Download Bayesian Estimation of Finnish Import Demand*

MINISTRY OF FINANCE
Discussion Papers 2/2013
Bayesian Estimation of Finnish Import Demand*
Mikko Sariolay
September 2013
Mikko Sariola
Ministry of Finance, Snellmaninkatu 1 A, Helsinki, P.O. Box 28,
FI-00023 GOVERNMENT. Tel. +358 400 12 77 43, [email protected]
y
*I am grateful to Mika Kuismanen, Jani Luoto and Meri Obstbaum for constructive and
helpful comments. The usual disclaimers apply.
This paper represents the views and analysis of the author and should not be thought to
represent those of the Ministry of Finance.
ISBN 978-952-251-500-1
…

          
         
          
           
          
             
          
          
          
          
           
           
         
Keywords: Bayesian Analysis: General, Time Series, Model Construction and
Estimation, Empirical Studies of Trade
JEL Codes: Bayesian
C11, C32, Analysis:
C51, F14 General, Time Series, Model Construction
Keywords:
and Estimation, Empirical Studies of Trade
JEL Codes: C11, C32, C51, F14
1
Introduction
Finland is a small open economy and its production structure has a relatively
strong bias towards manufacturing products with high value added. Due to the
small home market these products are mainly exported. The Finnish exports to
GDP ratio of 47 per cent1 at the outburst of the global …nancial crisis highlights
that feature. The majority of imported goods, over 70 per cent, are either
investment or intermediate goods (appendix, …gure 1), probably echoing the
needs of the exporting industry. The import ratio of goods and services to GDP
has increased over the years and reached its highest level of 43 per cent in 2008.
With these facts in mind coupled with high Finnish cost structure, Finnish
products should be highly specialized, and perhaps di¤erentiated, in order to
be competitive on the global market. Furthermore, given that Finnish export
goods are highly specialized, an interesting question is to what extent foreign
imports can be substituted to domestic inputs. Moreover, low subsitutability
of foreign and domestic inputs would suggest that the price elasticity of import
demand is also low.
A review of past empirical work on import elasticities at industry level can
be found for example in Erkel-Rousse and Mirza (2002). To gain a broader
than industry level2 view some studies are disussed here in short. Bergstrand
(1985) derives a model and estimates a gravity equation by using 15 OECD
countries. Bergstrand …nds that the elasticity of substitution between domestic
and import goods is below unity. Hooper et al. (2000) estimate and test the
stability of the income and price elasticities of import demand for G-7 countries.
They …nd that short run import price elasticity is not signi…cantly di¤erent
from zero in all but one country. The income elasticity ranges from unity (DE,
IT, JP, UK) to over 2 (US). Senhadji (1998) estimates a structural import
demand equation for a large set of countries and argues that the price elasticity
is di¤erent in the long run. The average price elasticity is close to zero in the
short run and slightly over unity in the long run. Senhadji also notes that price
elasticities are lower for industrialized countries. He extends his research to cover
income elasticities as well. Compared to the …ndings of Hooper et al, Senhadji’s
estimated income elasticities for G-7 countries are somewhat lower, ranging from
0.3 (JP) to 1.3 (DE). Erkel-Rousse and Mirza (2002) build their estimation on
monopolistic competition where they apply gravity type of equations. Their
…nding is that import price elasticities are low when estimated with OLS or
…xed e¤ect methods but the instrumental variable approach tends to increase
price elasticity. Furthermore, price elasticity is signi…cantly correlated with the
degree of product di¤erentiation.
In this paper Finnish imports and their substitutability is studied in a monopolistic competition framework. First, the import demand equation is derived
analytically. Second, the analytical log-linearized conditional factor demand
1 Total
exports to GDP in 2008.
a di¤erent perspective than this paper but yet interesting, Anderton et al. (2005)
investigate the substitution e¤ects of manufactured goods between intra- and extra euro area
imports. They …nd signi…cant substitution e¤ects due to changes in relative prices.
