Preliminary

Uncertainty of the Keynesian Multiplier in Chile:
Empirical Evidence from Stochastic Simulation

December 2009

Shinji Yoshioka*

Abstract
In the last quarter of 2008, the world economy fell into a deep recession and many governments, ofboth developed and developing countries, took fiscal measures to stimulate the economy. The
Government of Chile employed a sizable fiscal expansionary policy ahead of other developing countries.
It is definitely essential to understand the size of the Keynesian multiplier and the GDP gap when
adopting fiscal measures to stabilize or boost the economy. Therefore, this study focuses on Chilean
expansionary fiscal policy implemented in 2009 by adopting concepts of the Keynesian multiplier and its
uncertainty. Both deterministic and stochastic simulations, based on a small-sized IS-LM model, areperformed. The results of the former reveal the essence of the textbook-style multiplier, and those of thelatter, based on Monte Carlo simulations, indicate the uncertainty about the size of multipliers. Thisfinding supports the Lucas critique, and requires a more precisely designed fiscal policy.

Key Words: Econometric Model, IS-LM Analysis, Fiscal Policy, KeynesianMultiplier, Monte Carlo Simulation
JEL Classification: C32, C51, E17, E62, H31* Shinji Yoshioka, Faculty of Economics, Nagasaki University, s-ysk@nagasaki-u.ac.jp
The author is deeply grateful to Ms. Fujima, Secretary of Japanese Embassy in Santiago, Chile, forproviding useful information on Chile's fiscal policy and economic situation.

1. Introduction

Since late 2008, the world economy has fallen into a deep recession caused by the US subprime
bubble collapse and its aftermath, a financial crisis. In both developed and developing countries, strong
fiscal measures have been taken to boost the economy and to cover the large negative GDP gap to asizable extent. For example, figure 1.9 on p.16 of IMF (2009a) estimates that in 20010 the GDP gap in
Latin America will increase to around 4 percent. Recently, in many countries, fiscal measures, rather thanmonetary policies, have therefore been employed to attain economic stabilization and stimulation.
According to Hacienda (2009a) and Hacienda (2009b), the Government of Chile takes the following
package, which amounts to US$ 4 billion to boost the country's economy.

1) Stimulation of investment and demand (US$ 750 million increase in public investment, andtemporary elimination of the stamp tax, US$ 628 million in 2009, etc.)
2) Improved access to financing for business (temporary reduction of 15 percent in withholding tax
for small and medium businesses, and of 7 percent for larger businesses, up to US$ 460 million,
and increase of US$ 50 million in the CORFO 1 financing facility for bank and non-bank factoringfirms, etc.)
3) Economic support for individuals (special one-time payment of Ch$ 40 thousand (US$ 64) per
dependent for beneficiaries of the Chile Solidario system and recipients of the Subsidio Único
Familiar (SUF), etc., and early reimbursement of the 2010 income tax returns for individuals, etc.)
4) Promoting employment and training (creation of a subsidy to encourage formal employment for
low-income young workers between 18 and 24, etc.)

In the face of copper price shocks originating in volatile foreign demand, Chile adopted a structuralbalance fiscal rule. Marcel C. et al. (2001) develop the structural budget balance methodology that has
been adopted for use in the Chilean public sector since 2000. Kumhofl and Laxton (2009) also analyze
this rule, using IMF's Global Integrated Monetary and Fiscal Model (GIMF), and conclude that a 0.5%
surplus target, introduced in 2008, is desirable from a business cycle perspective. Fiess (2004) alsostresses the countercyclical role of Chilean fiscal rule in the context of Ehrlich-Becker insurance
framework.2 OECD (2009) concludes that Chilean fiscal resilience in 2007 is 2.42 and the highest in
Latin American countries as Figure 1 reports.

-1-

Figure 1: Fiscal Resilience Index
EMBIG Spreads

Debt/Exports

Government Budget Balance/GDP

3.0
2.5
2.0
1.5
1.0
0.5
0.0
1981200719812007198120071981200719812007198120071981200719812007
ARG

BRA

CHL

COL

DOM

MEX

PER

VEN

Note: For detailed information on EMBIG spreads, see Chambers and Kraemer (2005).
Source: OECD (2009) Figure 0.16 on p.46

