Noisy multipliers: the role of randomness in measuring the impact
of fiscal policy
Richard McManus
Canterbury Christ Church University
May 15, 2015
Abstract
There is significant variation in the estimates of fiscal policy multipliers which clouds both political
and academic debate. This paper uses simulated data generated from a medium scale DSGE model to
test the statistical accuracy of the methods applied to estimate these multipliers. We find that under
clinical experimental conditions, vector autoregressive techniques to identify the macroeconomic impact
of fiscal interventions perform well; however, when exposed to an estimated version of the model (which
allows for a closer empirical profile of fiscal policy), performance is significantly reduced and there is a
wide dispersion of estimates from a framework which provides the same underlying theoretical multiplier.
This suggests that noise within both fiscal and macroeconomic variables limits the reliability of fiscal
multiplier estimates and, subsequently, our results provide an intuitive explanation as to why such variety
in the literature exists.
1
Introduction
The recent global recession has encouraged renewed debate in both political and academic circles into the conduct and effectiveness of fiscal policy. The official justification for the American Recovery and Reinvestment
Act, the US fiscal stimulus in response to the global downturn, suggested that in the current circumstances
the fiscal multiplier was approximately 1.6 (Romer & Bernstein 2009), a claim vigorously refuted in Cogan
et al. (2010), as well as others. The former used (undisclosed) theoretical models building on research which
finds that under conditions of monetary accommodation the effectiveness of fiscal policy is stronger (see for
example Christiano et al. 2011), compared to estimates during ‘normal times’ (see for example Baxter & King
1993). As attention switched to growing levels of government debt, debate became between ‘austerity versus
stimulus’ with those advocating the former calling on research that identifies the possibility of expansionary
fiscal contractions (Giavazzi & Pagano 1990). As austerity became more widespread, growth continued to
1
be depressed and below expectations; Blanchard & Leigh (2013) presented evidence that forecast errors in
national output was correlated with the size of fiscal deficits within a given country, suggesting that multipliers used in forecasters’ calculations were lower than actual by a factor of 3, a finding which Ollie Rehn
(the then European Commissioner for Economic and Monetary Affairs) reportedly did not want disclosed.1
Fiscal debates, however, are not a modern phenomenon and occur for two main reasons: first, individuals
and policy makers are inclined towards ideological prejudices over the role of government which may influence
their interpretations of evidence; and second, the results from academic literature are rarely in unison
providing differing conclusions both with respect to quantitative, and frequently, qualitative impact of fiscal
interventions. These two barriers interact with one another where the presence of a wide range of academic
results allows individuals to select that which best suits their desired position. Romer & Romer (2010), for
example, estimate multipliers above one whereas Ramey (2011) identify much smaller fiscal effects. This
divergence derives from differing estimates of the path of private consumption following fiscal shocks where
Blanchard & Perotti (2002) find it increasing (reconciling with theoretical models such as Galı́ et al. 2007)
and Mountford & Uhlig (2009) find it decreasing (reconciling with the crowding out concepts of Baxter &
King 1993). Further contradiction also comes from the path of other seemingly important variables of wages,
employment and investment. Much of these differences relate to the wealth of identification methods applied
in the literature (see Section 2.1), however, it is hard to argue that this breadth of results aids policy making.
The innovation of this paper is to use simulated data generated from a medium scale dynamic stochastic
general equilibrium (DSGE) model to test the statistical accuracy of the methods applied to estimating fiscal
multipliers, in order to discern how much of the variation in the academic results is merely due to noise in
the underlying processes. That is, we test whether the variation in the observed empirical results is simply
due to randomness, and therefore we seek to identify how appropriate the models are which are used to
calculate the empirical effects of fiscal policy. Caldara & Kamps (2008) seek to reconcile the difference in
estimates through a stronger understanding of the identification methodologies involved in these different
processes; they show that after controlling for differences in the specification of the reduced form model, and
especially focusing on the size of automatic stabilisers, results from a variety of identification procedures are
in line with one another. We perform a similar task, but focus on the randomness in the data generating
processes. Even in the absence of fiscal shocks, there is variation in macroeconomic variables which will
cause movements in both output and government expenditures and revenues that will add difficulties to the
1 In a letter to finance ministers (available from http://ec.europa.eu/archives/commission 2010-2014/rehn/), Olli Rehn wrote
that the research ‘has not been helpful and has risked to erode the confidence that we have painstakingly built up over the past
years’.
2
process of estimating multipliers. Combine to this the perception that multipliers change over the course
of the business cycle (Auerbach & Gorodnichenko 2012), that multipliers are becoming smaller over time
(Blanchard & Perotti 2002), that different fiscal interventions will have differing levels of persistence and
repayment packages, and, that fiscal policy involves more than two instruments which themselves will have
differing degree of effectiveness, the scope for noise to enter the estimation is vast. It is the purpose of
this study to use simulated data to fully reflect on the impact of some of these sources of this noise on the
estimation process.
We find that in clinical experimental conditions conventional vector autoregressive techniques can estimate both spending and tax multipliers with a fair degree of accuracy. For example, when all fiscal instruments are calibrated to have the same degree of persistence in their shock processes, to have no response to
the business cycle, and where all instruments respond equally and sluggishly to lagged values of debt, ranges
in estimates of two year cumulative spending multipliers is 0.3 (the difference between the highest and lowest
result). However, when these fiscal parameters are estimated with Bayesian techniques applying UK data,
the results from the clinical experiments are significant amplified. Fiscal multiplier estimates become both
imprecise and very noisy. This, it is argued, may be the cause of the vast array of results in the literature,
and the source of debate in both political and academic circles.
