Evaluating the carbon-macroeconomy relationship

Transcription

1 Evaluating the carbon-macroeconomy relationship Julien Chevallier Université Paris Dauphine (CGEMP/LEDa) This presentation: February 2011

2 The Usual Suspects 115 EU 27 Seasonally Adjusted Industrial Production Index (Eurostat) 30 EUA Futures Price (ECX) EUR/ton of CO JAN05 NOV05 SEP06 JUL07 MAY08 MAR09 JAN10 5 JAN05 NOV05 SEP06 JUL07 MAY08 MAR09 JAN10 What is the impact of economic activity on the growth rate of carbon prices from an empirical point of view?

3 Outline 1. Literature review 2. Preliminary data analysis 3. Nonlinearity tests 4. Threshold autoregression and cointegration 5. Markov regime-switching VAR model

4 Related literature: Alberola et al. (2008, 2009) p t = α+β(l)p t +δbreak +νpsq i,t +ϕ(l)ngas t +γ(l)coal t +ι(l)elec t +κ(l)dark t +λ(l)spark t +σ Win07 +ς(l)cement t +τ(l)refin t +υ(l)coke t +ω(l)comb t +ξ(l)glass t +ψ(l)metal t +ζ(l)paper t +ρ(l)ceram t +χ(l)iron t +ǫ t p t = α+β i (L)p t +δbreak 1 +νpsq i,t +ϕ(l)ngas t +γ(l)coal t +ι(l)elec t +κ(l)dark t +λ(l)spark t +σ Win07 +ω sect i,j,t +ø sectpeak i,j,t +ǫ t σ t = α 0 +α + (L)ǫ + t α (L)ǫ t +β(l)σ t

5 Related literature: Chevallier (2009) σ 2 t = ω + Y t = θx t +ǫ t p q α i ǫ 2 t i + β j σt j 2 + i=1 j=1 r γ k ǫ 2 t kγ t k with X t = (Stock t,junk t,tbill t,exc.ret t,elec t,gas t,brent t,d APR06,D AUG07) k=1

6 22/04/ /09/ /02/ /07/ /12/ /05/ /10/ /03/ /08/ /10/2008 Stock 0.8 T bill /04/ /09/ /02/ /07/ /12/ /05/ /10/ /03/ /08/ /10/ Exc. Ret Bond /04/ /09/ /02/ /07/ /12/ /05/ /10/ /03/ /08/ /10/ /04/ /09/ /02/ /07/ /12/ /05/ /10/ /03/ /08/ /10/2008 Figure: Dividend yield, junk bond yield, T-bill rate, excess return variables from April 22, 2005 to October 1, 2008 Source: Thomson Financial Datastream, U.S. Treasury, Reuters

7 Related literature: Chevallier (2011) y it =λ 0i +λ i f t +γ i r t +ǫ it ( ft r t ) f t =Φ 1 f t Φ p f t p +ǫ f t ( ) ( = Φ ft r Φ ft p p t 1 r t p ) + ǫ f t with factors extracted from a broad dataset including macroeconomic, financial and commodities indicators.

8 2.5 Factor Factor Figure: Factor loadings for data series in the FAVAR model

9 0.5 EUA BNX SPOT 1 EUA ECX FUT Figure: Impulse response functions of carbon prices following a global economic shock in the FAVAR model

10 Descriptive Statistics EU27INDP ROD EUAF UT Mean Median Maximum Minimum Std. Dev Skewness Kurtosis JB Prob. JB LB test (p-value) ARCH test (p-value) Observations 67 67

11 Logreturns 0.03 EU 27 Seasonally Adjusted Industrial Production Index in Logreturn Form (Eurostat) 0.5 EUA Futures Price in Logreturn Form (ECX) JAN05 NOV05 SEP06 JUL07 MAY08 MAR09 JAN JAN05 NOV05 SEP06 JUL07 MAY08 MAR09 JAN10 there seems to remain some instability, especially for the EU 27 Industrial Production Index during May March 2009

12 OLS-CUSUM Tests OLS CUSUM of equation prod OLS CUSUM of equation eua Empirical fluctuation process Empirical fluctuation process Time Time the empirical fluctuation processes do not seem to indicate the presence of structural breaks

