The Analysis of Structural Breaks in Gasoline Demand Using a Bayesian Approach Koffi Akakpo a, Philippe Barla b and Stephen, Gordon c

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1 April 2017 The Analysis of Structural Breaks in Gasoline Demand Using a Bayesian Approach Koffi Akakpo a, Philippe Barla b and Stephen, Gordon c a Département d économique and CREATE, Université Laval, Quebec City, Canada b Département d économique and CREATE, Université Laval, Quebec City, Canada c Département d économique and CRREP, Université Laval, Quebec City, Canada Abstract: This paper focuses on the presence of structural breaks in the Canadian gasoline demand using a Bayesian approach. The analysis is carried on quarterly data from 1965 to Our methodology is based on Chib (1998) approach that allows for multiple and unknown breakpoints. The results suggest the existence of two breakpoints defining three regimes. In the first regime up to 1982, the price and income elasticities are in the range of values usually reported in the literature (-0.5 and 0.8 respectively). In the third regime after 1986, price reactiveness is close to zero while the income effect is much lower (0.2). The second regime provides results that are incoherent with economic theory and most likely corresponds to a transition phase disturbed by several shocks such as the oil crisis, a recession and fuel economy regulations. Keywords: Gasoline demand, price elasticity, income elasticity, structural breaks, Bayesian approach JEL-code: C11, Q41, Q Introduction Gasoline demand has been widely analyzed with hundreds of studies estimating price and income elasticities (Dahl and Sterner, 1991; Espey, 1998; Basso and Oum, 2007; Dahl, 2012). Based on a review of the literature, Basso and Oum (2007) conclude that the short run price elasticity of gasoline demand would be between -0.3 and -0.2 and the long run price elasticity between -0.8 and For income elasticity, the values would vary between 0.3 and 0.5 in the short run and between 0.9 and 1.3 in the long run. However, recent works suggest that the values of these parameters may have changed in recent decades. Hughes, Knittel and Sperling (2007) find a substantial decline in the short run price elasticities in the U.S. when comparing the 2001 to 2006 with the 1975 to 1980 periods. Small and Van Dender (2007) find a significant decline in the U.S. price and income elasticity over the 1966 to 2001 period. Park and Zhao (2010) estimate a smooth time-varying cointegration model using U.S. monthly data from 1976 to 2008 and conclude that price and income elasticities would have followed a cycle of increase-decrease ranging from to and 0.06 to 0.10, respectively. The authors link these changes to variation in the share of gasoline expenses in disposable income. Liu (2014) uses semi-parametric methods on a panel of U.S. states from 1994 to Contrary to the other studies, she finds that price and income elasticity would have slightly increased over this time period (from 0 to -0.1 and 0.12 to 0.2 respectively). Liu explains this time pattern by the dramatic fluctuations of gasoline prices as well as income growth. She also finds spatial variations in the elasticity values across states depending upon urbanization and income level. Overall, there are some evidence of changes in price and reactiveness of gasoline demand over time. However, none of the existing studies proceeds to formal structural break analysis. In this paper, we