The rest of the paper proceeds in the following way. Section 2 discusses the existing econometric methodologies applied to estimating fiscal multipliers and the theoretical model and techniques utilised in our paper
to test their accuracy. Section 3 uses a calibrated version of our theoretical model to calculate the probability
of both type 1 and type 2 errors under experimental conditions; that is, we identify both the spread of fiscal
estimates and the probability of incorrectly not identifying when two fiscal multipliers are different from one
another. Section 4 extends these experiments to an estimated version of the model using UK data. Finally,
Section 5 concludes.
2
2.1
Methodology
Existing econometric methodologies
This section discusses both the econometric strategies applied in the literature to estimate fiscal multipliers
and how the statistical accuracy of these is tested in this paper. For brevity, a full detailed discussion of
these processes is not included as this has been performed frequently elsewhere; see for example, Caldara &
Kamps (2008).
3
Much of the literature estimating the impact of fiscal policy recognises the endogeniety of the respective
processes whereby government expenditure and taxation effect output, and each other, and further variables
which also have an impact on GDP. As such it is common for a vector autoregression (VAR) to be estimated
of the form:
Xt = µ + µ1 t + A(L)Xt−1 + ut
(1)
where Xt is a vector of time series observations, including at least tax revenues (Tt ), government spending
(Gt ) and output (Yt ) and typically other variables such as the interest rate (Rt ) and inflation (πt ) (and,
occasionally, individual components of aggregate demand); A(L) is a lag polynomial (usually to a fourth order
with quarterly data); µ and µ1 t is a vector of constants and time trends, and ut are the observed residuals.
From this, the literature differs through the identification strategies employed to derive the structure shocks
εt from the estimated residuals ut . As Blanchard & Perotti (2002) highlight, the estimated residuals for
the fiscal variables will include three distinct components: first, the automatic response of spending and
taxes to the other variables within the VAR structure (most notably with respect to output these represent
the ‘automatic stabilisers’); second, policy maker’s systematic discretionary response to innovations in other
variables within (1); and third, the random or exogenous discretionary shocks to fiscal variables. It is
these final exogenous (‘structural’) shocks which the identification process is trying to identify in order to
appropriately estimate the fiscal multiplier.
The importance of isolating exogenous shocks to fiscal policy comes from the observation that were a
government to respond counter-cyclically to the business cycle, such that spending increases and/or taxes fall
in response to anticipated output declines, if implemented effectively this would lead to limited movements
in observed GDP at the cost of a budget deficit. In this case, fiscal policy would appear to be ineffectual
as the statistical process used to identify its impact failed to take into consideration the counter factual
of falling output. This is demonstrates the importance of isolating the exogenous shocks by removing
the endogenous components of policy; it is from these shocks that impulse responses are taken and fiscal
multipliers calculated.
Broadly speaking, there are four main identification strategies. The first of these is to perform a Cholesky
decomposition of the residual variance-covariance matrix which applies a recursive ordering to the causality
of shocks. The most significant disadvantage of this process is the required decision on an appropriate
ordering of the variables in the VAR to capture the dynamics of the structural shocks, for which there is
no uniform preferred method in the literature; for example, some order fiscal variables before GDP and
4
others after GDP, whereas Caldara & Kamps (2008) order government spending before output followed
by taxes. Within a Cholesky decomposition, ordering fiscal variables first is imposing that all automatic
responses of spending and taxes to all other variables within the VAR system are equal to zero; that is,
there are no automatic stabilisers in the economy. Ordering fiscal variables after output is imposing that
there is no contemporaneous response of GDP to fiscal intervention and that all effects are lagged; given
that government spending is a component of output, this is tantamount to imposing that all of an increase
in government expenditure is fully crowded out by the private sector, at least on impact.
It is for these short comings that Blanchard & Perotti (2002) suggested a structural VAR (SVAR)
approach whereby certain observations on the automatic elasticities of fiscal variables to output are estimated
outside the model and then subsequently imposed on the estimation of the structural shocks from the residuals
in (1). This improves upon the recursive approach above as it allows for automatic responses of government
spending and taxation to movements in other variables within the VAR system (most notably output),
but creates the difficulty of estimating these elasticities outside of the model.2 Moreover, this process still
assumes that there is no discretionary response of fiscal policy to the business cycle, which is argued to
be suitable in conjunction with the use of quarterly data, and the inertial decision making process of fiscal
authorities.
The main cited drawback of this approach is the predictability of the estimated structural shocks. As
discussed above, decision lags can aid the econometrician in identifying these shocks, however, implementation lags may generate a delay in the data between when agents within the economy are aware of the future
fiscal interventions and when these are observed. However, to the extent that this is an issue for the SVAR
approach, it is also an issue when applying a simple Cholesky decomposition, and the latter is rarely used
in the literature since Blanchard & Perotti (2002). Despite this, results of using either method tend to not
to diverge from one another, as the estimated elasticies are rarely sufficient to substantially alter the main
conclusions (see for example Caldara & Kamps 2008).
The third frequently applied identification procedure in the fiscal multiplier literature is the ‘sign restriction method’, whereby fiscal shocks are identified by imposing conditions on impulse response functions
(Canova & Paustian 2007, Mountford & Uhlig 2009). For example, a government spending shock would
be identified by imposing that the movement in spending, output and the budget deficit all move in the
same direction, for a fixed time horizon. This method neither requires the number of shocks processes to be
2 Procedurally, these elasticities are estimated by identifying each of the components of both government spending and
taxation and calculating their movements with the other variables under consideration; further, knowledge of the general
budgeting process can be applied. The aggregate elasticities are then calculated through taking a weighted average of these
individual elasticities from specific fiscal instruments.