2 From
1
form suggests that the data used in the estimation is stationary. Therefore,
the data is detrended using Hodrick-Prescott …lter before estimation. Then,
the import demand equation is estimated by Bayesian methods using …ltered
Finnish national accounts and quarterly national accounts data from 1976Q1
to 2008Q4. By choosing this time frame, we have excluded the global …nancial
crisis from the data set as being extremely rare event that does not represent
economy’s normal ‡uctuations. A novelty in the chosen approach is that the
unobservable variable for marginal costs of production is proxied by intermediate input de‡ator for …rms. The intermediate input de‡ator is a more correct
measure for …rms’marginal costs than for example the output de‡ator. These
two price concepts should be the same only if the …rms are operating in a perfect market but not in the monopolistic market. Therefore, using output prices
in estimating a monopolistic market leads to a severe conceptual con‡ict and
raises concerns about the reliability of such results.
The main results of the Bayesian estimation suggest that i) imports are
perfect complements in the production process. Therefore, Finnish production
technology has the Leontief form. The zero price elasticity of import demand
is in line with Senhadji (1998) and Hooper et al. (2000) results for short term
elasticity in industrialized countries. ii) Income elasticity is clearly above unity
(1.7) in Finland. Income elasticity of this magnitude is higher than in the most
G-7 countries according to Senhadji (1998) and Hooper et al. (2000). Hence,
the assumption of income elasticity equal to one does not hold for Finland and is
not an advisable choice for example when calibrating macro models representing
Finnish economy. Overall, the results suggest that import demand is driven by
income elasticity and not much by relative prices within this time period.
The paper is organized as follows. In chapter 2, the analytical import demand equation is derived and log linearized. In the third chapter the data is
introduced, the equation is estimated and the results are discussed. The last
chapter concludes.
2
2
Analytical model
In the model …nal composite good Yt is produced in the competitive market by
combining intermediate goods Yt (i)
21
Z
Yt = 4 Yt (i)
y
0
3
1
y
di5
(1)
It is assumed that intermediate goods are produced in the domestic market using constant elasticity of substitution (CES) production function. While
assuming perfect …nal goods market, value added is in fact created in the intermediate goods market where two inputs, one domestic and one imported, are
combined to produce the good i. Firms operating in the intermediate goods
market are assumed to be monopolistically competetive.
The constant elasticity of substitution (CES) production function for good
i in the intermediate goods market is following
Yt (i) = (1
,where
Y (i)
MH
MF
=
1
+1
)(MtH )
+ (MtF )
1=
(2)
production of good i
factor share of the foreign input used in production
domestic input
foreign input
elasticity of substitution
subsitution parameter, which determines the elasticity of substi-
tution.
Furthermore, elasticity of substitution between the inputs in CES leads to
1<
<0
) elasticity > 1
=0
0<
) elasticity = 1
<1
) elasticity < 1:
In order to get the conditional factor demand of the imported input, constrained cost minimization problem subject to technology constraint is applied
minPtH MtH + PtF MtF
MtF
s.t. Yt (i)
=
(1
)(MtH )
3
+ (MtF )
1=
Lagrangean is following
n
L = PtH MtH + PtF MtF
(1
)(MtH )
+ (MtF )
1=
o
Yt (i)
(3)
, where P H is the price of the domestic input and P F is price of the foreign
input.
First order condition, and the conditional factor demand, is then
MtF
=
1
+1
PtF
1
+1
Yt (i):
(4)
t
The …rst order condition, equation 4, can be rewritten using elasticity notation
MtF =
PtF
Yt (i):
(5)
t
Due to the monopolistic competition assumption, represents the marginal
cost of production. Finally, log linearization of the equation 5 around the steady
state yields equation 6. It is assumed that one representative …rm produces one
good. This will be the equation that is estimated.
b
m
bF
t = ( t
pbF
bt
t )+y
(6)
The constant elasticity of scale production function implies that if the elasticity, , is close to unity, then the production function reduces to Cobb-Douglas
form. If the elasticity is zero, then the factors of production are perfect complements and the production has Leontief form. More generally, if substitution
pareameter, , is negative, then factors are substitutes. Positive value of substitution parameter implies that factors are complements. By estimating the
import demand equation and elasticity parameter, it is simple to backtrack the
substitution parameter.