2 address this gap by testing for the existence of structural breaks in the Canadian demand for gasoline from 1965 to This analysis is particularly relevant in the current context of the fight against climate changes. The transportation sector is the second most important source of GHGs in Canada with a share of 19% just behind the production of gas and petroleum. In the transportation sector, gasoline account for over 68% of the sector total emissions (WWF-Canada, 2012). Policies have been adopted by Canadian authorities in order to curb road transportation emissions. However, the effectiveness of several interventions such as carbon tax or cap-and-trade is in fact directly related to the values of the price and income elasticities. Moreover, the assessment of measures such as the strengthening of new light duty fuel economy standards requires predicting how gasoline consumption would have evolved in the business as usual scenario. Though, the accuracy of this scenario depends very much on using appropriate price and income elasticities. The methods for detecting structural breaks have considerately evolved in recent years both under the classical and Bayesian paradigm. While the classical approach may be somewhat easier to interpret as it relies on classical regression techniques, the Bayesian approach offers several important advantages (Western and Kleykamp, 2004 and Spirling, 2007). Firstly, the Bayesian inference does not rely on asymptotic theory like the frequentist methods. This is particularly advantageous with structural breaks limiting the sample size. Secondly, the Bayesian approach allows for taking into account priors on the parameters as well as on the number and timing of the breaks. Thirdly, the Bayesian method does not impose a precise date on the breaks but rather estimate probability distributions for each break at each point in time. Specifically, we adopt the Bayesian approach initially proposed by Chib (1998) for modeling multiple unknown structural break problems. In this setting, the determination of the number of breaks becomes an exercise of model selection based on the Bayes Factor criteria. As already mentioned, the main particularity of this method is that it estimates, for each observation, a set of probabilities of being in each regime thereby allowing for drawing inferences on the dates of the breaks. For comparison purposes, we also report the results obtained using Bai and Perron (2003) methodology in the frequentist context. Our main results are based on a static specification where the per capita gasoline consumption depends upon gasoline price, disposable income and quarterly dummies. The results show that the model without any structural breaks is very unlikely. In fact, the Bayes factor procedure identifies the model with two structural breaks and thus three regimes as the model that best describes the data. The first break occurs around 1982 and the second around During the first regime from 1965 to 1982, the expected value of the posterior distribution of the price elasticity is (standard deviation at 0.077). For the income elasticity, the expectation is (0.048). In the last regime, after 1985, the expected price elasticity is close to zero. The income elasticity is also much lower with an expected value of (0.121). The period covering the second regime between 1982 and 1986 is characterized by a major recession caused by the second oil shock, the gradual implementation of the CAFE standards in the U.S. and its equivalent in Canada. During the eighties, the fleet of vehicles drastically evolved resulting in substantial fuel economy improvements. 1 Our results are robust to specifications changes and our conclusion holds when using Bai and Perron method. The rest of paper is organised as follows. Section 2 discusses the econometric issues to model structural breaks and presents the Bayesian approach with its particularities. Section 3 describes the data and provides some basic descriptive statistics. We outline the results and the robustness analysis in Section 4. Finally, Section 5 discusses the main conclusions and policy implications. 1 For the US, the fuel economy rose by up to 25% during this period (Bureau of Transportation Statistics, 2017). While comparable statistics are unavailable for Canada, the same phenomenon occurred in Canada (Beauregard- Tellier, 2004). 2

3 2. Methodology Most empirical studies of gasoline demand adopt a log-log linear functional of the following form: lng t = β 0 + β 1 lnp t + β 2 lny t + β 3 X t + ε t [1] where G t is gasoline consumption per capita at time t, P t is the retail price of gasoline per liter, Y t is the real disposable income per capita, X t represents others potential covariates and ε t is the error term. β 1 and β 2 measure respectively the price and income elasticities. Specification [1] does however assume that the parameters are constant over time. Thus, we use instead specification [2] which allows for an undetermined m number of shifts or breakpoints in the parameters values. with: r = 1,2,, m + 1 τ r 1 < t τ r and ε t ~N(0, σ k 2 ) lng t = β 0,r + β 1,r lnp t + β 2,r lny t + β 3,r X t + ε t [2] The m structural breakpoints define m+1 regimes. The number of breakpoints m is assumed to be unknown and thus need to be determined. The timing of the breaks is characterized by the parameters τ r that also need to be estimated. 2 For example, if the data revealed one breakpoint this means two regimes with different gasoline demand parameters. The number and timing of structural breaks is treated as a problem of model selection based on some information criterion. The optimal number of breaks may be determined either through a sequential (Yao, 1988; Liu et al., 1997; Giron et al., 2007; Lai and Xing, 2011) or simultaneous process (Chib, 1998; Bai and Perron, 1998). In this paper, we adopt the latter which has been shown to be more consistent (Bai et Perron, 1998). The estimation can be carried either within the classical (frequentist) or Bayesian framework. We adopt the latter as it does not rely on asymptotic theory and does not face the infinite samples problem (Chib, 1998; Maheu and Gordon, 2008). These issues are particularly worrisome with structural breaks that limit the size of the estimation samples (Raftery, 1994). In the Bayesian framework, the estimation is based on the maximization of the likelihood function which depends upon the way breaks are modeled. Some authors impose a constant probability of break (Chernoff and Zacks, 1964) while other allows the probability to vary by regime (Chib, 1998). But the estimation of these models may be burdensome with a long time series and a high number of breaks. Chib (1998) proposes a more straightforward parametrization based on a Markov process. A discrete random variable s t is introduced which captures the state of the system at each period t. This latent variable, which takes values from 1 to m+1, indicates from which regime an observation has been drawn. It is assumed to follow a Markov process with the transition matrix: P = p 11 p p 22 p p mm p m,m+1 ( ) where p ij = Pr(s t = j s t-1 =i) is the probability of moving from state i to state j at time t given that the state at t-1 is i. 2 Note however that, by definition, τ 0 = 0 and τ m+1 = T 3