13 ADF,PP,KPSS t-statistic Test critical values ADF test statistic PP test statistic LM-Stat. Asymptotic critical values KPSS test statistic linear unit root tests may not be adequate if the underlying process is nonlinear we also test explicitly for the presence of thresholds and unit root (Caner and Hansen (2001), Basci and Caner (2005))

14 Nonlinearity Tests: Results Evidence of Nonlinearity Keenan (1985) Tsay (1986) BDS (1987,1996) EU27INDPRODRET No Yes Yes EUAFUTRET No No Yes

15 Threshold Autoregression SETAR(2;p 1,p 2 ) model with delay d: { φ1,0 +φ Y t = 1,1 Y t φ 1,p1 Y t p1 +σ 1 e t, if Y t d r φ 2,0 +φ 2,1 Y t φ 2,p2 Y t p2 +σ 2 e t, if Y t d > r φ s are autoregressive parameters, σ s are noise standard deviations, r is the threshold parameter, and {e t } is a sequence of i.i.d (0,σ 2 ) random variables. the autoregressive orders p 1 and p 2 of the two submodels need not be identical. the delay parameter d may be larger than the maximum autoregressive orders.

16 Testing for Threshold Nonlinearity H 0 : AR(p) model vs. H 1 : two-regime SETAR model of order p with σ 1 = σ 2 = σ. Y t =φ 1,0 +φ 1,1 Y t φ 1,p Y t p +{φ 2,0 +φ 2,1 Y t φ 2,p Y t p } I(Y t d > r)+σe t I(.) is an indicator variable that equals 1 if and only if the enclosed expression is true. φ 2,0 represents the change in the intercept in the upper regime relative to that of the lower regime, and similarly interpreted are φ 2,1,...,φ 2,p. H 0 : φ 2,0 = φ 2,1 =... = φ 2,p = 0. Under H 0, r is absent. Assuming d p and the validity of linearity, the large-sample distribution of the test does not depend on d.

17 Testing for Threshold Nonlinearity (ctd.) LR test statistic: T n = (n p)log {ˆσ 2 } (H 0 ) ˆσ 2 (H 1 ) n p is the effective sample size, ˆσ 2 (H 0 ) the maximum likelihood estimator of the noise variance from the linear AR(p) fit, and ˆσ 2 (H 1 ) from the SETAR fit with the threshold searched over some finite interval. Chan (1991) derived an approximation method for computing the p-values of the test. The test depends on the interval over which the threshold parameter is searched (from the a 100 th percentile to the b 100 th percentile of {Y t }). choose a and b so that there are adequate data falling into each of the two regimes for fitting the linear submodels.

18 Testing for Threshold Nonlinearity: Results for EU27INDPRODRET with p=6 d Test Statistic p-value for EUAFUTRET with p=1 d Test Statistic p-value

19 Testing for Threshold Nonlinearity and Unit Root Caner and Hansen (2001), Basci and Caner (2005): y t =θ 1x t 1 +e t y t =θ 2x t 1 +e t if y t 1 y t m 1 < λ if y t 1 y t m 1 λ y t is the selected time series, x t 1 = (y t 1,1, y t 1,..., y t k ) for t = 1,2,...,T, e t is the i.i.d error term, m is the delay parameter and 1 m k, The threshold variable is the absolute value of y t 1 y t m 1, The threshold value λ is unknown and takes the value in the compact interval λ Λ = [λ 1,λ 2 ] where these values are picked P( y t 1 y t m 1 λ 1 ) = 0.15,P( y t 1 y t m 1 λ 2 ) = 0.85.

20 Testing for Threshold Nonlinearity and Unit Root (ctd.) Decompose the coefficients: ρ 1 θ 1 = β 1, θ 2 = α 1 where ρ 1 and ρ 2 are scalar, β 1 and β 2 have the same dimension as y t, α 1 and α 2 are k 1 vectors. ρ 2 β 2 α 2 (ρ 1,ρ 2 ) represent the slope coefficients on y t 1, (β 1,β 2 ) are the slopes on the deterministic components, and (α 1,α 2 ) are the slope coefficients on ( y t 1,..., y t k ) in the two regimes. Re-write the TAR model: y t = θ 1x t 1 1 { yt 1 y t m 1 <λ} +θ 2x t 1 1 { yt 1 y t m 1 λ} +e t with 1 {.} the indicator function.

