Universitat Autònoma de Barcelona Departament d Economia...

Universitat Autònoma de Barcelona

Departament d’Economia Applicada

WORKING PAPER The real business cycle of Vietnam.

Does it matter?

ABSTRACT We apply the Real Business Cycle theory to

Vietnam business phenomena using DSGE

modeling techniques. The findings are

twofold. On the one side, Vietnam economic

fluctuations have behaved similarly to other

emerging economies. Bayesian DSGE

analysis shows that the economy is subjected

to high investment adjustment costs and

strong habit persistence. Moreover, plain

RBC model cannot effectively understand

Vietnam business cycles over the Doi Moi

era (1986 - 2015), yet the explanatory power

of RBC theory has been reinforced in the

presence of real rigidity. The other side of

the coin is an evidence that public capital

appears to have a crucial role in explaining

the movements of Vietnam economy in the

last twenty years.

Pham Thai Binh PhD Student

Supervisor 1: Prof. Hector Sala

Supervisor 2: Prof. José I. Silva

2017-Jun

The real business cycle of Vietnam. Does it matter?

Binh Thai Pham1

Department of Applied Economics, Universitat Autònoma de Barcelona

Hector Sala2

Department of Applied Economics, Universitat Autònoma de Barcelona

Jose I. Silva3

Department of Economics, Universitat de Girona

Abstract

We apply the Real Business Cycle theory to Vietnam business phenomena using DSGE modeling techniques.

The findings are twofold. On the one side, Vietnam economic fluctuations have behaved similar to other

emerging economies. Bayesian DSGE analysis shows that the economy is subjected to high investment

adjustment costs and strong habit persistence. Moreover, plain RBC model cannot effectively understand

Vietnam business cycles over the Doi Moi era (1986 - 2015), yet the explanatory power of RBC theory has

been reinforced in the presence of real rigidity. The other side of the coin is an evidence that public capital

appears to have a crucial role in explaining Vietnam economic growth in the last twenty years.

1. Introduction

The year of 2015 celebrated an important milestone of Vietnam history as it has marked the seventy-years of

independence declaration, 40 years of reunification and 30 years since the launch of Doi Moi, that is the

seriously economic renovation program. To celebrate this ceremony, World Bank and Ministry of Planning

and Investment of Vietnam (subsequently World Bank) (2016) have extensively reviewed the development

of Vietnam economy in the past thirty-years. In that report, the World Bank has admiringly recognized the

success of Doi Moi reforms as Vietnam has maintained a high average growth rate of above 5% since 1992.

There is no Southern East Asia country performing better as shown in table 1.

Broadly speaking, Vietnamese economist community has agreed on four aspects of the Vietnam´s successful

renovation. At first, differing from the most formerly communist countries part of Eastern Bloc and Soviet

[email protected]. PhD [email protected]. Supervior [email protected]. Supervisor 2.

Preprint submitted to 1st year of Doctor of Philosophy in Applied Economics 9th June 2017

Author: Binh T. Pham (WPP 01/2017/UAB) 1. INTRODUCTION

Bloc, Vietnam had incrementally reformed the economy following the China approach. In the second place,

by transforming from the farming economy Vietnam has taken its relative advantage in supplying cheap labor.

Consequently, Vietnam focused on labor-intensive industries such as footwear, apparel and woodworking,

etc., and agriculture. Besides, the government has also concentrated on improving human-capital base with

an forward looking expectation that Vietnam would be the appealing destination of foreign investments

in high-tech and automation manufacturing. Finally, Vietnam has effectively participated in regional and

transcontinental trade agreements, leading to the highly opening economy. In 2015, Vietnam trade over GDP

reached above 178%, positioning the eighth in the top ten list of most trade countries in the world.

Although there have been impressive social-economic achievements in past decades, the concerns about the

quality of economic growth have been raised in recent years. Vietnam labor productivity improvement has

been on below the growing trend since beginning of the new millennium. The slow pace of technological

progress has occurred in most of the backbone industries, namely mining, finance and real estate among

others. Besides, the farming labor force has still accounted for half a workforce, roughly thirty million people

at the time World Bank report written.

Stemming from aforementioned intuitions, we study Vietnam business cycles under the prism of neoclassical

theory. Specifically, we attempt to investigate Vietnam economic phenomena in the last thirty-years using an

dynamic stochastic general equilibrium (henceforth DSGE) model as it is very interesting to understand how

the young economy like Vietnam has been decentralizing and emerging over the past decades. We question

on how Vietnam economy maintains relatively low oscillations around its noteworthy growth path in the

twenty-first century (see figure 1).

It is worth stressing that there is hardly any real business cycle (hereafter RBC) study about Vietnam

economy, although the RBC literature has extensively emerged since the seminal work of Kydland and

Prescott (1982). Also, there has not been much literature on business cycle about emerging countries in

general and ASEAN countries in particular. Perhaps, Aguiar and Gopinath (2007) is one of few RBC works

providing some evidences about three Asean economies, saying Malaysia, Thailand and Philippines, among

other Latin-America and small developed countries. On the contrary, García-Cicco et al. (2010) just focuses

on Latin-America emerging markets. Notably, the latter work made use of Bayesian DSGE methods to

estimate structural parameters as in such medium scale New-Keyesian DSGE model of Smets and Wouters

(2003), though it is merely a RBC model with financial friction.

In the perspective of Vietnam annual data, straight RBC model results in exceptionally high contemporaneous

correlations among aggregate variables so that our findings do not departure from the RBC literature for

reasons well-documented in King and Rebelo (1999). The presence of real frictions, however, do help closing

the gap between the real and simulated economy. Strikingly, Vietnam economy exhibits high investment

adjustment costs and strong consumption persistence. Our estimate reveals highly pure time preference

discount rate, about 14% per annum, implying the economy has experienced some hard time in the past.

Indeed, Vietnam inflation had soared to above 20% in the beginning of 2010s and been on the average of

11% per annum since 1992. It is evident that Vietnam economic growth was driven by significant total factor

2

Author: Binh T. Pham (WPP 01/2017/UAB) 2. GROWTH ACCOUNTING OF VIETNAM ECONOMY

productivity improvements in the 1990s, yet we do not observe this contribution again in the twenty-first

century. Instead, public investment has performed as the main growth engine.

For the transitional economy like Vietnam, it is potentially to introduce real rigidities such as habit formation,

investment costs and capital utilization into the plain vanilla RBC model. Put differently, dynamic stochastic

general equilibrium analysis framework gives rise to more than productivity shock in the model as what we

have done in the case. To the best of our knowledge, this paper is the first real business cycle investigation of

Vietnam economy employing DSGE methods.

The remainder of this paper is organized as follows. In section 2, we study factors contributing to Vietnam

economic growth. The business cycle analysis is represented in the third section, followed by RBC modelling

and estimation in the next two sections. Section 6 provides further analysis of model simulation whilst the

concluding remarks closes the paper.

2. Growth accounting of Vietnam economy

Vietnam real GDP per capita (hereafter rGDPpc) has constantly grown up from the low of $200 US per capita

per year in 1980 (constant 2005 USD) to the six times higher over the past three decades (see figure 1 below).

In term of current price, output of Vietnamese people has reached the high of above $2100 US per person in

year of 2015. It has remarked a highly successful of the Doi Moi program, a.k.a economic renovation.

Particularly, Vietnam rGDPpc growth rate has sustained at a higher level than the average of group Asean-5

countries, which consists of Indonesia, Malaysia, Philippines, Singapore and Thailand, as during the same

thirty-years of 1986-2015, i.e., the gap between Vietnam and Asean-5 output growth rate is about 1.10%

per annum. In the first and second half decade of the 21st century, this gap has been expanded to 1.56%,

whilst in the prior of 2000 Vietnam had merely growth at the moderate pace of 4.40% compared to 3.78%

of the Asean-5. Indeed, in the pre-2000 years Vietnam had still suffered from economic sanctions due to

aftermaths of the Vietnam War, but nevertheless the Bilateral Trade Argreement (BTA) between Vietnam and

USA signed in 2001 has totally flourished the economy until the end of 2008. The BTA with USA is very

important as it openned the gate for Vietnam joining to the WTO in 5 years later.

The columns 5 and 6 of table 1 report output growth rates of Vietnam and main Asean members in the two

five-year recession periods, saying the Asian financial crisis and the Global financial crisis respectively. In

the former, 1997 - 2001, Vietnam economy had kept the very impresive growth rate of 5.06% p.a, a two-third

percent point above the first-fifteen-year period average, namely 1986 - 2000. In that time, Thailand and

Indonesia economy had been depressed heavily as their five-year average growth rates were even negative,

and there is just Singapore maintained the output growth above a percent unit.

The same thing did not happen in the latter depression. In the period 2008 - 2012, the Asean-5 economy

had only fallen down 0.14% though it was not distributed evenly among five countries. Indonesia had led

the region as it grew above its long-run rate by 0.5%. Vietnam economy had lost 0.71% in the same years

marking the highest suffering economy within the Asean community.

3


Average real output growth rate, % Standard deviation, %

Country | Period 1986-2015 1986-2000 2001-2015 1997-2001 2008-2012 1986-2015 1986-2000 2001-2015

ASEAN-5 3.74 3.78 3.74 0.18 3.57 2.92 3.92 1.49

Indonesia 3.62 3.36 3.85 -1.38 4.34 3.91 5.56 0.73

Malaysia 3.54 4.08 3.20 0.52 2.52 3.59 4.55 2.38

Philippines 2.16 1.02 3.23 0.77 3.00 2.30 2.24 1.74

Singapore 3.78 4.88 2.88 1.45 2.73 3.88 3.57 3.93

Thailand 4.20 4.99 3.39 -0.85 2.96 4.12 5.29 2.34

Vietnam 4.84 4.40 5.30 5.06 4.59 1.74 2.27 0.79

Table 1: Output growth rate of Vietnam and Asean-5 (in percentage)

Besides, it can be seen from the table 1, Vietnam economy has been less volatility in the-second-fifteen-years

(2001 - 2015) than the-first-fifteen-years as output growth standard deviation of the former is roughly a third

of the latter. Among the Asean-5 countries, Indonesia has been the most stable economy in the twenty-first

century as its corresponding standard deviation is 0.73% compared to 5.56% in the pre-2000 years. On

the contrary, Singapore appears to the highest fluctuating economy even though it is the solely developed

economy in Southeast Asia region.

It is shown that Vietnam has attained significant achievements in the post-war decades with the removal

from the list of least developed economies a decade ago4. The aftermath of the 2008 global financial crisis

has however been presenting there as the economy does exhibit to stagnation in the couple of years. In

other words, Vietnam economic growth is experiencing below of its trend. It has raised concerns about

competitiveness of Vietnam economy in recent years.

This section provides an growth empiric of the Vietnam economy. We follow the growth accounting methods

as mentioned in Bosworth and Collins (2003) to investigate the motion of Vietnam total factor of productivity

(TFP) and its contributions to growth. The Cobb-Douglas production function, i.e., Yt = AtKαt L1−α

t , will

subsequently estimated with data extracted from Penn World Table 9.0 to obtain the estimated value of the

capital income share, α . This step is important due to the needs of the succeeding sections.

In the next section, basing on Uribe and Schmitt-Grohé (2017) and King and Rebelo (1999) as well as Cooley

and Prescott (1995) among many others, the business cycles of Vietnam economy shall be summarized

via log-quadratic and Hodrick-Prescott filtering processes. To be more specific, cyclic components of six

variables in the national account identity equation, Y = C+ I +G+X −M, have been isolated from the

trending counterparts. Consequently, the measures of Vietnam business cycles are observed through the

prism of deviations from trends.

4The UNCTAD officially removed Vietnam out of List of Least Developed Countries in 2003, 18 years from the outset of theDoi Moi.

4


1970 1975 1980 1985 1990 1995 2000 2005 2010 2015-10

-5

0

5

10

15%

, gro

wth

rat

e

200

400

600

800

1000

1200

Con

stan

t US

$ 20

05

VIETNAM

rGDPpc growth rate5% levelrGDP per capita

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015-10

-5

0

5

10ASEAN-5

rGDPpc growth rate

Figure 1: Vietnam and ASEAN-5 output per capita, 1970 - 2015, constant US$ 2005. Source: UNStad, 2016.

2.1. Factor share estimation

To estimate total factor productivity (TFP) we mainly rely on Penn World Tables (henceforth PWT) 9.0

database due to the lack of official Vietnam capital stock accumulation information and depreciation rates.

