Pascal Gastineau
email: pascal.gastineau@ifsttar.fr

Emmanuelle Taugourdeau
email: emmanuelle.taugourdeau@ens-cachan.fr.

Compensating for environmental damages

Keywords: Environmental Damage, Compensation, Welfare, Inequity

This paper examines a situation where a decision-maker determines the appropriate compensation that should be implemented for a given ecological damage. The compensation can be either or both in monetary and environmental units to meet three goals: i) minimization of the cost associated with the compensation, ii) no aggregate welfare loss, iii) minimal environmental compensation requirement. The findings suggest thatin some cases -providing both monetary and environmental compensation can be the cost-minimizing option. Minimal compensation constraints can increase total compensation costs but reduce individual gains and losses relative to the initial situation that arise from heterogeneous tradeoffs between income and environmental quality.

Introduction

This paper aims to analyze the choice of a policy-maker in charge of determining the scaling of compensation for accidental environmental damage. As a form of compensation, the policymaker may choose between prescribing a uniform amount of money to each individual and/or restoring a natural resource similar to the damaged one. Given the properties of the injured population (number of agents and heterogeneity in wealth or preferences), the policy-maker pursues a trade-off between two conflicting objectives: equity and efficiency. Here, equity refers to the idea that each agent does not suffer similarly from the damage and does not benefit similarly from the compensation. As a result, the pattern of compensation may either reestablish equity (no change in individual and aggregate welfare) or maintain a certain level of inequity resulting from the damage, since agents support any welfare losses whereas others benefit from welfare gain even if the aggregate welfare remains unchanged. We oppose this equity purpose to an efficiency one, here defined in terms of costs: an efficient compensation will be the one which ensures no aggregate welfare change together with a minimum level of costs.

Decision-makers are aware of the need to prevent and to remedy for environmental damage. This growing environmental awareness was notably embodied in various statutes such as the Comprehensive Environmental Response, Compensation, and Liability Act (CER-CLA) and the Oil Pollution Act of 1990 (OPA) in the U.S. and the Directive 2004/35/EC on Environmental Liability with regard to the prevention and remedying of environmental damage in the European Union. These texts highlight the role that authorities have to play in order to establish a common framework that any polluter may comply with.

In addition, there is a sharp debate on the best way to offset the damages on natural resources and services. Generally, two types of compensation are distinguished: environmental compensation and monetary compensation. The first one consists in providing an environmental restoration or implementing other actions that provide benefit to the restoration. The second one consists in an amount of money paid to the prejudiced people. Within the last couple of years, environmental compensation for the loss of environmental assets (whether the ecological damage is planned or accidental) gained popularity. Moreover, the resource-to-resource (R-R) or service-to-service (S-S) equivalence approaches are considered as a first option by the European Directive. Furthermore, this Directive precludes the use of direct monetary payments to victims.

Non-monetary methods such as equivalency analyses (EA) aim to implement actions that provide natural resources and/or services of the same type, quality and quantity as those of damaged ones (i.e. "in-kind" compensation) (Dunford et al., 2004; Zafonte and Hampton, 2007). 1 These techniques determine the necessary compensation to offset past, current and 3 future damages without directly valuing them in economic terms, by equalizing the amount of loss and gain of resources and services over time. To do so, they use a selection of proxies (metrics) representing the most important ecosystem services (English et al., 2009). 2 The presupposed advantages of S-S and R-R methods (i.e. "no net loss" principle) stand in contrast with drawbacks associated with well-known monetary valuation techniques. However, none of the methods are perfect and the reliability of the equivalency methods to measure the environmental damage and/or scale and to determine the appropriate compensation is under discussion. On the ecological side, while stressing the usefulness of the equivalency methods, Dunford et al. (2004) also emphasize their weaknesses: a high degree of uncertainty concerning estimates of compensatory restoration and their difficulty to consider complex impacts and phenomenon. Many attempts are made to improve ecological equivalency methods by focusing on specific issues: uncertainty (Moilanen et al., 2009), temporal dynamics (Bendor, 2009) or spatial analysis (Bruggeman et al., 2005; Bruggeman et al., 2008). 3 On the economic side, Zafonte and Hampton (2007) suggest that, under certain conditions, resource equivalency analysis (REA, i.e. R-R) provides an acceptable approximation of compensating wealth. By contrast, many authors argue that ecological equivalence specified in biophysical equivalents could fail to provide a satisfactory compensation in a welfare perspective (Flores and Thacher, 2002). Flores and Thacher (2002) also stress the potential economic inefficiencies that could occur when the money component is excluded from the analysis and thus recommend a case by case determination of the adequate compensation that would better consider distributional issues associated with compensatory projects.

In this paper, we go further in the analysis of compensation by showing that environmental and monetary compensations are not antinomic and may be implemented simultaneously.

Due to heterogeneous individual preferences (or income), compensation can result in some losers and winners relative to their initial (pre-injury or pre-project) utility. Therefore, careful attention must be paid to the characteristics and the size of the population affected by an environmental damage when determining the compensation to implement. Thus, we study habitat) are mitigated through the creation, restoration, or enhancement of another resource (or habitat).

2 When equivalency approaches can not be used, valuation scaling approaches (value-to-cost and value-tovalue) are recommended.

3 See Quétier and Lavorel (2011) for a synthesis.

how the decision-maker can combine both of them in order to determine the adequate compensation at minimal cost. Of course, this analysis is only relevant when an environmental compensation with a similar natural resource or service is feasible.

In line with Cole (2013), this paper allows us to investigate equity and cost efficiency issues associated with an enforced environmental compensation. We depart from Cole by considering equity issues for the prejudiced population instead of considering the society on its whole. Moreover, contrary to Cole (2013) who compares both compensation schemes, we allow for a mixed compensation in which both of the compensatory methods may be implemented at the same time.

To reach our goal, we propose a simple model of an economy with two goods, a composite good and a natural resource. In this model, we determine which type of compensation the decision-maker may enforce the polluter to implement given the magnitude of the damage, the number and the characteristics of the prejudiced agents, and the cost associated with each compensation scheme. Since we do not introduce any incentives in our model (prevention, mitigation), we focus on accidental or unanticipated damages. Moreover, our model refers to marginal damages in the sense that they do not alter the agents' preferences. For instance, these damages could be either an accidental release of hazardous-substance into the environment (soil or river) or unanticipated temporary damages to verges and footpaths due to road building processes. In these cases, environmental compensation could consist in replanting plants or restoring fish streams. To determine the optimal compensation scheme, the decision-maker pursues three goals:

• no welfare loss for the whole population impacted by the environmental damage;

• minimization of the cost of the compensation scheme, in line with recommendation of "reasonable cost" of the European directive 2004/35/EC;

• environmental compensation cannot be less than a given quantity defined by an EA criterion.

In doing so, the objective of the present paper is in line with the objective of the European Directive 2004/35/EC, namely "to establish a common framework for the [...] remedying of environmental damage at a reasonable cost to society". The aim of the introduction of an EA criterion, in accordance with the "no net loss" principle, is to ensure that the destruction or degradation of an environmental good is sufficiently offset. Considering an heterogeneous population, we show that the eligible compensation mechanism (which meets the three conditions) varies with the magnitude of the environmental impact, the design of heterogeneity and the number of agents that need compensation. We also show that enforcing a minimal non-monetary compensation not only implies ecological effects but also impacts the equity and cost efficiency issues associated with the compensation. More precisely, when the constraint is binding, an ecological constraint can reduce inequity at the expense of a rise in cost inefficiency.

The article is organized as follows. Section 2 presents the model. Optimal compensation schemes are derived in Section 3 according to two types of population heterogeneity: heterogeneity in preferences for goods and heterogeneity in wealth. The last section concludes and suggests future directions for additional works.

The Model

We consider a two-period economy composed by n heterogeneous agents in which the agent i 's lifetime utility is given by:

U i = u i1 (X i1 , q 1 ) + δu i2 (X i2 , q 2 )
where u it is the agent i 's utility in period t, δ characterizes the time-preference rate, X it measures the agent i's private consumption and q t the level of the environmental good or service measured in physical units at time t. Assuming that agents can lend in a perfect capital market, the intertemporal budget constraint writes

W i = X i1 (1 + r) + X i2 where r
is the interest rate. Then the lifetime indirect utility of agent i can be written:

V i = v i (W i , q 1 , q 2 ) (1)
where W i stands for the agent i's intertemporal income which is exogenously given.