21 Testing for Threshold Nonlinearity and Unit Root (ctd.) Under H 0 (unit root) : ρ 1 = ρ 2 = 0 1. if the time series follows a stationary threshold autoregressive pattern, the alternative of interest is H 1 : ρ 1 < 0,ρ 2 < there is the case of partial unit root: { ρ1 < 0 and ρ H 2 : 2 = 0 ρ 1 = 0 and ρ 2 < 0 If H 2 holds, then the time series is nonstationary, but we do not deal with a classic unit root. The test statistics for testing H 0 vs. H 1 and H 0 vs. H 2 are given by Caner and Hansen (2001). We test H 0 : ρ 1 = ρ 2 = 0 with a simple one-sided Wald as test statistic: R 1T = t 2 11 {ˆρ1<0} +t 2 21 {ˆρ2<0} with t 1, t 2 the t-ratios for respectively ˆρ 1 and ˆρ 2. In order to test H 0 vs. H 2, we use the negative of the the t statistics t 1, t 2.

22 Bootstrap p-values of threshold unit root tests Variable R 1T t 1 t 2 EU 27IN DP ROD 0.002* * EU AF U T 0.002* 0.014* Both R 1T statistics are significant at the 10% level. The one-sided Wald test (unit root vs. two-regime stationary nonlinear model) is rejected for both time series. EU27INDPROD: the rejection is due to the second regime, where the p-value for the t test is statistically significant. EUAFUT: the rejection is due to the first regime, where the p-value for the t test is statistically significant. ρ 1 = 0 and ρ 2 < 0 for EU27INDPROD. ρ 1 < 0 and ρ 2 = 0 for EUAFUT. Both time series are partially stationary threshold processes.

23 AIC of the SETAR Models for EU27INDPRODRET d AIC ˆr ˆp 1 ˆp for EUAFUTRET d AIC ˆr ˆp 1 ˆp

24 SETAR(2,4,1) Model with d=3 for EU27INDPRODRET Estimate Std. Error t-statistic p-value ˆd 3 ˆr Lower Regime (n 1 = 27) ˆφ 1, ˆφ 1, ˆφ 1, ˆφ 1, ˆφ 1, σ Upper Regime (n 2 = 35) ˆφ 2, ˆφ 2, σ

25 SETAR(2,1,4) Model with d=3 for EUAFUTRET Estimate Std. Error t-statistic p-value ˆd 3 ˆr Lower Regime (n 1 = 43) ˆφ 1, ˆφ 1, σ Upper Regime (n 1 = 17) ˆφ 2, ˆφ 2, ˆφ 2, ˆφ 2, ˆφ 2, σ

26 Thresholds Estimated by the SETAR Models EU27INDPROD Time EUAFUT Time Note: Solid (open) circles indicate data in the lower (upper) regime of a fitted threshold autoregressive model.

27 EU27INDPROD: data falling in the lower regime (with lag 3 values < ) are displayed as solid circles. The estimated lower regime corresponds mainly to the recession period. The estimated upper regime corresponds mainly to periods before/after the recession. EUAFUT: data falling in the lower regime (with lag 3 values < 0.059) are displayed as solid circles. The estimated lower regime corresponds to periods of decreasing prices. The estimated upper regime corresponds mainly to periods of increasing prices. From January 2009 onwards the carbon market is characterized by a lower regime, indicating a delayed adjustment to the financial crisis.

28 Model Diagnostics The dependence of the residuals necessitates the employment of a quadratic form of the residual autocorrelations: B m = n eff m i=1 j=1 m q i,jˆρ iˆρ j n eff = n max(p 1,p 2,d) is the effective sample size, ˆρ i is the i th lag sample autocorrelation of the standardized residuals, and q i,j model-dependent constants If the true model is a SETAR model, ˆρ i are likely close to zero and so is B m, but B m tends to be large if the model specification is incorrect. The quadratic form is designed so that B m is approximately distributed as χ 2 with m degrees of freedom.