4 The structure of this matrix is such that there is only one unknown parameter per row (the sum of the probability in each row being one). Moreover, the transition is only possible to the next state. This excludes jump across different regime as may occur in a Markov switching model. This setting corresponds to what is referred as a Hidden Markov Model (Chib, 1996). The model selection (i.e. the optimal number of breaks) is based on the Bayes factors method which compares the marginal likelihoods across models. Chib (1995) has developed a simple computation technique for the estimation of the marginal likelihood. 3 For a model with m breaks (M m ), the marginal likelihood is calculated as follows: L(lnG 1:T Z 1:T, M m ) = f (lng 1:T Z 1:T, β, P, M m )π(β, P M m ) π(β, P lng 1:T, Z 1:T, M m ) where Z 1:T contain all the explanatory variables, (β, P) is any point in the parameter space, f (. ) is the likelihood, π(β, P M m ) is a prior density on the parameters and π(β, P lng 1:T, Z 1:T, M m ) is a posterior density. The Bayes factor is then defined as the ratio between likelihood of two models M m and M n : B m,n = L(lnG 1:T Z 1:T, M m ) L(lnG 1:T Z 1:T, M n ) The choice of model is then determined by a set of rules proposed by Jeffreys (1961). Essentially, a value of B m,n larger than one means that model M m better fits the data than M n. Moreover, the larger is this ratio the stronger is the conclusion. The evidence is considered decisive if log 10 (B m,n ) is greater than 2. The log-likelihood function is Now given Chib (1998) transition matrix, we have: Ln f(lng 1:T Z 1:T, M m ) = Ln f(lng t Z t, β, P) m Ln f(lng t Z t, β, P) = Ln f(lng t Z t, β, P, s t = k) Pr(s t = k lng 1:t 1, Z 1:t 1, β, P) k=1 where f(lng t Z t, β, P, s t = k) is the density of observation a time t conditional to the state s t = k, Pr(s t = k lng 1:t 1, Z 1:t 1, β, P) is the marginal probability to be in regime k which is defined through Bayes rule : Pr(s t = k lng 1:t 1, Z 1:t 1, β, P) = k T t=1 Pr(s t = k s t 1 = i)pr(s t 1 = i lng 1:t 1, Z 1:t 1, β, P) i=k 1 The estimation is carried by simulation of the posterior distribution of the parameters through an iterative process. The likelihood function is computed using the Expectation-Maximization (EM) algorithm. This iterative estimation technique is particularly efficient when the objective function contains latent variables. In fact, the EM algorithm generalizes the maximum likelihood principle. It takes place in two main stages: 1. Expectation which consists in simulating the latent variables given the priors; 2. Given the value of the latent variables, the likelihood is maximized with respect to the parameters. The estimated parameters are then used to update the value of the latent variables. 3 An alternative is the algorithm of Dufays et al. (2014) with stepping stone techniques. 4

