Back Matter
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund

References

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VIII. Appendix

A. Motivation for heterogeneity

In this section, we motivate the discussion on the potential correlation between the spending multiplier and the level of government expenditure via a stylized model that assumes an active stabilization role on behalf of the government and accounts for productive government services. Optimal government spending, where government spending is an input to private production, has been considered by Barro (1990) and (Corsetti and Roubini, 1996) in the context of endogenous growth models. Our focus here is on the dependence of optimal government spending on the fiscal multiplier. Assume an economy with a unit measure of identical representative agents, each of which maximizes the following utility function:

𝔼tΣi=0ρiU(ct+i),(19)

where 𝔼t is the conditional expectation operator based on information available at time t, ρ ∈ (0,1) is the subjective discount factor, U(·) is a utility function that is strictly increasing in the consumption of the final good ct, with U″(·) < 0.

Each agent maximizes expression (19) subject to the following budget constraint:

ctwtltτt,(20)

where lt denotes the labor supply of the representative agent, wt is the real wage and τt denotes a lump-sum tax. For simplicity, we assume that the representative agent is endowed with one unit of labor which is supplied inelastically Government expenditure is fully financed by the lump-sum tax, so that gt = τt for every period t. The final consumption good, yt, is produced by a competitive representative firm using the following aggregate production function:

yt=ztH(gt)lt,

where zt is a technology shock, and H(gt) satisfies H′(·) > 0 and H″(·) < 0. For simplicity, we adopt the functional form H(gt)=gtϵ, where ε is a parameter reflecting the productivity of government spending.

Profit maximization suggests that wt=ztgtϵ, i.e., the real wage is equated to the marginal productivity of labor. In equilibrium, households and firms optimally choose consumption and production, so that the budget and aggregate resource constraints are binding:

ct=ztgtεgt,(21)

where equilibrium output is given by:

Yr=ztgtε.(22)

The government spending multiplier βM is obtained by totally differentiating the above equation and rearranging terms to get:

βMdytdgt=εyt1gt1.(23)

Based on expression (23), we can conclude that:

Result 1: The government spending multiplier is increasing in the productivity of government spending.

We now proceed to characterize optimal spending on behalf of the government based on its output stabilization objective.24 More specifically, the government chooses gt to minimize deviations of output yt from a target level of output yP, which can be the natural or potential output level:

minwrt{gt}(ytyp)2.(24)

Substituting (22) into (24), we obtain the following optimal value for government expenditure:

gt*=(ypzt)1β,(25)

where β ≡ ε denotes the elasticity of output with respect to government spending.25 Note also, using equation (23), that:

gt*=(ypzt)yt1βMgt1.(26)

Equations (28) and (26) yield our central proposition:

Proposition 1. Optimal government spending is decreasing in (i) the output elasticity β, (ii) the government spending multiplier βM, and (iii) the technology shock zt.

Hence, governments with higher multipliers, defined either as ratios (i.e., βM) or elasticities (i.e., β), require relatively lower levels of expenditure to meet a certain target level of output compared to governments with lower multipliers. This induces a negative correlation between the level of government spending and the level of the fiscal multiplier. Additionally, optimal government spending is declining with the magnitude of the stochastic productivity shock (i.e., zt). This result is consistent with counter-cyclical spending rules, where governments reduce expenditure during periods of high growth and increase spending during recessions (see, for example, Fève, Matheron, and Sahuc (2013) and Galí and Perotti (2003)).

Denoting upper case letters as the logarithmic transformations of each variable, we obtain the log-form of equation (22) as:

Yt=Zt+εGt.(27)

We also obtain an equivalent expression for the optimal government spending rule of equation (25), which in log-form is equal to:26

Gt*=YpZtβ.(28)

B. Identification and estimation of the CRC model

Our model, with one regressor and an intercept, is given by:

Yit=Gitbit(Ai,Uit).(29)

As noted earlier, Assumptions (1.1)(1.3) imply that:

𝔼[bit(Ai,Uit)|Gi]=𝔼[bi*(Ai,Uit)|Gi]+𝔼[dit(U2,it)|Gi](30)
=𝔼[bi*(Ai,Ui1)|Gi]+𝔼[dit(U2,i1)](31)
=βi(Gi)+δt,t=1,,T.(32)

where δ denotes the 2(T−1) × 1 vector of aggregate shifts in the random coefficients over time. δ1 is normalized to be equal to zero. Next, we define a matrix of time shifters, W. As described by Graham and Powell (2012), W is a T × 2(T -1) block diagonal matrix containing the regressors corresponding to the aggregate time shift coefficients (with the first row containing a vector of zeroes due to the normalization).