29 Model Diagnostics: Results SETAR(2,4,1) Model with d=3 for EU27INDPRODRET Standardized Residuals ACF of Residuals P values

30 Model Diagnostics: Results (ctd.) SETAR(2,1,4) Model with d=3 for EUAFUTRET Standardized Residuals ACF of Residuals P values

31 Linear Johansen Cointegration Rank Tests for LOG(EU27INDPROD) and LOG(EUAFUT) Lags interval (in first differences): 1 to 2 Trace Test Hypothesized Trace 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None* At most 1* Max-Eigen Test Hypothesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None* At most 1*

32 Linear VECM Estimates EU27INDPROD t EUAFUT t Cointegrating Vector µ *** (0.0326) (0.0326) w t *** (0.2753) (0.2753) EU27INDPROD t *** (0.0545) (0.0545) EUAFUT t (0.0135) (0.0135) ( ) ( ) EU27INDPRODt EU27INDPRODt 1 = µ+αw EUAFUT t 1 +Γ +u t EUAFUT t t 1 with w t 1 = EU27INDPROD t 1 βeuafut t 1.

33 Two-Regime Threshold Cointegration Let x t be a p-dimensional I(1) time series which is cointegrated with one p 1 cointegrating vector β. Let w t (β) = β x t denote the I(0) error-correction term. Following Hansen and Seo (2002): { A x t = 1 X t 1 (β)+u t, if w t 1 (β) γ A 2X t 1 (β)+u t, if w t 1 (β) > γ A 1 and A 2 are the coefficient matrices governing the dynamics of the regimes. The error u t is assumed to be a vector martingale difference sequence with finite covariance matrix = E(u t u t). The notation w t 1 (β) and X t 1 (β) indicate that the variables are evaluated at generic values of β.

34 Two-Regime Threshold Cointegration (ctd.) All coefficients (except β) switch between the two regimes. The threshold effect only has content if 0 < P(w t 1 γ) < 1, otherwise the model simplifies to linear cointegration. Therefore, we assume that: π 0 P(w t 1 γ) 1 π 0 π 0 > 0 is a trimming parameter set to π 0 = 0.05 (see Andrews (1993), Andrews and Ploberger (1994)). The model is estimated by maximum likelihood under the assumption that the errors u t are i.i.d Gaussian (algorithm developed by Hansen and Seo (2002)). Let the estimates be denoted by ( β,ãi, ) with ũ t the residual vectors.

35 Testing for Threshold Cointegration To test linear vs. threshold cointegration, Hansen and Seo (2002) suggest to use the LM test statistic proposed by Davies (1987): SupLM = sup γ L γ γ U LM( β,γ) [γ L,γ U ] is the search region, so that γ L is the π 0 percentile of w t 1 and γ U is the (1 π 0 ) percentile. As the function LM( β,γ) is non-differentiable in γ, it is necessary to perform a grid evaluation over [γ L,γ U ]. The LM statistics are computed with heteroskedasticity consistent covariance matrix estimates.

36 LM Statistic for the Two-Regime Threshold Cointegration Model 60 LM Statistic as function of Gamma Gamma SupLM test (estimated β) with 300 gridpoints the p-values are calculated by the parametric bootstrap

37 LM Tests Results for Threshold Cointegration Lagrange Multiplier Threshold Test Statistic (Asymptotic).05 Critical Value Bootstrap.05 Critical Value (Asymptotic) p-value Bootstrap p-value all p-values were computed with 5,000 simulation replications.

38 Concentrated Negative Log-Likelihood of the Two-Regime Threshold Cointegrated Model LConcentrated Negative Log Likelihood Negative Log Likelihood Negative Log Likelihood Gamma Beta

39 Threshold VECM Estimates Threshold Estimate Cointegrating Vector Estimate First Regime EU27INDPROD t EUAFUT t µ *** (0.0317) (0.0700) w t *** (0.2718) (0.7057) EU27INDPROD t *** (0.0594) (0.0395) EUAFUT t (0.0116) (0.0183) Percentage of Observations Second Regime EU27INDPROD t EUAFUT t µ *** (0.2897) (0.2339) w t *** (4.2598) (3.2803) EU27INDPROD t *** *** (0.1453) (0.0806) EUAFUT t ** (0.1125) (0.2155) Percentage of Observations