After breaking the fiscal rule and adopting stimulus, however, it is essential to measure the impact ofthe policy, in particular, the size of the Keynesian multiplier3 and of the GDP gap. For example, Restrepo
and Rincón (2006) estimate structural VAR and structural VEC models to evaluate the fiscal policy
impact on Chile and Colombia and note that Chilean policy is more effective than Colombian, since
public finances are well under control. Cerda et al. (2006) also find empirical evidence of theeffectiveness of Chilean fiscal policy, using structural VAR.
This study also considers Chilean fiscal policy, estimating the size of the Keynesian multiplier, byemploying small-sized IS-LM model simulations. Among existing studies, Serven and Solimano (1991)
construct, estimate, and simulate a macroeconomic model for Chile, to explore the effects of domestic
policies and external shocks, including a balanced-budget fiscal expansion. Corbo and Tessada (2003) use
both macroeconometric and nonstructural VAR models to disclose the Chilean growth accounting anddynamic forecast, etc. On the other hand, this study constructs a small IS-LM model to evaluate theuncertainty of fiscal multipliers by focusing on the consumption function, which is a key factor for themultiplier. Both deterministic and stochastic simulations are conducted to reveal the uncertainty about thesize of Chilean fiscal multipliers.
First, it is helpful to confirm that simple and textbook-style Keynesian multipliers are calculateddifferently for the three types, regardless of whether there is a closed or open economy, or whether thegovernment sector, which does or does not collect taxes, is include. The model is described as follows:(EQ-1)

Closed economy excluding the government sector (both tax revenue and imports are
exogenous)

MULT =

1
1− c

-2-

(EQ-2)

Closed economy including the government sector (tax revenue is endogenous)

MULT =(EQ-3)

1
1 − (1 − t )c

Open economy including the government sector (both tax revenue and imports areendogenous)

MULT =

1
1 − (1 − t )c + m

where

MULT

multipliercmarginal propensity to consumetmarginal tax ratemmarginal propensity to import

According to the textbook-style explanation, the multiplier in a closed and government-excluded4
economy seems the largest of the three, and its size declines as the model assumption is expanded toinclude the government sector collecting taxes, and to open the economy that introduces the foreign sector,
particularly imports. This study then estimates and compares the size of the Chilean multiplier bysimulating three types of models based on (EQ-1), (EQ-2), and (EQ-3). Finally, to reveal the uncertaintyof the size of the Keynesian multiplier, three types of stochastic simulations are performed on the basis ofa Monte Carlo methodology.
The contents of the paper are as follows. The introduction overviews the main fiscal policy adopted
by the Government of Chile in 2009 and some existing literature on Chilean fiscal policy, and presents
basic or textbook-style concepts of the Keynesian multiplier. The next section reveals the structure of asmall-sized IS-LM model of Chile, which includes four structural equations, estimated using thetwo-stage least squares method, and two definitions. The third section reports the deterministic, i.e.,
non-stochastic, simulation results of the model, which indicate a tendency to decline the size ofmultipliers, according to the expanding model sectors such as the government and the foreign sectors. The
fourth section shows the stochastic, i.e., Monte Carlo, simulation results, including some uncertainty inthe consumption function in both the disturbance term and the estimated parameter for disposable income.
The final section briefly concludes the paper and presents some remaining issues. EViews V6 isemployed for both estimations and simulations.

2. Structure of the Small-Sized IS-LM Model

This study employs a small-sized IS-LM Keynesian model in which demand factors determine theoutput level, and the production sector is not included. The following equations in (EQ-4) depict thestructural functions, while (EQ-5) reports the definitions. All structural functions are estimated for theperiod 1986-2008, and the two-stage least squares method is employed to avoid estimation bias, whereconstant term and one-year lagged explanatory variables, etc., are included as instrument variables. The
data adopted in the model are at current prices on an annual basis, and taken from IMF (2009b) except for-3-

tax rate, which is calibrated. R2 and SE denote the adjusted R square and the standard error of theequation, respectively. The numbers within parentheses under the equations indicate the standard errors ofeach parameter. Table1 reports the list of model variables.

(EQ-4)

Structural equations(1) Consumption function
C = 1.209 + 0.264 ( GDP – TAX ) – 0.0108 R + 0.673 C-1 + ERROR(1.219) (0.0668)(0.0378)(0.116)2

R = 0.998, SE = 0.0621(2) Investment function
I = 0.623 + 0.280 ( GDP + M ) – 0.203 ( GDP-1 + M-1 ) – 0.0077 R + 0.459 I-1(1.240) (0.0908)(0.104)(0.0380)(0.229)R2 = 0.977, SE = 0.800(3) Import function
M = -1.461 + 0.358 GDP(0.609)(0.0138)

