Discounting the distant future: How much does model selection affect the certainty equivalent rate?

Discounting the distant future: How much does model selection affect the certainty equivalent rate? Ben Groom∗Phoebe Koundouri†Ekaterini Panopoulou‡ Theologos Pantelidis§ December 20, 2004 Abstract Evaluating investments with long-term consequences using discount rates that decline with the time horizon, (Declining Discount Rates or DDRs) means that future welfare changes are of greater consequence in present value terms. Recent work in this area has turned towards operationalising the theory and establishing a schedule of DDRs for use in cost benefitanalysis. Using US data we make the following points concerning this transition: i) model selection has important implications for operationalising a theory of DDRs that depends upon uncertainty; ii) misspecification testing naturally leads to employing models that account for changes in the interest rate generating mechanism. Lastly, we provide an analysis of the policy implications of DDRsinthecontextofclimatechangefortheUS and show that the use of a state space model can increase valuations by 150% compared to conventional constant discounting. JEL classification: C13, C53, Q2, Q4 Keywords: long-run discounting, interest rate forecasting, state-space models, regime-switching models, climate change policy. ∗Department of Economics, University College London. †Department of Economics, University of Reading, UK and Department of Economics, University College London, UK. ‡Department of Banking and Financial Management, University of Piraeus, Greece and Department of Economics, National University of Ireland Maynooth. Correspondence to: Ekaterini Panopoulou, Department of Economics, National University of Ireland Maynooth, Co.Kildare, Republic of Ireland. E-mail: [email protected]. Tel: 00353 1 7083793. Fax: 00353 1 7083934. §Department of Banking and Financial Management, University of Piraeus, Greece. Acknowledgements: We are grateful to Christian Gollier, Cameron Hepburn, Dimitrios Malliaropulos, David Pearce, Nikitas Pittis, participants in the 2003 Royal Economic Society Conference, the 2004 Applied Environmental Economics Conference Royal Society, the First Hispanic Portuguese Congress of Environmental and Natural Resource Economics, the 13th Annual EAERE Conference, and seminar participants at the University College London and Reading University for helpful comments and suggestions. Panopoulou and Pantelidis thank the EU for financial support under the “PYTHAGORAS: Funding of research groups in the University of Piraeus” through the Greek Ministry of National Education and Religious Affairs. 1 1 Introduction Thedramaticeffects of conventional exponential discounting on present values of costs and benefits that accrue in the distant future along with the issues of intergenerational equity that arise are well documented (see e.g. Portney and Weyant 1999, Pearce et al. 2003). The emergence of a long-term policy arena containing issues as diverse as climate change, nuclear build and decommission, biodiversity conservation, groundwater pollution, and the use of social Cost Benefit Analysis (CBA) to guide decision-makers in this arena has brought the discussion of long-run discounting to the fore. Discount rates that decline with the time horizon (Declining Discount Rates or DDRs) have often been touted as an appropriate resolution to what Pigou (1932) described as the ‘defective telescopic faculty’ of conventional discounting, and there has been much discussion about the moral and theoretical justification for such a strategy (see e.g. Dybvig et al. 1996, Sozou 1998, Weitzman 1998, 2001, Portney and Weyant 1999, Gollier 2002a). Of particular interest are the declining yet socially efficient discount rates resulting from the analysis of Weitzman (1998, 2004) and Gollier (2002a, 2002b, 2004) both of which appear to offer a theoretical path through the ‘dark jungles of the second best’ (Baumol 1968) and the intergenerational equity-efficiency trade-offcontained therein. If these theoretical solutions offer even a partial resolution of the problems of conventional discounting then it is clearly important that they can be operationalised and a schedule of DDRs can be determined. In the case of Gollier (2002a) and Weitzman (1998) it is uncertainty that drives DDRs, with regard to future growth of consumption and the discount rate respectively, thus the question of implementation is one of characterising the uncertainty of these primals in some coherent way. However, of these two approaches it is Weitzman (1998) that has proven to be more amenable to implementation mainly because