40 The estimated threshold is ˆγ = with the error-correction term defined as w t = EU27INDPROD t 0.118EUAFUT t. Typical regime: EU27INDPROD t 0.118EUAFUT t , i.e. when the EU industrial production index is less than 13.3 percentage points above the carbon futures price. (94% obs.) Extreme regime: EU27INDPROD t > 0.118EUAFUT t , i.e. when the gap is above 13.3%. (6% obs.) The EU industrial production index impacts positively the EUA futures price in regime 2 at the 1% significance level. the carbon-macroeconomy relationship goes from the EU industrial production index (lagged one period) to the carbon futures price, with a coefficient equal to The EUA futures price has no statistically significant effect on the EU industrial production index in either of the regimes.

41 EU27INDPROD governs most of the adjustment from the short-run to the long-run equilibrium of the model: its coefficients for w t 1 are highly significant in both regimes. the magnitude of the response for EU27INDPROD is between 1.33 (regime 1) and 18 (regime 2) times greater than the coefficient of EUAFUT. EUAFUT: the error-correction term is not statistically significant. The 4 error-correction coefficients (for EU industrial production and carbon futures prices in both regimes) are either negative or insignificantly different from zero if positive.

42 Model Response to the Error Correction 3 Model Response to Error Correction EU27INDPROD EUAFUT 2 1 Model Response Error Correction This figure plots the error-correction effect, i.e. the estimated regression functions of EU27INDPROD t and EUAFUT t as a function of w t 1 by holding other variables constant.

43 RHS of the threshold: strong error-correction effect for EU 27IN DP ROD and minimal error-correction effect for EUAFUT. LHS of the threshold: minimal error-correction effect for EU 27IN DP ROD and flat near-zero error-correction effect from EUAFUT. Asymmetry for EU27INDPROD: there is a stronger error-correction effect in the extreme regime. The w t 1 coefficient for EU27INDPROD is especially important (-23.58) in the extreme regime (at 1% level), which implies a mean-reverting dynamic behavior of the gap between the two time series once the threshold (13.3 percentage points) has been reached. A value of the gap above 13.3 percentage points in one month will produce a downward pressure on the industrial production index in the subsequent month in order to restore the long-run equilibrium relationship. Industrial production leads the nonlinear mean-reverting behavior of the carbon price, but not vice versa.

44 Markov-Switching VAR Consider an n-dimensional vector y t (y 1t,...,y nt ) which is assumed to follow a VAR(p) with parameters: y t =µ(s t )+ p Φ i (s t )y t i +ǫ t i=1 ǫ t N(0,Σ(s t )) The parameters for the conditional expectation µ(s t ) and Φ i (s t ), i = 1,...,p, as well as the variances and covariances of the error terms ǫ t in the matrix Σ(s t ) all depend upon the state variable s t which can assume a number q of values (corresponding to different regimes).

45 Markov-Switching VAR (ctd.) The general idea behind the class of Markov-switching models is that the parameters and the variance of an autoregressive process depend upon an unobservable regime variable s t {1,...,M}, which represents the probability of being in a particular state of the world. A complete description of the Markov-switching model requires the formulation of a mechanism that governs the evolution of the stochastic and unobservable regimes on which the parameters of the autoregression depend. Once a law has been specified for the states s t, the evolution of regimes can be inferred from the data.

46 Markov-Switching VAR (ctd.) Typically, the regime-generating process is an ergodic Markov chain with a finite number of states defined by the transition probabilities: p ij = Prob(s t+1 = j s t = i), M p ij = 1 i,j {1,...,M} j=1 The transition probabilities of the Markov-process determines the probability that volatility will switch to another regime, and thus the expected duration of each regime.