Equation (29) can be rewritten as:

𝔼[Yi|Gi]=Wiδ+Giβi(Gi).(33)

A variance weighted within group transform, which essentially differences away the unobserved heterogeneity, βi(Gi), is obtained using the residual making matrix, MiΦi=ITGi[Gi'Φi1Gi]1Gi'Φi1 where Фi =Var(Yi|Gi). Using the fact that MiΦiGi=0, we get:

𝔼[Y˜i|Gi]=MiΦiWiδ+MiΦiGiβi(Gi)(34)
=W˜iδ,(35)

where Y˜i=MiΦiYi and W˜i=MiΦiWi.

The pair of moment restrictions that identify the parameter are given by:

𝔼{WiΦi1MiΦi(YiWiδ)Gi1,Φi1(YiWiδ)βi}=0.(36)

For estimation, we proceed in two steps:

Estimation step 1: The first step in the estimation procedure consists in using the moment restrictions to identify δ in the following way:

δ=𝔼[W˜iΦi1W˜i]1𝔼[W˜iΦi1Y˜i].(37)

Estimation step 2: In the second step, with an estimate of δ, Chamberlain (1992) shows that the average partial effect, β, is identified by the (population) mean of the unit-specific generalized least squares fits:

β^i=(Gi'Φi1Gi)1Gi'Φi1(YiWiδ).(38)

i.e., β is given by:

β=𝔼[(Gi'Φi1Gi)1Gi'Φi1(YiWiδ)].(39)
*

We are grateful to Gustavo Adler, Jaebin Ahn, Céline Allard, Jorge Ivan Canales-Kriljenko, Tiago Cavalcanti, Benedict Clements, Giancarlo Corsetti, Mai Dao, Mitali Das, Xavier Debrun, Oliver DeGroot, Bill Dupor, Eric Gautier, Ruy Lama, Nan Li, David Newbery, Michael Plante, Issouf Samake, Sampawende Jules Tapsoba, Anne Villamil, and participants at various seminars for helpful comments.

1

International Monetary Fund;

CREST (ENSAE).

1

See, for example, Blanchard and Leigh (2013) for a discussion on the policy implications of the forecasters’ under-estimation of fiscal multipliers at the early stages of the recent financial crisis.

2

This aspect has been recently brought to attention by several authors who have argued that fiscal multipliers vary systematically with features of the economy or the business cycle that are potentially also correlated with government spending, such as the phase of the business cycle, the exchange rate regime, the degrees of trade openness and government indebtedness, as well as the extent to which the zero lower bound on nominal interest rates is binding. Supportive evidence for the range of these conditional estimates can be found in Auerbach and Gorodnichenko (2012), Baum, Poplawski-Ribeiro, and Weber (2012), Blanchard and Leigh (2013), Christiano, Eichenbaum, and Rebelo (2011), Corsetti, Meier, and Müller (2012), Corsetti and others (2013), Erceg, Lindé, and Erceg (2014), Favero, Giavazzi, and Perego (2011), and Ilzetzki, Mendoza, and Végh (2013).

3

In the Appendix (Section VIII.A), we analytically motivate the finding of the negative correlation between the multiplier and the level of government spending by showing that countries with higher fiscal multipliers will optimally choose a lower level of government spending to minimize a given output gap.

4

Kraay (2012) uses a first-differenced instrumental variable estimator that is also, potentially, consistent under correlated heterogeneity, to identify the effect of government spending shocks on output growth for a sample of 29 low income countries. The effects are, however, assumed to be homogeneous.

5

Later, we consider a more general model, Yit = βit (Ai, Uit)Git + αit(Ai,Uit), where we allow the coefficient and the intercept to be functions of time-invariant unit specific heterogeneity, Ai, and a time-varying disturbance Uit.

6

A burgeoning literature exploits cross-sectional variation in government spending, mostly across sub-national units, to secure identification though an instrumental variable strategy. See, for example, Acconcia, Corsetti, and Simonelli (2014) and Serrato and Wingender (2010). An IV approach would also require strong conditions to estimate the average partial effect. In a cross-section case, (Heckman, Urzua, and Vytlacil, 2006) show that, in general, an IV strategy cannot identify the average partial effect when the heterogeneous coefficients are correlated with the endogenous variable even when the instrument is separately orthogonal to each. In Section II, we show how additional variation through panel data can help identify the effect.

7

In this section, for ease of notation, our model contains only one regressor and a constant. However, in our empirical model we consider a more general specification with additional conditioning covariates.