the informational requirements stop at the characterisation of uncertainty, and do not extend to specific attributes of future generations’ risk preferences as would be unavoidable in the case of Gollier (2002a, 2002b).1 1Weitzman (1998) assumes risk neutral agents for exposition, but this represents a special case of his general point. For realistic scenarios, determination of DDRs a la Gollier (2002a, 2002b) requires knowledge of the 4th and 5th derivatives of utility functions, something that he admits is very far from 2 Weitzman’s Certainty Equivalent Discount Rate (CER) is derived from the expected discount factor and is therefore a summary statistic of the distribution of the discount rate. The level and behaviour over time of this statistic is clearly dependent upon the manner in which uncertainty is characterised and the two applications that exist have taken different approaches stemming from different interpretations of uncertainty. Weitzman (2001) defines uncertainty by the current lack of consensus on the appropriate discount rate for the very long term. His survey of professional economists results in a Gamma probability distribution for the discount rate which leads to the so-called ‘Gamma discounting’ approach, a version of which can also be seen in Sozou (1998). Apart from uncertainty his model has persistence in-built, the assumption being that each individual discounts the future at their preferred constant rate, that is each of the responses that make up the probability distribution remain constant over time. More recently, Newell and Pizer (2003) (N&P, henceforth) suggest that while we are relatively certain about the current level of discount rates, there is considerable uncertainty in future. From this standpoint they assume that the past is informative about the future and characterise interest rate uncertainty by the parameter uncertainty typically found in any econometric model. They choose to describe the behaviour of the US longterm real interest rate with a reduced-form model. Their model is the direct analogue of the Vasicek (1977) model for the term structure of interest rates in the sense that only the conditional mean equation is specified and the conditional variance is held constant. In this respect, the authors get a working definition of the CER based upon an econometric model and estimation of the CER schedule comes from a forecasting simulation. Weitzman (2004) goes one step further and builds a “statistical optimal growth model” by combining a neoclassical economic model of optimal growth under uncertainty with a fully integrated Bayesian statistical model of estimating, updating and predicting the outcome of this uncertainty. His model is able to produce persistent uncertainty in the interest rate and as a result DDRs stemming mainly from the uncertainty over future technological progress. From a different point of view, mainly driven by the existing finance literature being accomplished. 3 on the term structure of interest rates, Gollier (2004) reaches similar conclusions. He, specifically, finds that a positively correlated growth process leads to a decreasing yield curve in the case of a prudent representative agent due to increased uncertainty for the distant future. He also links his model with second order stochastic correlation and as a result to the Cox, Ingersoll and Ross model (1985) (CIR, henceforth) of the finance literature, introducing the analogue of heteroscedasticity in his process for the interest rate. In two simulation experiments, one including discrete jumps in the growth of consumption and the other parameter uncertainty, he provides evidence of DDRs and suggests that the discount rate should be as low as 1% for periods exceeding 400 years. The aforementioned studies bring to light some interesting issues concerning the characterisation of the future path of interest rates. It is mainly persistence combined with uncertainty that leads to decline in discount rates over time. In the theoretical studies of Gollier and Weitzman, persistence is generated by the economy itself, while in N&P, the existence of persistence is an empirical question and it is the degree of persistence in the series that determines the rate of decline of the CER. In particular, N&P specify a simple AR(p) model of interest rate uncertainty, which limits the characterisation of uncertainty to a process in which the distribution of the permanent and temporary stochastic components is constant for all time. Such a process guarantees declining CERs, but it takes into account only the evolution of the mean of the process. As already mentioned their model is a discrete time version of the