The Cobb-Douglas production function has form:

Yt = AtKαt L1−α

t

where capital letters Y, K and L are aggregating values of outputs, capital stocks and labor forces employed,

respectively; At is total factor productivity which has trend λ t, i.e., At = A0ελ t . It is more convenient if we

work with labor intensive values such that:

Yt

Lt= A0ε

λ t(

Kt

Lt

)α

Taking logarithm both sides, the equation reads:

yt = a0 +λ t +αkt

where a0 ≡ logA0, yt ≡ log YtLt

and kt ≡ log KtLt

. Next, we estimate the simple OLS model with per worker

data as follows:

yt = β0 +β1× trend +β2× kt + εt (1)

5


The Solow residuals (TFP) have been computed from the econometric model above such as: T FPt =

yt − β2× kt with α ≡ β2 and λ ≡ β1.

It is worth noting that real GDP (rgd pna) and capital stocks (rkna) in PWT 9.0 are compiled in term of

constant prices across countries and times, so it may not be totally compatible with real GDP data from

conventional sources, e.g., World Bank or General Statistical Office (GSO) of Vietnam because of the

conversion factors implied. For the case of Vietnam, the database has only covered forty-five years, from

1970 to 2014, and it does not provides working population data, albeit employed people is available. We

hence compute per labor capital stock and output using 16+ - 64 ages population ratio of World Bank. We

estimate the equation (1) in five periods5, namely 1970 - 2015, 1986 - 2015, 1992 - 2015, 1986 - 2000 and

2001 - 2015. The second and third period are important as because the latter refers to the year of 1992, at

which the private sector having been fully recognized by the amended constitutional law, while the former

1986 is the year of Doi Moi program initiated. Table 2 below reports all coefficient estimations.

Coverage 1970-2015 1986-2015 1992-2015 1986-2000 2001-2015

λ ≡ β1 0.0207*** 0.0192*** 0.0170*** 0.0197*** 0.0249***

(0.000918) (0.00331) (0.00742) (0.00484) (0.00789)

α ≡ β2 0.307*** 0.325*** 0.337*** 0.365*** 0.245**

(0.0176) (0.0438) (0.0908) (0.0696) (0.1043)

a0 ≡ logA0 4.544*** 4.466*** 4.480*** 4.112*** 4.887***

(0.115) (0.225) (0.431) (0.381) (0.532)

Obs 45 30 23 15 14

Standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1

Table 2: Factor shares without accounting for augmented human capital (Vietnam, PWT 9.0)

It is highly likely that the average of capital share α in range [0.250, 0.340] inducing the labor income share

could be in between 66% - 75%. The coefficient of trend, λ , is about 2.10% in the whole period 1970 - 2014,

it even exhibits a higher level of 2.49% per annum during 2001 - 2015. The magnitude of capital income

share has been declined over years from the high of 36.5% in 1986 - 2000 to the low of 24.5% in the 21st

century, implying the structural changes of Vietnam economy in the last fifteen years.

Nonetheless, the figure 2 demonstrates the fluctuations of linear detrended TFP components are much

different among periods and among countries. Vietnam TFP’s standard deviation in the first-fifteen-year of

Doi Moi has been three-fold of the post-2000 period, as of 2.34% and 0.78%, respectively. Five years after

Doi Moi, Vietnam had enjoyed the eight consecutive years of significantly improving of TFP as shown in

table 3 the growth rate in 1992 - 2015 is of 2.19% compared to 1.84% and 2.10% of 1986 - 2000 and 2001 -

2015, respectively. Since the Asian financial crisis, TFP has however exhibited to the sluggish growth rates

below the trend in two-third of the times, i.e., 2.10% average growth rate against the trend of 2.50%. This

5As noticed, PWT 9.0 does not update to year of 2015, hence, the corresponding periods will not contain data of the last year.

6


1970 1980 1990 2000 2010-20

0

20Indonesia

1970 1980 1990 2000 2010-20

0

20Malaysia

1970 1980 1990 2000 2010-50

0

50Philippines

1970 1980 1990 2000 2010-20

0

20Singapore

1970 1980 1990 2000 2010-20

0

20Thailand

1970 1980 1990 2000 2010-10

0

10Vietnam

Figure 2: Linear detrended TFP of Vietnam and Asean-5 countries

could express one of intrinsic weaknesses of Vietnam economy, i.e., the economy has primarily relied upon

exporting labor-intensive goods such as footwear and wearing apparel, crude oils and raw materials, rice,

land-based and sea-based aquaculture and low value-added agriculture products.

Statistics 1970-2015 1986-2015 1992-2015 1986-2000 2001-2015

TFP growth rate, % 1.96 1.97 2.19 1.84 2.10

Linear detrended TFP standard deviations, % 2.91 1.76 1.35 2.34 0.78

Table 3: Vietnam TFP growth and variations

Taking into account of human capital improvement such as years of schooling being internalized in every

worker per se, the Cobb-Douglas production function has been augmented by incorporating variable H,

capital human index. The model shall be written as:

Yt = AtKα(HtLt)1−α

where H ≡ eφ(s) with s is years of schooling, h is thus logH ≡ φ(s). Applying the same transformation pro-

cedure as above, we alternatively write the difference equation (2) to accounting for growth decomposition.

∆yt = α∆kt +(1−α)∆ht +∆logAt (2)

PWT 9.0 provides Vietnam human capital index (hc) starting from year of 1970, which is mainly based on

interpolation and extrapolation of Barro and Lee (2013) dataset. PWT has made some assumptions about

7


returns on education by specifying the rates of returns to education as of 13.4%, 10.1% and 6.8% for 0 to 4,

above 4 to below 8, and above 8 years of schooling, respectively. Also, in much of the growth literature, the

returns on education of developing countries has been assumed of 7% to 13% 6 per schooling-year. Lee and

Hong (2012) have made the choice of flat rate 8% for some developing Asia countries including Vietnam.

We do keep the default setting of PWT 9.0 to figure out the contributions of labor augmenting technology to

output growth.

Before reaching output growth decomposition, the capital share,say α , needs to be explicitly identified. From

the previous estimates, we already know the Vietnam capital share of output falls in range of 25% - 34%

depending on period estimated. Growth literature generally suggests the choice of one-third for α , Bosworth

and Collins (2003) alternatively make use of 0.35 across countries in the sample. Gollins (2002) claimed

that capital share should be varying among countries as the author provided evidence for Vietnam after

adjustments as of 80.2% in the year 1989 indicating the very low level of physical capital at that time. This

can be understood if we take into account the fact that Vietnam’s initial capital stock per capita is very low

due to the consequences of thirty-five years of war against USA. Figure (3) demonstrates the movement of

capital stock accumulation as it has remarkably sped up in the past three decades. On the contrary, Lee and

Hong (2012) used constant α = 0.4 for Vietnam and other Asia countries. Simply put, the capital share of

Vietnam apparently lies in between 0.2 and 0.4.

1970 1975 1980 1985 1990 1995 2000 2005 2010 20150

1

2

3

US

D (

PP

P)

104 Capital stock per labor

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015-10

-5

0

5

10

%

Growth rate

Labor productivity growthOutput per capita growthCapital per labor growth

Figure 3: Capital stock revolution and economic growth per worker

6Bosworth and Collins (2003) noted that returns to schooling is estimated as of 10%, 12% and 13% for Latin America, Asia andAffrica, respectively.

8


It is necessary to cross validate the estimate of α by using actual wage with labor income share (LIS)

formula: LIS≡ Wt LtYt

7. Unfortunately, information about average labor income is not available in all

evaluating periods, there are merely few data points for recent years as shown in the table 4. On the average

of six years observed, the labor income share varies in the range of 67.5% - 72.0 % which has agreed the

feasible range of α ≡ [0.32, 0.38] for the long period of forty-five years.

Year 2009 2010 2011 2012 2013 2014 2015

Average salary (×106 VND / year) 25.27 30.29 37.33 45.08 49.51 53.70 55.87

Employment (Million people) 47.7 49.0 50.4 51.4 52.2 52.7 52.8

Nominal Output (×109 VND) 1,809,149 2,157,828 2,779,880 3,245,419 3,584,262 3,937,856 4,192,862

Labor income share (%) ≡ (1−α) 0.667 0.689 0.676 0.714 0.721 0.719 0.704

Capital income share, α 0.333 0.311 0.324 0.286 0.279 0.281 0.296

Table 4: Labor income share (LIS). Data source: GSO-Vietnam, CEIC and ADB.

2.2. Growth decomposition

With empirics on hand, the choice of α = 0.35 is neither unrealistic nor far from econometric estimations.

The contributions of TFP and human capital to Vietnamese worker’s output growth can be easily decomposed

by utilizing the equation (2). Building on the assumptions of returns to schooling, the total contributions of

TFP and human capital to Vietnamese worker’s output growth in the last twenty years is approximately to

one-half, of which the proportion of TFP has been lower than one-fourth over the years. It would seem that

the average TFP growth rate in the past twenty years is nearby 40% lower than the coinciding human capital

build-up rate showing the persistence of slow technological progress. The TFP was relatively high in the first

ten years of Doi Moi could be explained by the very modest educational status and the rural economy at that

time. It is thus very likely that over the past two decades the main driver of Vietnam economic growth is

attributed to capital accumulation as returns on capital accounting for approximately 60% of the real GDP

per labor growth since 1992. Again, this reveals the gray color of Vietnam economy structure in the present

owing to the fact that Vietnam has focused on exporting low added-value products and raw materials for a

long time.

7The formula is derived by assuming that perfect market has forced wage equating marginal return on labor.

9

Author: Binh T. Pham (WPP 01/2017/UAB) 3. VIETNAM BUSINESS CYCLES

Average growth rate, % p.a. 1986-2015 1992-2015 2001-2015

Real GDP per capita, y 4.82 5.52 5.30

Real GDP per worker, yworker 4.00 4.46 4.37

Capital stock, k 6.64 7.93 7.45

Human capital growth, h 0.96 1.34 1.41

Total factor productivity, T FP 1.06 1.01 0.84

Contributions to labor productivity growth

Physical capital 49.35 63.48 60.51

Human capital 16.66 19.43 21.42

TFP 33.98 17.08 18.06

Table 5: Growth’s decompositions

3. Vietnam business cycles

Isolating business cycles (cyclic components) from their trending counterparts is a basic procedure in most

of the RBC studies. The two previously mentioned filters, saying linear-quadratic and Hodrick-Prescott,

have been customarily employed in the literature. It is worth mentioning that the reasons for applying two

different filters have been noted in the recent literature8 as these authors stated that original HP filter may

introduce spurious dynamic into the cyclical series, it hence necessarily introduces the one-sided version

(only backward looking) as suggested by Stock and Watson (1999), and restated in Mehra (2004).

In what follows, the decomposition of output variances are computed and compared between those filtering

techniques. Also, the corresponding macroeconomic aggregate of the ASEAN-5 countries are arranged as

the benchmarks.

The linear-quadratic filter is nothing-else but an OLS estimation of the simple equation as below:

zt = γ0 + γ1t + γ2t2 + εct

where zt is the time-series variable in logarithm form and t is time trend variable. The cyclic component is

defined as zct ≡ εct = zt − zt , with z being fitted values (also known as trending parts).

On the contrary, HP filter is actually the optimization process with respect to the loss function:

T

∑t=1

(zct )

2 +λhp

T

∑t=1

[(zg

t+1− zgt )− (zg

t − zgt−1)

]2where zg and zc are trend and cyclic components, respectively. The multiplier λhp is chosen so that the

cyclical series is minimal. If λhp→ ∞ then zgt is the linear trend, while λhp = 0 gives us the series itself. In

8See DeJong and Dave (2011) and Hamilton (2016).

10


practice, one can set λhp = 100 for annual data or λhp = 1600 for quarterly series. Ravn and Uhlig (2002)

suggest λhp = 6.25 for yearly data while the literature also recognizes the usage of λhp = 400 or 10 and 25.

Importantly, HP filter consists of both backward-looking and forward-looking elements, it is possible that the

spuriousness may appear during the detrending process by the HP imposed pattern per se (Harvey and Jaeger,

1993). For the purpose of eliminating future values in each data point estimated, Stock and Watson (1999)

proposed an one-sided HP version which only makes use of backward data. It thus estimates the trending

series τt from the equations in the following:

zt = τt + εt ,

(1−L)2τt = ηt ,

where τt is unobserved trend component, εt and ηt are two uncorrelated white noises which have relative

variances such that q = var(ηt)var(εt)

.