We assume that the natural resource is accidentally damaged in the first period and compensated in the second one according to a compensating rule decided by a policy-maker.

The compensation is twofold: a monetary compensation identical for each agent whatever his type, and an environmental compensation.

Leaving the utility of an individual unchanged following an environmental damage implies:

dV i = ∂v i ∂W i dW i + ∂v i ∂q 1 dq 1 + ∂v i ∂q 2 dq 2 = 0 (2)
where dq 1 < 0 stands for the accidental damage, dq 2 > 0 represents the environmental compensation and dW i is the monetary compensation.

The individual willingness to accept a monetary compensation for the environmental damage is defined as:

4 W T A W i = ∂v i ∂q 1 ∂v i ∂W i (-dq 1 ) (3)
It expresses how much money the individual i is willing to accept in exchange for the loss dq 1 .

Assuming that the environmental good q 1 is normal, the income elasticity of the willingness to pay is positive. 5 As a result, in line with Brekke (1997), a rich agent is inclined to require a higher amount of MC to compensate the environmental damage than a poor agent.

Using the same reasoning, it is possible to express a WTA in terms of environmental units:

W T A q i = ∂v i ∂q 1 ∂v i ∂q 2 (-dq 1 ). (4)
Note that both willingnesses to accept depend positively on the magnitude of the environmental impact.

When determining the compensation pattern, the decision-maker aims to account for three criteria: minimize the costs involved by the implementation of the whole compensation, leave the aggregate welfare unchanged and comply with a minimal environmental compensation requirement.

The program of the decision-maker writes:

min MC,dq 2 C(dq 2 , MC) (5)
4 W T A W i is the value of dWi obtained by equation (2) stating that dq2 = 0. W T A W i is identified with the compensating variation. The absence of environmental damage is the reference state for most people. W T A is the better measure to use (Knetsch, 2007).

5 See Ebert (2003) for an exhaustive analysis on the effect of the distribution of income on the marginal willingness to accept.

subject to dW = 0 (6)

dq 2 ≥ -dq 1 σ (7) MC ≥ 0 (8)
where MC = dW i ∀i is the monetary compensation, dq 2 the environmental compensation, and C is the cost function associated to the compensation. W = n i=1 V i stands for the aggregate welfare of the n victims and constraint ( 6) characterizes the fact that the compensating policy must leave the aggregate welfare unchanged. Combined with (2), this constraint implies a clear trade-off mechanism between both compensations for a given environmental damage. Constraint ( 7) with σ > 1 specifies that the environmental compensation must at least be equal to a given value larger than the initial damage. This value corresponds to the one that would be determined when using Equivalence Approaches (EA) in their simplest formulation, i.e. the "discounted" environmental gain equalizes the "discounted" environmental loss. In this expression, σ is the discount parameter associated to the EA constraint. 6

Note that no ex-post redistribution of monetary compensation between losers and gainers is feasible.

Compensation scheme

The Lagrangian associated to this program is given by

L = C (dq 2 , MC) + λ 1 [dW] + λ 2 [dq 2 + dq 1 σ] + λ 3 [MC]
where λ 1 is the Lagrangian multiplier associated to constraint (6), λ 2 to (7) and λ 3 to (8).

The conditions arising from solving the Lagrangian are:

∂L ∂MC = - ∂C ∂MC + λ 1 ∂dW ∂MC + λ 3 = 0 ∂L ∂dq 2 = - ∂C ∂dq 2 + λ 1 ∂dW ∂dq 2 + λ 2 = 0 6
The determination of the appropriate discount rate is still controversial in the literature. In practice, a 3 percent rate is recommended for equivalency analysis in the US (NOAA, 1999).

∂L ∂λ

1 = dW = 0; ∂L ∂λ 2 = dq 2 + dq 1 σ ≥ 0; ∂L ∂λ 3 = MC ≥ 0
Four regimes can be distinguished from this program, that determine the pattern of the compensation:

• Regime 1: (monetary compensation [R 1 ]) : λ 2 > 0; λ 3 = 0 => dq 2 = -dq 1 σ; MC > 0.
In this case both compensations are implemented but the level of the environmental compensation being the minimal one defined by the EA constraint, we call this case "monetary compensation". Without the EA constraint, the environmental compensation would be between 0 and -dq 1 σ. This case leads to the relation

∂dW ∂MC ∂dW ∂dq 2 > ∂C ∂MC ∂C ∂dq 2 (9)
One unit spent on monetary compensation generates more welfare than one unit spent on environmental compensation. Then the decision-maker should favor monetary compensation in order to compensate at minimal cost.

• Regime 2: (mixed compensation [R 2 ]): λ 2 = λ 3 = 0 => dq 2 > -dq 1 σ; MC > 0.
There exists a couple of compensation (MC * , dq * 2 ) such that:

∂dW ∂MC ∂dW ∂dq 2 = ∂C ∂MC ∂C ∂dq 2 (10)
The ratio of the marginal differences in utility equals the ratio of the marginal costs.

In other words, there exists a couple (MC * , dq * 2 ) such that the gain of welfare from an additional unit of MC or dq 2 per fund spent is the same due to the trade-off mechanism resulting from constraints (2) and ( 6).

• Regime 3: (environmental compensation [R 3 ]): λ 2 = 0; λ 3 > 0 => dq 2 > -dq 1 σ; MC = 0, which implies ∂dW ∂MC ∂dW ∂dq 2 < ∂C ∂MC ∂C ∂dq 2 (11)
This is the opposite case to Regime 1. The decision-maker should promote environmental compensation.

• Regime 4:

(minimal compensation [R 4 ]) λ 2 > 0; λ 3 > 0 => dq 2 = -dq 1 σ; MC = 0.
This regime does not fulfill constraint (6). The EA constraint applies and overcompensates the loss of the social welfare.

Three remarks can be made here concerning the choice between regimes 1, 2 and 3.

First, assuming that the marginal cost of the monetary compensation is equal to the number of victims ∂C ∂MC = n and that the marginal cost of environmental compensation does not depend on n, the frontiers between the three regimes depend on the number of victims (n). It is particularly clear when agents are perfectly homogeneous which imply identical

willingnesses to accept (W T A W i = W T A W ∀i and W T A q i = W T A q ∀i). Then ∂dW/∂MC ∂dW/∂dq 2 = W T A q W T A W and ∂C/∂MC ∂C/∂dq 2 = n ∂C/∂dq 2 .
Obviously, Regime 1 applies for a low number of victims whereas Regime 3 applies for a large one. A higher damage directly increases the EA constraint and consequently shifts the limits of the regimes for higher n. The introduction of a degree of heterogeneity does not change the qualitative results. 7

Second, the choice between regime 1, 2 or 3 crucially depends on the magnitude of the environmental impact (-dq 1 ) since it affects the EA constraint together with the willingnesses to accept (W T A W i and W T A q i ).

Third, when EA constraint no more exists, regimes 1 and 4 disappear and only regimes 2 and 3 remain.

Cost and welfare analysis

If the compensation mechanism leaves the aggregate welfare unchanged, it is not necessarily true for individual ones when agents are heterogeneous. Under each regime, we can determine which agent is inclined to loose or win from the willingnesses to accept together with equations ( 2) and ( 6).

Compensation implies a loss (no change, gain) for the agent whose willingnesses to accept verify:

• For Regime 1: W T A W i > (=, <) MC 1-σ W T A q i
7 For instance, when heterogeneity between agents such that W T A q i = W T A q ∀i and ∂v

i ∂q 2 = ∂v j ∂q 2 ∀i, j is introduced, we have ∂dW/∂MC ∂dW/∂dq 2 = 1 n W T A q W T A W i .
The choice between regimes is still determined by threshold levels of n.

• For Regime 2:

W T A W i > (=, <) MC 1+ 1 W T A q i (-dq 2 )
• For Regime 3: W T A q i > (=, <) dq 2

Let us consider the case where agents have identical W T A q i . In Regime 3, the compensation is fully granted in environmental units and leaves each individual welfare unchanged.

When the compensation includes a uniform monetary component (R 1 and R 2 ), compensation results in losers and winners. If individuals are only differentiated by their income, then a rich agent looses and a poor wins. If they are only differentiated by their preferences for the environmental good, we can intuitively assume that W T A W i increases with the preference for the environmental good. An agent who values more (less) the environmental good looses (wins) from compensation. When W T A q i differs between agents, Regime 3 implies a gain (loss) of individual welfare for agents with a high (low) W T A q i . In regimes 1 and 2, agents characterized by high (low) willingnesses to accept support a loss (gain) of welfare.

