
2
5000 10000 15000
0
SE_e7 (m2)
0.04
0.5
0.05
0.02
0
0.1
1
0.05
0.5
0
5000 10000 15000
5000 10000 15000
0
-3
SE_e8 (Interval)
0
SE_e9 (Interval)
SE_e8 (m2)
x 10
4
5000 10000 15000
0
SE_e9 (m2)
0.4
1
0.2
0.2
0
5000 10000 15000
-5
SE_e8
(m3)
x 10
5000 10000 15000
57
5000 10000 15000
SE_e9 (m3)
0.4
5000 10000 15000
5000 10000 15000
2
2
0
-4
SE_e6
(m3)
x 10
SE_e7 (m3)
0.1
5000 10000 15000
5000 10000 15000
1
1
0
5000 10000 15000
0
5000 10000 15000
SE_e10 (Interval)
SE_e10 (m2)
0.4
0.02
2
0.2
0.01
1
0
5000 10000 15000
0
SE_e11 (Interval)
5000 10000 15000
0
SE_e11 (m2)
0.4
0.4
1
0.2
0.2
5000 10000 15000
0
SE_e12 (Interval)
5000 10000 15000
0
SE_e12 (m2)
0.2
0.1
0.5
0.1
0.05
5000 10000 15000
0
SE_e13 (Interval)
1
2
0.5
5000 10000 15000
0
0.04
2
0.02
1
5000 10000 15000
0
0.04
2
0.02
1
0
5000 10000 15000
SE_e13 (m3)
5000 10000 15000
0
x 10beta1 (m2)
4
0
5000 10000 15000
-6
x 10beta1 (m3)
2
5000 10000 15000
0
-4
beta11 (Interval)
5000 10000 15000
0.5
-4
beta1 (Interval)
0
0
SE_e13 (m2)
4
0
5000 10000 15000
5000 10000 15000
SE_e12 (m3)
1
0
5000 10000 15000
SE_e11 (m3)
2
0
-3
SE_e10
(m3)
x 10
beta11 (m2)
x 10
2
5000 10000 15000
-6
beta11
(m3)
x 10
1
5000 10000 15000
58
0
5000 10000 15000
beta2 (Interval)
0.02
4
0.01
2
0
5000 10000 15000
0
0.04
1
0.02
0.5
5000 10000 15000
0
0.1
4
0.05
2
5000 10000 15000
0
beta3 (Interval)
0.05
4
5000 10000 15000
x 10beta6 (m2)
5000 10000 15000
x 10beta8 (m2)
5000 10000 15000
0
0.04
2
0.02
1
5000 10000 15000
0
0.015
4
0.01
2
5000 10000 15000
0
1
5000 10000 15000
-4
x 10beta3 (m2)
0
1
0
5000 10000 15000
-7
x 10beta6 (m3)
5000 10000 15000
-5
x 10beta8 (m3)
5000 10000 15000
-5
x 10beta3 (m3)
0.5
5000 10000 15000
0
x 10beta4 (m2)
4
5000 10000 15000
-6
x 10beta4 (m3)
2
5000 10000 15000
0
-5
beta5 (Interval)
0.005
5
-4
beta4 (Interval)
0
0
0.5
2
0
-7
x 10beta2 (m3)
2
-4
beta8 (Interval)
0
4
-4
beta6 (Interval)
0
-5
x 10beta2 (m2)
x 10beta5 (m2)
2
5000 10000 15000
-7
x 10beta5 (m3)
1
5000 10000 15000
59
0
5000 10000 15000
beta7 (Interval)
0.05
4
-4
x 10beta7 (m2)
2
0
5000 10000 15000
0
0.04
2
0.02
1
5000 10000 15000
0
0.04
2
0.02
1
5000 10000 15000
0
rho2 (Interval)
0.01
1
0.005
0.5
0
5000 10000 15000
0
0.1
1
0.05
0.5
5000 10000 15000
0
x 10beta9 (m2)
4
5000 10000 15000
0
5000 10000 15000
-6
x 10beta9 (m3)
1
5000 10000 15000
x 10rho1 (m2)
0
4
5000 10000 15000
-6
x 10rho1 (m3)
2
5000 10000 15000
-5
x 10rho2 (m2)
0
2
5000 10000 15000
-8
x 10rho2 (m3)
1
5000 10000 15000
x 10rho3 (m2)
5000 10000 15000
x 10tau1 (m2)
2
0
2
0
5
0
-4
tau1 (Interval)
0.05
0
-3
rho3 (Interval)
0
5000 10000 15000
-4
rho1 (Interval)
0
0.5
-4
beta9 (Interval)
0
1
-5
x 10beta7 (m3)
1
5000 10000 15000
-5
x 10rho3 (m3)
5000 10000 15000
-5
x 10tau1 (m3)
0.5
5000 10000 15000
60
0
5000 10000 15000