5 For comparison purposes, we also estimate the model using a frequentist approach. Specifically, we use Bai and Perron (2003) techniques which are suitable for both pure and partial structural change models. This empirical method is inspired by their theoretical model on the estimation of multiple structural changes by least squares. In each regime, a minimal number h of observations is imposed. The optimal number and timing of breaks as well as the parameter values are derived by minimizing the following function: m+1 τ k k=1 t=τ k LS(β, Γ) = (lng β Z) (lng β Z) = [lng t β k Z t ] 2 Bai and Perron (2003) propose a new efficient algorithm to find β the vector of parameters and Γ the vector of breaks which minimize the function LS given m. The estimation is basically based dynamic programming and the use of the Bellman principle. The constraint on the minimum duration of a regime considerably decreases the number of potential regimes to be tested. 4 The optimal number and timing of the regimes is based on the comparison of the sum of the squares of the residuals (SSR), Bayesian Information Criteria (BIC) or log-likelihood. The BIC is generally considered to be the best criteria Data and descriptive analysis The main source of our dataset is Statistics Canada. Quarterly gasoline consumption is approximated by domestic gasoline sales. They include all type of gasoline designed for internal combustion engines other than aircraft and exclude exports and inter-company sales. They likely somewhat overestimate gasoline consumption for transportation but this is the only long term time series available. The price variable is constructed using the average retail prices for unleaded gasoline available starting in 2006 with a gasoline consumer price index which is available from The average price before 2006 is thus backtracked using the price index. Quarterly disposable income statistics are only available up to 2012Q2. Our dataset covers therefore the 1965Q1 to 2012Q2 period providing 190 observations. All dollars values are expressed in real Canadian dollar of Annex 1 details the sources of the variables used in the empirical analysis. Panel A of Figure 1 shows per capita gasoline consumption and real gasoline price over the analysis period. Panel B shows per capita gasoline consumption and real disposable income. From 1965 to about 1980, per capita gasoline consumption exhibits a strong positive trend with an average annual growth rate of about +2.8%. This is also a period of low gasoline price and high income growth. The second oil shock marks a drastic change in the trend of gasoline consumption: from 1980 to 1990, gasoline consumption declined by -1.9% per year on average. Obviously, this decline could be due to the large price hike as well as the deep recession that Canada experienced in the early 80 s. But this decline is also associated with the sharp improvement in the fuel economy of vehicles during this period in part because of the implementation of the U.S. Corporate Average Fuel Economy Standards (CAFE) and its equivalent in Canada the Company Average Fuel Consumption (CAFC) goals. Contrary the CAFE, the Canadian targets were not mandatory until 2011 (McCauley, 2011). However, given the deep integration of the US and Canadian automotive market, the Canadian targets have always been similar to the US standards and most manufacturers have voluntary met the targets. From 1979 to 1982, the average fuel consumption of new vehicles declined by about 25% due to technological advances but also a 20% decline in horsepower and a 15% drop in vehicle weight (BST, 2017). In fact, there was a rapid shift from large gas guzzlers to smaller fuel efficient automobiles. These events may have caused a break in gasoline demand. During that period, higher gasoline prices could have led to increase price reactiveness but improved energy efficiency could have had the opposite impact. 4 For more details see appendix 1. 5 The Bayesian results are obtained with Matlab while the classical results are derived by R. 6 A Canadian dollar was worth 0.87 U.S. dollar in

6 Panel C of Figure 1 shows the share of gasoline expenses in disposable income which may affect the price response. This share significantly increases after the second oil shocks but decline rapidly after 1985 before reaching historical lows in the nineties then trending up in the 2000s up to the financial crisis. During the last decade, gasoline prices have fluctuated with significant peaks in 2001, 2008 and The rise of disposable income has been slow in the 1990s and has picked up in the next decade up to the financial crisis. The increase in per capita gasoline during this period is often associated with the development of large, more powerful vehicles and especially the growing popularity of SUVs. Obviously, it is difficult to conclude based on these descriptive evidence if and when structural breaks have occurred in gasoline demand. We thus turn next to the econometric analysis results. Figure 1 : Gasoline consumption, disposable income and average retail price Panel A: Gasoline consumption and average retail price Panel B: Gasoline consumption and disposable Panel C: Share of gasoline expenses in relation to income 6