2

R = 0.968, SE = 1.677(4) Money demand function
R = 25.123 + 6.868 log(GDP) – 10.826 log(M3)(14.180) (9.366)(6.270)2

R = 0.641, SE = 6.501(EQ-5)

Definitions(5) Gross domestic product
GDP = C + I + II + G + X – M(6) Tax revenue
TAX = GDP * RTAX

Table1: List of Model Variables in Alphabetical Order
Notation
Type
Definition
C
endogenous
household consumption
G
exogenous
government consumption
GDP
endogenous
gross domestic productI
endogenous
gross fixed capital formation
II
exogenouschanges in inventories
M
endogenous
imports of goods and services
M3
exogenous
money supply, M3
Rendogenous
lending rate
TAX
endogenous
central government taxes revenue
RTAX
exogenous
rate of taxes on GDP
X
exogenous
exports of goods and services
Sources: Author and IMF (2009b)

-4-

Unit
billion pesos
billion pesos
billion pesos
billion pesos
billion pesos
billion pesos
billion pesos
percent per annum
billion pesos
decimalbillion pesos

IFS code22896F..ZF...
22891F..ZF...
22899B..ZF...
22893E..ZF...
22893I..ZF...
22898C..ZF...
22859MC.ZF...
22860P..ZF...
228a11..CG...
n.a.
22890C..ZF...

Before proceeding to the model’s final test results, we must focus on the following four points:

1) The money demand function of equation (4) in (EQ-4) is set to be solved for the interest rate as aninverse function.
2) Since data for tax revenue at IMF (2009b) are available only for years 2005-2008 and theiraverage proportion to GDP is 20.1 percent for four years, tax data are calibrated as 20 percent of
GDP value and the tax definition equation (6) in (EQ-5) is set as proportional to GDP, whichmeans both marginal and average tax rates are exogenously set at 20 percent before 2005, and(GDP-TAX) is taken as a proxy for disposable income.
3) Imports and money demand functions are estimated on the basis of a normal and well-known
assumption, and Yoshioka (2001) provides some background information for estimating theconsumption and investment functions.
4) Since the Keynesian multiplier is defined in a short run, which means that prices are constant, allvariables in the model are denoted as nominal.

A final test of the model is completed, and reports that the Root Mean Square Percentage Error(RMSPE) of GDP is around 0.40 percent. Figure 2 depicts both the actual GDP, and GDP as solved
dynamically in the model. The model simulation results seem to have some tendency to overestimation inrecent two years.

Figure 2: Actual and simulated GDP in the final test of the model
100
80
60
40
20

1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008

0

ActualSimulation

Note: The unit of the vertical axis is trillion Chilean pesos.
Source: Author’s estimation-5-3. Deterministic Simulation Results

Before proceeding to stochastic simulations, some deterministic simulations are performed. Here,"deterministic" means that neither the elements of uncertainty, nor those of stochasticity, are included inequations or parameters. According to (EQ-1), (EQ-2), and (EQ-3), three types of simulation are
performed to confirm the decline in the multipliers when the sectors are included in a model. The model
is simulated through its estimation period 1986-2008 with constant terms adjusted to the actual values ofeach variable for baseline simulations. A sustainable shock of 1 percent of GDP is added to government
consumption for the final three years, from 2006 to 2008, to provide an impact simulation. Evidently, themultipliers are derived with the deviation between baseline and impact simulations. Table 2 reports thedetails of the model assumptions for estimated functions. Therefore, the smaller the number of cases, thelarger the size of the multipliers is in a macroeconomics textbook.

Table 2: Model Assumptions
Functions

Case 1Case 2

Case 3

Consumption function

endogenous

endogenous

endogenous

Investment function

endogenous

endogenous

endogenous

Tax revenue definition

exogenous

endogenous

endogenous

Import function

exogenous

exogenous

endogenous

Note: Tax revenue is treated as exogenous at its value for Case 1 and endogenous forCases 2 and 3 while tax rate is exogenous in both cases.
Source: Author

While the imports function is not included in the model in Cases 1 and 2, it is included in Case 3. On
the other hand, tax revenue is treated as exogenous at its value in Case 1 and endogenous in Cases 2 and 3,
where the tax rate is exogenous. For all simulation cases, the money demand function5 is always treatedas endogenous, which means that the LM curve is not flat. According to Friedman (1973), the slope of the
LM curve determines the size of the multiplier, and Romer (2000) discusses the requirement of the LM
curve in a Keynesian framework. Apparently, Case 1 corresponds to (EQ-1), which excludes thegovernment tax revenue definition and assumes a closed economy, and Case 2 is equivalent to (EQ-2),
where the closed economy leaks some expanding fiscal policy effect to taxes. Case 3, the open economy,
spills the stimulus to both taxes and the foreign sector through imports. Most economists seem to regard
that the multiplier of Case 1 is the largest of the three cases and that of Case 3 is the smallest, whereas
Case 2 lies midway between Cases 1 and 3. This argument is empirically proven in this paper. Figure 3
depicts the Chilean fiscal multipliers of three cases in Table 2.