5
equal to the number variables in the VAR system and, further, it does not impose linear restrictions linking
estimated residuals and structural shocks. Another advantage of this approach is that it is not subjected to
the criticism of not being able to incorporate anticipated shocks, however, at the same time this is due to
the process not being able to identifying the timing of shocks, only the timing of the impacts. Further, the
sign restriction process could be argued to be imposing conditions which are too prohibitive and rule out
the possibility of specific fiscal episodes. For example, under the sign restriction method there is no scope
for ‘expansionary fiscal contractions’ (Giavazzi & Pagano 1990), or other unusual effects; in this respect,
this identification process may over estimate fiscal multipliers as it ignores ‘non-Keynesian’ responses. However, as with the two identification strategies above, Caldara & Kamps (2008) demonstrates that the sign
restriction and SVAR approaches delivery similar results.
The final commonly applied process to identify exogenous fiscal shocks is to consider the narrative record
accompanying fiscal interventions to isolate those which are truly independent of the business cycle. Initially,
studies such as Ramey & Shapiro (1998) used military expenditures in the build-up to war time periods, and
found modest results for the value of the fiscal multiplier. This approach requires few identifying assumptions
and providing the episodes are truly exogenous, a reduced form expression can be estimated through applying
dummy variables for specific identified time horizons. However, there are limited time periods with which to
estimate the fiscal multiplier, other fiscal activity may have accompanied these build-ups (for example tax
rises to pay for the expenditures), and, moreover, a further concern of this approach is that it imposes that
the economy is no different during periods of war than in ‘normal times’.
Romer & Romer (2010) and Cloyne (2013) extend this approach by studying political records associated
with tax changes in the US and UK respectively to categorise these into those which are made as a response to
the business cycle and those made for other reasons (for example, ideological motives, to reduce government
debt, and long run concerns); through applying this approach larger multipliers are found compared to both
Ramey & Shapiro (1998) and the SVAR approach. Again, a reduced form expression can applied for these
approaches; however, the method is beholden to the accurate identification of genuinely exogenous shocks.
This relies in turn on the sincerity of political statements (budgets speeches and committee transcripts) to
describe the motives behind tax movements.
2.2
Theoretical model
The theoretical model applied in order to generate simulated data is a medium scale DSGE model similar to
Smets & Wouters (2003) and Christiano et al. (2005) featuring: nominal rigidities in price and wage setting;
6
real frictions in adjustment costs and monopolistic competition; distortionary taxation on labour, capital
and consumption and a social security contribution from the employers; a complete set of fiscal spending
instruments including consumption, investment, public employment and transfers; and both fully rational
and ‘rule-of-thumb’ households. A full description of the model can be found in Appendix A.
To provide a summary of the model, however, two types of households are assumed. Fully rational agents
generate income through their labour, capital and one period risk-free bond holdings, and gain utility from
consumption and disutility from labour. Rule-of-thumb agents (with a weight of λ) simply consume their
period disposable income due to their restricted access to either capital or bond markets. Households are
subject to shocks to their intertemporal preferences and labour supply, and fully rational households own
capital and therefore are also subject to shocks to the investment cost function. There is a continuum
of intermediate good producers who apply labour and private and public capital using a Cobb-Douglas
technology function, subject to productivity shocks. Differentiated goods are produced and these agents are
subject to a Calvo price setting restriction. A trade union provides labour to these firms using differentiated
labour inputs; their pricing process is also subject to a Calvo pricing structure. The intermediate goods are
then aggregated into a final product in a perfectly competitive sector. Monetary policy is conducted using
a Taylor rule responding to both deviations in output and inflation, with inertia. Policy is also exposed to
exogenous shocks to interest rates.
Although the model framework is standard for the literature and is therefore discussed fully in an appendix, the fiscal sector is of particular importance to this paper. The operation of the fiscal authority is
similarly standard whereby the government budget constraint requires that total expenditure on government
consumption, Gt , public investment, ItG , public employment, LG
t , and transfers, Tt , be paid through either
taxes or transactions bonds (Bt ):
Gt +
ItG
+ (1 +
τter ) wt LG
t
+ Tt
=
(1 + it−1 )Bt−1
Bt −
+ τtc Ct
πt
+ τtl + τter wt Lt + τtk rk,t ut Kt−1
(2)
where it and πt are the interest rate and inflation rate and where τtc , τtl , τter and τtk represent distortionary
taxes on consumption (Ct ), labour income (wt Lt ) and capital income (rk,t ut Kt−1 ) respectively. Public
employment (KtG ) is assumed to be completely wasteful, as in government consumption, both of which
neither enter the utility function of agents or the production function of firms. Public capital, however, does
7
enter the Cobb-Douglas production function of firms with an elasticity represented by αG and accumulates
according to:
G
KtG = (1 − δkG )Kt−1
+ ItG
(3)
which is equivalent to the accumulation of private capital but without costs to adjustment (as is common in
the literature) and where δkG represents depreciation specific to public capital.
Each fiscal variable moves with respect to three stimulates: first, there is a cyclical, automatic response,
of fiscal instruments to output (Yt responding with pararmeter ϕy,x ); second, to ensure that the solvency
condition is maintained, fiscal instruments respond to the level of debt (ϕb,x ); and third, each fiscal variable
has its own exogenous shock component (η x ). Each instrument responds to these factors, maintaining a level
of persistence (ρx ):
x ρ x
xt
t−1
=
x
x
Yt−1
Y
ϕy,x Bt−1
B
ϕb,x
ηtx
(4)
where x = τ c , τ k , τ l , τ er , G, I G , LG , T and where variables with no time subscript represent relevant steady
state values.
3
Experimental results
This section reviews results from applying the experimental procedures outlined above in a calibrated version
of the model; these results are then extended in Section 4 where Bayesian estimations of the model are made
using UK data. The calibration is standard for the literature and is outlined in Table 1. In particular,
the calibration is performed in such a way to reduce many of the identification issues highlighted in the
literature and discussed in Section 2.1. This is achieved through careful selection of the fiscal policy rules
and their associated parameters; in particular, there is no automatic response of fiscal instruments to output
(ϕy,x = 0) and the response to debt is lagged and mild (ϕb,x ). Further, this response to debt is the same for
all parameters, as is the persistence of all fiscal shocks (ρx ).