4
3
3.1
Empirical estimation of the model
Data
In the model estimation, data from the national accounts (NA) and quarterly
national accounts (QNA) is used. All the data is acquired from Statistics Finland and quarterly data is seasonally adjusted. It is easy to select observable
variables for Yt , PtF and MtF but for t the choice is more open for judgement.
Yt , PtF and MtF are real value added of total economy, de‡ator of imported goods
and services and real import of goods and services respectively. The marginal
cost of production, t , is unobservable variable but is proxied by the intermediate input de‡ator for …rms. Recently, Statistics Finland has adopted double
de‡ation standard in calculation of value added. This means that there are
separate de‡ators for output and intermediate inputs. Hence, the de‡ator for
intermediate inputs should serve as a proxy for marginal cost of production.
These two price concepts should be the same only if the …rms are operating in
perfect market where price equals marginal cost of production. But this is not
the case in monopolistic market. Therefore, using output prices in estimating
monopolistic market leads to a severe conceptual con‡ict and raises concerns
about the reliability of such results. The downside of this variable is its availability. Data (NA) is only available in annual terms, therefore annual observations
are disaggregated into quarterly series. Hence the quarterly series of marginal
cost of production is more rigid than perhaps in reality it would be. The four
series are represented in appendix (…gure 2)
In order to have stationary series as the log linearization around steady
state in equation 6 requires, the data is …ltered with Hodrick-Prescott …lter. In
…ltering the data, conventional smoothing parameter of 1600 is applied. The
properties of the stationary variables for the period 1976Q1-2008Q4 are presented in table 1. As expected, the import price de‡ator is more persistent than
the import volume (…gure 1). Perhaps for that reason the import volume appears to be twice as volatile than the import de‡ator (appendix, …gure 3). Total
value added is very persistent and has small variance compared to imports. It
is noteworthy, that it seems the import volume has become less volatile after
Finland entered the EU. The proxy for marginal costs is very persistent partly
due to the data interpolation.
Table 1. Properties of the variables
Maximum
Minimum
Autocorrelation
Std. Dev.
Skewness
Kurtosis
Observations
m
bF
0.160
-0.202
0.511
0.055
-0.094
4.488
132
pbF
0.073
-0.056
0.769
0.028
0.267
2.557
132
yb
0.058
-0.048
0.821
0.020
0.495
4.206
132
5
b
0.036
-0.047
0.939
0.019
-0.506
2.491
132
import volume
total value added
.2
.2
.1
.1
.0
.0
-.1
-.1
-.2
-.2
1980
1985
1990
1995
2000
2005
1980
import deflator
1985
1990
2000
2005
'MC', intermediate input deflator for firms
.2
.2
.1
.1
.0
.0
-.1
-.1
-.2
-.2
1980
1985
1990
1995
2000
2005
1980
1985
1990
b
Figure 1. Deviations from the steady state for m
b F ; ybt ; pbF
t and t
3.2
1995
Bayesian estimation
In this section we carry on with Bayesian estimation. By this, we seek to incorporate prior knowledge to estimation and to increase understanding of the
parameter distribution. Also we have a reason to believe there might be heteroscedasticity present and want to take care of the t-distributed errors appropriately by Bayesian approach3 .4 The selected approach will not only a¤ect
3 Another Bayesian tradition approach involving CES production structure is Luoma and
Luoto (2011), who apply Bayesian methods in system estimation of CES production function
and pro…t maximizing …rst order conditions.
4 For the interest of a classical econometrician, table 1 in the appendix presents the theorical
model results when residuals are assumed normal. The coe¢ cient for is very close to zero
6
1995
2000
2005
the shape of the posterior distribution but may also change the posterior mode
when compared to a linear model with normal errors.
As usual, in the following notation, letters with underscores are priors and
letters with bars are posteriors. Parameter vector is presented in equation
7.
is as in equation 6 and
denotes the coe¢ cient of the value added yb.
As a starting point, prior propability density function p( ) is assumed to be
multinormal and prior density function p(h) is assumed gamma5 .