Vasicek (1977) continuous-time model in which the drift of the process is linear and mean-reverting, while the diffusion function is held constant. Since the seminal contribution of Vasicek (1977), an immense literature on the term structure of interest rates has produced interesting insights as to what drives efficient discount rates. The basic extensions mainly come from the specification of the variance of the process, namely the diffusion function. For example, CIR model the diffusion function as a linear function of the level of the interest rate, while Chan et al. (1992) allow the diffusion function to be any power function of the level of the interest rate. However, the aforementioned one-factor models display time-homogeneity, i.e. their parameters remain constant over time. It is reasonable to expect that the instantaneous return and volatil4 ity slowly evolve over time. In this respect, various efforts have been made to produce time-dependent models, such those of Ho and Lee (1986), Black et al. (1990), Hull and White (1990) and Black and Karasinski (1991). These models specify both the drift and the diffusion process of the instantaneous stochastic rate via time-varying functions of the level of interest rates. The empirical issues stemming from the environmental literature on declining discount rates along with the development of an econometric model, versatile enough to reproduce the empirical regularities typically encountered in interest rate data are the main concern of this paper and we build upon the following points. Firstly, it is clear that if we believe that the past is informative about the future, it is important to characterise the past as accurately as possible. Indeed, the selection of the econometric model is of considerable moment in operationalising a theory of DDRs that depends upon uncertainty and defines the CER in statistical terms. Each specification differs in the assumptions made concerning the time series process, hence the forecasts of the interest rate and the attributes of the resulting schedule of the CER will differ accordingly. Secondly, the prescription of CBA will differ markedly depending upon the empirical schedule of discount rates employed, particularly for projects with a long time horizon such as climate change prevention. Moreover, model selection is also an empirical question. Typical misspecification testing and comparisons among various econometric models based upon their out-of-sample forecasting performance should guide model selection for the practitioner. We revisit these issues for US interest rate data and show that misspecification testing generates a natural progression away from the simple AR(p) specification towards models which account for second-order dependence and explicitly consider changes in the time series process over time. We employ, for comparison purposes, the same data set of the US interest rates with N&P and show the policy implications of interest rate uncertainty and model selection in the value of carbon damages or sequestration. The paper is organised as follows. In Section 2, we introduce the theory of the CER offered by Weitzman (1998), our methodology for model selection and the econometric models employed to replicate the stochastic nature of US interest rates. The results of 5 the estimation and the simulations are presented in Section 3. Section 4 draws policy implications for model selection in the case of the value of carbon mitigation and Section 5 concludes the paper. 2 From Theory to Practice 2.1 The Certainty Equivalent Discount Factor and Rate Discounting future consequences in period tback to the present is typically calculated using the discount factor Pt,where Pt=exp(− t P i=1 ri).When ris stochastic, the expected discounted value of a dollar delivered after tyears is: E(Pt)=EÃexp(− t X i=1 ri)!(1) Following Weitzman (1998) we define (1) as the certainty equivalent discount factor,and the corresponding certainty-equivalent forward rate for discounting between adjacent periods at time tas equal to the rate of change of the expected discount factor: E(Pt) E(Pt+1)−1=ert(2) where ertis the forward rate from period tto period t+1at time tin the future, or the marginal discount rate. Gollier (2002a) shows that the certainty equivalent rate is the socially efficient discount rate in a risk neutral world −risk neutral agents are only concerned with the expected value of the discount factor rather than higher order moments −by showing that an arbitrage exists if this is not the case.2In effect this represents the economic theory underlying Weitzman’s definition, however the behaviour of ˜rtover time is dependent upon the nature of the uncertainty surrounding the discount rate. Weitzman (1998) and N&P show that ertas defined in (2) is a declining function of time provided that there is sufficient persistence in the series over time.3This makes it clear 2Strictly, Gollier deals with the average certainty equivalent rate, however the same arguments hold as t→∞. His proof follows Dybvig et al. (1996). 