8

Equation (32) is equivalent to a common-trends assumption, i.e., the differences in the coefficient values between two time periods are equal, and equal to the difference between the aggregate time trends, regardless of the regressor histories. Formally, consider two regressor histories Gi and Gi:𝔼[bit(Ai,Uit)|Gi]𝔼[bis(Ai,Uis)|Gi]=𝔼[bit(Ai,Uit)|Gi]𝔼[bis(Ai,Uis)|Gi]=δtδs.

9

The growth in total government spending is scaled by the lagged level of real GDP rather than by the lagged level of total government spending, i.e., ΔGit=gitgit1yit1, where g and y represent the levels of total government spending and real GDP respectively.

10

For many years, the U.S. has adopted a very low fuel taxation policy. To verify that the fuel taxes in the U.S. are the lowest among the industrialized countries, see Tables 8, 9 and 10 of IEA (2013, pp. 297–299).

11

The ‘pump price’ is the retail price of gasoline. Further details on the data used are provided in Section III.

12

See, also, Brückner, Chong, and Gradstein (2012) for a similar definition of the oil price shock.

13

Note that we include the current oil price shock as a control variable while using its first lag as an instrument. This specification is consistent with the findings of Brückner, Chong, and Gradstein (2012) who find that only the first lag of an oil price shock has a significant and positive effect on change in government spending while its impact and lead effects are statistically insignificant. In contrast, they find that current oil price shocks have a significant positive effect on output growth on impact but its lead and lagged effects are insignificant. We show in Section IV that we obtain similar results. Therefore, we exclude further lags of the oil price shock in the second stage not only because they have an insignificant effect on output growth, but also because they weaken the instrument set, as only the first lag of the oil price shock is informative in predicting change in government spending.

14

Murtazashvili and Wooldridge (2008) note that these conditions are most likely to apply when the endogenous explanatory variables are continuous, as in our context.

15

(Garen, 1984) requires Assumptions (2.1)(2.2) to hold as before, but adds the additional restrictions 𝔼[βiM|Z¨it]=0,𝔼[ϵ¨it|Δ¨Git,Z¨it]=λGΔ¨Git+λZZ¨it, and 𝔼[βiM|Δ¨Git,Z¨it]=ψGΔ¨Git+ψZZ¨it.. Effectively, the combination of all these assumptions imply that 𝔼[ϵ¨it|Δ¨Git,Z¨it]=λGΔ¨Git and 𝔼[βiM|Δ¨Git,Z¨it]=ψGΔ¨Git. Replacing these assumptions into equation (1) produces the convenient formulation of equation (16).

16

The variance of the disturbance term in equation (16) is a function of Δ¨𝔾it and (Δ¨𝔾it)2. To obtain consistent estimates, we first regress Δ¨𝔾it and (Δ¨𝔾it)2 on the squared residuals from equation (16) and use this to estimate the variance matrix of disturbances used in weighted least squares.

17

Access to the data is provided via the GIZ publication International Fuel Prices (www.giz.de/fuelprices) and the World Development Indicators database maintained by the World Bank.

18

The price data and the BP Statistical Review of World Energy are available at www.bp.com/statisticalreview.

19

More specifically, GIZ (2012) distinguishes between the high and the low fuel taxation categories depending on whether the retail price of gasoline (or diesel) is above the price level of the United States and above the lowest price that can be found among EU countries.

20

With only two random coefficients and three years of data, our model is overidentified. In principle, one can use all the available time periods to estimate the CRC model. However, adding more time periods than necessary results in the model becoming heavily overidentified and the structural parameter becoming a more complicated function of the underlying reduced form parameters.

21

Brückner, Chong, and Gradstein (2012) analyze data between 1960–2007 compared to our analysis which spans 1992–2010.

22

Although unlikely, note that the conditional IVQR estimates are identified even if some country does not vary in its position in the distribution of growth over time. Mathematically, the quantity is identified as long as government spending and other covariates change over time.

23

Figure 5 is plotted on a large scale (y-axis) compared to Figure 4. The conditional IVQR estimates are similar in both figures barring some minor differences in the vector of supporting covariates.

24

A similar output stabilization problem is explored in Dixit and Lambertini (2003).

25

Fiscal multipliers are often defined as elasticities of output with respect to government spending (i.e., β), as well as ratios of changes in output over changes in government spending (i.e., βM). Our results are shown to hold under either definition.

26

To obtain the optimal government spending rule in a log-form, we assume that the government minimizes the log-deviations of output from its potential level, i.e., the minimization problem becomes minwrt{Gt}(YτYP)2

Estimating Fiscal Multipliers with Correlated Heterogeneity
Author: Emmanouil Kitsios and Manasa Patnam