For each experiment we derive 200 quarters of data from simulations (after dropping the first 200 observations) which represents 50 years which is towards the upper end of that used in the literature. Despite the
fact that many studies reflect on a potential structural shift from the start and the end of the sample period
(typically between 1960 and 2010) this, for now, is not imposed on the simulation. Moreover, note that the
8
Table 1: Calibration
Share/parameter
Expenditure shares
C/Y
G/Y
I/Y
I G /Y
Preferences
β
σl
θ
h
Technology
δk
δkG
α
φk
κ
$
$W
γp
γw
s
ν
Φ
Monetary policy
ρ
ρπ
ρy
Fiscal policy
τc
τk
τl
τ ee
τ er
B/Y
αG
wLG /Y
ρx
ϕy,x
ϕb,x
Description
Value
Private consumption to GDP
Public consumption to GDP
Private investment to GDP
Public investment to GDP
0.65
0.1
0.11
0.02
Discount factor Ricardian
Inverse Frisch elasticity
Share of non-Ricardian households
Habit persistence
0.99
2
0.3
0.0
Depreciation rate: private capital
Depreciation rate: public capital
Share of capital in production
Investment adjustment cost parameter
Capital utilisation adjustment parameter
Stickiness in prices
Stickiness in wages
Price indexation
Wage indexation
Elasticity of substitution in consumption
Elasticity of substitution in labour
Fixed costs in production
0.025
0.02
0.31
5
0.6
0.75
0.5
0.3
0.3
6
6
0.15
Monetary policy persistence
Inflation Taylor rule weight
Output Taylor rule weight
0.8
1.8
0.125
Steady state consumption tax
Steady state capital tax
Steady state labour income tax
Steady state employee social security tax
Steady state employer social security tax
Government debt to annual GDP
Elasticity of public capital in production
Share of public to total employment
Persistence of all fiscal shocks
Fiscal instrument response to output
Fiscal instrument response to debt
0.2
0.4
0.18
0.05
0.07
0.6
0.02
0.15
0.85
0.0
0.1
simulated data from the results is stationary and therefore considerations over stochastic or deterministic
time trends, and the further noise this creates, is unnecessary; in this section, the framework and calibration
is deliberately designed to aid the estimation and identification process.
3.1
Dispersion of fiscal multipliers
Figure 1 presents the spread of results on the cumulative fiscal multiplier when applying different fiscal
shock calibrations in order to isolate the specific outcomes. In this respect, the process is identifying the
9
distribution of the significance level of multipliers in the model and, conversely, the prospect of a type 1
error as in all simulation true multipliers are the same; in each case results are based on 500 iterations.
Figure 1: Cumulative multiplier estimates
Fan charts illustrating the range of cumulative fiscal policy multipliers from 500 simulations of the model, using the calibration
as outline in Table 1 and where in each simulation 200 quarters of data is used, after dropping the first 200 quarters of results.
In each case the fiscal shock identification system is a Cholesky decomposition ordering variables as: government spending,
tax revenue, output, inflation and interest rates. This identification method was used for presentation as it provides the most
accurate result. In each plot, the dashed line represents the multiplier from the theoretical multiplier and the solid white line
represents the median estimate, and the fan the respective deciles. In each pane fiscal shocks are calibrated to represent 20% of
the variance of output in the first quarter of a variance decomposition, and those instruments which make up these are signified
by the headings in each pane.
The top left hand pane of Figure 1 presents the spread of estimates of the government spending multiplier
when government consumption is the only fiscal instrument shocked, and where the magnitude of this shock
is calibrated to fit variance decomposition on output of 20% on impact, which is standard in the literature
(see for example Christoffel et al. 2008); the results presented are from using a Cholesky decomposition
identification process on a five variable VAR using government spending, tax revenue, output, inflation
and interest rates (in this order), as this provides the best estimation of the true multiplier under this
calibration, and all other calibrations in Figure 1.3 As can be seen under these clinical conditions, where
3 With the steps taken to reduce any of the specific identification issues discussed in the literature, the result that a Cholesky
decomposition provides the most accurate results is intuitive. Under these conditions, the automatic elasticities of fiscal
10
only one fiscal instrument is being shocked and where parameters are calibrated to avoid identification
problems, the estimation is reasonably precise; at a horizon of eight quarters, for example, the middle 95% of
results has a range of 0.3. The second pane performs a similar procedure whereby all government spending
instruments are calibrated to have an approximately equal share on the variance decomposition of output,
calibrated to be 20% (and where tax instruments are not shocked). Here, the dispersion of results are in line
with those when only one instrument is being shocked, and the median estimate compares favourably, again,
with that from the theoretical model. In both scenarios, the precision of estimation deteriorates when the
time horizon being measured is extended. For example, the middle 90% of the distribution for three year
cumulative multipliers have a range of 1, and this is despite the fact that through focusing on cumulative
multipliers, past values (which are estimated with considerable precision) are also within the calculation.
This demonstrates the severe weakening of the estimation procedure as time moves on.
These results are in line with those obtained from just shocking labour taxes and just shocking all
tax instruments, the fourth and fifth panes of Figure 1 respectively, with similar degrees of dispersion
across all time horizons. However, with lower theoretical values for tax multipliers, this dispersion becomes
more as a proportion of the underlying multipliers, which provide more problematic results with respect
to interpretation. For example, when utilising all tax instruments, results statistically insignificant from
zero at the 95% level are observed after 10 quarters. Moreover, the estimation of tax multipliers tends
to be less accurate compared with the true multiplier values compared with spending statistics. These
differences are driven by two main factors. First, there is more of a range in the multipliers of individual tax
instruments compared with individual spending instruments and therefore including further fiscal variables
into the analysis of an aggregate measure includes more scope for variation. Note, also, that the VAR itself
is not designed to estimate a multiplier directly, but to estimate the paths of variables within the system
from which the cumulative multipliers are calculated. As taxes are distortionary, the paths of the variables
differ more with different taxation instruments compared with spending instruments, where the latter do
not directly contribute to agents’ actions. The second cause of this relative performance of the estimation of
tax versus spending multipliers is derived from the different instruments’ elasticity with respect to output.