Let us now compare the costs associated to the different regimes. We denote by CS * R i with i = 1, 2, 3 the cost associated with the compensation scheme under R 1 , R 2 and R 3 . We also denote by CS 0 the scheme that combines monetary and environmental compensation without EA constraint. Finally, we introduce two other compensation schemes that could be referred as benchmark cases: Full environmental compensation (CS F env ) and Full monetary compensation (CS F mon ). They are characterized as follows:

CS F env : dq 2 > 0 and MC = 0 ∀n CS F mon : MC > 0 and dq 2 = 0 ∀n Note that CS F env is fixed and do not vary with n.

Due to the characteristics of the cost function and the characterization of each compensation scheme, we can clearly deduce the following relationships:

• CS F env > CS * R i ≥ CS 0 for i = 1, 2
and the values of n corresponding to regimes 1 and 2

• CS F env = CS * R 3 = CS 0 < CS F mon
for the values of n corresponding to Regime 3.

• CS F mon < CS * R 1 for sufficiently low values of n in Regime 1.

• CS F mon > CS * R 2 CS F env > CS * R 2   
for the values of n corresponding to Regime 2.

From a cost minimization perspective, we deduce that for low values of n the compensation scheme described by Regime 1 is not the less costly possible option. The EA constraint imposes an additional cost. Without this constraint, there would exist two better options:

Full monetary compensation and both compensations without EA constraint. CS * is the less costly option jointly with CS 0 for the values of n corresponding to Regime 2 and with CS 0 and CS F env for the values of n corresponding to Regime 3.

When regime 4 applies (no monetary compensation and a minimal environmental compensation driven by the EA constraint), the change of the aggregate welfare is positive. In this particular case, the compensation cost is higher than the one which would leave the aggregate welfare unchanged. As a result, the cost associated with this regime (CS EA ) is constant and higher than the cost associated with the other schemes except for the pure monetary compensation with a high number of victims.

Application

We now specify both the cost and the utility functions. We assume a lifetime log linear utility function of the form

U i = α i ln X 1i + (1 -α i ) ln q 1 + δα i ln X 2i + δ(1 -α i ) ln q 2
where α i is the agent i s preference for the private good.8

The arbitrage in private consumption between period 1 and 2 gives the relation between both private consumptions X 2i X 1i = δ i (1 + r) that combined with the intertemporal budget constraint gives the demand for private goods. The indirect utility writes

V i = α i ln W i (1 + δ) (1 + r) + (1 -α i ) ln q 1 + δα i ln δ (1 + δ) W i + δ(1 -α i ) ln q 2 (12)
and the willingnesses to accept given by ( 3) and (4) are:

W T A W i = (1 -α i )W i q 1 α i (1 + δ) (13) W T A q i = 1 δ q 2 q 1 = W T A q ∀i (14)
The willingness to accept a monetary compensation is decreasing with the income and the preference for the public good while the willingness to accept an environmental compensation is identical for each agent whatever the nature of heterogeneity.

Finally, we assume that the cost function for compensation is given by:

C (dq 2 , MC) = nM C + 1 {MC>0} CF MC + a (dq 2 ) b (15)
The cost function is decomposed in three parts: a lump sum part (nM C) which characterizes the monetary compensation granted uniformly to all agents, a fixed cost (CF MC ) associated to the implementation of a monetary compensation and a cost proportional to the ecological restoration which depends on the type of the good that should be restored (b > 0 can be either ≥ 1 or < 1 ).9 The greater a and b, the higher the weight of environmental compensation in the whole cost. The fixed cost component (CF MC ) may characterize the cost of conducting a study relying on the use of monetary valuation methodology. This cost can significantly vary according to the survey mode (mail, telephone, face-to-face surveys).

When the monetary compensation is not chosen, the fixed cost associated to the monetary compensation disappears and the cost function reduces to the cost associated to the ecological restoration. Since the cost function is not continuous in MC = 0, the comparison of costs under each scenario determines the best compensation scheme. It is straightforward that the program is quasiconvex in MC whereas it is quasiconvex in dq 2 for b 1. Due to the form of the cost function and the objective to limit the cost of compensation while maintaining the level of the social welfare, it is intuitive that it is more relevant to implement a monetary compensation for a low level of victims and an environmental compensation for a high level of victims. Indeed, while the marginal cost of monetary compensation is equal to n, the marginal cost of environmental compensation is increasing with dq 2 and does not depend on the number of victims.

Heterogeneity in preference for goods

We assume that agents are only differentiated by their preference for goods, α i . The aggregate welfare function writes:

W = W v 1 (W, q 1 , q 2 ), . . . , v n (W, q 1 , q 2 )
Solving the program described by Equations ( 5) to (8) gives the following values for MC and dq 2 in the different regimes (see Appendix A.):

• Regime 1:

dq 2 = -dq 1 σ and MC = -dq 1 W (1-α) α(1+δ) 1 q 1 -δ q 2 σ • Regime 2: dq 2 = (1-α)nW δ α(1+δ)q 2 ab 1 b-1 and MC = (1-α)W -dq 1 q 1 (1+δ)α -δ(1-α)W q 2 (1+δ)α b b-1 n ab 1 b-1 • Regime 3: dq 2 = -q 2 q 1 1 δ dq 1 and MC = 0
• Regime 4: dq 2 = -dq 1 σ and MC = 0 where α = 1 n α i is the mean preference for the private good.

Given the cost function and the relation between both compensations, we are able to distinguish two different cases according to the value of b: b ≥ 1 or b < 1.

Proposition 1 For b ≥ 1 the optimal compensation scheme is of the following form:

1. When σ < q 2 q 1 1 δ (a) if CF MC ≥ CF , Regime 1 applies for n ≤ n and Regime 3 applies for n ≥ n (b) if CF MC < CF , Regime 1 applies for n ≤ n, Regime 2 applies for n < n < n and Regime 3 applies for n ≥ n

2. When σ ≥ q 2 q 1 1 δ , Regime 4 applies ∀ n with CF = a (-dq 1 ) b q 2 q 1 1 δ b + (b -1) σ b -q 2 q 1 1 δ bσ b-1 n = a(-dq 1 ) b q 2 q 1 1 δ b -σ b -CF MC -dq 1 (1-α)W α(1+δ) 1 q 1 -δ q 2 σ ; n = ab (1+δ)α (1-α)W q 2 (-σdq 1 ) b-1 δ
and n is solution of the equation F (n) = 0 with

F (n) = n (1-α) - dq 1 q 1 W (1+δ)α -n b b-1 a (b -1) (1-α)W δ α(1+δ)q 2 ab b b-1 + CF MC -a q 2 q 1 1 δ (-dq 1 ) b Proof. See Appendix B.
Regime 4 crucially depends on the discount parameter (σ) in the EA constraint. Especially, if we consider that δ = 1 σ then this regime applies as soon as q 2 < q 1 which seems to be consistent in case of a damage in period 1. 10 The three other regimes occur when the discount parameter is relatively high compared to the marginal rate of substitution between the environmental good in period 1 and 2

q 2 q 1 1 δ > σ .
Proposition 1 highlights the role of the fixed cost in the choice of the regime and gives the threshold values of n which determine the switch between one regime to another one.

Figure 1 illustrates Case 1 of Proposition 1. 11 Under Case 1.b., the value of n increases with α, (-dq 1 ), a and b, and decreases with W and δ. An agent who values more the future expects a lower level of compensation so that the switch from Regime 1 to Regime 2 occurs for a lower n. Conversely a lower weight for the environmental good in the utility (high α) implies a lower need for compensation and the limit between both regimes is shifted for a higher n. The interval (n, n) on which Regime 2 applies is the larger for CF MC = 0 and decreases with CF MC . The discontinuity of the levels of dq 2 and MC between regimes 2 and 3 is due to the fixed costs in the cost function. Under Case 1.a., the fixed costs are too high (CF MC > CF ) and Regime 2 never applies since it is always too costly compared to Regime 3. CF is increasing with a and b whereas it is not affected either by ᾱ and W . In addition, Regime 1 is reduced ( n < n) because it is very costly to implement a monetary compensation.