=
(7)
Furthermore, we expect that prior vector is independent of prior h. This
implies we know neither joint posterior distribution for and h nor do we know
their marginal posteriors. Therefore, Monte Carlo integration method is not
available. In addition, we want to make sure that heteroscedastivity is taken
into consideration and use the independent Student-t linear model. However,
with these independent Normal-Gamma priors and Student-t regression model
we resort to Metropolis-Within-Gibbs sampler in order to carry out Monte Carlo
Markov Chain posterior simulation.
We use relatively informative priors. ; the prior for the relative price variable is assigned to zero with high standard deviation. This prior expected value
and standard deviation was chosen while our theoretical model does not give
any advice whether negative or positive priors should be used. However, the
zero price elasticity prior is in line with Senhadji’s (1998) results for short term
elasticity in industrialized countries. On the contrary in case of , the prior
for the value added6 is set at 1 as suggested by our theoretical model. However, we do not constrain to unity but apply somewhat smaller prior standard
deviation compared to : So, it is set
0
1
=
0:52
0
var( ) = V =
(8)
0
0:32
(9)
and t-test reveals it is not statistically di¤erent from zero. Hence implying the foreign imports
are perfect complements to domestic inputs. However, White test reveals heteroscedasticity
with 1 % risk level. Residuals seem to be heteroscedastic even if assumptions concerning the
theoretical model are relaxed and parameter restriction to unity for the coe¢ cient of the value
added is lifted (appendix …gure 4 and 5). The pre 1995 era appears far more volatile than the
time period after EU accession.
Residuals of the OLS estimation where coe¢ cient restriction for value added is relaxed and
not restricted to unity were tested also. Normality assumption was rejected with p-value 0.01.
This may raise some concerns of the robustness of the results if the classical OLS estimation
principle is violated. Residuals of the unrestricted model are not autocorrelated with 1 % risk
level in Breusch-Godfrey LM test with lags 2 to 4.
5 More on density functions in question, see for example Koop (2003, p. 60).
6 This can be viewed as a scaling variable.
7
Other prior hyperparameters are prior degrees of freedom (13) and prior
expected value for h (0.012 ). In total we have 132 observations in the data set
meaning we are attaching around 10 % weight to our prior. Initial draw in the
Gibbs procedure for the error precision is set at the inverse of 0.012 :7
The Metropolis-Within-Gibbs algorithm proceeds in a following manner.
First in the Metropolis algorithm section we use random walk chain to get
degrees of freedom parameter, , for the vector e
h that is used as a weight to
transform the data. e
h is drawn from the 2 -distibution. These are the hierarcical priors of the model. In the random walk chain a new draw is taken
from a normal distribution and added to the old accepted value . Then, the
old value and and the new candidate value are evaluated at the conditional
posterior density kernel of :
If the acceptance probability is positive and is higher than a random number
drawn from a uniform distribution [0,1], then the candidate value is accepted
to the chain. After the Metropolis algorithm the observables are reweighted by
the e
h and the standard Gibbs sampling is carried out. This Metropolis-WithinGibbs loop is run S times. Value 0.01 is chosen for prior . The interested
reader on methodology is referred to Geweke (2005, p. 205-208) and Lancaster
(2004, p.185-164).
In posterior simulation we take 500 000 draws and throw away …rst 30 000
draws as burn in period. According to Geweke’s CD statistics both chains
have converged while test values are well below 1.96.8 By using central limit
theorem, we can say that test statistic CD follows normal distribution N(0,1) and
test value below 1.96 implies that there is 95 % propability remaining series is
converged to its stationary distribution. Relevant prior and posterior properties
are summarized in table 3. The posterior distribution of the degrees of freedom
parameter ( ) for the vector e
h that is used as a weight to transform the data is
reported in appendix (…gure 6). The acceptance rate in the Metropolis algorithm
is 0.49.
7 This value is roughly at the same magnitude as the OLS model variance reported in the
appendix table 1.
8 Also chain for
has converged as CD statistics is below 1.96.