Whereas government spending instruments vary little with the business cycle (given the calibration of the
fiscal parameters) tax revenues are naturally cyclical as a rise in economic activity leads to a rise in revenues.
All government spending instruments do not respond to output directly (and only respond very mildly
to debt); the only spending variable which indirectly responds to output is that of public employment,
instruments with respect to other variables in the VAR (which a Cholesky decomposition does not estimate) are negligible, and
statistical power is not wasted on estimating these.
11
whereby although the total hours of employment remains constant the wage rate will be mildly cyclical. The
comovement between output and government revenues results in greater variation in the estimation of tax
multipliers, as the exogenous shock is harder to isolate from the cyclical component.
In the last column of Figure 1 all fiscal instruments included in the model are shocked and the estimation
of both tax and spending multipliers becomes wider with different iterations of the simulated model. Tax
multipliers are now clearly estimated with less precision than spending multipliers, however, the variation in
both is more considerable than in the more isolated experiments above; moreover, for tax instruments there
appears to be less symmetry in the distribution of results. This suggests, therefore, that although VARs are
suitably accurate at estimating relationships in isolated cases, especially in the presence of fiscal instruments
which do not vary significantly with the business cycle, in a more reasonable framework with many fiscal
instruments, this accuracy is reduced. This is despite the steps taken to ensure that the identification
issues discussed in the literature are reduced, and further, through subjecting the data to many different
identification schemes, whereby the presented results are those which most accurately represent the point
estimate of the fiscal multiplier with the lowest variation. However, over short horizons the estimation
performs reasonably well, but these are specifically clinical conditions; the next section develops these results
for a more empirical framework.
3.2
Statistical power from a calibrated model
Using the distribution of multipliers identified above from a model where the true values are fixed, this section
can test the statistical power when comparing two estimates derived from an underlying model which has
different levels of fiscal effectiveness. It does this through simulating the model with varying values of λ, the
share of non-Ricardian households. These agents spend entirely out of their disposable income and as such
do not adhere to the permanent income hypothesis, nor do they crowd out their private consumption when
government spending is increased. As such, higher proportions of these agents are matched by both higher
spending and tax multipliers. From simulations of the theoretical model at different calibrations of λ fiscal
multipliers can be estimated and using the statistical distribution of results derived in Section 3.1 a test can
be made as to what proportion of estimates derived from the new model would be statistically significantly
different from that derived from a benchmark model. Conversely, the process is identifying the chance of
a Type 2 error, that is of not correctly rejecting a null hypothesis of no difference between the multiplier
estimates.
Results from this process are presented in Figure 2 for both the tax and spending multiplier, using the
12
Figure 2: Statistical power of cumulative multipliers
‘Statistical power’ represents the probability of correctly identifying that the difference between two estimates are significantly
different from underlying models calibrated to provide theoretical multipliers which differ; these differences are signified on the
x-axis. This is performed for the ‘all instruments’ experiments in Figure 1 at three time horizons (one, two and three year
cumulative multipliers) and for three levels of statistical significance; 50%, 90% and 95%.
same five-variable VAR from Figure 1, testing the statistical power at three levels of significance, and at three
time horizons for the cumulative fiscal multiplier: one, two and three years. The calibration of shocks is the
same as in the last column of Figure 1 where by each fiscal instrument contributes approximately equally
towards a variance decomposition on output for all fiscal variables of 20%. When the one year cumulative
government spending multiplier from the true model is higher than 0.25 (or increased by 50%) there is more
than a 50-50 chance of identifying a statistically significant result at the 90% confidence level. Intuitively,
as the horizon over which the cumulative multiplier is measured is increased, lower powers are obtained,
because as is seen in Figure 1 the spread of results also increases. Considering the statistical power with a
one standard deviation improvement in the fiscal multiplier at different time horizons (a common measure
in the statistical literature), the one, two and three year cumulative government spending multipliers have a
power of less than 25% at the 90% and 95% confidence level.
Similar results for the path of tax multipliers are observed. Taking the same one standard deviation
13
improvements, a statistical power of 50% is observed at the 95% confidence level.
Even under these clinical conditions whereby cumulative rather than point multipliers are estimated and
whereby the calibration is chosen specifically to reduce specific identification issues, the statistical power to
identify reasonably large changes in effectiveness of fiscal policy is relatively small. From a policy perspective,
some of these changes in multipliers can have quite important outcomes. Fiscal multipliers first have to be
shown to be statistically significant from zero, and then, preferably, bigger than one such that they are genuine
multipliers; subsequently policy makers would like to know what fiscal action is most effective, and then may
also be targeting specific components of aggregate demand. However, the performance of multiplier estimates
are not unreasonably large; the following section performs similar steps using an estimated calibration of the
model to see how much the experiment design from above is improving the results.
4
Results from estimated model
The results from above are now extended to an estimated version of the model using UK data. This is to
test the robustness of the results to a more empirical framework, where the strategy to reduce both the noise
and potential identification issues is removed.
We apply per capita data on GDP, private consumption, private investment, hours, wages, inflation, government consumption, government investment, transfers, and effective tax revenue on consumption, labour
and capital, over the period 1987:Q2 to 2010:Q1, where this range is determined by the availability of tax
data. The estimation procedure is as standard in the literature as are the prior distributions applied for
parameters. Of particular importance are the results on fiscal parameter estimates, as these will contribute
to the accruacy of fiscal multipliers from this model; these estimates are presented in Table 2.