For Regime 2, the impact of the environmental damage (-dq 1 ) on the monetary compensation is obviously positive while the positive effect of (-dq 1 ) on dq 2 is offset by the trade-off effect between MC and dq 2 due to the quasi linearity of the cost function. The impact of wealth on compensation can be clearly explained by equation ( 10) which can be rewritten here as W T A q W T A W = n ab(dq 2 ) b-1 where W T A represents the average willingness to accept. A rise in W diminishes the ratio of the average willingness to accept so that environmental compensation becomes more efficient in terms of costs. It tends to increase the environmental compensation whereas the impact on monetary compensation depends on the willingness to accept effect (W T A W ) relatively to the trade-off effect between both compensations. The willingness to accept effect diminishes with n so that the monetary compensation decreases with an increase of the wealth for a relatively large number of victims. 12 The impact of the mean preference for the private good (α) operates through the same channels. A raise of α increases the environmental compensation and decreases the monetary compensation for a relatively high number of victims. 13

R 3 R 2 R 1 n n (b) CF MC < CF
The effect of the time preference is clear: the more the second period is valued in the utility, the higher is the level of required environmental compensation. The impact of δ on MC is also unambiguously negative through both the willingness to accept effect and the trade-off effect.

Figure 2 stresses the case without any EA constraint. As stipulated in the general case, regimes 1 and 4 disappear and only regimes 2 and 3 remain. Under Regime 2, the compensation scheme leads to an increasing level of dq 2 and a decreasing level of MC. Under both regimes, dq 2 > 0 whatever the value of n. Nevertheless, the level of environmental compensation is low for small values of n. Proposition 2 For b < 1, the optimal compensation scheme is the following 1. If σ < q 2 q 1 1 δ Regime 1 applies for n ≤ n and Regime 3 applies for n ≥ n

2. If σ ≥ q 2 q 1 1 δ Regime 4 applies ∀ n with n = a(-dq 1 ) b-1 q 2 q 1 1 δ b -σ b -CF MC W (1-α) α(1+δ) δ q 2 q 2 q 1 1 δ -σ Proof. See Appendix C.
With b < 1 the limit between regimes 1 and 3 is given by n. As previously explained, a higher (resp. lower) level of n goes in favor of the use of environmental (resp. monetary) compensation. Contrary to the case with b > 1, there is no more optimal mixed compensation and regime 1 switches directly to Regime 3 with the increase in n since only corner solutions enable to minimize the cost. Since condition ( 6) is not fulfilled, the trade-off mechanism does not work anymore and Regime 2 disappears.

Turning to cost and welfare analysis, first recall that for a slightly high discount parameter, the compensation scheme reduces to Regime 4 (no monetary compensation and a minimal environmental compensation driven by the EA constraint whatever the level of n).

The change of the aggregate welfare is positive as well as every individual welfare variation. 14 The agent that values the environmental good the most (lowest α i ) wins the most.

Figure 3 depicts the costs associated with the different compensation schemes (CS 0 , CS F env , CS F mon and with CS * R i with i = 1, 2 and 3) for the case b > 1 and σ < q 2 q 1 1 δ . 15 We can clearly observe the ranking of costs described in the general case. This cost analysis must be put in perspective with the welfare analysis derived from the minimization program. Clearly it results in losers and winners in regimes 1 and 2. Since W T A q i is identical Figure 4: Individual welfare gain/loss for two different levels of α i when σ < q 2 q 1 1 δ

14 dVi = (1 -αi)dq1 -1 q 1 + δ q 2 σ > 0 ∀i under Regime 4. 15 CS0 is decomposed in two parts: • dq2 = (1-α)nW δ α(1+δ)q 2 ab 1 b-1 and MC = (1-α)W -dq 1 q 1 (1+δ)α -δ(1-α)W q 2 (1+δ)α b b-1 n ab 1 b-1 n < n (Regime 2) • MC = 0 and dq2 = -q 2 q 1 1 δ dq1 iff n > n (Regime 3) CSFenv: dq2 = -q 2 q 1 1 δ dq1 and MC = 0 ∀ n CSFmon: MC = W (1-α) α(1+δ) -dq 1 q 1 and dq2 = 0 ∀ n

19

Both cost and welfare analyses highlight that regime 1 is worth in terms of cost compared to a compensation scheme without EA constraint (CS 0 ) but better in terms of equity. As suggested by figures 4.a and 4.b, when the EA constraint applies, it limits the gains for the winners but also the losses for the losers. In the trade-off between efficiency and equity, the EA constraint diminishes the cost efficiency of the compensation but also lowers inequity between agents. In that context, while the primary justification of the EA constraint is based on environmental criteria, it may also be supported for equity purposes. Finally, under Regime 3, every individual welfare loss from the damage is offset by the environmental compensation (W T A q i = W T A q ∀i). From a welfare perspective, a Full environmental compensation is the most appropriate solution since there is no welfare loss at aggregate and individual levels. Nevertheless, Figure 3 shows that for a low n the cost of the Full environmental compensation is definitely higher than the cost associated with other compensation schemes.

When agents highly weight the gains associated to the future environmental good respectively to the gains associated to the present environmental good (high σ), Regime 4 applies.

This case is depicted by Figure 5. 16 Under Regime 4, whatever the level of α i , agents win from compensation (except for α i = 1). In addition, the agents who value more the environmental goods win more, as shown in Figure 6. 17

Heterogeneity in wealth

In this section, we assume that agents are differentiated according to their wealth, W i . The aggregate welfare function writes:

W = W v(W 1 , q 1 , q 2 ), . . . , v(W n , q 1 , q 2 )
Solving the program described by Equations ( 5) to (8) leaves regimes 3 and 4 unchanged while the values for MC and dq 2 in regimes 1 and 2 are:

16 For the numerical simulation the new value of σ is 1.62. q 1 1 δ

17 ∂dV i ∂α i = dq1 -1 q 1 + 1 q 2 σ δ < 0 since σ > q 2 q 1 1 δ
• Regime 1: dq 2 = -dq 1 σ and MC = (-dq 1 ) (1-α) α(1+δ)

W I W 1 q 1 -δ q 2 σ • Regime 2: dq 2 = n(1-α)δ abα(1+δ)q 2 W I W 1 b-1 and MC = (1-α) α(1+δ) W I W -dq 1 q 1 -(1-α) α(1+δ) W I W δ q 2 b b-1 n ab 1 b-1 where 1 n n i=1 W W i = I W ≥ 1
is a measure of the average wealth inequality in the society. An increase in I W implies a greater wealth inequality in the society (I W = 1 means no inequality). 18

18 When considering the special case where dq2 = 0, in analogy with Medin et al. (2001), MC corresponds to the per person 'benefit' when equal marginal utility of the environmental good is assumed. It is defined by MC =

21

Similarly to the heterogeneous preferences study, we distinguish two different cases according to the value of b with respect to 1.

Proposition 3 For b ≥ 1 the optimal compensation scheme is of the following form:

1. When σ < q 2 q 1 1 δ (a) if CF MC ≥ CF , Regime 1 applies for n ≤ n and Regime 3 applies for n ≥ n (b) if CF MC < CF , Regime 1 applies for n ≤ n, Regime 2 applies for n < n < n and Regime 3 applies for n ≥ n

2. When σ ≥ q 2 q 1 1 δ , Regime 4 applies ∀ n with n = a(-dq 1 ) b q 2 q 1 1 δ b -σ b -CF MC -dq 1 (1-α) α(1+δ) W I W 1 q 1 -δ q 2 σ ; n = ab (1+δ)α (1-α) I W W q 2 (-σdq 1 ) b-1 δ
and n is solution of the equation G (n) = 0 with

G (n) = n (1-α) - dq 1 q 1 W I W (1+δ)α -n b b-1 a (b -1) (1-α) W I W δ α(1+δ)q 2 ab b b-1 + CF MC -a q 2 q 1 1 δ (-dq 1 ) b Proof. See Appendix D.
The comments about each regime are quite similar to those for heterogeneous preferences.

Here we concentrate on the distinctions between both cases. The values of MC and dq 2 show that the heterogeneity in wealth introduces expression W /I W instead of W with no heterogeneity. This expression highlights two different elements in the wealth heterogeneity: the value of the average wealth (how rich the society is), and the distribution effect (how unequal the society is).

In Regime 2, the impact of W can be compared to the impact of W in the previous case.

The impact of I W is of opposite sign. Due to the concavity of the indirect utility function in wealth, a more inequal society implies a lower average monetary willingness to accept. Then all the mechanisms that operate with W still remain but go in the opposite side. As already stressed, monetary compensation will be in favor of individuals that value money the most. As shown in Figure 7.a the poorest individuals (W i < W /I W ) are the winners. 19 Under Regime 4, every individual wins from the minimal environmental compensation. In addition, the gain from the environmental compensation is the same for each individual whatever his wealth. Indeed, heterogeneity only impacts the welfare through the monetary compensation which is here null.