-6-

Figure 3: Sizes of Multipliers
3

2.69
2.25

2.12
1.87

2
1.36

1.40

1.26

1.22
0.87

1

Case 1Case 2

2008

2007

2006

2008

2007

2006

2008

2007

2006

0

Case 3

Note: The unit of the vertical axis is an absolute number, since the values shown are ofmultipliers.
Source: Author’s estimation

According to tax definition and imports function reported in (EQ-4), the marginal propensity toimport is around 0.36, while the calibrated marginal tax rate is 0.20. Hence, the leakage of fiscal stimulus
to taxes and the foreign sector is large in size. This is also confirmed by actual data of GDP components
shown in Figure 4, which depicts imports' and taxes' shares to GDP.

Figure 4: Imports' and Taxes' Shares to GDP.

45
40
35
30
25
20
15
10
5
0
2005

2006

2007

Imports

Note: The unit of vertical axis is percent to GDP.

-7-

Tax

2008

Source: Author's calculation based on IMF (2009b)

4. Stochastic Simulation Results

In addition to the previous deterministic, i.e., non-stochastic, simulations of the model, a few Monte
Carlo simulations are completed for Case 3 in Table 2 where both imports and taxes are treated as
endogenous. As mentioned above, this study focuses on the uncertainty of the consumption function.
Thus two types of stochastic elements are employed and three types of Monte Carlo simulations are
executed as follows:

1) Case A includes the uncertainty in the entire consumption function, i.e., a disturbance term that isthe product of the equation standard error given as SE1 in ensuing equation in (EQ-6) and a normal
random number is added at the equation's error term:2) Case B includes the uncertainty in the parameter for the disposable income at the consumptionfunction, i.e., a disturbance term that is the product of the parameter's standard error given as SE2
in ensuing equation in (EQ-6) and a normal random number is added in the parameter for
disposable income; and,3) Case C includes both uncertainties in 1) and 2) above.

(EQ-6)

Consumption Function Including Uncertainty

C = 1.209 + 0.264 × (1 + SE 2 * RAND 2 )× (GDP − TAX ) − 0.0108 × R + 0.673 × C −1 + SE1* RAND1
where

SE1

Standard error for entire consumption function

SE2Standard error for parameter of disposable income

Here, normal random numbers are calculated from uniform random numbers according to Box-Müller
transformation in Box and Müller (1958), which generates two sets containing 1000 random numbers that
follow N.I.D.(0,1).6 These two sets of normal random numbers are stored and identical random numbers
are employed for all three cases, which is one of the most remarkable features of the Monte Carlosimulations in this paper.7 Figure 5 depicts the histograms of these two sets of normal random numbers.
The sample means are almost equal to zero and the sample standard deviations are almost identical to
one.

-8-

Figure5: Histograms of Normal Random Numbers(1) RAND1: Random Numbers for whole Consumption Function
250
200

Sample mean: -0.037227
Sample std. dev.: 1.02769

150
100
50
0
-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

over

3

over

(2) RAND2: Random Numbers for Coefficient of Disposable Income
250
200

Sample mean: 0.0029948
Sample std. dev.: 0.98241

150
100
50
0
-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Source: Author's transformation

After completing the Monte Carlo simulations for each case 1000 times, the values of standard
deviations of impact simulations in last three years for each case are derived from the results, andcompared with deterministic simulation results of Case 3 in Table 2 and Figure 3. Figure 6 reports
multipliers in each case. The center bars of each year depict the multiplier derived from deterministic
simulation results of Case 3 in Table 2 and Figure 3, while the left-side and right-side bars indicate thoseto which is added or deducted, respectively, a value of one unit of the standard deviation derived fromeach Monte Carlo simulation case.