Before discussing Vietnam business stylized facts, it should be recalling that business cycle stylized facts

have been intensively analyzed and reviewed over the past decades. Undoubtedly, it is difficult to iterate the

endless list of authors in the RBC literature, yet one can easily point out some typical facts about US economy

as in King and Rebelo (1999), King et al. (1988a, b), and world economy from Uribe and Schmitt-Grohé

(2017)9 among others. Loosely speaking, academia has agreed consumption volatility depending on the kind

of goods purchased, i.e., consumption on durable goods is more fluctuating than output, while the contrary

has been observed for non-durable consumption. Investment generally subjects to very high vibration with

respect to output. Government spending is usually acyclical and oscillates below the level of output variations.

In reference to cross-border business activities, literature suggests that exports and imports are both positive

correlated with output but the trade balance and current account movements are reversed or countercyclicality.

In other words, the components of aggregate demand excluding government purchases tend to be procyclical.

As noted in Romer (2012), business fluctuations do not follow any regular patterns, and are distributed

unequally among output’s components, but their variances may change significantly over time.

9Uribe and Schmitt-Grohe (2017) presents the estimates of 122 countries, excluding Vietnam due to data limitations.

11


Country VIETNAM ASEAN-5

Period Measure y c g i x m y c g i x m

1970

-201

5 Mean 4.06 3.40 4.27 6.85 7.33 7.37 3.72 3.48 3.85 4.58 5.88 5.99

SD 3.03 3.18 4.02 10.65 8.07 8.31 2.65 2.35 2.8 8.56 5.71 7.33

Min -5.80 -5.79 -6.33 -12.78 -13.26 -12.04 -9.25 -6.33 -4.56 -37.66 -9.89 -13.84

Max 10.55 10.55 10.55 41.34 29.33 29.02 6.67 8.94 10.33 17.45 13.98 17.84

1986

-201

5 Mean 4.84 3.86 5.18 9.02 9.73 9.78 3.74 3.57 3.34 4.15 6.64 6.37

SD 1.74 2.39 3.58 12.17 8.48 8.81 2.92 2.54 2.34 9.75 5.71 7.85

Min 0.27 -0.21 -6.33 -12.78 -13.26 -12.04 -9.25 -6.33 -4.56 -37.66 -9.89 -13.84

Max 7.40 8.40 10.55 41.34 29.33 29.02 6.67 8.94 8.38 17.45 13.98 17.84

1986

-200

0 Mean 4.40 2.48 3.94 10.86 9.36 9.20 3.78 3.66 2.31 3.9 8.93 8.16

SD 2.27 1.95 4.61 16.11 11.83 11.56 3.92 3.54 2.59 13.63 4.42 8.3

Min 0.27 -0.21 -6.33 -12.78 -13.26 -12.04 -9.25 -6.33 -4.56 -37.66 -0.9 -13.69

Max 7.40 7.15 10.06 41.34 29.33 29.02 6.67 8.94 4.86 17.45 13.98 17.84

2001

-201

5 Mean 5.29 5.24 6.43 7.19 10.09 10.37 3.69 3.48 4.38 4.41 4.34 4.57

SD 0.82 1.99 1.42 6.35 2.92 5.14 1.53 0.91 1.54 3.32 6.06 7.21

Min 3.99 1.24 4.26 -9.22 3.99 2.93 0.43 0.98 1.19 -1.24 -9.89 -13.84

Max 6.36 8.40 10.55 20.71 14.87 23.46 6.56 4.77 8.38 8.9 13.52 16.4

ASEAN-5 countries: Indonesia, Malaysia, Philippines, Singapore and Thailand.

Table 6: Growth rate (in percentage, %) of six national income variables (y, c, g, i , x and m)

Regarding to Vietnam macroeconomic data as represented in table 6, it is not possible to work with

quarterly frequency over the very long periods. The most recently quarterly output data covers only post-

2000 years, say from 2000 to 2016. Therefore, we provide all estimations below using annual data from

United Nation Statistical Division (UNSTAD) since the timing coverage has been long enough to get the

meaningful business cycles. Table 7 below reports standard deviations of cyclical components (denoted as

σ j for j = y, c, i, g, xand m) of all national income identity variables and their relative ratios with respect

to σy. Noticing that the variables are natural logarithm of level data so that the cylical components are

deviations from trends which can successively be mapped to the percentage of trends. Mathematically, if

an arbitrary variable Z is measured in level then its cyclical component in percentage can be expressed as

zct ≡ log(Zc) = log( Zt

Zgt) = log(Zt)− log(Zg

t ), with Zgt being the trending counterpart.

From the table 7, the linear-quadratic filter reports the larger standard deviations for both of Vietnam and

ASEAN-5 economic fluctuations compared to the two Hodrick-Prescott filters. As shown in the upper

plot of figure 4, it should be noticed that standard HP filters extract much more secular information from

the time series, whereas the gap produced by one-sided HP with linear-quadratic trend (see the lower plot

of the same figure) is much closer to the linear cyclic components indicating that one-sided filter have a

propensity to retain as much intertemporal information as possible. The figure 7 in Appendix 1 demonstrates

the patterns of all six macroeconomic variables in term of deviations from trends. Visibly, the black curves,

12


Filter Linear-quadratic Hodrick-Prescott One-sided HP

Standard deviations Vietnam ASEAN-5 Vietnam ASEAN-5 Vietnam ASEAN-5

σy 4.56 4.22 2.82 3.09 2.40 2.81

σk/σy 0.95 - 0.80 - 1.35 -

σc/σy 0.92 0.85 1.06 0.90 1.03 0.88

σi/σy 4.97 3.54 3.33 3.37 3.86 3.38

σg/σy 1.18 1.77 1.47 0.90 1.49 1.02

σx/σy 2.45 2.59 1.98 1.80 2.21 1.87

σm/σy 2.50 2.53 2.06 2.59 2.27 2.49

Table 7: Business cycle’s statistics for Vietnam and ASEAN-5, 1970-2015.

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015-10

-5

0

5

10

HP trend less linear trend

Linear-quadratic cycle

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015-10

-5

0

5

10

One-sided trend less linear trend

Linear-quadratic cycle

Figure 4: Trend gaps of different filters and linear-quadratic cyclical components

i.e., linear-quadratic cyclical series appear to be more aggressive movements. Meanwhile, standard and

one-sided HP filters markedly differ from each other in four out of six time series, they do have somewhat

similar pattern but the blue line, say one-sided series, appears to tracking on the linear movements.

Vietnam output growth in the forty-five years, from 1970 to 2015, oscillates around the trends by 4.56% per

year given the linear-quadratic estimation, roughly 60% higher of the alternatives, saying the gaps of 1.74%

and 2.16% with respect to standard and one-sided HP filters. The gap between linear-quadratic and HP filters

are considerably narrower in the case of ASEAN-5, as below of 1.4% per annum. These differences lead to

some contradictory facts drawn from the estimates. For the sake of convenience, we contrast Vietnam and

ASEAN economy’s facts with evidences of emerging countries extracted from Uribe and Schmitt-Grohe

(2017) in the following table:

13


Linear-quadratic Hodrick-Prescott

Volatility w.r.t output Vietnam ASEAN-5 Emerging Vietnam ASEAN-5 Emerging

Private consumption Lower Lower Lower Higher Lower Higher

Government consumption Higher Lower Higher Higher Lower Higher

Capital stock Lower - - Mixed - -

Investment Higher Higher Higher Higher Higher Higher

Exports Higher Higher Higher Higher Higher Higher

Imports Higher Higher Higher Higher Higher Higher

Table 8: Co-movements of output and other aggregate variables

The results suggest that Vietnam business facts are much consistent with the literature in the light of

Hodrick-Prescott detrending processes. The linear filter produces a bit different figure. Investment and

foreign trade activities are by far the most volatility components, but nevertheless Vietnam capital stock

and household consumption is merely more or less fluctuating than output. The former coincides with the

previous growth facts as Vietnam has strong demands on imported goods in past years due to the needs

of high-tech manufacturing equipment, by-products for fabricating and assembling industries, electronic

devices, automobiles, etc, and huge investments in public infrastructures. The evidence of capital stock is

mixed as one-sided filter generating a ratio greater than one. Vietnam economy is young so that capital stock

fluctuation is considerably close to output variation.

Consumption of Vietnamese household generally fluctuates higher than output although linear filter does

not support this conclusion. Literature on emerging economies also provides the same conclusion. The

great success of Doi Moi is to make the whole society benefited from economic growth. The living standard

of Vietnamese people has been much improved over thirty-years as “more than 40 million people escaped

poverty over the course of two decades”, World Bank (2016) concluded. People become rich, they tend to

consume more. Yet, the consumption patterns are different between Vietnam and Asean-5 as we discuss in

next paragraphs.

Also, computed from table 6, Vietnam exports and imports growth rates are all high, and have sustained at the

notable pace of 10% per year since 2001, almost always double of the Asean-5 peer, leading to the openness

index has surged from 1.13 in the beginning of the period to the height of 2.39 in the end of 2015. However,

the Vietnam trade balance always negative over years indicating the less competitive and vulnerability of the

economy to adverse shocks.

From the standard HP filtering perspectives, Vietnam exports and imports seems to be experienced acyclical

as their lagged and contemporaneous correlations with output are very low, 0.358 and 0.424 respectively,

compared to Asean-5 as shown in column 6 from the left of table 9. Their autocorrelation coefficients almost

die out after one year suggesting that Vietnam foreign trades are highly sensitive to international economic

shocks. Meanwhile, Asean-5 exports and imports are moderate stable and strong procyclicality.

Probably, Vietnam household consumption, investment and government consumption are weakly procyclical.

14

Author: Binh T. Pham (WPP 01/2017/UAB) 4. THE DSGE-RBC MODEL

Their contemporaneous correlations with output are all below 0.600 but investment interestingly exhibits

“time-to-build” effect as its first and second-order correlation are much higher as of 0.704 and 0.712,

respectively. Correspondingly, lead correlations between investment and output are poor such the second-

order deems to be uncorrelated. On the contrary, Asean-5 private consumption and investment are extremely

procyclical and moderately persistent, as their contemporaneous correlations are all close to one, and

first-order autocorrelations are much above two-third.

Correlation with yc Lags of variables Leads of variables Auto-correlation

Country Variables -3 -2 -1 0 +1 +2 +3 -1 -2 -3

Vietnam

yc 0.017 0.451 0.812 1.000 0.802 0.455 0.056 0.774 0.437 0.053

cc -0.460 -0.150 0.234 0.566 0.622 0.530 0.250 0.587 0.149 -0.239

ic 0.491 0.712 0.704 0.590 0.338 -0.005 -0.376 0.515 0.390 0.085

gc 0.379 0.591 0.627 0.513 0.231 -0.231 -0.519 0.732 0.407 0.123

xc -0.026 0.135 0.338 0.358 0.261 0.344 0.160 0.201 0.074 -0.112

mc -0.056 0.104 0.342 0.424 0.360 0.441 0.221 0.201 0.038 -0.185

Asean-5

yc 0.037 0.338 0.727 1.000 0.716 0.340 0.062 0.682 0.304 0.053

cc -0.119 0.222 0.649 0.951 0.730 0.352 0.044 0.689 0.286 -0.013

ic 0.056 0.365 0.720 0.981 0.741 0.294 -0.042 0.708 0.278 0.001

gc 0.036 0.308 0.623 0.832 0.646 0.359 0.052 0.708 0.298 0.054

xc 0.105 0.400 0.618 0.820 0.603 0.246 0.181 0.507 0.148 -0.045

mc -0.023 0.373 0.671 0.928 0.681 0.253 0.069 0.575 0.215 -0.024

Table 9: Correlation with contemporaneous yc and yc, cc, ic gc, xc and mc. HP filter with λ = 100. Period 1986 - 2015.

In Vietnam, public demands for goods and services are more persistent than private ones. First-order

autocorrelation of government consumption is as high of 0.732 compared to 0.587 and 0.708 of domestically

private consumption and Asean-5 counterpart, respectively. Yet, Asean-5 government consumption exhibits

to procyclicality.

To sum up, from the beginning it is evident that Vietnam economic growth has been led by capital accumula-

tion. The contributions of TFP to output growth have been relatively low, saying below one-fifth, since the

Asian financial crisis. It is probable that Vietnam economy has been exposing to technological regress in

recent years. As being a low-middle income country, the characteristics of Vietnam business movements are

totally consistent with much of the real business cycle literature. The next sections will set up a dynamic

stochastic general equilibrium RBC model then estimate its structural parameters. Our goal is to point out

how much RBC theory can explain Vietnam aggregate data.