3Weitzman (1998) gives a proof for a general but time invariant distribution function of hrt.Weitzman (2001) estimates this distribution empirically as a Gamma distribution. Pearce et al. (2003) provide a 6 that operationalising this theory is an empirical question, requiring the determination of thestochasticnatureof ert. 2.2 Parameterisation of Real Interest Rates N&P employed a simulation method to forecast discount rates in the distant future, which was properly designed to account for uncertainty in the future path of interest rates and was mainly based on the estimation results of two econometric models, namely an autoregressive Mean-Reverting (MR) model and a Random Walk (RW) model. They estimated the following AR(p)model for rt: rt=η+et(3) et= p X i=1 aiet−i+ξt where ξt∼N(0,σ2 ξ),η ∼N¡η,σ2 η¢and p P i=1 ai<1for the MR model, while p P i=1 ai=1for the RW model. The authors prove that in the case of an AR(1) model, the CER takes the following form: ert=η−tσ2 η−σ2 ξf(ρ, t)(4) where ηis the unconditional mean discount rate, ρis the autoregressive coefficient, f(ρ, t)=1−ρ2−2log(ρ)ρt+1(1+ρ−ρt+1) 2(1−ρ)3(1+ρ)for MR and f(ρ, t)= 1 12(1 + 6t+6t2)for RW. It is straightforward to see that (4) is a declining function of t(See N&P for details). This model, although simple, is successful in capturing the basic features of the underlying Data Generation Process (DGP) which lead to DDRs, namely persistence and uncertainty. However, given the abundance of models already designed to capture the dynamics of the interest rate data either in discrete or continuous time, it is hard to believe that simply modelling the mean of such a process is an adequate parameterisation of reality. As early as 1985, CIR introduce second-order dependence in the stochastic process of the interest rate by letting the conditional variance vary with the level of the numerical example of the decline of the certainty equivalent discount rate for a uniform distribution. 7 interest rate.4The simpler discretised diffusion model motivated by the CIR model is the GARCH (1,1) model, in which the conditional variance depends on its own lag as well as the lag of squared innovations. However, when fitting a GARCH model to interest rates, one often finds that the parameter estimates imply that the conditional variance process is either integrated or explosive. Engle et al. (1987, 1990), Hong (1988), Harvey (1993) and Kees et al. (1997) document such a behaviour mainly for the US short term interest rates. In such cases, proper statistical testing usually cannot reject the hypothesis that the conditional variance of the process follows an integrated GARCH process (IGARCH). In our study, we employ the AR(p)-GARCH(l, m)model to account for both mean and volatility effects in the US interest rate process. Specifically our model is as follows: rt=η+et et= p X i=1 aiet−i+ξt ξt=h1/2 tzt(5) ht=c+ m X i=1 βiξ2 t−i+ l X i=1 γiht−i where htis the conditional volatility of ξt(given all available information at time t−1)and zt∼IIDN(0,1). In the case that m P i=1 βi+ l P i=1 γi=1,we have an AR(p)-IGARCH(l, m) model. Both the AR(p)and AR(p)-GARCH(l, m)models assume that the parameters driving the stochastic process are constant over the sample period, i.e. they are timehomogenous. This is likely to be an unrealistic assumption for a period of 200 years and certainly for forecasting the CER over the long-term policy horizon in hand which, following N&P, extends for 400 years. It is well known that the behaviour of interest rates is strongly affected by the economic cycles as well as shocks destabilising them, i.e. periods of economic crisis. For example, in the US, during the period 1979 through 1982, the Federal Reserve Bank (FED) stopped its usual practice of targeting interest rates and decided to use non-borrowed reserves as a target instrument for monetary policy. As a 4Chan et al. (1992) extend the CIR model to include any power function for the diffusion function. 