7 m breaks 4. Results First, we analyse the results of the most basic specification with gasoline consumption depending upon price, disposable income and quarterly dummies to account for seasonable effects. Table 1 shows the log 10 (B m,n ) namely the Bayes Factors in its log version for up to four breakpoints. In fact, beyond four breaks, the marginal likelihood function is continuously decreasing with m. The values in Table 1 allow comparing the likelihood of models with m breaks (M m ) with respect to models with n breaks (M n ). A positive value means that M m is more likely than M n while the opposite holds for negative values. Thus, it appears that M 2 fits best the data. However, Jeffreys rule indicates that the log 10 (B m,n ) should be higher than 2 in order to decisively conclude that M m is better than M n. In our setting, this means that we cannot clearly select M 2 over M 1 as log 10 (B 2,1 ) is well below two. Table 1 : Log of the Bayes Factor for comparing the basic specification model with different number of breakpoints Model n breaks Thus, we analyse the results of both models. Figure 2 and 3 plot the posterior density of regime change (Panel A) and the posterior probabilities of states (Panel B) for M 1 and M 2 respectively. Figure 2 : Posterior density and probabilities M 1 Panel A Panel B Figure 3 : Posterior density and probabilities M 2 Panel A Panel B 7

8 The posterior density of regime change for M 1 is concentrated on the period running from 1982 to 1985 with a maximum around 1984Q1. Panel B shows a transition from regime 1 to regime 2 in less than ten years. When considering two breakpoints, Figure 3 shows a first break around the last quarter of 1982 and a second break at the end of This pattern is coherent with the idea that the 1979 oil shock initiated a transition phase (i.e. regime 2) from regime 1 to regime 3. Figure 4 illustrates the posterior density of the price and income elasticities for M 1 and M 2. We also show the results for M 0 as it is the model most often estimated in the literature. Table 2 presents summary statistics on the distribution of these elasticities. We find that the average price elasticity obtained in the no break model is consistent with the findings in the literature (-0.33) while the income elasticity is somewhat on the low side (0.34). The model with one break shows an average price and income elasticity in the first regime that is very much in line the central values in the literature. In the second regime however, the average price elasticity is positive but very close to zero while the income effect is lower than in the first regime. In the model with two breaks, the results in the first regime and third regime are very close to those obtained with two breaks. In the second regime, the densities of the elasticities are very spread indicating that the price and income effect are very blurry. Recall that during this period of time several shocks affected the economy at large and the automobile market in particular. This second regime may thus be viewed as a transition phase. Figure 4 : Posterior density of price and income elasticities Panel A : No break Panel B : One break Panel C : Two breaks 8

9 Table 2 : Summary statistics on the posterior distribution of the elasticities Bayesian results - Basic Model Type of regime Statistics Price Income Intercept Q2 Q3 Q4 Mean Regime 1 St.dev Lower Upper One break Mean Regime 1 St.dev Lower Upper Mean Regime 2 St.dev Lower Upper Two breaks Mean Regime 1 St.dev Lower Upper Mean Regime 2 St.dev Lower Upper Mean Regime 3 St.dev Lower Upper Robustness analysis In order to access the robustness of our results, we examine several alternative specifications. First, we estimate a model with additional covariates namely the level of unemployment and the interest rate on Canadian treasury bills. These variables are commonly used in the literature and capture the macroeconomic conditions. The results on the elasticities (see Table 3) are very similar to those obtained without these covariates. Second, we estimate the basic model but with an interaction term between the price and income variable as an additional covariate. The idea is that the price reactiveness is decreasing with the level of income. Ignoring this possibility could eventually lead to the erroneous detection of structural breaks. Table 3 results shows that the inclusion of the interaction terms do not change drastically the value of the elasticities. Moreover, the data continues to indicate two structural breaks at around the same time than in the initial specification. 9