Instrument
Government consumption
Government investment
Transfers
Public employment
Consumption taxes
Labour taxes
Capital taxes
Employer social security
Table 2: Estimated fiscal parameters
Persistence Response to debt Response to output
0.87
-0.11
-0.15
0.74
-0.56
-0.01
0.86
-0.11
-0.01
0.87
-0.15
0.07
0.78
0.03
0.00
0.83
0.11
0.02
0.89
0.08
-0.07
0.91
-0.01
-0.07
Shock size
0.09
0.01
0.09
0.01
0.16
0.13
0.23
0.01
Estimates of fiscal parameters from a Bayesian estimation of the model using UK data. Note that for spending parameters a
negative response to both debt and output represents an aversion to debt and a countercyclical response respectively.
14
As is demonstrated in Table 2 there is mild variation in the estimate of fiscal variables across the different
instruments, however, these differences are small. Further, there is only small (generally countercyclical)
responses of instruments to output and only a mild degree of aversion to debt. Therefore, in relation to the
values used in the calibrated version of the model above, there are no extreme observations that may have
pose a significant bias on results.
Using the estimated version of the model for both fiscal and structural parameters to perform the same
procedures as in Section 3.1, results in Figure 3 are derived. Estimates become more disperse; for example,
the middle 95% range for the two year cumulative government spending multiplier is nearly 2, compared
with 0.3 above. Moreover, estimates become less accurate in providing the true mean value of multipliers;
this is particularly true of tax instrumented which are estimated to have the ‘wrong’ effect more often than
not (higher taxes lead to higher output), possibly owing to the estimated procyclical response to the business
cycle. This, combined with the fact that tax revenues naturally decrease in poor economic conditions, and
are more intertwined with the health of the economy than spending instruments, provides both a wide range
of estimates and quite imprecise estimation, where tax multipliers, especially in the short run, are estimated
to be positive.
Figure 3: Cumulative multiplier estimates
Fan charts illustrating the range of cumulative fiscal policy multipliers from 500 simulations of the model, using an estimation
of the model, with the fiscal parameters outlined in Table 2 and where in each simulation 200 quarters of data is used, after
dropping the first 200 quarters of results. In each case a Blanchard & Perotti (2002) SVAR identification system is used with
a VAR that contains the same five variables as in Figure 1. This identification method was used for presentation as it provides
the most accurate result. In each plot, the dashed line represents the multiplier from the theoretical multiplier and the solid
white line represents the median estimate, and the fan the representative deciles.
15
These results suggest that under more empirically appropriate conditions, using VAR techniques to
estimate fiscal multipliers does not provide reliable results. This is independent of the identification procedure
used, where the results in Figure 3 represent the most accurate set from a wide variety of methodologies
incorporating different variables and variable ordering. This is not, however, that surprising. Now the
empirical regularities of different fiscal instruments being used in different ways by policy makers is adding
sufficient noise to the estimation process. Fiscal expenditures and revenues are highly dependent on different
economic circumstances. Their use is vast and disparate, and their impact on the economy is heterogeneous
from one another.
Finally, it should be highlighted that many other procedures which could add noise to this estimation
process have not been incorporated into these results. The prospect of a structural change over the 50 year
time horizon of simulated data has not been allowed for, and the prospect of the impact of fiscal policy
to change given different economic conditions is also removed. Given that these two factors are frequently
observed in data, including these omissions should only act to amplify the still somewhat prudent results
above.
5
Conclusions
Our findings provide the intuitive result that traditional methods of identifying the effectiveness of fiscal
policy are largely unreliable. That is to say, there is no evidence of any bias in the results necessarily,
however, the range of potential results is broad. This is intuitive for two main reasons: first, the range
of empirical estimates currently in the literature suggests that something in these results are sensitive to
specific elements in the estimation process; and second, there is a large scope for noise to enter this estimation
process. Not only is there noise in the macroeconomy in general, the way in which specific fiscal episodes
interact within this is highly heterogeneous. Fiscal policy is made up of different instruments which are
applied in very different ways. Moreover, truly exogenous fiscal shocks announced by policy makers are
performed under very different circumstances.
These results suggest that the use of fiscal multiplier estimates from VAR analysis to aid policy making
is a dangerous activity. Further, over reporting the importance of these results only adds to the idealogical
debates which occur within this field. The degree of accuracy of estimates required for fully informed policy
making goes beyond that possible using the methods applied above. Not only do fiscal multipliers need
to pass a test of being statistically different from zero, they need to go further and preferably be greater
16
than one, and ideally, results comparing the suitability of specific instruments is required. Although recent
advancements go some way towards providing alternative ways of measuring fiscal multipliers, these are still
done with a high degree of statistical uncertainty.
17
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334.
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Bureau of Economic Research.
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government spending and taxes on output’, Quarterly Journal of Economics 117(4), 1329–1368.
Caldara, D. & Kamps, C. (2008), ‘What are the effects of fiscal policy shocks? A VAR-based comparative
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12(3), 383–398.
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evidence’, NBER Macroeconomics Annual 4, 185–216.
Canova, F. & Paustian, M. (2007), ‘Measurement with some theory: using sign restrictions to evaluate
business cycle models’, Working Paper.
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Christiano, L. J., Eichenbaum, M. & Evans, C. L. (2005), ‘Nominal rigidities and the dynamic effects of a
shock to monetary policy’, Journal of Political Economy 113(1), 1–45.
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spending multipliers’, Journal of Economic Dynamics and Control 34(3), 281–295.
18
Erceg, C. J., Guerrieri, L. & Gust, C. (2006), ‘Sigma: A new open economy model for policy analysis’,
International Journal of Central Banking 2(1).
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price contracts’, Journal of Monetary Economics 46(2), 281–313.
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American Economic Review 95(3), 739–764.