8 result, the volatility of US interest rates increased dramatically during that period. Other periods of high volatility of the US interest rates were the OPEC oil crisis (1973-1975), the October 1987 stock market crash and wars involving the US. Such turbulent periods are likely to induce persistence in volatility, which is often an artifact of the changes in the economic mechanism generating the interest rate (see Gray 1996). Lamourex and Lastrapes (1990) show that any structural shift in the unconditional variance is likely to lead to unreliable estimates of the GARCH parameters such that they imply too much persistence in volatility. In this sense, regime shifts are mistaken for periods of volatility clustering. Consequently, studies in the term structure literature have modelled discrete regime shifts in the spot interest rate process (Hamilton 1988, Das 1994, Gray 1996 and Naik and Lee 1997). These models typically posit a spot interest rate process that can shift randomly between two or more regimes (for example a low-mean and a high-mean regime). The diffusion and drift functions are kept the same but the specific parameter values are different in each regime. This makes the process time-heterogeneous. Each regime incorporates a different speed of mean-reversion to a different long-run mean and adifferent unconditional variance. Specifically, in our study we consider the following Regime-Switching (RS) model with two states: rt=ηk+et(6) et= p X i=1 ak iet−i+ξt where ξt∼IIDN(0,σ2 k),k=1,2for the first and second regime, respectively. At any particular point in time there is uncertainty as to which regime we are in. The probability of being in each regime at time tis specified as a Markov 1 process, i.e. it depends only on the regime at time t−1.We define the probability that the process remains at the firstregimeasP, while the probability that the process remains at the second regime is Q. The matrix of the transition probabilities is assumed to be constant.5 5We define the following matrix of transition probabilities: Pr ob(Rt=1|Rt−1=1)=P, Pr ob(Rt=2|Rt−1=2)=Q Pr ob(Rt=2|Rt−1=1)=1−P, Pr ob(Rt=1|Rt−1=2)=1−Q 9 CER increases slightly due to some overshooting during the first 40 years. Except for this overshooting, the RS model regains its quick declining path for the rest of the period reaching a rate of 0.7% after 400 years. The highest terminal rate is produced by the SS model, which projects a rate of 1.6%, followed by MR at 1.4%. {INSERT TABLE 2 HERE: 2: CERs} In summary, the forecasts of the alternative models differ substantially. In this respect, we need to evaluate the models with respect to their predictive ability. Typical misspecification testing has shown that a constant coefficient model may not be able to fully capture the dynamics of the US interest rates over the period examined. Along this line of reasoning, we suggested two time-varying coefficient models (RS and SS), one accommodating abrupt changes and the other allowing for a gradual change over time in the generating mechanism of the interest rates. These two models seem eminently preferable to the constant coefficient models. In the following subsection, we perform an out-of-sample forecast exercise to select among the various models. 3.4 Model Selection Evaluating the out-of-sample forecasting performance of the models under consideration for the long run is impossible due to limitation of data, as forward rates exist for a maximum period of 30 years. However, we attempt to discriminate between these models on the grounds of their forecasting performance over a 30-year horizon using available real data. We specifically make use of annual forward rates suggested by the term structure of the inflation-indexed US government bonds. Then, we calculate the commonly-used Mean Square Forecast Error (MSFE) and judge the models by this criterion. Alternatively, we calculate four modified MSFE criteria by incorporating four kernels15 which weigh observations by their relevant proximity to the present. The results are presented in Table 3. {INSERT TABLE 3: Average MSFEs} Interestingly, the various specifications of the MSFE criterion unanimously rank the 15The Bartlett(B), the Parzen(P), the Quadratic-Spectral (QS) and the Tukey-Hanning (TK) kernels are the weighting functions used in our evaluation. 16 SS model first followed by the RS model in most of the cases. The AR-IGARCH model ranks third followed by MR and then RW. In sum, if we select a model on the basis of its ability to characterise the past and its accuracy concerning forecasts of the future, we are inclined to accept the SS model as the best model (among the estimated models) to describe the US real interest rates. Our second best choice would be the RS model. 