Proposition 4 For b < 1, the optimal compensation scheme is the following 1. If σ < q 2 q 1 1 δ Regime 1 applies for n ≤ n and Regime 3 applies for n ≥ n

2. If σ ≥ q 2 q 1 1 δ Regime 4 applies ∀ n with n = a(-dq 1 ) b-1 q 2 q 1 1 δ b -σ b -CF MC W I W (1-α) α(1+δ) δ q 2 q 2 q 1 1 δ -σ
Proof. Similar to Proposition 2 with the comparison of the cost under regimes 1 and 3 that yields:

C 3 < C 1 ⇐⇒ n > W I W a (-dq 1 ) b-1 q 2 q 1 1 δ b -σ -CF MC (1-α) α(1+δ) δ q 2 q 2 q 1 1 δ -σ = n 19
The following parameter set was used for numerical simulation: (W = 400000, IW = 1.5, α = 0.8, δ = 0.67, q1 = 10000, q2 = 10000, dq1 = -200, a = 300, b = 1.75, σ = 1.34).

For b < 1, the level of n which seperates both regimes 1 and 3, i.e. ñ, decreases with I W . Then heterogeneity in wealth goes in favor of an environmental compensation since the borders of this regime are extended.

Concluding remarks

While the European Directive 2004/35/EC precludes the use of monetary compensation in response to an environmental damage, this article reintroduces the monetary compensation as a potential compensating tool complementing an environmental compensation. We explore which satisfactory compensation can be provided at a minimal cost under an ecological constraint (here EA constraint). The results feature that the best way to provide compensation for ecological damage at a minimal cost may be sensitive to several parameters: nature of heterogeneity, number of victims, relative costs of monetary and environmental compensations.

More precisely, we show that when the population affected by the environmental damage is small, the equivalency constraint implies the use of a minimal natural resource quantity that would not be provided without this constraint for cost reason. But this constraint enables to diminish the inequity generated by the environmental damage on the heterogeneous population. Although the main purpose of enforcing an ecological constraint is an environmental one (i.e. "no net loss" principle) it also has welfare and cost implications. In that sense, a key result of our paper is to find the optimal balance between equity and cost efficiency considerations.

However, to go further, some results of our paper may be linked to prevention issues.

For instance, we show that a poor population (low mean income) values more the monetary compensation than a rich and as a consequence, accepts a lower level of money to compensate the damage it supports. This mechanism extends the use of monetary compensation.

Moreover, if this poor population is relatively small, the polluter can consider that the cost of compensation that he should support in case of damage is sufficiently low to not undertake any prevention measure that could avoid any environmental damage.

Moreover, as shown in this paper, the use or not of an ecological constraint crucially modifies the optimal compensation scheme. Without such a constraint, a mixed compensa-tion is desirable for a relatively small population of victims. Finally, as often mentioned in the literature devoted to the Equivalency Analysis, the choice of the value attributed to the discount rate is determinant for the determination of the optimal compensation. According to this value, the compensation can be either the one resulting from the Equivalent Analysis method or a more complex one depending on the number of victims.

Work still remains to be done to get a better understanding of all the implications of providing compensation for an environmental damage. In particular, a better consideration of natural resource dynamics as well as a deeper study of redistributive effects of the tradeoff between money and nature should be considered in a next step. Either time preference issues and discount rates issues would be relevant topics for further research in a dynamic perspective.

with n > n ⇐⇒ CF MC < a (-dq 1 ) b q 2 q 1 1 δ

b -(1 -b) σ b - q 2 q 1 1 δ bσ b-1 = CF Now, let us compare C 2 to C 3 C 2 < C 3 ⇐⇒ n (1 -α) -dq 1 q 1 W (1 + δ) α +n b b-1 a (1 -b) (1 -α)W δ α(1 + δ)q 2 ab b b-1 +CF MC < a q 2 q 1 1 δ (-dq 1 )
b we define

F (n) = n (1 -α) -dq 1 q 1 W (1 + δ) α -n b b-1 a (b -1) (1 -α)W δ α(1 + δ)q 2 ab b b-1 + CF MC -a q 2 q 1 1 δ (-dq 1 ) b F (n) = (1 -α) -dq 1 q 1 W (1 + δ) α -n 1 b-1 ab (1 -α)W δ α(1 + δ)q 2 ab b b-1 < 0 ⇐⇒ n > ab -dq 1 q 1 b-1 q 2 δ b (1+δ)α (1-α)W = n Then F (n) increases on [0, n] F (n) = CF MC > 0 F (n) = a (-dq 1 ) b b q 2 σ b-1 q 1 δ -σ b (b -1) - q 2 q 1 1 δ b + CF MC < 0 ⇐⇒ CF MC < a (-dq 1 ) b σ b (b -1) + q 2 q 1 1 δ b -bσ b-1 q 2 q 1 δ = CF
Then if CF < CF , there exists a n ∈ [n, n] such that F n = 0 (C 2 = C 3 ) and if CF > CF we have C 2 > C 3 ∀ n > n.

C. Proof of Proposition 2

Rewriting the cost function in MC for MC > 0 according to (17) gives

C (dq 2 , MC) = nM C + CF MC + a -dq 1 q 1 -MC α(1 + δ) (1 -α)W q 2 δ b
Which is clearly concave in MC for b < 1. Minimizing the cost leads to set MC = 0 (condition (8)). The value of dq 2 is then derived from (18) which corresponds to Regime 3 if σ < q 2 q 1 δ and to Regime 4 otherwise. 20 20 Condition σ < q 2 q 1 δ ensures dq2 > -dq1σ for dq2 = -dq 1 q 1 q 2 δ Rewriting the cost function in dq 2 for MC > 0 according to (17) gives C (dq 2 , MC (dq 2 )) = nW (1 -α) α(1 + δ)

-dq 1 q 1 -δ q 2 dq 2 + a (dq 2 ) b which is clearly concave in dq 2 for b < 1 so that the only solution which minimizes the cost is again a corner solution. According to condition (7) minimizing the cost requires dq 2 = -dq 1 σ. The value of MC is derived from (17), which corresponds to Regime 1 if σ < q 2 q 1 δ and to Regime 4 otherwise. 21 We now compare Regime 1 and Regime 3.

Under Regime 3, the cost reduces to C 3 (dq 2 , MC) = a -dq 1 q 1 q 2 δ b whereas under Regime 1, the cost reduces to

C 1 (dq 2 , MC) = n (-dq 1 ) W (1 -α) α(1 + δ) 1 q 1 - δ q 2 σ + CF MC + a (-σdq 1 ) b C 3 < C 1 ⇐⇒ n > q 2 α(1 + δ)a (-dq 1 ) b-1 q 2 q 1 1 δ b -σ b -CF MC W (1 -α)δ q 2 q 1 1 δ -σ = n

D. Proof of Proposition 3

Similarly to Proof of Proposition 1, conditions ( 7) and (8) imply:

dq 2 > -dq 1 σ ⇐⇒ n > α (1 -α) (1 + δ) δ q 2 ab I W W (σ (-dq 1 )) b-1 = n MC > 0 ⇐⇒ n < ab (1 + δ) α (1 -α) I W W q 2 δ b -dq 1 q 1 b-1 = n
both conditions can be fulfilled iff

n > n ⇐⇒ q 1 q 2 < 1 δσ
The comparison of costs gives

C 1 < C 3 ⇐⇒ n < a (-dq 1 ) b q 2 q 1 1 δ b -σ b -CF MC -dq 1 (1-α) W I W α(1+δ) 1 q 1 -δ q 2 σ = n
21 Condition σ < q 2 q 1 δ ensures MC > 0 when MC = -dq1W (1-α) α(1+δ) 1 q 1 -δ q 2 σ 29 with n < n ⇐⇒ CF MC > CF and

C 2 < C 3 ⇐⇒ n (1 -α) -dq 1 q 1 W I W (1 + δ) α +n b b-1 a (1 -b) (1 -α)W δ α(1 + δ)q 2 ab b b-1 +CF MC < a q 2 q 1 1 δ (-dq 1 ) b With G (n) = n (1 -α) -dq 1 q 1 W I W (1 + δ) α -n b b-1 a (b -1) (1 -α)W δ α(1 + δ)q 2 ab b b-1 + CF MC -a q 2 q 1 1 δ (-dq 1 ) b G (n) = (1 -α) -dq 1 q 1 W I W (1 + δ) α -n 1 b-1 ab (1 -α)W δ α(1 + δ)q 2 ab b b-1 < 0 ⇐⇒ n > n G (n) = CF MC > 0 and G (n) < 0 ⇐⇒ CF MC < CF

10

  This situation corresponds to the case where the discount rate (here (σ -1)) equals the time preference rate ((1 -δ)/δ).11 The following parameter set was used for numerical simulations: (W = 372000, α = 0.8, δ = 0.67, q1 = 10000, q2 = 10000, dq1 = -200, a = 300, b = 1.75, σ = 1.34)

Figure 1 :

 1 Figure 1: Optimal Compensation Scheme as a function of the population size

Figure 2 :

 2 Figure 2: Compensation scheme without EA constraint (CS 0 )

Figure 3 :

 3 Figure3: Costs associated with the four compensation schemes when σ < q 2 q 1 1 δ

  Figures 4.a

  and 4.b also show that the monetary compensation (CS F mon ) is the worst in terms of equity compared to the other compensation schemes.