Mankiw, N. (2000), ‘The savers-spenders theory of fiscal policy’, American Economic Review 90(2), 120–125.
Mountford, A. & Uhlig, H. (2009), ‘What are the effects of fiscal policy shocks?’, Journal of applied econometrics 24(6), 960–992.
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in ‘Carnegie-Rochester Conference Series on Public Policy’, Vol. 48, Elsevier, pp. 145–194.
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measure of fiscal shocks’, American Economic Review 100, 763–801.
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Perspectives 12(3), 131–150.
19
A
A.1
The Model
Households
Utility for both types of household is assumed to be the same and evolves according to:
∞
X
b
i t
E0 εt β U ln Cti (h) − Ht −
t=0
1+σl
1
Lit (h)
1 + σl
(5)
where E0 is the expectation operator, β ∈ (0, 1) is the discount factor, σl denotes the inverse of the Frisch
labour supply elasticity, Ct and Lt denote consumption and Labour respectively, Ht denotes an external
habit variable such that Ht = hCt−1 , and εbt represents a first-order autoregressive exogenous shock process
to preferences. Superscript i differentiates variables between Ricardian (i = R) and non-Ricardian (i = N R)
households.
A.2
Ricardian Households
Each period Ricardian household, h, faces a budget constraint which states that the household’s total
expenditure on consumption, CtR , investment in physical capital, It , and accumulation of a portfolio of
riskless one-period contingent claims, Bt , must equal the household’s total disposable income:
(1 + τtc ) CtR (h) + It (h) + Bt (h)
=
1 − τtl wt (h) LR
t (h) + divt (h)
+ 1 − τtk rk,t ut (h) − a(ut (h)) K̄t−1 (h)
+
(1 + it−1 )Bt−1 (h)
+ Tt (h)
πt
(6)
where τtc , τtl represent taxes on consumption and labour income, and wt the real wage; divt represents
dividends paid out of firms’ profits; τtk is a tax on capital, rk,t the real return on capital services, ut the
capital utilisation rate where the cost of capital utilization is given by a(ut )K̄t−1 , and K̄t−1 the stock of
physical capital; it−1 represents the net nominal interest rate on one-period bonds, πt the gross inflation rate,
and the gross nominal interest rate is given by Rt = 1 + it ; and Tt represents a lump sum transfer. Following
Christiano et al. (2005), we assume complete markets for the state contingent claims in consumption and
capital but not in labour, which implies that consumption and capital holdings are the same across Ricardian
households: consequently, CtR (h) = CtR , KtR (h) = Kt .
20
In line with most of the existing literature, we maintain that physical capital accumulates in accordance
with:
i It
K̄t = (1 − δk )K̄t−1 + 1 − S εt
It
It−1
(7)
where we follow Schmitt-Grohe & Uribe (2006) and define the cost of investment adjustment function
as S (It /It−1 ) = (φk /2)(It /It−1 − 1)2 and εit is an investment specific first-order autoregressive shock
process.4 Each Ricardian household maximises utility (5) subject to the flow budget constraint (6), the
capital accumulation function (7), and the labour demand from the labour unions discussed below.
A.3
Non-Ricardian households
As discussed above, non-Ricardian households are credit constrained agents who simply consume current
after-tax income which comprises of after-tax labour income and transfers.5 This behaviour can be rationalised in a setting where non-Ricardian households are more impatient than Ricardian households,
β R > β N R , and can default on their debt up to the value of their collateral (see, for example, Iacoviello
2005). With no durable goods in the non-Ricardian utility function, impatience prohibits the accumulation
of collateral and as such non-Ricardians are prohibited from engaging in bond and capital markets.6 The
budget constraint of non-Ricardian households is therefore:
R
(1 + τtc ) CtN R = 1 − τtl wt LN
+ Tt
t
(8)
Following Erceg et al. (2006) we assume that each non-Ricardian household sets its wage equal to the average
wage of optimising households (discussed below). Given that all households face the same labour demand,
the labour supply and total labour income of each Ricardian and non-Ricardian households will be the same:
by extension, consumption for all rule-of-thumb agents will also be the same (CtN R (h) = CtN R ).
A.4
Wage-setting behaviour
As in Erceg et al. (2000) we consider a competitive labour union that transforms households’ differentiated
labour into composite labour which is subsequently supplied to private intermediate firms and the public
S (1) = S 0 (1) = 0, and S 00 (1) = φk > 0 are assumed for the adjustment cost function process: as a result the steady
state does not depend on parameter φk .
5 In what follows the terms ‘non-Ricardian’, ‘rule-of-thumb’ and ‘credit constrained’ are used interchangeably.
6 Existing literature provides two sources of motivation for introducing rule-of-thumb consumers; first is the lack of evidence
for consumption smoothing in the face of income fluctuations (see, for example, Campbell & Mankiw 1989); and second the
observation that an important fraction of households have near-zero net worth (see, for example, Wolff 1998, Mankiw 2000).