47 Markov-Switching VAR: Results (1/2) Log-likelihood µ (Regime 1) *** (0.0009) µ (Regime 2) *** (0.0006) Equation for EU27INDP RODRET EU27INDP RODRET EUAF UT RET φ 1 (Regime 1) * (0.0777) (0.0043) φ 1 (Regime 2) (0.3383) (0.0213) φ 2 (Regime 1) *** (0.1178) (0.0136) φ 2 (Regime 2) *** (0.5488) (0.0559)

48 Regime 1 (expansion): output growth per month is equal to 0.14% on average. Regime 2 (recession): the average growth rate amounts to -0.48%. the effects of the recessionary shock are found to be quite strong. AR(2) seems to describe the autocorrelation structure of EU27INDPRODRET. No statistically significant impact of carbon futures on EU industrial production.

49 Markov-Switching VAR: Results (2/2) Equation for EUAF UT RET EU27INDP RODRET EUAF UT RET φ 1 (Regime 1) * (1.1483) (0.0760) φ 1 (Regime 2) (8.7970) (0.5497) φ 2 (Regime 1) *** ** (0.1893) (0.0151) φ 2 (Regime 2) *** (0.2448) (1.4546) Standard error (Regime 1) Standard error (Regime 2) Transition Probabilities Matrix Regime 1 Regime 2 Regime *** * (0.1700) (0.3111) Regime (0.0815) (0.4326) Regime Properties Prob. Duration Regime Regime

50 For EUAFUTRET, the process seems characterized by an AR(1). EU industrial production (variable EU27INDPRODRET) has two kinds of delayed impacts on carbon futures: positive during Regime 1 (as φ 2 is equal to 0.78 and highly significant), negative during Regime 2 (as φ 2 is equal to and highly significant). delayed impact of macroeconomic activity on carbon markets.

51 During an expansionary phase, the series are most likely to remain in Regime 1 (88.73%). The probability that the series switch from Regime 1 to Regime 2 is equal to 12.11%. Once the economy finds itself in a depression, the probability that it will be in a depression the following month is 49.27%. If the economy is in the recessionary phase, the probability that it will change directly to a growth regime is equal to 51.89%. Regime 1 is assumed to last 8.60 months on average. Regime 2 is assumed to last 1.96 months on average. The economy would spend about 80% of the time spanned by our sample of data in regime 1. Regime 2 has an ergodic probability of about 20%.

52 Smoothed Transition Probabilities Regime 1 Regime Smoothed Transition Probabilities Jan. Apr.05 Apr. Jun.06 Oct.08 Apr JAN05 NOV05 SEP06 JUL07 MAY08 MAR09 JAN10 the smooth probability, which is the probability of a particular state in operation at time t conditional on all information in the sample.

53 Regime Transition Probabilities Regime 1 Regime Regime Transition Probabilities Jan. Apr. 05 Apr. Jun. 06 Oct. 08 Apr JAN05 NOV05 SEP06 JUL07 MAY08 MAR09 JAN10 the regime probability at time t is the probability that state t will operate at t, conditional on information available up to t 1.

54 Model Diagnostics Markov-switching VAR LR Statistic p-value Symmetry test p-value Distributional Characteristics EU27INDP RODRET EUAF UT RET Mean Median Maximum Minimum Std. Dev Skewness Kurtosis The carbon-macroeconomy relationship is better described by a two-regime Markov-switching model than by the random walk model. We reject the hypothesis of symmetry of the Markov transition matrix (which implies symmetry of the unconditional distribution of the growth rates) at the 5% level. The Markov-switching model produces both the degree of skewness and the amount of kurtosis that are present in the original data.

55 Key Messages 1. macroeconomic activity is likely to affect carbon prices with a lag, due to the specific institutional constraints of this environmental market 2. the carbon-macroeconomy relationship seems adequately captured by two-regime threshold error-correction and two-regime Markov-switching VAR models compared to linear models as main competitors

56 Further Work Dynamic correlations between industrial production and carbon prices in the DCC-MIDAS model (Colacito et al. (2010), Baele et al. (2010)) Smooth transition error-correction models (Martens et al. (1998), (1998)) Markov error-correction models (Psaradakis et al. (2004)) Time-varying transition-probability Markov-Switching models (Filardo (1994)) Markov-Switching GARCH models (Marcucci (2005), Henry (2009), Janczura and Weron (2010))

57 Thanks for your attention! Contact: julien [dot] chevallier [at] dauphine [dot] fr sites.google.com/site/jpchevallier/