Figure 5 :Figure 6 :

 56 Figure5: Costs associated with the alternative compensation schemes when σ > q 2 q 1 1 δ

  ). If equal marginal utility of income is assumed (i.e IW = 1 in our case),

Figure 7 :

 7 Figure 7: Individual welfare gain/loss for two different levels of W i

This option is preferred to "out-of-kind" compensation in which the adverse impacts to one resource (or

Following Leroux (1987), this specification allows the environmental good to be a normal good and the properties of the willingness to accept with respect to the income apply.

On one hand, the marginal cost of providing environmental goods is decreasing for a levee that could be moved back to create tidal marsh (b < 1). It may have significant environmental benefits without substantially raise the cost of the compensation. On the other hand, when lands are being purchased and managed for conservation, the marginal cost of environmental compensation is likely to be increasing (b > 1).

Acknowledgements

We warmly thank two anonymous referees for their comments and helpful suggestions. We also thank Vincent Martinet and Gilles Rotillon for their comments and advices on previous drafts of the paper. We are grateful to the audiences at 14 th BIECON, ISEE conference 2012 and GREQAM seminar for helpful discussions.

Appendix

A. Values of dq 2 and MC for each regime

The aggregate welfare function can be rewritten as

or

Rewriting the cost function in dq 2 according to (17) for MC > 0 gives

which is clearly quasi-convex in dq 2 if and only if b ≥ 1.

Minimizing this cost function gives

and condition (8) gives the value for

we can deduce regime 1: dq 2 = -σdq 1 and MC is derived from (17) regime 2: dq 2 and MC are given by ( 19) and (20) regime 3: MC = 0 and dq 2 is derived from (18) regime 4: MC = 0 and dq 2 = -dq 1 σ

B. Proof of Proposition 1

Under Regime 2, conditions ( 7) and (8) imply

The interval on which Regime 2 may apply is reduced to n ∈ ]n, n[ . Both conditions will be fulfilled iff

q 1 1 δ , which implies n > n, none of condition ( 7) and ( 8) are fulfilled so that both compensations are implemented at their minimal level whatever the level of n, i.e. MC = 0 and dq 2 = -dq 1 σ (Regime 4).

To check which regime (1, 2 or 3) is optimal to implement, we have to compare the costs associated with the different regimes. The optimal regime is the one which implies the lowest cost.

Under regime 1 the cost reduces to

and under Regime 3


Laurent Ferrara

-Clément Marsilli
email: clement.marsilli@banque-france.fr.

-Juan-Pablo Ortega

Keywords: JEL codes -C53, E37 GDP forecasting, financial volatility, MIDAS approach

The Great Recession endured by the main industrialized countries during the period 2008-2009, in the wake of the financial and banking crisis, has pointed out the major role of the financial sector on macroeconomic fluctuations. In this paper, we reconsider macrofinancial linkages by assessing the leading role of the daily volatility of two major financial variables, namely commodity and stock prices, in their ability to anticipate US GDP growth. For this purpose, an extended MIDAS model is proposed that allows the forecasting of the quarterly growth rate using exogenous variables sampled at various higher frequencies. Empirical results

show that using both daily financial volatilities and monthly industrial production is helpful at the time of predicting quarterly GDP growth over the Great Recession period.

Introduction

In the wake of the financial and banking crisis, virtually all industrialized countries experienced a very severe economic recession during the years 2008 and 2009, generally referred to as the Great Recession. This recession has shed light on the necessary re-assessment of the contribution of financial markets to the business cycle. In this respect, there is growing volume of work in the literature that underlines their leading role in the forecasting of macroeconomic fluctuations. For example, [START_REF] Kilian | The economic effects of energy price shocks[END_REF] reviewed the impact of energy prices shocks, especially oil prices, on macroeconomic fluctuations; [START_REF] Hamilton | What is an oil shock?[END_REF] put forward a non-linear Markov-Switching model to predict the US Gross Domestic Product (GDP) growth rate using oil prices. [START_REF] Stock | Forecasting output and inflation: The role of asset prices[END_REF] have proposed a review on the role of asset prices for predicting the GDP while [START_REF] Claessens | How do business and financial cycles interact[END_REF] have empirically assessed interactions between financial and business cycles. [START_REF] Bellégo | Macro-financial linkages and business cycles: A factor-augmented probit approach[END_REF] have also proposed a factor-augmented probit model enabling to summarize financial markets information into few synthetic factors in order to anticipate euro area business cycles.

Nevertheless, there are only very few studies in the literature dealing with the impact of financial volatility on macroeconomic fluctuations. Among the rare existing references, [START_REF] Hamilton | Stock market volatility and the business cycle[END_REF] have shown evidence of relationships between stock market volatility and US industrial production through non-linear Markov-Switching modeling; [START_REF] Ahn | Volatility relationship between stock performance and real output[END_REF] have estimated bivariate VAR models with GARCH errors for both industrial production and stock indices in five industrialized countries. [START_REF] Chauvet | What does financial volatility tell us about macroeconomic fluctuations? Working Paper[END_REF] have recently analyzed the predictive ability of stocks and bonds volatilities over the Great Recession using a monthly aggregated factor. Indeed, they estimate a monthly volatility common factor based on realized volatility measures for stock and bond markets. They show that this volatility factor largely explains macroeconomic variables during the 2007-2009 recession, both in-sample and out-of-sample.

When dealing simultaneously with daily financial variables and quarterly macroeconomic variables, a standard way to proceed is to temporally aggregate the high frequency variable in order to assess dependence at the same frequency between both types of variables. Alternatively, instead of aggregating data, the MIxed DAta Sampling approach (MIDAS) introduced by Ghysels and his coauthors has proved to be useful (see [START_REF] Ghysels | The MIDAS touch : Mixed data sampling regression models[END_REF] and [START_REF] Ghysels | MIDAS regressions: Further results and new directions[END_REF]). In the forecasting framework, several empirical papers have shown the pertinence of financial information at the time of predicting macroeconomic fluctuations using a MIDAS-based approach, mainly in the US economy (see for example [START_REF] Clements | Macroeconomic forecasting with mixed-frequency data[END_REF]) but also as regards the euro area (see [START_REF] Marcellino | Factor MIDAS for nowcasting and forecasting with ragged-edge data: A model comparison for German GDP[END_REF] for Germany or [START_REF] Ferrara | Financial variables as leading indicators of GDP growth: Evidence from a MIDAS approach during the Great Recession[END_REF] for euro area countries). Indeed in the presence of various sampling frequencies, the MIDAS approach avoids the need for data temporal aggregation with the associated loss of information.

In this paper, our objective is twofold. First, we propose an extended MIDAS model enabling to mix daily and monthly explanatory variables in order to predict quarterly US GDP growth. We focus on two well-known daily financial ingredients, namely, commodity and stock prices, combined with a monthly industrial production index. Then, we assess through this extended MIDAS model the gain in forecasting, for various forecasting horizons, stemming from financial volatility variables estimated on a daily basis. We empirically prove that this approach would have increased the predictive accuracy during a period that includes the last Great Recession.

The econometric model

In this paper, we aim at assessing the predictive content of the daily volatility of financial variables regarding the broadest output growth measure (GDP) by avoiding the temporal aggregation step. In addition, it is well known that hard data, generally sampled at monthly frequency, convey additional information to anticipate GDP. To achieve this goal we propose an extended MIDAS model that allows us to explain a quarterly sampled variable using both monthly and daily sampled variables.