4. The DSGE-RBC model

The 1980s of the last century was the golden time of real business cycle theory. Starting with the very

influential work by Kydland and Prescott (1982) and subsequent development by Long and Plosser (1983)

15


and Hansen (1985) as well as King et al. (1988a, b), RBC theory urges on the dominant role of technology

shocks in explaining economic fluctuations, implying the cause mainly comes from exogenous shocks to

Solow residuals. Needless to say, money has no position in RBC analysis, yet it should be noticed that the

original objective of the RBC research program is to establish a theoretical linkage between the dynamics

of neoclassical model and economic movements. Hence, Prescott (1986) and King et al. (1988a) argued

that people may want to understand the real fluctuations prior to the money matters, and such RBC theory

actually lies in the more general framework.

Despite of the recent dynamic stochastic general equilibrium models promoting the New Keynesian analytical

framework, it is not uncommon observing that RBC analysis and its variants have widely been used as a

useful toolkit to study economic “business phenomenon”10. As its name self-explanatory, the so-called

DSGE-RBC models are structural in the context of underlying economic interpretations. In other words, each

equation has been underpinned by economic theory. General equilibrium implies micro-founded features of

the model as the welfare of representative agent bound to some non-trivial constraints has been optimized.

That economy is viewed as being in continuous equilibrium in which an atomic agent, saying the household,

makes optimal decisions (both in the present and future) without repeating any persistent errors. Here, one

critical assumption has been imposed is the rationality of household.

Nonetheless, the most interesting aspect of DSGE modeling is the presence of numerous stochastic processes

in the system of equations. The first generation of RBC model in 1980s were only concerns about productivity

shock, i.e., if we are employing Cobb-Douglas production technology then productivity shock is nothing else

but unanticipated variations of Solow residuals. Adopting Hick neutral technological progress, one can write

aggregate output, Yt as follows:

Yt = AtF(Kt−1, Ht) (3)

Yt ≥Ct + It (4)

where Ct and It are aggregate consumption and investment, respectively. At is commonly supposed following

an AR(1) process so that,

logAt+1 = (1−ρA)logA+ρAlogAt + εA,t+1

with εA,t ∼ N(0,σ2A) and A denotes TFP level in the long-run (steady-state), Ht and Kt−1 are quantity of labor

employed and level of physical stock, respectively.

The capital stock in RBC is dated at which it has been determined. Strictly speaking, it is time-to-build effect

due to Kydland and Prescott (1982) as physical capital requires some installation periods to be productive.

Following end-of-period convention, the law of motion for capital in much of the DSGE literature is as:

Kt = (1−δ )Kt−1 + It (5)

Equation (5) implies new capital stock, It , needs one period to be in service so that the economy output at the

10Prescott (1986) suggested the usage of “business phenomenon” refering to business cycles.

16


end of time t is actually generated by utilizing physical assets given at the beginning of time t11. For that

reason, it can be said that capital stock is predetermined by construction.

Following Cooley and Prescott (1995) notations, the decentralized RBC model considers a symmetric closed-

economy which is inhabited with a unit mass of identical households. We are assuming in that economy

people live forever and enjoy the same preference over consumption and leisure at each date. Furthermore,

the identical preference has been presumed to be additively separable so that the period utility function of an

arbitrary individual jth is described as below:

u(c j,0, c j,1, . . . ,c j,∞) =∞

∑t=0

βtU(c j,t , l j,t) (6)

where small ct and lt represent for individual consumption and leisure. The admissible utility function has

to be satisfied some regularity conditions. It is assumed that U(.) is continuously increasing, concave and

twice differentiable. Mathematically, the conditions are Uc ≡ ∂U(.)∂ct

> 0 and Ucc ≡ ∂ 2U(.)

∂C2t

< 0; as well as

Ul ≡ ∂U(.)∂ lt

> 0 and Ull ≡ ∂ 2U(.)

∂ l2t

< 0. Because of Ucc < 0 and Ull < 0, household is generally impatient. He

hence values future utility less than current one at the discount rate θ > 0 so that 0 < β = 11+θ≤ 1 is time

discount factor. Also, we assume U(·) is separable in term of its arguments leading to Ucl =Ulc = 0.

The household in the economy chooses the sequences of consumption and leisure,

c j,t , l j,t∞

t=0, to maximize

his welfare, subjecting to period-by-period budget constraints:

c j,t + i j,t = w j,tht + rtk j,t−1 (7)

and law of motion for capital:

k j,t = (1−δ )k j,t−1 + i j,t (8)

The left hand side of (7) indicates the use of income whereas the right hand side gives the source of

household’s earnings. For the sake of convenience, one often normalize individual time endowment to a

unity such that l j,t +h j,t = 1 where h j,t is time devoting to work. Again, as household being identical, the

economy subjects to aggregating rules as follows: Ct =´ 1

0 c j,td j, Ht =´ 1

0 h j,td j and Kt =´ 1

0 k j,td j. Notice

that a unit mass of identical households results in Ct = c j,t , Ht = h j,t and Kt = k j,t .

Households are also assumed holding physical capital so that non-human agents, a.k.a firms, rent capital

from them. As specified previously, symmetric information sweeps out any abnormal returns so that earnings

from capital rent, rt , and labor supply, wt , are at marginal. Providing homogeneous of degree one technology

in the economy, we read:

rt = Fk(·) = At∂F(Kt−1, Ht)

∂Kt−1(9)

wt = Fh(·) = At∂F(Kt−1, Ht)

∂Ht(10)

Equation (9) and (10) show that if everything else is equal then a positive shock to total factor productivity

11Accordingly, level of capital stock at the end of t−1 is the same as at the beginning of time t.

17


pushes up wage and returns on invested capital.

Alternatively, we can formally let firms maximize their period profits, their optimizing problem is static as

below:

Πi,t = AtF(ki,t−1, hi,t)− rtki,t−1−wthi,t (11)

From the first order conditions of (11), one reaches the same results as above.

In equation (11) we are indexing firms with small letter i denoting a continuum of price-taking firms in our

economy so that the same aggregating rule can be applied.

Kt =

ˆ 1

0k j,td j =

ˆ 1

0ki,tdi

We now turn back household problem defined in (6). The individual Lagrangian function can be written

L j =∞

∑t=0

βt[U(c j,t , 1−h j,t)−λt(c j,t + k j,t − (1−δ )k j,t−1−w j,tht +−rtk j,t−1)

Taking into account for transversality condition, i.e., limt→∞

β tλtk j,t = 0 where λt = Uc(·). The first order

condition offers:

Uc(c j,t ,1−h j,t)

Uc(c j,t+1,1−h j,t+1)= β (1−δ + rt+1) (12)

wt =Uh(c j,t ,1−h j,t)

Uc(c j,t ,1−h j,t)= Fh(Kt−1, Ht) (13)

along with equations(7)−(9) and the feasibility constraint AtF(kt−1, ht)≥ ct + it , we reach the most standard

RBC model by Hansen (1985). Given constant returns to scale Cobb-Douglas technology and aforementioned

assumptions we can rewrite these equations in term of aggregate variables.

As stated elsewhere in the paper, we employ DSGE techniques to study real fluctuations of Vietnam over the

thirty-years after the Doi Moi initiative. In the subsequent section, we augment above RBC model with real

rigidities described in King and Rebelo (1999) and Christiano, Eichenbaum and Evans (2005). Also, the

model shares many features discussed in Leeper, Walker and Yang (2010).

4.1. Setting the environment

To be consistent with the previous analysis on Vietnam business cycles, we build up a non-steady growth

economy comprising of three economic agents, namely household, firms and government. Household

consumes one good, invests in private capital, and contributes taxes to the government. But he has received a

lump-sum transfer from the government. Typically, the economy is assumed to be symmetric and endowed

with constant returns to scale Cobb-Douglas technology so that firms are identical in terms of technological

aspect. We allow for productivity motion being exogenously stochastic.

Over the past decades, Vietnam government has made huge investments in infrastructures and public facilities

18


as public investment roughly accounts for 40% of total investment12. For instance, the new highway or

bridge would help reduce transportation costs, or sufficient water and power supply have also lured and

fostered investments in rural regions13. Firms enjoy infrastructure improvements as they can make extra

profits by making use of public inputs. We therefore model two roles of government in the economy: as the

tax levying entity by imposing three kind of taxes to households, saying taxes on labor and capital income

and consumption tax, and as the public input supplier.

Besides, the model has contained components reflective real inflexibility of the economy. It is worth

considering capital-related rigidities such as capital utilization as in King and Rebelo (1999) and quadratic

type of investment adjustment costs as in Christiano, Eichenbaum and Evans (2005) since the authors claimed

rigidness provides somewhat better explanations to US business cycles than the vanilla RBC. Furthermore, the

persistence of household consumption has been considered as it is now a typical feature of DSGE modelling.

Boldrin, Christiano and Fisher (2001) among many others argued that the existence of consumption habit has

been supported by empirical facts.

There is a shock attached with household’s decision as we want the time discount factor β varying over

time instead of a constant. In addition, we design two shocks linking with fiscal consumption and public

investment as we do permit stochastic fluctuations in government fiscal schedule. However, in this paper

we do not consider shock to investment quality as those in Justiniano, Primiceri and Tambalotti (2010) and

shock to labor supply as in Smets and Wouter (2003) among others.

Households

It has not been much different from the previous RBC specification about the household welfare problem.

Simply put, the additive utility function of a representative household has form:

maxE0

∞

∑t=0

βtzp

t

[log(CP,t −hCP,t−1 +πCG,t)−χ

H1+σLt

1+σL

](14)

where β and zpt are time discount factor and time preference shock, respectively. Ht : labor supply in term of

hours-work; 1σL

is Frisch labor elasticity; h: internal habit formation coefficient of household. CP,t denotes

private consumption, whilst CG,t is government purchases being given exogenously. Time endowment is

normalized so that Ht +Lt = 1, with Lt representing for leisure time. χ is dis-utility labor parameter to be

calibrated such that Ht satisfies empirical hour-work. The time preference shifter zPt is assumed following

AR(1) process as logzPt = ρPlogzP

t−1 +σPεPt with εP

t ∼ N(0,1).

12Data from General Statistics Office of Vietnam.13“Access to basic infrastructure in Vietnam has improved substantially. Significant progress were charted from 1993 to 2012.

For example, at least 99 percent of the population now use electricity as their main source of lighting compared to 14 percentmore than twenty years ago. More than 67 percent of the rural population now enjoy access to sanitation facilities, and morethan 61 percent have access to clean water, compared to only 36 percent and 17 percent, respectively, two decades earlier.”,http://www.worldbank.org/en/country/vietnam/overview, (World Bank, 2017).

19


The involvement of government consumption and public capital in RBC model have been well-established in

Aschauser (1985, 1989), Aiyagari et al. (1992) and Baxter and King (1993) among others. Baxter and King

(1993) did specify the function Γ(CG,t , KG,t−1) with KG representing for public capital though the authors

argued that KG has no direct influence on household’s decisions. The variable CG,t is normally assumed as an

uncontrollable stochastic process (Christiano and Eichenbaum, 1992). Parameter π governs the elasticity of

substitution between private and public consumption in final consumption. Basically, there is no constraint

on sign of π , yet setting of 0≤ π < 1 implies government purchases crowding out private consumption.

The household only consumes on what he earns so that his budget should be even in every period. We also

assume physical capital owned by household, the inter-temporal budget constraint reads:

(1+ τc)CP,t + IP,t = (1− τw)WtHt +(1− τk)rk,tutKP,t−1 +Gt +(1− τk)Pro f itst (15)

where Wt is wage; Kp,t is private capital. Household pays consumption tax τc, capital gain tax (or return on

capital rental, τk) and labor tax bracket τw but he receives a lump-sum transfers Gt from the government

along with net extraordinary profits due to free access to public infrastructures.

Household chooses optimal rate of capital service ut in each period and level of investment which subject to

costs of over-utilization and investment congestion.

KP,t = (1−δP−δ (ut))KP,t−1 + IP,t

[1−S

(IP,t

IP,t−1

)](16)

S(

ItIt−1

)is a function representing investment adjustment costs, which commonly has a quadratic form as

S( IP,t

IP,t−1

)= κ

2

( IP,tIP,t−1− 1)2, where κ = ψ

1−ψis defined as in Smets and Wouters (2003). In the steady-state

adjustment costs do not exist, we therefore collect the following claims S(1) = 0, S′(1) = 0 and S′′(1)> 0.

δP denotes depreciation rate of private capital. The function δ (u) represents the rate of accelerated capital

depreciation, hence it should be an increasing function with respect to ut such that δ ′(ut)> 0, δ ′′(ut)> 0.

Adopting the quadratic form: δ (u) = δ1(ut −1)+ δ22 (ut −1)2 similar to Schmitt-Grohé and Uribe (2012), it

thus gives δ (u) = δ (1) = 0.