-9-Figure 6: Multipliers Derived from Monte Carlo Simulations(1) Case A: Including uncertainty in whole consumption function
4

Case A

3
2

3.34

2.60
1.58

1.40

1.22
0.87

1

0.16
0
-0.15
-0.53

-1
2006

2007
ave + std

ave

2008
ave - std

(2) Case B: Including uncertainty in coefficient of disposable income at consumption function
5

3

2.08

2
1

Case B

3.66

4

4.96

1.40

1.22

0.87

0
-0.34

-1

-1.21

-2

-2.16

-3
2006

2007
ave + stdave

- 10 -

2008
ave - std

(3) Case C: Including uncertainty both in whole consumption function and in coefficient ofdisposable income at consumption function
6

Case C

5

5.58

4.11

4
3

2.32

2
1

1.40

1.22

0.87

0
-1

-0.58

-2

-1.66

-3
2006

2008

2007
ave + std

ave

-2.78

ave - std

Source: Author's estimation.

It is not difficult to see at a glance that the multiplier deviations of the stochastic simulation from eachdeterministic case grow larger as time passes, i.e., from 2006 to 2008. And deviations in case A, B, and C
grow larger in this order, according to the amount of uncertainty included in the consumption function.
One of the most impressive results derived from these simulation results should be stressed: The deviationof the parameter uncertainty case is far larger than that of the equation uncertainty case. This is partly due
to small significance or large standard error of the parameter, but it also reminds the Lucas critiquepresented in Lucas (1976), which insists that the estimated parameters of models are not necessarily
structural, i.e., not policy-invariant or policy-variant so that they may change whenever policy is changed.
This also suggests that policy conclusions based on those models would therefore potentially be
misleading. Since the model system of this study reported in (EQ-4) does not adapt to Lucas critique, theMonte Carlo simulation is employed for approximation. The estimation results reported Figure 6 revealthat the Keynesian fiscal multiplier may possibly show a large negative decline when the model includes
some uncertainty in the consumption function and, in particular, in its parameter for disposable income,which is usually called the marginal propensity to consume.

5. Conclusion and Remaining Issues

Many economists consider that Romer and Bernstein (2009) were unveiled to provide an empiricalbasis on which to promote fiscal stimulus not only in the United States, but also all over the world,
including Chile. On the other hand, some critical literature, including Cogan et al. (2009) and Davig andLeeper (2009), has also been presented, while Smets and Wouters (2007), adopting Bayesian DSGE
approach, insist that the multiplier is around 1.0-0.4 and far smaller than that of Romer and Bernstein- 11 -

(2009).
This study too is inspired by Romer and Bernstein (2009); however, it employs a different approachfrom other literature that adopts stochastic Monte Carlo simulations, which reveal that the Keynesianfiscal multiplier may become negative if sizable uncertainty is included in consumers' behavior and, inparticular, in marginal propensity to consume. The uncertainty of the Keynesian multiplier emerges inboth deterministic and stochastic simulations. Even in deterministic simulations, the Keynesian multipliervaries, depending on whether the government and foreign sectors are included in the model or not. Of
course, stochastic simulation results unveil large uncertainty about the size of multipliers. From anotherpoint of view, the results of this study support Lucas critique, and suggest that more precisely designed
fiscal policy is required in both developed and developing countries, since developing countries alsoutilize fiscal policy as autonomous tool for stabilizing national economy after the recent recession.
A few remaining issues should also be noted. First, more accuracy in equation estimations is required.
Partly due to limited data availability of the general government's tax revenue in developing countries, theparameter for disposable income in the consumption function, which defines the marginal propensity toconsume, may contain some degree of bias. Second, more comprehensive model is required. The model
constructed in this paper consists of only four structural functions and two definitions. On the other hand,a more comprehensive model does not necessarily bring more precise simulation results. Third, quarterly
models, including the model built in Romer and Bernstein (2009), are employed in most developed
countries. In developing economies, however, it is difficult to obtain seasonally adjusted quarterly data. It
is not easy to resolve these remaining issues quickly, but further efforts to investigate developing areas
will be made.

(Endnote)
1

CORFO is Production Development Corporation, a government entity for industrial promotion in Chile.
For further information, see its web site (http://www.corfo.cl/).
2
See Ehrlich and Becker (1972).
3
Romer and Bernstein (2009), at p.12, suggest that the size of multiplier based on Government purchases
for the US is around 1.5-1.6.
4
In the government-excluded economy here, both government expenditure and tax revenue are treated as
exogenous, and the word "government-excluded" does not mean the absence of the government or its
expenditure.
5
The money demand function in the model is solved for an interest rate function as a reverse function.
This seems usual approach in econometric models.
6
For this calculation, EXCEL is employed.
7
Hence the Monte Carlo simulations completed in this paper are reproducible.

- 12 -

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