4 Policy Implications of Model Selection The foregoing has established the importance of model selection in determining a schedule of declining discount rates for use in CBA. The differences that arise from alternative specifications of the time series process have been revealed and a method for selecting one model over another has been proposed. In this section we highlight the policy implications of declining discount rates and the impact of model misspecification by considering the same case study as N&P, that is, climate change and the value of carbon sequestration.16 We establish the present value of the removal of 1 ton of carbon from the atmosphere, and hence the present value of the benefits of the avoidance of climate change damages for each of the specified models. To understand what follows it is important to be familiar with the profile of benefits resulting from the removal of 1 ton of carbon from the atmosphere. We use the estimates taken from the DICE model of Nordhaus and Boyer (2000) shown in Figure 3 (Appendix B). Table 4 shows the present value per ton of carbon emissions when evaluated using the schedule of discount rates associated with each of the models described in Section 3.2. {INSERT TABLE 4 HERE} The RS model gives the lower valuations followed by the conventional 4% discounting. Interestingly, the SS model gives the highervaluationfollowedbytheRWmodel. For example, the present value of carbon emissions reduction is over 150 % larger in the case of the SS model compared to the case of constant discounting at 4 %. On the other hand, the present value of the removal of 1 ton of carbon emissions from the atmosphere increases 16See N&P for the assumptions concerning the modeling of carbon emissions damages. 17 by only 12 % based on the MR’s forecasts compared to the constant rate discounting approach. The preceding discussion has argued that the RS and SS models are to be preferred over the others since they allow for changes in the interest rate generating process and have desirable properties. From the policy perspective we have established that both these models provide well specified representations of the interest rate series. However, the RS model provides roughly equivalent values of carbon to the constant discounting rate values (there is a 9% difference), while the SS model produces values that are up to 150% higher than those of the constant rate. The disparity between the RS and the SS models, and the proximity of the carbon values generated by the former to those generated by conventional constant discounting represents a clear signal of the policy relevance of model selection in determining the CER. It is crucial from a policy perspective to make a clear judgment as to which of the two models (RS and SS) is most appropriate to the case in hand. Our forecasting exercise reveals that the SS model is preferable to the RS model due to its lower MSFE for the 30-year horizon. Hence in the context of SS the carbon values are increased by 150% compared to conventional discounting and 40% compared to N&P’s approach. In short, in the US context, the selection of econometric models on the basis of forecasting performance, and the preferred schedule of discount rates makes climate change prevention a more desirable investment. 5 Conclusions In response to the need to appraise projects over very long time horizons, a number of theoretical discussions have arisen concerning the appropriateness of discount rates that fall with the time horizon considered. Such Declining Discount Rates (DDRs) would add greater weight to the costs and benefits that accrue to future generations and thereby at least partially address the issue of inter-generational equity that so often besets the long term policy arena. Weitzman’s (Weitzman 1998) theoretical justification for DDRs depends upon un18 certainty of the discount rate and therefore the operationalising of this theory is highly dependent upon the manner in which one interprets and characterises uncertainty. Weitzman (2001) suggested that it was the lack of consensus about the correct discount rate to employ in the far distant future that was the source of uncertainty and his estimated Gamma distribution provided the means of operationalising this theory and determining the declining Certainty Equivalent Rate (CER). Newell and Pizer (2003) (N&P) took an alternative view, accounting for the uncertainty through an econometric forecasting approach. This paper builds on N&P’s approach in determining DDRs and it makes the following points concerning the model selection and the use of DDRs in general. Firstly, N&P’s approach is predicated upon the assumption that the past is informative about the future and therefore characterizing uncertainty in the past can assist us in forecasting the future and determining the path of CERs. We have argued that if one subscribes to this view it is important to characterise the past as well