Household maximizes (14) by choosing the sequence CP,t , Ht , KP,t , ut∞t=0 subject to (15) and (16) taking

as given other stochastic processes and suitable initial conditions. The Lagrangian can be written as:

L ≡maxE0

∞

∑t=0

βt

zp

t

[log(CP,t −hCP,t−1 +πCG,t)−χ

H1+σLt

1+σL

]

−λt

[(1+ τc)CP,t + IP,t −

((1− τw)WtHt +(1− τk)rk,tutKP,t−1 +Gt +(1− τk)Pro f itst

)](17)

−µt

[KP,t − (1−δp−δ (ut))KP,t−1− IP,t

(1−S

(IP,t

IP,t−1

))]If we define the ratio qt =

µtλt

as the Tobin´s Q, with a little bit algebra one collects from solving (17) the

system of equations defining household’s optimal choices in the equilibrium:

20


qt =(1− τk)rk,t

δ ′(ut)=

(1− τk)rk,t

δ1 +δ2(ut −1)(18)

(1+ τc)zPt χHσL

t

(1− τw)Wt= zP

t (CP,t −hCP,t−1 +πCG,t)−1−βhE0

[zP

t+1 (CP,t+1−hCP,t +πCG,t+1)−1]

(19)

1 = qt

[1− κ

2

(IP,t

IP,t−1−1)2

−κ

(IP,t

IP,t−1−1)

IP,t

IP,t−1

]+βE0qt+1κ

(IP,t

IP,t−1−1)(

IP,t

IP,t−1

)2[ zPt+1Wt

zPt Wt+1

(Ht+1

Ht

)σL]

(20)

qt = βE0

[zP

t+1Wt

zPt Wt+1

(Ht+1

Ht

)σL][

qt+1(1−δP−δ (ut+1))+(1− τk)rk,t+1ut+1

](21)

Our first observation is that if S(·) = 0, then qt = 1 implying λt = µt and δ ′(ut) = (1− τk)rk,t so that the

relationship between δ1 and rental rate is known in the stationary state.

Firms

We assume identical firms exhibit to the constant returns to scale Cobb-Douglas technology with respect to

all production factors. In the presence of public capital, our assumption differs from Baxter and King (1993)

and Leeper, Walker and Yang (2010) as these authors only constrain private factors implying increasing

returns to scale. Firm’s management optimizes period-profits by setting level of physical capital rented from

households and labor inputs.

Yt = zAt (utKP,t−1)

α1Kα2G,t−1Hα3

t (22)

where KG denotes public capital such as infrastructures, public utilities, etc. By construction, we have

α1 +α2 +α3 = 1 therefore firm earns extraordinary economic profits from KG by a factor share α2.

Profit function can be written as

Πt = zAt (utKP,t−1)

α1Kα2G,t−1Hα3

t − rk,tKP,t−1−WtHt

In perfect information environment, firm makes no profit with respect to private factors. That is marginal

returns on private capital and labor input equal zero. There is no market price for public inputs, firm therefore

earn extra profits by an amount Pro f itst =∂Πt

KG,t−1= α2

YtKG,t−1

, which then returns to household net of capital

gain tax, say (1− τk)Pro f its. Since capital utilization has already been set by household, firms just choose

level of capital and labor to rent so that by taking 1st-order derivative w.r.t Ht and KP,t−1, we collect two

trivial relations:∂Πt

∂Ht= 0⇒Wt = α3

Yt

Ht(23)

∂Πt

∂KP,t−1= 0⇒ rk,t = α1

Yt

KP,t−1(24)

21


Government

Government raises fund by imposing taxes on households, then makes lump-sum transfers, investments and

spending for operational activities. The budget constraint is given by:

τwWtHt + τk(rk,tut −δP−δ (u))KP,t−1 + τkPro f itst + τcCP,t = Gt +CG,t + IG,t (25)

Differing from Leeper, Walker and Yang (2010), we merely adopt a basic law of motion for public capital:

KG,t = (1−δG)KG,t−1 + IG,t (26)

The national income identity has been modified to take into account for capital utilization rate by adding

costs or subtracting gains.

Yt =CP,t +CG,t + IP,t + IG,t +δ (ut)KP,t−1 (27)

where

CG,t = zCGt ζCGYt (28)

IG,t = zIGt ζIGYt (29)

with ζCG and ζIG are public consumption and public investment over GDP, respectively. We let CG and IG

expose to AR(1) shocks capturing variations in government expenditure schedule.

logzst = ρslogzs

t−1 +σsε(s)t (30)

where s ∈ IG,CG, ρs and σs are persistent coefficient and standard deviations of shocks, respectively. As

usual, we note εst ∼ N(0,1).

4.2. Linearization and model solution

4.2.1. Solution algorithm

From above equilibrium conditions, our economy has been determined by the set of seventeen equations

which comprise of four exogenous stochastic process (zAt , zP

t , zCGt , zIG

t ) and thirteen equations defining

corresponding endogenous variables (CP,t ,CG,t , Ip,t , IG,t , KP,t , KG,t ,Wt , Ht , ut , rk,t , Gt , Pro f itst , qt).

The above system apparently subjects to high non-linearity so it seems to be that an analytical solution

cannot be found. The textbook strategy is a two-stage approach. In the first stage, numerical method such

as high-order Taylor expansion or log-linearization à la Uhlig (1999) is employed to linearize equation by

equation. The latter technique is actually the first-order Taylor approximation when variables are in logarithm

while the former can be generalized from second-order to third- or even fifth-order14 though we very often

do not gain much from the above second-order. However, Uhlig (1999) has widely been used as it is easy to

understand and abstract from tedious calculus computation.

14Dynare++ program provides solution to the fifth-order approximation.

22


The result of first-stage is the so-called rational expectation linear system15. It is a non-trivial linear system

as the equations do contain both of backward-looking and forward-looking endogenous variables. There are

many algorithms can provide the solution for the class of RE linear system but the first and very influential

algorithm is due to Blanchard and Kahn (henceforth BK) (1980) since the authors have also proposed

conditions for which the existence and uniqueness of the RE linear solution can be verified.

To make the paper be self-contained, we briefly explain BK algorithm and its extension by Klein (2000)

before log-linearizing our system. Let us consider a linear RE model in the state-space representation16 as:

Γ0

[xt+1

Etyt+1

]︸︷︷︸

(n+m)×1

= Γ1

[xt

yt

]+Ψεt (31)

where xt and yt are (n×1) vector of state or predetermined variables and (m×1) jump or control variables,

respectively; εt is (k× 1) vector of exogenous random processes which have zero means. The order of

equations in the system is important as variables attached with expectation operator should be put in the

bottom.

The weakness of BK algorithm is the assumption of Γ0 being invertible because this assumption is easily

violated in large system. Basically, BK method bases on eigen-decomposion of matrices Ξ ≡ Γ−10 Γ1 and

Ω≡ Γ−10 Ψ. Truly, matrix Ξ can be decomposed into PΛP−1 where Λ is a diagonal matrix of Ξ’s eigenvalues,

and P is matrix of corresponding eigenvectors.

The BK condition for the existence and uniqueness of the solution is that the number of eigenvalues greater

than one in absolute values, m, must equal to number of jump (or forward-looking) variables, m. Otherwise,

we are facing two extreme cases:

• If m > m then there are infinite solutions or indeterminacy matter.

• If m < m then the solution does not exist.

Alternatively, Klein (2000) makes use of generalized Schur decomposition to solve (31). Given any square

matrix Γ0 and Γ1, the generalized Schur decomposition of Γ0 and Γ1 are the upper triangular matrix S and T

such that:

QΓ0Z = S and QΓ1Z = T

where QQ′ = I and ZZ′ = I so that Z−1 = Z′. Obviously, Klein do not make any assumptions about Γ0 but

he does impose a feasible assumption about a partitioned matrix as we will see in below.

With respect to Klein (2000) notation, the multivariate linear RE model (31) can be rewritten in the following

form:

SZ′[

xt+1

Etyt+1

]= T Z′

[xt

yt

]+QΨνt (32)

15We only refer to first-order approximation when dealing with solution algorithm.16This notation has been taken from McCandless (2008).

23


where νt is an zero-mean VAR(1) system which has matrix of coefficient Ω, i.e., νt = Ωνt−1 + εt .

Partition S, Z and T into conformable blocks,[S11 S12

021 S22

][Z′11 Z′12

Z′21 Z′22

][xt+1

Etyt+1

]=

[T11 T12

021 T22

][Z′11 Z′12

Z′21 Z′22

][xt

yt

]+

[Q11 Q12

Q21 Q22

][Ψ1

Ψ2

]νt

(33)

where S22 is block of unstable roots, it thus has size of (m×m). S11 should be (n× n) matrix that only

constains stable eigenvalues. That means S and T are sorted conformably with Q and Z before partitioning.

The sorting rule is simple as the stable eigenvalues should come first. Among Klein’s assumptions, the

invertible matrix Z11 (the author claimed that it is very likely guaranteed) plays the key role as it implies

the BK condition - the number of predetermined variables equates to the number of stable eigenvalues. The

algorithm is very straightforward as we firstly solve the bottom block in (33) then the upper block will

be derived from the solution of the first17. We rewrite system (33) in more compact form by absorbing

Z′[

xt yt

]′into

[st ut

]′where st and ut are vector of stable and unstable components of the system,

respectively. Thus, [S11 S12

021 S22

]Et

[st+1

ut+1

]=

[T11 T12

021 T22

][st

ut

]+

[Q1

Q2

]Ψνt

At the end, we are looking for the policy rule that has a representation in the state space form:

xt+1 = Axt +Bεt (34)

yt =Cxt +Dεt

where εt is assumed a Gaussian vector white noise process satisfying Eεt = 0, Eεtε′t = I and Eεtεt− j = 0 for

any j 6= 0. The A B C D matrices are defined as below:

C =(Z22)−1Z21

D =(Z22−Z21Z−111 Z12)M

A =NZ21Z−111

B =Z11S−111 T11Z−1

11 Ψ+L

with vec(M) = [(Ω′⊗S22)− Iν ⊗T22]−1 vec(Q2Ψ), and N = Z11S−1

11 T11Z−111 , where Iν is square identity ma-

trix compatible with number of shocks ν . The final matrix is L=−NZ12M+Z11S−111 [T12M−S12MΩ+Q1Ψ]+

Z12MΩ.

4.2.2. Linearization and stationary equilibrium

For the sake of simplicity, we follow Uhlig (1999) to derive the linear approximations of non-linear equi-

librium conditions. It is convenient to write all variables in term of deviation from its steady-state value.

The simple rule is to define xt = logXt − log(Xt), so that ext = elogXt−log(Xt). Rearrange it we get a formula

17We summarize in depth typical solution algorithms in the appendix.

24


Xt = Xext , xt is however small so that Xt ≈ Xt(1+ xt). An another rule is frequently used ext+ayt ≈ 1+ xt +ayt .

One can interpret xt as the percentage deviation from its steady-state value.

In the steady-state, all stochastic variables zA,P,CG,IGt = 1 and Xt+1 = Xt = Xt−1 providing X be endogenous

variable. We thus remove time subscript t and denote over-bar letters as corresponding endogenous variables

in stationary equilibrium. We have IG = ζIGY , CG = ζCGY and Y (1−ζIG−ζCG) = CP + IP. Law of motion

for public capital gives KGY = ζIG

δG, whereas wage W = α3

YH and capital rental rate rk = α1

YKP

. By construction,

δ (u) = δ (1) = 0, δ ′(u) = δ1 and S( IP

IP≡ 1)= 0 so that we collect terms as below:

rk =

1β+δP−1

1− τk(35)

δ1 = (1− τk)rk =1β+δP−1 (36)

The implication from (36) is that accelerated depreciation rate should be calibrated to net rental rate in

equilibrium.

Cobb-Douglas production function provides additional relations:

Y = Kα1P Kα2

G Hα3 = H

[(KP

Y

)α1(

KG

Y

)α2] 1

α3

= H

[(α1

rk

)α1(

ζIG

δG

)α2] 1

α3

(37)

FromχHσL

(1− τw)W=

11+ τc

[(CP−hCP +πCG

)−1−βh(CP−hCP +πCG

)−1]

with Ip = δpKp so that CpY = (1−ζIG−ζCG)−δp

KpY = (1−ζIG−ζCG)−δp

α1rk

.

Substituting out, we may write the relationship between H and other deep parameters as follows:

H =

α3(1− τw)(1−βh)

χ(1+ τc)

[(1−ζIG−ζCG−δp

α1rk)(1−h)+πζCG)

]

11+σL

(38)

In RBC literature, it is much more convenient if hour-work H has been set to a constant level of 0 < H∗ < 1

and χ is correspondingly calibrated then we can compute Y directly from production function (37) given all

deep parameters having been set. As a consequence, other steady-state values are calculated easily.