Let (Y Q t ) be a quarterly sampled stationary variable that we aim at predicting, (X M t ) is a vector of N M stationary monthly variables, and (X D t ) is a vector of N D stationary daily variables. We propose the following extended MIDAS model enabling the mixing of daily and monthly information:

Y Q t = α + N D i=1 β i midas K D (θ i ) X D i,t + N M j=1 γ j midas K M (ω j ) X M j,t + φ Y Q t-1 + ε t , (1)
where (ε t ) is a white noise process with constant variance and α, β, θ, γ, ω are the regression parameters to be estimated. We also include a first order autoregressive term in the expression (1)

as it has been showed that it generally improves forecasting accuracy based on leading indicators (see for example [START_REF] Stock | Forecasting output and inflation: The role of asset prices[END_REF]). [START_REF] Andreou | Should macroeconomic forecasters use daily financial data and how?[END_REF] has proved that this extended MIDAS mode that integrates different frequencies in the right hand side of the equation is useful in the context of macroeconomic forecasting.

The midas K (•) function in equation ( 1) prescribes the polynomial weights that allow the frequency mixing. The main idea behind the MIDAS specification consists of smoothing the past values of each covariate {X κ i,t } by using polynomials midas K (b) of the form

midas K (b) := K k=1 f ( k K , b) K k=1 f ( k K , b) L (k-1)/κ , ( 2
)
where K is the cardinality of the data set window on which the regression is based and κ is the number of realizations of (X κ i,t ) during the period [t -1, t]; for example, in equation ( 1), κ = M ≡ 3 for X M j,t . Also L is the lag operator such that L s/κ X κ t = X κ t-(s/κ) , and f (•) is the weight function that can be chosen out of various parametric families. As in [START_REF] Ghysels | MIDAS regressions: Further results and new directions[END_REF]

, f (•) is given in our case by f (z, b) = b (1 -z) b-1 .
(3)

The window size K is in general chosen by the user but the parameter θ is part of the estimation problem. By construction, equation ( 2) is only influenced by the last K sample values. Other parametrizations of the weight function have been used in the literature, but we have chosen the particular specification for f (•) given by equation (3) because it constitutes a parsimonious solution for which the weights are automatically positive and show a slow decrease over time.

As one of the main objectives of our work consists of providing evidence of the macroeconomic predictive content of financial volatilities, a crucial issue is the estimation of volatility. Given that volatility is not directly observable, several methods have been developed in the literature to estimate it. The most straightforward approach to this problem relies in the use of the absolute value of the returns as a proxy for volatility; unfortunately, the results obtained this way are generally very noisy (see [START_REF] Andersen | Answering the skeptics: Yes, standard volatility models do provide accurate forecasts[END_REF]). This difficulty can be partially fixed by using an average of this noisy proxy over a given period; this method yields one of the most widely used notion of volatility, namely the realized volatility (as used, for example, in [START_REF] Chauvet | What does financial volatility tell us about macroeconomic fluctuations? Working Paper[END_REF]). For example, for a given quarter t, the realized volatility RV t can be estimated by

RV t = nt s=1 r D 2 s 1/2 , (4)
where (r D s ) are the daily returns and n t is the number of days for the quarter t.

However, when using daily financial volatility in the prediction of quarterly economic growth, the most convenient approach appears to be the volatility filtered out of a GARCH-type parametric family [START_REF] Engle | Autoregressive Conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[END_REF][START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF]). The AR(p)-GARCH(1,1) specification is given by

         r D t = ψ 0 + ψ 1 r D t-1 + • • • + ψ p r D t-p + w t , w t = v D t η t , (v D t ) 2 = c + a (v D t-1 ) 2 + b w 2 t-1 , (5)
where ψ 0 is a constant, where ψ = (ψ 1 , . . . , ψ p ) is a p-vector of autoregressive coefficients and where {η t } ∼ WN(0, 1). In order to ensure the existence of a unique stationary solution and the positivity of the volatility, we assume that a > 0, b ≥ 0 and a + b < 1. Estimated daily volatilities (v D t ) stemming from equation ( 5) will be considered as explanatory variables of the macroeconomic fluctuations using the MIDAS regression equation (1), with

X D i,t = vD i,t .
Finally, when using general regression models for forecasting purposes at a given horizon h > 0, forecasters can either predict covariates or implement direct multi-step forecasting (see for example [START_REF] Chevillon | Direct multi-step estimation and forecasting[END_REF] for a review on this point). The idea behind direct multi-step forecasting is that the potential impact of specification errors on the one-step-ahead model can be reduced by using the same horizon both for estimation and for forecasting at the expense of estimating a specific model for each forecasting horizon. In our work we adopt the direct multi-step forecasting approach and assume that the predictor Y Q t+h|t of the GDP quarterly growth rate, for any forecasting horizon h, is given by

Y Q t+h|t = α(h) + N D i=1 β(h) i midas K D ( θ(h) i ) vD i,t + N M j=1 γj midas K M (ω (h) j ) X M j,t + φ(h) Y Q t , (6)
where h) ) are the non-linear least squares estimators (this yields 2N D + 2N M + 2 parameters that need to be estimated).

(α (h) , β(h) 1 , . . . , β(h) N D , θ(h) 1 , . . . , θ(h) N D , γ(h) 1 , . . . , γ(h) N M , ω(h) 1 , . . . , ω(h) N M , φ (

Empirical results

In this section we focus on the US GDP growth prediction. We implement the model previously introduced in equation ( 6) in order to assess the forecasting ability of the volatility of two financial variables, namely commodity and stock prices (i.e. N D = 2) in comparison with the monthly industrial production (i.e. N M = 1).

The variable that we want to predict is the quarterly growth rate of the real US GDP (expressed in percentage), denoted (GDP t ), as released by the BEA in May 2012. We consider as explanatory daily variables the CRB index of commodity prices and the S&P500 as an index of stock prices.

For both daily returns of financial variables, we estimate their volatility over the period January 1967-December 2010 by using the AR-GARCH specification given by the equation ( 5). Given the low autocorrelation exhibited by these time series at lags higher than 2, a first order AR model is sufficient to whiten residuals. Since we are using a standard maximum likelihood estimator for the GARCH process and not a robust one we have smoothed out the Black Monday outlier in the S&P500 series occuring the 19th October 1987 (see for example, [START_REF] Charles | Outliers and GARCH models in financial data[END_REF], or [START_REF] Carnero | Estimating GARCH volatility in the presence of outliers[END_REF]). Estimates of daily volatility for both variables, denoted (v D t,CRB ) and (v D t,SP ), are presented in Figure 1. It is worth noting that as it usually happens with financial time series, periods of high volatility are clustered in time; nevertheless, the high volatility clusters do not occur at the same time for both time series. The volatility of stock prices presents a huge peak during the recent financial crisis, as well as several smaller peaks related to specific events (first oil shock in 1974-75, Asian crisis, burst of the internet bubble, etc.). The commodity volatility exhibits two main peaks; the first one corresponds to the first oil shock while the second one is also related to the recent financial crisis. Some specific events also drive commodity volatility dynamics such as the second oil shock in the early 1980s. The information conveyed by both volatilities does not seem redundant in spite of a recent increase in their correlation (see e.g. [START_REF] Creti | On the links between stock and commodity markets' volatility[END_REF]) and both variables are potentially useful in explaining GDP growth. 6), we use the Industrial Production Index (IPI), well known by practitioners to reflect macroeconomic evolutions as a whole, with a high degree of correlation. We consider the monthly IPI growth rate, denoted (IP I t ). The time series dataset used in this empirical study is described in Table 2.

In our study we carry out an in-sample analysis over the period 1967q1-2006q4 and then we implement an out-of-sample experience over the period 2007q1 -2010q4 that includes the Great Recession.

Though we do not carry out a true real-time analysis, we take into account the specific lag structure in data publication. Indeed, while financial variables are available at the end of each month, we assume that IPI figures are available with a lag of one month and that GDP figures of the previous quarter are only available four months before the target date. Concerning the forecasting experiment, given that financial data are always available the last working day of any given month, we suppose that forecasts for a specific quarter are computed at the end of each month, for 12 horizons that range from h = 0 (nowcasts computed at the end of the last month of the reference quarter) to h = 11/3 (forecasts computed 11 months before the end of the reference quarter). For any time t either in the in-sample or in the out-of-sample periods, the MIDAS regression optimally takes advantage of the fluctuations of the last K M = 12 for the monthly series (IPI) t and K D = 200 for financial covariates (v D t,CRB ) and (v D t,SP ). As we have chosen a direct multi-step forecasting approach, model parameters are estimated separately for each prediction horizon h, as in equation ( 6).