8
Table 3. Bayesian estimation of (unrestricted) theoretical model
m
b F = (bt pbF ) + ybt
t
t
Sample: 1976Q1 2008Q4
Included observations: 132
Prior and posterior results (standard deviations in parentheses)
Prior
0 (0.5)
1 (0.3)
Posterior
-0.22 (0.1)
1.76 (0.13)
Geweke’s CD
-0.03
-1.26
Simulated posterior mean for is very close to zero and is not two standard
deviations away from zero. Increasing the number of draws does not change the
results. The prior and posterior distributions can be seen in …gure 3 (top). As
zero is within two standard deviations from the posterior, this has straightforward economic implication suggesting foreign imports are perfect complements
to domestic inputs (Leontief production). The implied zero price elasticity is
in line with Senhadji (1998) and Hooper et al. (2000) results for short term
elasticity in industrialized countries. To check whether the result is robust to
the choice of marginal cost variable, two other proxy marginal cost variables
are tested. When using other proxies for marginal costs, intermediate input
de‡ator for manufacturing and intermediate input de‡ator for total economy,
simulated posterior mean remains within two standard deviations from zero9 .
Hence, result seems to be robust to the choice of marginal cost proxy.
Finally, simulated posterior mean for the scale variable is 1.7. The prior
and posterior distributions are reported in …gure 3 (bottom). is clearly above
unity and in this case all the propability mass is located above one. Based on
this we do not recommend restricting the model’s scale variable to unity. Higher
than one income short term elasticity is in line with results by Hooper et al.
(2000) for G-7 countries. Senhadji’s (1998) …ndings for G-7 are somewhat lower
ranging from 0.3 (JP) to 1.3 (DE). The results suggest that import demand is
driven by income elasticity and not much by relative prices within this time
period.
9 Especially, when using intermediate input de‡ator for manufacturing, 17 % of the probability mass lays above 0. When Intermediate input de‡ator for total economy is used as proxy
for marginal costs, results are in line with the baseline simulation.
9
Figure 3. Prior and posterior distributions.
10
4
Conclusions
In this paper the import demand equation is …rst derived analytically by assuming constant elasticy of scale production function and then the import demand
equation is estimated by Bayesian methods. Prior knowledge suggested by the
theoretical model and empirical studies is introduced in the Bayesian estimation.
The simulated posterior distribution suggests that imports are used as complements in the production process within this time period. Interestingly, the
posterior distribution implies that they are perfect complements and hence the
production technology would be of the famous Leontief form. The implied zero
price elasticity is in line with Senhadji (1998) and Hooper et al. (2000) results
for short term elasticity in industrialized countries. To assess the robustness of
the …ndings, we tested that results hold when changing the unobservable marginal cost variable to other candidate proxy variables. What might then explain
the perfect complementarity result? Intuitively, it is easier to substitute butter
than brent. A small country producing highly specialized export products has
limited options to substitute specialized import inputs needed in the production
process to domestic ones. In the case of Finland, over 70 % of imported goods
are investment, energy and raw materials (Customs Finland, 2009).
In addition to price elasticity, the income elasticity of import demand was
estimated. The simulated posterior mean for income elasticity is clearly above
unity (1.7) and higher than in the most G-7 countries according to Senhadji
(1998) and Hooper et al. (2000). Moreover, the whole posterior distribution lies
above one. Hence, the assumption of income elasticity equal to one does not
hold for Finland and is not an advisable choice for example when calibrating
macro models representing Finnish economy. The results suggest that import
demand is driven mainly by income elasticity and less by relative prices within
this time period.
This line of work could be continued in several ways. The method could be
extended to cover export elasticities to gain a broader view on Finnish external trade in macro perspective. Another way forward could be estimating an
equation at a more disaggregated level in order to capture substitution possibilities in the production technologies of for example consumption, investment
or intermediate goods. The subsitutability or complementarity might namely
depend on the type of produced good. Another interesting direction could be
to break the data set into two and exploit the pre-EU era dataset in setting up
the priors. Then Bayesian estimation could be carried out with EU era dataset
to capture the potential change of market and competition setting.
11
References
[1] Anderton, R., Baltagi, B., Skudelny, F. and Sousa, N. (2005). "Intra and
Extra Euro Area Import Demand for Manufacturers". ECB Working Paper
Series 532.
[2] Bergstrand, J. (1985). "The Gravity Equation in International Trade: Some
Microeconomic Foundations and Empirical Evidence". The Review of Economics and Statistics 67 (3), p. 474-481.