We close this section with the set of seventeen log-linearized equations that describes our real economy.

The four stochastic processes has form logzt = ρlogzt−1 + εt , rewriting log zezt = ρlog zezt−1 + εt . In

equilibrium z≡ 1, it leads to

zt = ρ zt−1 + εt (39)

National income identity

Y yt = CPcp,t +CGcg,t + IP iP,t + IG ig,t +δ1utKP,t−1kp,t−1 (40)

25


Public consumption and investment

CGcg,t = ζCGY (zcg,t + yt) (41)

IG ig,t = ζIGY (zcg,t + yt) (42)

Government budget constraint

Gct +CGcg,t + IG ig,t =τwW H(wt + ht)+ τkKP,t−1

[rk(ut + rk,t + kp,t−1)−δPkp,t−1−δ1ut

]+ τkPro f its ˜pro f itst + τcCPcp,t (43)

Law of motion of public capital

KGkg,t = KG(1−δG)kg,t−1 + IG ig,t (44)

Firm

wt = α3(yt − ht) (45)

rk,t = α1(yt − kp,t−1) (46)˜pro f its = y− kg,t−1 (47)

yt = za,t +α1ut +α1kp,t +α2kg,t−1 +α3ht (48)

Household

(1− τk)rk,t = qt +δ2

δ1ut (49)

(1+ τc)zPt χHσL

t

(1− τw)Wt= zP

t auxt −βhE0

[zP

t+1auxt+1

](50)

where auxt = (CP,t −hCP,t−1 +πCG)−1 and auxt =

(hcp,t−1−cp,t)CP−πCGcg

aux , then applying first-order Taylor

expansion we get the linearized relationship:

χ(1+ τc)HσL

(1− τw)Wt

(zP

t +σLht − wt)= aux(zP

t + auxt)+βhauxE0[zP

t+1 + auxt+1]

(51)

Let dI =IP,t

Ip,t−1⇒ dI = ip,t − ip,t−1 rewrite equation (20) as follows:

1 = qt[1−κ(dI−1)2−κ(dI−1)dI

]+βκE0qt+1(d3

I −d2I )

[zP

t+1Wt

zPt Wt+1

(Ht+1

Ht

)σL]

It can easily see that d3I −d2

I = dIexp(3dI−2dI) = d(1+ dI), applying first-order Taylor expansion to write:

26

Author: Binh T. Pham (WPP 01/2017/UAB) 5. PARAMETER ESTIMATION

ip− ip,t−1 =1

κ(1+β )q (52)

From equation (21), we expand δ (ut+1) = δ1(ut+1−1)+0.5δ2(ut+1−1)2 to have:

qt = βE0

[zP

t+1Wt

zPt Wt+1

(Ht+1

Ht

)σL]

qt+1

[(1−δp+δ1−

12

δ2)+(δ2−δ1)ut+1−12

δ2u2t+1

]+(1−τk)rk,t+1ut+1

It is easily to see that G≡ zPt+1Wt

zPt Wt+1

(Ht+1Ht

)σL

= 1+ zPt+1− zP

t + wt− wt+1 +σL(ht+1− ht)≡ G. It subsequently

deduces Gqt+1 = G+ qt+1, Grk,t+1ut+1 = rk

(G+ rk,t+1 + ut+1

)and Gu2

t+1 = G+2ut+1.

The last log-linearized equation is as:

q = βE0

[1+δp + rk(1− τk)

]G+(1+δp)qt+1 +

[(1− τk)rk−δ1

]ut+1 +(1− τk)rkrk,t+1

−1 (53)

5. Parameter estimation

5.1. Overview

In the past decade, it has been witnessed a vast majority of economic literature employing Bayesian-based

techniques, and in particular to the class of Markov Chain Monte Carlo (MCMC) algorithms. Within the

field of macroeconomic modelling, Smets and Wouters (2003, 2007) were probably the most influential

medium-scale DSGE models estimated with Bayesian computational methods, as the authors heavily made

use of Random Walk Metropolis-Hastings (RWMH) simulators. In the following paragraphs we merely

represent the stylized estimation strategy used in Smets and Wouters (2007), which is now implemented

in DYNARE and similar packages. Our short discussion has been inspired from the excellent practical

algorithm implementations due to Blake and Mumtaz (2012), and the review of Fernández-Villaverde (2010).

Also, we are presumably working on log-linearized DSGE model, such the Bayesian algorithms can afford

non-linear system is however beyond our attention in this paper, we refer readers to Herbst and Schorfheider

(2015) for a standard treatment of non-linear matter.

At the heart of Bayesian estimation techniques the Bayes rule for the conditional distribution of parameter

set θ ∈Θ given observational data Y is written as:

π(θ |Y ) = f (Y |θ)π(θ)f (Y )

(54)

where π(θ |Y ) is the so-called posterior probability distribution function of θ (or posterior distribution shortly)

conditional on observed data Y ; the prior distribution p(θ) is the unconditional probability distribution of θ ;

the likelihood function f (Y |θ) is the same as in classical econometric methods. The last component f (Y ), i.e.

marginal likelihood, is defined as a constant such that f (Y ) =´

f (Y |θ)π(θ)dθ to ensure´

π(θ |Y )dθ = 1.

27


By construction marginal likelihood is independent of estimating parameter set θ , it is convenient to rewrite

equation 54 in the following form:

π(θ |Y ) ∝ f (Y |θ)π(θ) (55)

Noting that π(θ |Y ) in equation 55 is not integrated to 1, it nonetheless retains all distributional characteristics

as in the original form. To be honest, posterior distribution is the beauty of Bayesian technique since it

embraces information before knowing the data, i.e. π(θ), and information contained in data observed,

f (Y |θ ). In Bayesian view, the prior has been updated by the likelihood function when receiving new data.

Thus, equation 55 explicitly requires us specifying prior knowledge about DSGE model parameters and

evaluation of likelihood function for every data point. In practice we work with log-likelihood value so that

the posterior is the sum of two right-hand-side components.

While the prior always has tractable form as we prefer to do so, the most challenging task is to compute

likelihood function because of highly dimensional of parameter space and unobserved state variables.

Fortunately, providing state space representation of log-linearized DSGE model Kalman filter is the delightful

method for estimating likelihood function. In what follows, we recall the random walk MHMC algorithm

employed in Smets and Wouters (2003, 2007) among many others, which has been studied extensively in An

and Schorfheide (2007). Avoiding lengthiness, we represent in the appendix how likelihood function can be

estimated via Kalman filter.

Random-walk Metropolis-Hastings algorithm for DSGE model estimation

1. Step 1: Finding posterior mode θ ∗ by maximizing log-likelihood function: maxθ

logπ(θ |Y )=log f (Y |θ)+ logπ(θ).

2. Step 2: Compute Hessian matrix Σ of logπ(θ ∗|Y ) . Let Σ = Σ−1.

3. Step 3: Draw an initial parameter set θ 0 from normal distribution N (θ ∗, c20Σ) or import

directly.

4. Step 4: Loop over i = 1 . . .L times. Draw a new set of parameter θ (i) = θ (i−1)+ ε , where

ε ∼N (θ ∗, c20Σ) .

Compute the probability of accepting θ (i) as such pa =min[exp(log f (Y |θ (i))−log f (Y |θ (i−1))), 1

].

Draw u∼Uni f orm(0, 1), if u < pa then θ (i) is accepted for the next iteration.

5. Step 5: Compute expected value of posterior function of θ by approximating E(h(θ) =1

L−B ∑L−Bj=1 h(θ ( j), where B is burn-in draws.

5.2. Estimation

Theoretically, prior distribution could be any known distribution basing on our informative beliefs in advance

about estimating parameters. Practically, we should only choose prior distribution such that its support could

afford the parameter boundary. For instance, capital share in Cobb-Douglas production function is confined

in the range of 0 and 1 so that beta distribution is subjectively a candidate prior. Taking time discount rate θ

28


as another example, we assume the positive time discount factor β being at most of 1, implying θ ≥ 0 as

β = 11+θ

. The gamma distribution has the support of [0,+∞) is naturally a good choice of θ .

While specifying prior distributions of estimating parameters is an easy task, setting up prior’s moments,

namely mean and variance is more difficult. Possibly, this is a trickiest part of DSGE estimation when dealing

with large set of parameters because everything could be wrong or likely be wrong if one pines down too

narrow or too widen prior’s parameters. Indeed, the rule of thumb for this step is due to Smets and Wouters

(2003, 2007) and Del Negro and Schorfheide (2008).

We are going to estimate the first group of eight deep parameters Θ1 = θ , h, η , χ, π, ψ, α1, α3 where

η being Frisch elasticity as we already know σL = 1η

. Literature says η lies in between 0 and 1, we

specify η ∼ Beta(0.5, 0.2), where first parameter of prior distribution denotes mean, the other is standard

deviation. Habit formation and investment adjustment costs have been set similar to Smets and Wouters

(2003) as h ∼ Beta(0.7, 0.1) and ψ ∼ Beta(0.8, 0.05), respectively. Stressing that the prior of ψ is fairly

tight stemming from our belief that investment costs in Vietnam is relatively higher than most of the advanced

economies.

As earlier discussed, π is assumed to be positive and unbound from above so its prior should be Gamma(1, 0.5).

That is we do not limit π within [0, 1]. Factor shares are easier thing as both of α1 and α2 having been elicited

tighter than common practice as such Beta(0.3, 0.05) and Beta(0.65, 0.15) respectively, since we do have

their empirics in the previous section. Constant returns to scale implies α2 belonging to Beta distribution

since α2 = 1−α1−α3. Finally, χ is labor dis-utility parameter related to H, hour-work in steady-state as

shown in equation 38. It is often calibrated so that H roughly closes to 13 , however, we estimate it with other

parameters with prior being approximate to value computed from 38, i.e., χ ∼ Gamma(9, 3).

The second set consists of parameters that characterize the DSGE model’s endogenous propagation mecha-

nism, namely Θ2 = ρA, σA, ρP, σP, ρIG, σIG, ρCG, σCG. However, their priors are very straightforward as

we follow aforementioned authors to jot down ρA, ρP, ρIG, ρCG∼Beta(0.75, 0.15) and σA, σP, σIG, σCG∼Inverse−Gamma(0.01, 0.10). Put simply, AR(1) coefficients ρ(·) ∈ [0, 1] are assumed to be nonexplosive,

meanwhile, standard deviation has support of [0,+∞). All parameter specifications are summarized in the

column 3 to 6 of table 10.

The hardly remaining parameters such as depreciation rates, government spending-related ratios and tax

codes will not be estimated. We have to calibrated them based on RBC literature and other empirical

evidences. It is obvious to us that public capital is normally expected to be depreciated at the lower pace

than private counterpart, and government spending schedule should fluctuate around a fixed ratio to GDP.

In general, the choice of depreciation rate in RBC analysis varies from 3% to 12% depending on specific

sector and modeled economy. Cooley and Prescott (1995) argued that steady-growth economy should be

depreciated by 5% per annum but higher rate has been recommended for zero-growth economy. As we

have no information about compositions of either public or private capital, hence, we are considering the

much lower depreciation rate of 4% for public capital while private capital is assumed wearing out at 8% per

annum of its book value.

29


Vietnam government consumption is round 6% in 2000s yet its investment has been much higher. Statistical

yearbooks have reported Vietnam public investment accounting for 40% or higher of gross investment over

past decades. As gross investment over GDP in the thirty-years period had dramatically increased from the

low of 14% - 17% in years before 1992 to the high of 33% - 39% in the mid of 2010s, we thus calibrate

public consumption and public investment ratios as ζCG = 6% and ζIG = 10.8%, respectively.

Tax codes are such difficult things for the young economy like Vietnam. Before Asian financial crisis,

Vietnam government had not imposed VAT tax on goods and services, since then an identical consumption

tax of 10% is applied. Labor income tax bracket has mainly been social contributions for a long time. It is

worth noting that there are two personal income tax bases, namely basic wage and actual wage, in Vietnam.

The basic wage is only used as tax base for social contributions while the progressive income tax is levied on

actual income net of deductibles. Since, family deductibles is relatively high compared to Vietnam labor

wage, “de-factor” revenue from high income person is actually as low as 5% of total fiscal revenue in the

most recent years18. Corporate income tax has decreased considerably from 28% in 2000s to the level of

22% by 2014 but tax on capital gain such as leasing is straight as of 20%. Consequently, we assume a flat

corporate and labor income tax as of 25%.