4 Where
21
sector. The technology used in this transformation is defined by:
ν
 1
 ν−1
Z
ν−1
Lt =  (Lt (h)) ν dh
(9)
0
where ν > 0 is the elasticity of substitution among the differentiated labour inputs and Lt the aggregate
labour index. The union takes every household’s wage, Wt (h), as given and maximises profit ΠU
t :
ΠU
t = Wt Lt −
Z
1
Wt (h)Lt (h)di
(10)
0
where Lt (h) denotes the amount of labour supplied by household h to the union, and Wt (h) is the corresponding wage rate for the labour: Wt is the aggregate wage index. Profit maximisation results in the
following demand for household h’s labour:
Lt (h) =
Wt (h)
Wt
−ν
Lt
(11)
Setting the profits of labour unions to zero, due to the prevailing perfect competition in the composite labour
market, results in the aggregate wage index:
Z
Wt =
1
(Wt (h))
1−ν
1/(1−ν)
di
(12)
0
Nominal wages are set in a staggered-price mechanism as in Calvo (1983), where every period, each Ricardian
household faces a fixed probability (1 − $W ) of being able to adjust the nominal wage. The household then
sets nominal wages to maximize expected future utility subject to labour demand from firms. Those who
cannot reoptimize set wages in accordance with the indexation rule, Wt = πtγw Wt−1 , where γw ∈ h0, 1i is
a parameter that measures the degree of wage indexation. The objective is to maximise the following with
respect to W̃t :
Et
∞
X
(β$w )
l



1
− 1+σ
L
W̃t Xtl
Wt+l
−ν

 +λτ 1 − τ l W̃t X
t+l
t+l Pt+l t,l
l=0
1+σL
Lt+l
W̃t Xtl
Wt+l
−ν



Lt+l
(13)


where Xtl = πt × πt+1 × ... × πt+l−1 for l ≥ 1 and Xtl = 1 for l = 0. The maximisation results in:
Et
∞
X
l=0
(
l
(β$w )
Lτt+l λt+l
ν
W̃t Xtl Uc,t+l
Ul,t+l
−
c
l
Pt+l 1 + τt+l
(1 − ν) 1 − τt+l
22
)
=0
(14)
The first-order condition implies that Ricardian households set their wages so that the present value of the
marginal utility of income from an additional unit of labour is equal to the markup over the present value
of the marginal disutility of working. When all households are able to negotiate their wage contracts each
period, the prevailing wage is W̃t /Pt = (ν/ (1 − ν))(Ul,t (1 + τtc ) /Uc,t 1 − τtl ). Finally, the wage index can
be transformed into the following:
"
1−ν
Wt = (1 − $w ) W̃t
A.5
+ $w
Pt−1
Pt−2
1
1−ν # 1−ν
γw
Wt−1
(15)
Production
A competitive final good producer purchases differentiated goods from intermediate producers and combines
them into one single consumption good. The final good, YtP , is produced by aggregating the intermediate
P
goods, Yj,t
, with technology:
YtP
Z
1
P
Yj,t
=
s−1
s
s
s−1
dj
(3.8)
0
Profit in the final good sector, ΠF
t , can be stated as:
ΠF
t
=
Pt YtP
Z
−
1
P
Pj,t Yj,t
dj
(3.9)
0
where Pj,t is the price of the intermediate good j. Standard demand functions for intermediate goods and
a zero profit condition for prices can be derived, as was performed for labour unions.
The intermediate good production sector is populated by monopolistic firms indexed by j that use the
following production function:
P
Yj,t
= εat (Kj,t−1 )
α
LP
j,t
1−α
G
Kj,t−1
αG
−Φ
(16)
where KG denotes public capital, Φ represents a fixed cost of production, and εat represents total factor
productivity shock that follows a first-order autoregressive process. Firms rent capital services Kj,t−1 , and
incur a cost of labour equal to (1 + τter ) Wt where τter denotes employers social security contributions. As
is standard in the new-Keynesian framework, intermediate-good sector firms face three constraints: the
production function, a demand constraint, and price rigidity determined by a Calvo (1983) mechanism.
Each firm acts to minimise its total costs, (1 + τter )Wt LP
j,t + Rk,t Kj,t−1 , subject to the production function
23
(16). The nominal marginal cost is represented by the following:
Pt mct =
1
1−α
1−α α
−1 −αg
1
1−α
α
εA
Kg,t−1 ((1 + τter ) Wt )
(Rk,t )
t
α
(17)
Intermediate goods producers act as price setters where each period a given firm faces a constant probability,
(1 − $), of being able to reoptimise its nominal price. Those who can, maximize expected future profits at
these prices:
"
#
∞
X
Pet Xtl
l
− mct+l Pt+l Yj,t+l − Pt+l mct+l f c
Et (β$) λt+l
Pt+l
(18)
l=0
subject to the standard demand (Yj,t = (Pj,t /Pt )s Yt ) and maximisation results in:
"
#
∞
X
Pet Xtl
s
l
Et (β$) λt+l
mct+l Pt+l Yj,t+l = 0
−
Pt+l
1−s
(19)
l=0
In the case that all firms are allowed to reoptimise their prices, the above condition reduces to, Pet =
(s/(s − 1))Pt mct , which indicates that the optimised price is equal to a markup over the marginal costs.
In addition, (β$)l λt+l denotes a discount factor of future profits for firms. Here λt denotes the Lagrange
multiplier on the Ricardian household’s budget constraint and is treated by firms as exogenous. The price
index can be rewritten as:
"
Pt = (1 − $) Pet1−s + $
A.6
Pt−1
Pt−2
1
1−s # 1−s
γp
Pt−1
(20)
Monetary policy
As standard in the literature, the monetary authority sets nominal interest rates (Rt ) by following a Taylor
rule which responds to both output and inflation with some persistence:
Rt
=
R
Rt−1
R
ρ πt ρ π
π
Yt ρy
Y
1−ρ
ηR,t
(21)
where ρ is the interest rate smoothing parameter and ηR,t represents an i.i.d. shock to the nominal interest
rate: all other variables are as defined earlier.
24
A.7
Market clearing
Total output is the sum of private and public sector output where the equilibrium conditions are given by:
Yt = Ct + Gt + It + ItG + a(ut )K̄t−1 − (1 + τter ) wt LG
t
(22)
G
Lt = LP
t + Lt
(23)
where Ct and Lt denote aggregate consumption and employment which are given by the weighted averages
of the consumption and employment of Ricardian and non-Ricardian households. Similarly, the market for
capital and bonds are in equilibrium when demand equals supply.7
7 The non-stochastic steady state of the model is solved, and the perturbation method in Dynare is used to apply a secondorder approximation of the model. The stochastic simulations are also computed using Dynare.
25