In a first step, we assess the specific impact of both financial volatilities on the GDP growth process through a standard MIDAS model that relates daily variables with a quarterly variable. Thus, the first model that we estimate, denoted Model M d , contains as regressors the daily volatilities of both financial series, namely vD t,CRB and vD t,SP :

GDP t+h|t = α(h) + β(h) 1 midas K D ( θ(h) 1 ) vD t,CRB + β(h) 2 midas K D ( θ(h) 2 ) vD t,SP + φ(h) GDP t . (M d )
Parameter estimates are presented in Table 3. It turns out that the slope of the commodity volatility tends to be remarkably stable as h varies, being always negative. This means that, as expected, an increase in financial volatility has a negative impact on economic growth. The slope of the stock price volatility is higher in absolute value, but goes from positive to negative values for h ≤ 2.

This reflects the negative short-term influence of stock volatility on output. Also, the autoregressive coefficient clearly increases for h ≤ 4/3. In fact, this corresponds to the moment when the GDP realization of the previous quarter is included in the regression model.

The second model, denoted Model M m , contains only as regressors the monthly growth rate of the IPI:

GDP t+h|t = α(h) + γ(h) midas K M (ω (h) ) IPI t + φ(h) GDP t . (M m )
Explaining GDP growth using industrial production is standard in the empirical literature on shortterm macroeconomic forecasting, especially when using bridge equations (see for example [START_REF] Diron | Short-term forecasts of euro area real GDP growth: an assessment of real-time performance based on vintage data[END_REF], or [START_REF] Barhoumi | Monthly gdp forecasting using bridge models: Application for the french economy[END_REF]). However, the monthly IPI series is generally aggregated before using it in quarterly equations. Here, by using a standard MIDAS equation, we allow for different weights concerning the contribution of monthly IPI to GDP growth, adding thus more flexibility to the model. Parameter estimates in Table 3 reveal that the industrial production strongly contributes to anticipate GDP growth, especially in the short-run, γIP I being greater than one for h = 0 and h = 1/3. At the same time, the contribution of the autoregressive part is less important. when forecasting GDP, for all horizons. On this latter point we are in agreement with much of the literature on short-term macroeconomic forecasting and nowcasting that points out the usefulness of either combining information (for example through dynamic factor models, see e.g. [START_REF] Giannone | Nowcasting: The real-time informational content of macroeconomic data[END_REF]) or combining forecasts (see e.g. [START_REF] Timmermann | Forecast combinations[END_REF]). In fact, the gain of using the Model M dm in forecasting becomes important as soon as h ≤ 1. We also note that between h = 4/3 and h = 7/3, the contribution of financial volatilities to the forecasting accuracy is remarkable, as RMSFE(h) stemming from Model M dm and Model M d are similar. This result is interesting for practitioners in the sense that using industrial production to predict GDP with a lead of four to seven months does not appear useful; only financial volatilities help in this range of horizons. When we are close to the target date, that is during the quarter before the release (i.e. h ≤ 1), the IPI tends to increase its impact on the forecast. This stylized fact has been also observed in empirical papers pointing out the increasing role of hard variables on macroeconomic forecasts when we are close to the release date, while financial variables have a stronger impact for longer horizons (we refer for example to [START_REF] Angelini | Short-term forecasts of euro area GDP growth[END_REF]). Our study shows that the information contained on the industrial output series can not replace the one associated to financial volatility; both sources of information are playing an important role, but at different forecast horizons.

Conclusion

In this paper, we assess the predictive content of financial volatility on the US economic growth during the 2007-2010 period, that includes the Great Recession. In this respect, we implement an extended MIDAS model that integrates explanatory variables at both daily and monthly frequencies to predict the quarterly GDP growth. We show that adding daily financial volatility of stock and commodity prices increases the forecasting accuracy in comparison with a benchmark model that includes only industrial production as an explanatory variable. Moreover, our results indicate that an extended MIDAS model that includes both financial volatilities and industrial production is the most adequate approach for all forecasting horizons. Additionally, we note that between 4 and 7 months before the target date, there is no predictive gain to include industrial production, the information conveyed by financial volatilities predominates.

A Description of variables involved in the MIDAS regressions

Figure 1 :

 1 Figure 1: Volatilities estimated by GARCH from 1967q1 to 2010q4, CRB on the top and SP500 on the bottom

Figure 2 :

 2 Figure 2: RMSFE for the three MIDAS models over the period 2007q1-2010q4

Table 2 :

 2 Description of indicators and covariates B Estimated parameters

		Forecasting horizons h
		0	1/3	2/3	1	4/3	5/3	2	7/3	8/3	3	10/3	11/3
	Model M d
	θSP	2.13	3.09	5.22	8.40 12.51 46.77 310.03 721.14	1.00	1.65	1.94	3.09
	θCRB	12.64 33.31 212.47	2.66	3.51	8.43	38.53	79.84	2.26	3.16	4.80	7.21
	βSP	-0.15	-0.15	-0.16	-0.18	-0.15	-0.12	-0.11	-0.01	0.17	0.18	0.17	0.14
	βCRB	-0.09	-0.10	-0.11	-0.08	-0.08	-0.11	-0.11	-0.12 -0.12 -0.12 -0.12	-0.13
	α	0.89	0.89	0.96	0.94	1.01	1.04	1.02	0.94	0.76	0.75	0.76	0.92
	φ	0.21	0.21	0.15	0.14	0.12	0.02	0.01	0.03	0.06	0.06	0.05	-0.09
	Model Mm
	ωIPI	3.34	5.34	13.53 22.61 34.82 13.24	1.00	1.00	1.00	1.00	1.00	3.92
	γIPI	1.40	1.04	0.77	0.45	0.15	0.14	-0.12	-0.15 -0.27 -0.23 -0.20	0.00
	α	0.46	0.38	0.56	0.57	0.61	0.70	0.72	0.72	0.71	0.71	0.72	0.82
	φ	-0.13	-0.05	0.03	0.10	0.16	0.03	0.09	0.09	0.14	0.12	0.11	-0.08
	Model M dm
	θSP	6.90 16.80	18.93 21.12 56.81 15.87	92.29	1.00	1.02	1.27	1.27	2.20
	θCRB	22.06 52.18	71.12 95.18	3.16	8.22	38.99	62.11	2.07	3.01	4.53	7.33
	ωIPI	4.32	8.58	13.58 21.69	1.00 16.65	13.76	1.00	1.02	1.10	1.00 21.52
	βSP	-0.04	-0.04	-0.02	-0.12	-0.14	-0.05	-0.11	0.09	0.09	0.09	0.09	0.08
	βCRB	-0.06	-0.09	-0.07	-0.05	-0.10	-0.12	-0.11	-0.13 -0.12 -0.12 -0.12	-0.13
	γIPI	1.30	1.16	0.74	0.43	-0.12	0.09	0.05	-0.16 -0.31 -0.25 -0.21	-0.06
	α	0.70	0.80	0.70	0.79	0.94	0.97	1.01	0.86	0.85	0.83	0.84	0.97
	φ	-0.13	-0.10	0.03	0.07	0.17	0.01	0.00	0.08	0.12	0.11	0.09	-0.09

Table 3 :

 3 Estimated parameters of the three MIDAS models M d , M m and M dm , over the period1967-2006.

The third model, denoted Model M dm , contains as regressors both daily volatilities and the monthly IPI:

Parameter estimates of this Model M dm in Table 3 are consistent with those of the previous two models, which can be seen as a proof of robustness. Indeed, slope parameters present the same sign as in the previous two models. We only note that the impact of the stock prices volatility has been reduced in the short run.

To assess the forecasting accuracy of each model, we compute the usual root mean square forecasting As expected, RMSFE(h) for all models decrease when h tends to zero, reflecting the use of an information set of increasing size. Indeed, RMSFE(h) are more than halved when h goes from 8/3 to zero. Especially, when 2/3 ≤ h ≤ 4/3, we observe a strong negative slope, visible on all models. This is due to the integration of the newly available GDP growth figure for the previous quarter.

From Figure 2, we note that the Model M dm , based on daily financial volatilities and monthly IPI, provides the best outcomes for all horizons h. This result underlines the fact that combining the information coming from both macroeconomic and financial sources appears to be a good strategy