Prior Posterior

Parameter Description Mean SD Mode Dist. Mean Mode 5% 95%

θ = 1β−1 Time discount rate 0.08 0.03 0.069 Gamma 0.138 0.094 0.181

h Habit formation 0.70 0.10 0.722 Beta 0.812 0.737 0.882

η Frisch elasticity 0.50 0.2 0.500 Gamma 0.731 0.535 0.940

χ Labor dis-utility 9.0 3.0 8.000 Gamma 8.935 4.149 13.413

π Public spending elasticity 1 0.26 0.750 Gamma 0.620 0.148 1.094

ψ Investment costs 0.8 0.05 0.810 Beta 0.910 0.874 0.946

α1 Private capital income share 0.35 0.075 0.295 Beta 0.221 0.164 0.280

α3 Labor income share 0.60 0.20 0.692 Beta 0.601 0.384 0.801

ρA TFP persistent 0.75 0.15 0.844 Beta 0.802 0.685 0.925

ρP Preference persistent 0.75 0.15 0.844 Beta 0.435 0.248 0.616

ρCG Government spending persistent 0.75 0.15 0.923 Beta 0.787 0.566 0.998

ρIG Public investment persistent 0.75 0.15 0.844 Beta 0.813 0.717 0.915

σA Productivity shock 0.01 0.10 InvGamma 0.013 0.009 0.016

σP Cons. preference shock 0.01 0.10 InvGamma 0.121 0.080 0.163

σIG Public investment shock 0.01 0.10 InvGamma 0.064 0.048 0.080

σCG Government spending shock 0.01 0.10 InvGamma 0.022 0.018 0.027

Table 10: Estimating parameter priors and posteriors

The last step is to point out measurement equations in order to construct likelihood function described as in

18Data from Statistical Yearbook of Vietnam, 2015.

30


equation (55). We reuse four annual data series which are aggregate output, private consumption, government

consumption and investment in section 3. The fifth series is aggregating hours-work per year, being gathered

from The Conference Board19. They all are normalized in term of per capita and detrended with one-sided

Hodrick-Prescott filter because the model belongs to the class of zero-growth.

In DSGE estimation we must have as many exogenous shocks in the model as the number of observation

series, otherwise we are in trouble with singularity matter. The model has four shocks, thus we cope with

singularity by incorporating an measurement error. At the end of the day we define five measurement

equations as what follows:

Yobs =

yt − y

cp,t − cp

cg,t − cg

it − i

ht − h

+

0

0

0

εime

0

(56)

where Yobs is vector of cyclic components of Yt ,CP,t ,CG,t , It , Ht, small letter denotes variable in logarithm

and over bar indicates steady-state value. It should be noticed that the first block of above equation expresses

variables in the sense of deviation from their trends.

The model is estimated over the period of 1986 - 2015 by simulating posterior distributions through 100,000

draws from Metropolis-Hastings algorithm. The estimation outputs are put in Appendix B. The last four

columns in table 10 report the estimates of posterior mean, mode and highest posterior density interval,

respectively.

The first observation is that most of the posterior distributions can be distinguished from the priors (see

appendix), showing that those estimates are informative except for parameter ρCG as it has failed to identifi-

cation test due to Iskrev (2010). We get π=0.620 demonstrating government purchase produces crowding

out effects on household consumption. Nonetheless, the household does exhibit strong habit persistent as

h = 0.811 compared to 0.57 of the euro zone (Smets and Wouters, 2003).

The estimated value of θ is 13.8% per annum entailing β = 0.879. The pure time discount rate is fairly

high implying Vietnam economy exposed to some high inflation periods. Truly, Vietnam had experienced

hyperinflation during the period of 1985 - 1995 and GDP deflator has dramatically soared up to 375% over

20 years20, viz., 1996 - 2015.

More interestingly, private capital income share and labor income share are estimated by 22.1% and 60.1%,

respectively, leading to public capital income share is about 17.8%, which communicates the important

role of public capital in Vietnam economy. The implication is that public investments being made over

thirty-years have a crucial contribution to Vietnam economic growth and living standard improvements.

The estimate of investment costs κ = 0.911−0.91 = 10 is very high compared to 4−6 in the literature for US and

19https://www.conference-board.org/data/economydatabase/20GDP deflator increases from the low of 30.6 in 1996 to the high level of 145.8 in 2015.

31


Eurozone. To a moderate extent, it is relevant as Vietnam economic environment is not very favorable for

starting business because of high hidden costs, overloading infrastructures and weakly institutional indicators.

Regarding to persistent shocks, household preference is much less persistent, whereas the remains, saying

TFP, government spending and public investments are adequately high in term of yearly AR(1) process. Time

preference persistence ρP is considerably low, saying 0.435, but the standard deviation σP is notably large as

of 12.1%, indicating time discount factor β has strongly oscillated throughout the assessment period. It can

be seen that the triplet of β , h and ρP has well characterized Vietnam household consumption behavior as in

the lower chart of figure 5, the movement of private consumption is mainly driven by shocks to consumption

preference shifter.

Although the estimates of ρA and ρIG are nearly identical as of 0.802 and 0.813 respectively, their variances

are much different as σIG = 6.4% versus σA = 1.3%. The ratio of σIGσA

= 4.92 interestingly closes to that

of linear-quadratic cyclical components in section 3, one hence should take that ratio into account when

calibrating Vietnam economy over Doi Moi era. The values of ρCG and σCG are 0.787 and 2.2% respectively,

not differing from government consumption’s cyclical statistics.

One advantage of DSGE estimation techniques is a capability to analyze propagation mechanisms through

Kalman disturbance smoothing. The contribution of each exogenous process to data variations could have

been computed and visualized as in figure 5. One may be interested in how much output fluctuations be

explained by shocks to TFP and investments as well as the role of time preference shifter in the model. The

upper chart in figure 5 illustrates the driven of output motion is mainly due to TFP shocks (green bar) but

other shocks, i.e., light blue and red bars representing for time preference and public investment shocks

respectively, have also had considerable impacts in particular periods. In periods 12 to 16 (corresponding to

Asian financial crisis, 1997 - 2002) Vietnam economy did appear to structural changes. Truly, the contribution

of TFP shocks flipped back to negative side in year 17th, saying 2002, meanwhile preference and public

investment shocks help explain output vibrations in years 1997 - 2015. Public investment plays negative

role in Asian financial crisis, yet in the late middle of 2010s the economy witnessed the strong expansion of

public investment. There was a real estate market boom in 2008 - 2011 at the time of government stimulus

package in effective. The balloon asset has nevertheless been burst in the end of 2011 deepening the economy

in next two years. The expansion of fiscal spending actually caused a big side-effect reflected by the last

three periods of the chart, as Vietnam government has followed tighter monetary policy and been trying to

balance budget deficits since the end of 2012.

It seems to be that the Vietnam economy has been growing below the trend originated by the low contributions

of TFP to labor productivity as shown in table 5, TFP only accounts for 18% of worker productivity growth

during 2000 - 2015. Strong preference to consumption but weak productivity in the 2010s are mostly

responsible for Vietnam prolonged stagnation in the post-2008 of global financial crisis.

We close the section with discussion about time preference shocks to Vietnam economy. It is worth reminding

that positive shock to zPt causes an increasing in β and vice versa, which in turn implies a decreasing or

increasing in pure time discount rate θ , respectively. If people put less weight on future consumption (lower

32


θ ) they spend more today. Shock decomposition of private consumption suggests that high level of household

consumption in 2010s could be understood by the fairly favorable business atmosphere at that time. In

2001, the Vietnam - US Bilateral Trade Argreement was signed and effective in a year later tempting large

investments in both private and public sectors. Vietnam joined WTO21 in 2007 but the global financial crisis

occurred in the next year had induced the economy into sluggish growth until 2015.

Output

0 5 10 15 20 25 30-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

Initial values

eps_zcg

eps_zig

eps_za

eps_zp

Private Consumption

0 5 10 15 20 25 30-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

Initial values

eps_zcg

eps_zig

eps_za

eps_zp

Figure 5: Shock decomposition of output and private consumption. Light blue denotes shock due to time preference shifter; green isshock to TFP , red and dark blue are shocks to public investment and government consumption, respectively.

21Abbreviate for World Trade Organization

33

Author: Binh T. Pham (WPP 01/2017/UAB) 6. SIMULATION AND DISCUSSION

6. Simulation and discussion

The common practice in RBC analysis is to compare moments generated by modeled economy with actual

economy. It is necessary to understand how good RBC theory help explain the target economy, saying

Vietnam as an example. In this section, we calibrate four models A, B, C and basic RBC then contrast their

moments, correlation coefficients with actual counterparts.

The first model, called model A, is the one that we developed in section 5, model B is a bit less inflexible than

model A by having no habit formation and disutility of government consumption. We switch off investment

adjustment costs in model C while the plain vanilla RBC (Basic) is similar to one described in King and

Rebelo (1999).

The parameter set for all three augmenting models has been taken from results above excepting for model

Basic which comprises of less deep parameters and only a shock to productivity. Notice that we are disabling

model features by setting the corresponding parameters close to zero instead of rebuilding up from scratch.

Taking model B for instance, assuming no habit formation and no disutility of government consumption are

to impose h = 0 and π = 1, respectively. In model C we simply let ψ → 0 lead to κ = ψ

1−ψ= 0. Besides,

one thing differs between model A, B and C is the values of χ as we have to calibrate it to match H ≈ 0.33,

that is hour-work is assumed to be eight hours per day or one-third of time endowment in stationary state.

Put simply, we reports new updates of parameters in table below:

Parameter χ h π ψ α1 α2 α3 β ρA δG δP ζCG ζCG

Model A 8.8 0.812 0.633 0.910

0.221 0.178 0.601 0.879 0.802 0.04 0.08 0.06 0.108Model B 6.7 0 1 0.910

Model C 8.8 0.812 0.633 0

Basic 9.5 - - - 0.400 - 0.600 0.879 0.802 - 0.08 - -

Table 11: Some specific calibrated parameters

Taking shock standard deviation σAε = 1.76% to TFP from table 3, we simulate 1000 economy realizations,

each run generates 1045 data points which is used to compute statistics after dropping 1000 burn-in periods

to match simulated periods to original data (1970 - 2015). The next step is to adjusted simulated series using

one-sided Hodrick-Prescott filter and compute their moments with respect to the last thirty data points.

From table 12, we observe basic RBC model reproduces around 86% of actual output variation, σbasicy =

1.75% compared to σactualy = 2.03%. It however cannot explain variations in hour-work and final consumption

as the estimates are fairly low. Crucially, aggregate variables in the plain model exhibit to be strong correlated

with output as contemporaneous correlations are all above 0.850. Actual data in table 13 (both of standard

and one-sided Hodrick-Prescott filtered cyclic components) report much lower contemporaneous correlation

with output. That suggests basic RBC has failed to capture the dynamic Vietnam business cycles.

Turning to augmenting RBC models, without investment adjustment costs model C nearly replicates output,

investment and hour-work variations, as σCy equals to 91.5% of σactual

y . As similar to model A , it produces a

34


bit lower relative standard deviations of household and government consumption in term of output variations

(or ratios for short), namely 0.80 and 1.40 respectively, meanwhile, the corresponding figures of actual

components are of 1.05 and 1.93. Private consumption in model C appears to be uncorrelated to output

movement, but government consumption is moderately procyclical. It can be also seen that model C and A

are similar in the sense of first order autocorrelation although model A have a considerably higher output

autocorrelation.

Model A only explains 70% of actual output fluctuations, yet its relative ratios are in acceptable range. The

presence of habit formation makes private consumption less variability as we compare the relative ratios

of model A and C to model B as around of 0.75 to 1.11, respectively. Perhaps, the estimated values of

h = 0.822 is actually high so that household consumption in model A or C has been over-smoothed, yet in

model B, standard deviation of hour-work is markedly low, say 37% of σBy , suggesting the important role of

habit formation in modeling Vietnam economy. Moreover, the absence of habit persistence induce the lower

investment variations as shown in basic model and B, of which σiσy≈ 4.0 in comparison with the estimate of

4.5 for model A and the high of 5.5 for model C and empirical data.

To understand the role of investment costs in our models, the easiest things is to contrast the estimated

moments of model A and C. High investment costs such as κ = 10 reduces 2.05% and 0.43% investment

and output variations, respectively, implying labor demand in the unfavorable business environment is likely

to be more volatility.

In general, the augmenting RBC models have done good jobs in explaining Vietnam business cycles during

the thirty years of 1986 - 2015 though they are still not perfect as concluded in much of the RBC literature.