as possible by correctly specifying the model of the time series process. This is particularly so when dealing with lengthy time horizons where the accuracy of forecasts is important. Indeed the selection of the econometric model is of considerable moment in operationalising a theory of DDRs that depends upon uncertainty, because econometric models contain different assumptions concerning the probability distribution of the object of interest. We have shown that when modelling the US interest rate data, the econometric model should allow for changes over time in the data generating process and that state space and regime switching models are likely to be appropriate. Our estimations, simulations and case study bear out this assertion. The path of the CER differs considerably from one model to another and therefore each places a different weight upon the future. The policy implications of these estimates is revealed in the context of a case study that calculates the present value of carbon emissions reduction. The utilisation of a state space model to estimate the discount factors results in an increase of 150% in the present value of carbon emissions reduction compared to a constant rate discounting approach. 19 References [1]Ait-Sahalia, Y. (1996). Testing Continuous-time Models of the Spot Interest Rate. Review of Financial Studies 9, 385-426. [2]Baumol, W.J. (1968). On the Social Rate of Discount. 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Certainty Equivalent Discount Factors Model 4% Mean Random AR Regime State Year Constant Reverting Walk IGARCH Switching Space 10.96154 0.96154 0.96154 0.96154 0.96154 0.96154 20 0.45639 0.45906 0.46177 0.45876 0.45390 0.56424 40 0.20829 0.21661 0.22917 0.21250 0.19576 0.33136 60 0.09506 0.10471 0.12480 0.10062 0.08458 0.20296 80 0.04338 0.05150 0.07777 0.04894 0.03700 0.12889 100 0.01980 0.02567 0.05082 0.02455 0.01647 0.08408 150 0.00279 0.00476 0.02333 0.00529 0.00238 0.03132 200 0.00039 0.00095 0.01830 0.00178 0.00041 0.01255 250 0.00006 0.00022 0.01119 0.00104 0.00010 0.00526 300 0.00001 0.00006 0.00890 0.00086 0.00003 0.00227 350 0.00000 0.00002 0.00715 0.00080 0.00002 0.00100 400 0.00000 0.00001 0.00669 0.00078 0.00001 0.00044 Table 2. Certainty Equivalent Discount Rates Model Mean Random AR Regime State Year Reverting Walk IGARCH Switching Space 14.00 4.00 4.00 4.00 4.00 20 3.91 3.85 3.96 4.22 2.79 40 3.76 3.46 3.88 4.31 2.59 60 3.65 3.08 3.74 4.26 2.38 80 3.58 2.60 3.60 4.18 2.23 100 3.51 2.17 3.42 4.09 2.10 150 3.36 1.39 2.75 3.79 1.91 200 3.16 0.94 1.62 3.31 1.79 250 2.87 0.75 0.65 2.46 1.72 300 2.43 0.56 0.23 1.83 1.67 350 1.87 0.43 0.09 0.95 1.64 400 1.41 0.34 0.04 0.70 1.61 23 Table 3. Average MSFEs Model Mean Random AR Regime State Criterion Reverting Walk IGARCH Switching Space AMSFE 2.058 2.171 2.102 2.323 1.832 AMSFE (B) 1.692 1.724 1.692 1.687 1.499 AMSFE (P) 1.725 1.746 1.720 1.683 1.426 AMSFE (QS) 0.842 0.870 0.848 0.879 0.760 AMSFE (TH) 1.769 1.797 1.765 1.738 1.550 Notes: The weighting functions are as follows: Bartlett(B), Parzen(P), QuadraticSpectral (QS) and Tukey-Hanning (TK). Table 4. Value of Carbon Damages Carbon Values Relative to Relative to Relative to Model ($/tc) Constant Rate Mean Reverting Random Walk Regime-Switching 5.22 -9.0% -18.8% -49.4% Constant (4.0%) 5.74 –-10.7% -44.4% AR-IGARCH 6.37 11.0% -0.9% -38.3% Mean Reverting 6.43 12.0% –-37.7% Random Walk 10.32 79.8% 60.5% – State Space 14.44 151.6% 124.6% 39.9% 24 Appendix A: Tables Table A.1: Unit Root Tests Test Lags /Bandwidth t-stat. 5% critical value Decision ADF 13 -2.314 -2.877 non-stationary Phillips-Perron 12 -2.016 -2.876 non-stationary DF-GLS 13 -0.473 -1.942 stationary ERS Point-Optimal 12 19.733 3.170 non-stationary Ng-Perron 12 -0.824 -8.100 non-stationary KPSS 15 1.158 0.463 non-stationary Notes: SIC is employed to determine the lag-length of the series. The kernel sum-ofcovariances estimator with Parzen weights is used, while the bandwidth is determined based on the Newey-West bandwidth selection method. Table A.2: Estimation Results PanelA:AR(3)-IGARCH(1,1)model Coefficient Estimate Std. Error t-stat. n1.330 0.104 12.811 a11.951 0.085 23.033 a2-1.322 0.156 -8.472 a30.355 0.080 4.441 c0.000 0.000 3.236 β10.442 0.092 4.805 Panel B: Regime Switching model Coefficient Estimate Std. Error t-stat. n11.189 0.128 9.327 a1 11.589 0.078 20.36 a1 2-0.660 0.086 -7.630 n21.714 0.238 7.206 a2 11.787 0.050 35.55 a2 2-0.800 0.049 -16.395 σ2 10.004 0.001 5.651 σ2 20.000 0.000 6.070 P0.867 0.058 14.934 Q0.917 0.035 25.976 Panel C: State Space model Coefficient Estimate Std. Error t-stat. n0.510 0.082 6.185 n10.990 0.002 494.9 ln(σ2 e)-9.158 1.324 -6.917 ln(σ2 u)-6.730 0.144 -46.63 25