[3] Customs
Finland
(2009).
"Foreign
Trade
in
Finland
2009:
Finnish
Trade
in
Figures".
http://www.tulli.…/en/releases/ulkomaankauppatilastot/tiedotteet/
kuluvavuosi/pocketstatistics2009/index.html
[4] Erkel-Rousse, H. and Mirza, D. (2002). "Import Price Elasticities: Reconsidering the Evidence". Canadian Journal of Economics 35 (2), p. 282-306.
[5] Geweke, J. (2005). "Contemporary Bayesian econometrics and statistics".
Wiley series in probability and statistics.
[6] Hooper, P., Johnson, K. and Marquez, J. (2000). Trade Elasticitities for
the G-7 Countries. Princeton Studies in International Economics.
[7] Koop, G. (2003). "Bayesian Econometrics". John Wiley, ltd.
[8] Lancaster, T. (2004). "An Introduction to Modern Bayesian Econometrics".
Blackwell Publishing Ltd.
[9] Luoto and Luoma (2011). "A Critique of the System Estimation Approach
of Normalized CES Production Functions". HECER Discussion Paper No.
336.
[10] Senhadji, A. (1998). "Time Series Estimation of Structural Import Demand
Equations: A Cross-Country Analysis". IMF Sta¤ Papers 45 (2).
12
A
Appendix
Figure 1.Imports by use of goods in 2009.
Source: Foreign trade in 2009 (Customs Finland, 2009)
13
import volume
total value added
20,000
40,000
35,000
16,000
30,000
12,000
25,000
8,000
20,000
4,000
15,000
0
10,000
1980
1985
1990
1995
2000
2005
1980
import deflator
1985
1990
1995
2000
2005
'MC', intermediate input deflator for firms
120
120
100
100
80
80
60
60
40
40
20
20
1980
1985
1990
1995
2000
2005
Figure 2. Un…ltered series of Yt , PtF ; MtF and
14
1980
t
1985
1990
1995
2000
2005
total value added
20
25
16
20
Frequency
Frequency
import volume
12
8
4
15
10
5
0
0
-.2
-.1
i
.0
.1
.2
-.2
import deflator
-.1
.0
.1
.2
'MC', intermediate input deflator for firms
16
12
10
12
Frequency
Frequency
8
8
6
4
4
2
0
0
-.2
-.1
.0
.1
b
Figure 3. Distributions of m
b F ; ybt ; pbF
t and t
15
.2
-.2
-.1
.0
.1
.2
.2
.1
.0
.15
-.1
.10
-.2
.05
.
-.3
.00
-.05
-.10
-.15
76
78
80
82
84
86
88
90
Residual
92
94
Actual
96
98
Fitted
Figure 4. Unrestricted linear model with normal residuals assumed.
16
00
02
04
06
08
Figure 5. Squared residuals of unrestricted linear model with normal residuals assumed.
17
Figure 6. Posterior distribution.
18
Table 1. OLS estimation of theoretical model
m
b F = (bt pbF ) + ybt
t
t
Sample: 1976Q1 2008Q4
Included observations: 132
Normal residuals assumed
R-squared
S.E. of regression
Durbin-Watson stat
Coe¢ cient
-0.117
Std. Error
0.143
t-Statistic
-0.817
0.413
0.042
1.394
S.D. dependent var
Akaike info criterion
Log likelihood
0.055
-3.492
231.454
19
Prob
0.415
Ministry of Finance
Discussion Papers
1/2009 Mika Kuismanen – Ville Kämppi
The effects of fiscal policy on economic activity in Finland
2/2009 Mikko Sariola
Monetary policy and exchange rate shocks: effects on foreign trade in Finland
3/2009 Juha Itkonen
Päästökauppajärjestelmien linkittämisen ilmastopoliittiset vaikutukset
1/2010 Samuli Pietiläinen
The Bayesian Estimation of Private Investment in Finland
1/2011 Meri Obstbaum
The Finnish unemployment volatility puzzle
1/2012 Ilari Ahola
Kuntien finanssipoliittiset säännöt ja niiden toimivuus
1/2013 Marja Tuovinen
Terveysmenojen kasvu