Indeed, fine-tuning parameters could help account for 80% of observed moments of output and other

aggregate variables. The big weakness of our augmenting RBC models is failing to capture totally the

dynamic characteristics of empirical data.

Moments Actual Simulated

Std.Dev, % One-sided Model A Model B Model C Basic

σy 2.03 1.42 1.66 1.86 1.75

σ f c/σy 0.99 0.77 1.11 0.74 0.61

σhc/σy 1.05 0.81 1.15 0.80 -

σgc/σy 1.93 1.65 1.50 1.40 -

σi/σy 5.58 4.52 3.90 5.55 4.00

σh/σy 1.10 1.45 0.37 1.06 0.23

Table 12: Simulation with σAε = 1.76% of orthogonized TFP shock. Model A, B and C are specified as in section 5. Model A denotes

full configuration; Model B is without habit formation and disutility of government consumption; Model C has no investment costsbut habit formation and disutility of government consumption; Basic is standard RBC as such Model A without any real rigiditiesand no government as well as public capital. All variables are in term of deviation from trend.

35


Variables Actual First-order autocorrelation Actual Contemporaneous corr. with output

HP 1s A B C Basic HP 1s A B C Basic

Y 0.774 0.801 0.519 0.485 0.387 0.552 1.000 1.000 1.000 1.000 1.000 1.000

C f c 0.550 0.694 0.731 0.445 0.772 0.745 0.624 0.493 0.396 0.626 0.019 0.890

Chc 0.587 0.698 0.727 0.446 0.758 - 0.566 0.463 0.308 0.582 -0.080 -

G 0.732 0.680 0.489 0.490 0.432 - 0.513 0.271 0.604 0.651 0.701 -

I 0.515 0.760 0.466 0.486 0.423 0.445 0.590 0.642 0.767 0.441 0.848 0.915

H 0.593 0.736 0.276 0.487 0.327 0.436 0.304 0.020 0.363 0.317 0.465 0.855

Table 13: AR(1) coefficients and contemporaneous correlation. Model A, B and C are specified as in section 5. Model A denotesfull configuration; Model B is without habit formation and disutility of government consumption; Model C has no investment costsbut habit formation and disutility of government consumption; Basic is standard RBC as such Model A without any real rigiditiesand no government as well as public capital. All variables are cyclical components.

Figure 6: Impulses - Responses (in percentage). TFP shock σAε = 1.76%

Impulse and response analysis

Providing that simulated moments are somewhat close to empirical ones, we are interested in exploring the

time path of output, investment and other aggregate variables given an positively orthogonalized TFP shock

with σAε = 1.76% at the outset. That means we are observing the reaction of each evaluating variables given

one time total factor productivity shock. To fit all four model together, we only report hour-work in addition

36


to the canonical triplet as depicted in figure 6 above.

As shown in the top-left panel, output in full real rigidity mode responds slowly and only reaches the

maximum in period 3. From period 4, all three models A, B and C exhibit the similar path although model

B and C have been slightly steeper than A. Basic RBC appears to strongly move as its output jumps to

2.04% and steadily dies out. The halfway of output in full flexible economy is approximately seven years.

While model B and C roughly give a clear figure as of five years halfway, model A has demonstrated a bit

complicated movement. The latter takes three years climbing to the peak of 1.17%, since then it fades at a

leisurely pace until reaching halfway point at the ninth period. Probably, the hump shape of model A’s output

is a result of high investment adjustment costs and household habit persistence.

The reaction of other aggregate components, namely final consumption, investment and hour-work, are

much different from each other and from the output. At first, we discuss the time path of model B’s final

consumption as it does not have a common hump shape. Without habit formation but enjoying extraordinary

earnings from utilizing public capital, household in economy B strongly favors to consume given the raise

of his/her income due to exogenously technological improvement. In basic model, we do not model public

capital the household thus has to save more to maintain relative capital stock level over output. In the

presence of internal habit formation, final consumption in model A and C share the same hump-shaped curve

with the basic RBC.

Looking at bottom-right panel of figure 6, investment in model C behaves identically with the one in the

frictionless and standard economy. On the one side, investment adjustment costs significantly slowdown

investment activities in both model A and B. Yet, investment quickly backs to its stationary value or even

negative level after seven periods in model basic and C instead of smoothing out until the 15th period as in

the other two is the other side. We have also seen the matching of blue line curves in the panels of final

consumption and investment. Households in the economy B consume more so that they invest less.

A sizable TFP shock to an economy like A has caused the temporarily plunges of labor demand. It takes

three periods to get back the steady-state from the lowest of −1.65%. Without habit formation, the labor

demand has been much less volatility since investment in that economy adapt immediately to TFP shock.

It should be reminded that the higher internal habit persistence coefficient h leads to the lower marginal

utility of consumption at the present but increasing it in the next period. In model C, there is an absence of

investment costs the negative labor demand from trend are more persistent as it merely dies out when going

beyond the 10th period. Put differently, labor demand in the perfect and simple economy like the basic RBC

has favorably responded to positive technological shock.

Providing salient impulse-response functions of modeled economies with internal habit formation and

investment adjustment costs, it seems necessarily to be extending standard RBC model with state-of-the-art

DSGE features when modeling Vietnam economy.

37

Author: Binh T. Pham (WPP 01/2017/UAB) 7. CONCLUDING REMARKS

7. Concluding remarks

In a nutshell, we find evidences showing that Vietnam economic growth has been led by capital accumulation

in the second-half of 1986 - 2015. The contributions of TFP to output growth have been relatively low, saying

below 20%, since the end of Asian financial crisis. In comparison with other emerging economies, Vietnam

has however been on the similar path as its stylized facts are in line with the literature.

Regarding to the real business cycle analysis, our augmenting RBC model exhibits a fairly good performance

in capturing Vietnam economic moments. Apparently, we do find the significant contribution of public

investment to fostering economic growth in Vietnam in the course of Doi Moi epoch. Taking advantage of

stochastic dynamic general equilibrium modelling and Bayesian estimation, it appears that real rigidities,

namely capital utilization, habit formation and investment adjustment costs are all relevant to help explain

business phenomenon of Vietnam in the post-war years. Put simply, we suggest that Vietnam economy

should not come along with the basic RBC theory.

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Appendix 1: Data

Vietnam Business Cycles

1985 1990 1995 2000 2005 2010 2015-10

-5

0

5

10

%

Output

One-sidedHP filterLinear

1985 1990 1995 2000 2005 2010 2015-5

0

5

10

%

Private consumption

1985 1990 1995 2000 2005 2010 2015-20

-10

0

10

20

%

Government consumption

1985 1990 1995 2000 2005 2010 2015-40

-20

0

20

40

%

Investments

1985 1990 1995 2000 2005 2010 2015-40

-20

0

20

%

Exports

1985 1990 1995 2000 2005 2010 2015-40

-20

0

20

%

Imports

One-sidedHP filterLinear

Figure 7: Comparisons of three detrending techniques. Variables are detrended using Linear-quadratic (black), Hodrick-Prescottfilter (red) and One-sided Hodrick-Prescott filter (blue).

40


Appendix 2: Bayesian estimation outputs

0 0.02 0.040

100

SE_eps_za

0 0.2 0.40

100

SE_eps_zp

0.02 0.06 0.1 0.140

100

SE_eps_zig

0.01 0.03 0.050

100

SE_eps_zcg

0.2 0.40

20

SE_EOBS_ln_ivm

0 0.1 0.2 0.30

10

theta

0.5 10

5

10h

0 0.5 10

2

eta

0 10 20 300

0.1

chi

0 2 40

1

ppi

0.8 10

10

psi

0 0.2 0.40

5

10

alpha1

0 0.5 10

2

alpha3

0.5 10

5

rhoA

0 0.5 10

2

rhoP

0 0.5 10

2

rhoCg

0.5 10

5

rhoIg

Figure 8: Priors and Posteriors

41


0.05 0.1360

380

400SE_eps_ime

0.02 0.03360

380

400SE_eps_hme

0.006 0.01 0.014360

380

400SE_eps_za

0.05 0.1360

380

400SE_eps_zp

4 6

10-3

390

392

394SE_eps_zig

0.02 0.03360

380

400SE_eps_zcg

0.4 0.6 0.8385

390

395eta

10 15 20385

390

395chi

0.06 0.1 0.14360

380

400theta

log-post log-lik kernel

0.15 0.2 0.25 0.3300

350

400alpha1

0.5 0.6 0.7 0.8 0.9380

390

400alpha3

0.5 0.6 0.7 0.8 0.9-500

0

500h

0.5 1391

392

393ppi

0.8 0.85 0.9 0.95-500

0

500psi

0.8 0.9360

380

400rhoA

0.2 0.4385

390

395rhoP

0.7 0.8 0.9360

380

400rhoIg

log-post log-lik kernel

Figure 9: Mode checks

42


5 10

104

4

6

810-3SE_eps_za (Interval)

5 10

104

0

0.5

110-5SE_eps_za (m2)

5 10

104

0

510-8SE_eps_za (m3)

5 10

104

0.05

0.1SE_eps_zp (Interval)

5 10

104

0

0.5

110-3SE_eps_zp (m2)

5 10

104

0

0.5

110-4SE_eps_zp (m3)

5 10

104

0.02

0.03

0.04SE_eps_zig (Interval)

5 10

104

0

1

210-4SE_eps_zig (m2)

5 10

104

0

510-6SE_eps_zig (m3)

5 10

104

6

8

1010-3SE_eps_zcg (Interval)

5 10

104

0.5

1

1.510-5SE_eps_zcg (m2)

5 10

104

0

0.5

110-7SE_eps_zcg (m3)

5 10

104

0.02

0.03

0.04SE_EOBS_ln_ivm (Interval)

5 10

104

1

2

310-4SE_EOBS_ln_ivm (m2)

5 10

104

0

510-6SE_EOBS_ln_ivm (m3)

5 10

104

0.06

0.07

0.08theta (Interval)

5 10

104

0.5

1

1.510-3theta (m2)

5 10

104

0

0.5

110-4theta (m3)

Figure 10: Univariate convergence diagnostics for the Metropolis-Hastings. The first, second and third columns are respectively thecriteria based on the eighty percent interval, the second and third moments.

43


5 10

104

0.05

0.1

0.15h (Interval)

5 10

104

1

2

310-3h (m2)

5 10

104

0

2

410-4h (m3)

5 10

104

0.2

0.3

0.4eta (Interval)

5 10

104

0.01

0.015

0.02eta (m2)

5 10

104

0

510-3eta (m3)

5 10

104

6

8

10chi (Interval)

5 10

104

5

10

15chi (m2)

5 10

104

0

100

200chi (m3)

5 10

104

0.6

0.8

1ppi (Interval)

5 10

104

0

0.1

0.2ppi (m2)

5 10

104

0

0.05

0.1ppi (m3)

5 10

104

0.04

0.06

0.08psi (Interval)

5 10

104

0

0.5

110-3psi (m2)

5 10

104

0

2

410-5psi (m3)

5 10

104

0.08

0.1

0.12alpha1 (Interval)

5 10

104

0

1

210-3alpha1 (m2)

5 10

104

0

1

210-4alpha1 (m3)


44


5 10

104

0.2

0.3

0.4alpha3 (Interval)

5 10

104

0.01

0.015

0.02alpha3 (m2)

5 10

104

2

3

410-3alpha3 (m3)

5 10

104

0.15

0.2

0.25rhoA (Interval)

5 10

104

5

6

710-3rhoA (m2)

5 10

104

6

8

1010-4rhoA (m3)

5 10

104

0.2

0.3

0.4rhoP (Interval)

5 10

104

0.01

0.015

0.02rhoP (m2)

5 10

104

1

2

310-3rhoP (m3)

5 10

104

0.3

0.35

0.4

0.45

0.5rhoCg (Interval)

5 10

104

0.01

0.02

0.03

0.04rhoCg (m2)

5 10

104

2

4

6

8

1010-3rhoCg (m3)

5 10

104

0.14

0.16

0.18

0.2rhoIg (Interval)

5 10

104

3

3.5

4

4.5

510-3rhoIg (m2)

5 10

104

3

3.5

4

4.5

510-4rhoIg (m3)


45


1 2 3 4 5 6 7 8 9 10

104

6

8

10Interval

1 2 3 4 5 6 7 8 9 10

104

5

10

15m2

1 2 3 4 5 6 7 8 9 10

104

0

50

100m3

Figure 13: Multivariate convergence diagnostics for the Metropolis-Hastings. The first, second and third rows are respectively thecriteria based on the eighty percent interval, the second and third moments. The different parameters are aggregated using theposterior kernel.

46

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