10 Table 3 : Summary statistics on the posterior distribution of the elasticities different specifications Model Basic Model Model with covariates 7 Price-Income 8 Interaction Model Type of regime Statistics Price Income Price Income Price Income Mean Regime 1 St.dev Lower Upper Mean Regime 2 St.dev Lower Upper Mean Regime 3 St.dev Lower Upper Third, we also estimate the basic specification with the classical approach. In this setting, we also find that the model with two breaks and three regimes fits best the data (see Table 4). The timing of the first break is very similar than in the Bayesian approach (i.e. in the early eighties). The second break is however placed later than in the Bayesian approach (1991 rather than 1985). This difference is however due to the imposition of a minimum number of observations by regime in this estimation approach. 9 For the values of the elasticities, the first regime results are very comparable to those obtained in the Bayesian setting. The values in the second regime are also inconsistent with economic theory. Once again, this regime likely reflects a transition phase. In the third regime, the price elasticity is negative and statistically significant but very small at while the average elasticity in the Bayesian setting is close to zero. The income effect is somewhat larger than in the Bayesian setting but remain smaller than in the first regime. These differences may also be explained by the constrained imposed on the sample size in the second regime that thereby also affects the third regime sample. Overall however, the results obtained in both approach lead to similar conclusions: the price and income reactiveness of gasoline demand has declined after the second oil crisis. Moreover, the transition seems to have been rapid i.e. about a decade. Table 4 : Comparison of models with Classical approach m breaks RSS BIC LogLik The covariates are rate of unemployment and interest rate of Treasury-Bill. 8 The elasticities are calculated on the average of each regime. The statistics are computed by evaluating the posterior distribution for both parameters on average point. 9 In the classical approach, we fixed the parameter h at 0.20 meaning that each regime should have at least 20% of the total number of observations in the sample. 10

11 Table 5 : Classical approach - results for basic model for m=2 m=2 {1982-Q1 ; 1991-Q3} Type of regime Price Income Intercept Q2 Q3 Q4 Regime *** *** *** *** *** *** (0.0446) (0.0221) (0.1817) (0.0101) (0.0106) (0.0101) Regime *** *** 01367*** *** *** (0.0705) (0.1360) (1.3776) (0.0145) (0.0196) (0.0165) Regime * *** *** *** 01122*** *** (0.0284) (0.0568) (0.3891) (0.0067) (0.0068) (0.0068) *** p< 0.01, ** p< 0.05 and * p< 0.1. Values in parenthesis are the standard errors of parameters. 5. Conclusion This paper proposed an investigation of Canadian gasoline demand from 1965Q1 to 2012Q2 using a Bayesian approach to capture breakpoints. Different specifications have been tested in order to check robustness. We also use a frequentist approach with multiple breakpoints. The analysis shows the present of two structural breaks one after the second oil shocks around 1982 and the other around The elasticity values before 1982 are consistent with the consensus values reported in the traditional literature. Our analysis also confirms that there has been a decline in price and income reactiveness in recent decade. We find a price effect that is essential null. Our analysis suggests that the shift occurs rapidly during a transition phase in the eighties. During this transition, the fuel economy of vehicles drastically improved because of high gasoline price and fuel economy regulations. The recent strengthening of the fuel economy standards in the US and Canada could therefore very well lead to an additional reduction in the price and income reactiveness of future gasoline demand. In future research, we would like to add dynamic in our model in order to be able to distinguished short and long term changes. It would also be valuable to apply the same approach on US data. 11

12 Appendix 1 The dynamic programming algorithm and the principle of Bellman We describe briefly the principle of Bellman used by Bai and Perron (2003). Let u(i,j) be the recursive residual at time j obtained with sample starting at date i ; SSR (i, j) as the sum of the squares of the residues between dates i and j with i <j. The recursive relation between residuals is SSR (i, j) = SSR (i, j-1) + u(i,j) 2. Let the function SSR(T r,n ) be the sum of the squares of the residuals for an optimal number of structural breaks on the first n observations. The recursive problem is defined as follows: SSR(T m,t ) = Min mh j T h { SSR(T m 1,j ) + SSR(j + 1, T)} Where T is size of data, m is optimal number of breaks and h is a parameter for limiting a minimum size of each regime. Annex 1 Table 6 : Source of data, Statistics Canada Name Table Frequency footnotes Gasoline consumption Monthly data from 1965:1 to 2012:6 Disposable income Quarterly data from 1965:1-2012:2 Average retail prices for Monthly data from 2006:1 unleaded gasoline to 2012:6 Treasury bills: 3 months Monthly data from 1965:1 to 2012:6 Government bonds Monthly data from 1965:1 to 2012:6 Population Quarterly data from 1965:1 to 2012:2 Consumer Price Index Monthly data from 1965:1 to 2012:6 Gasoline Consumer Index Monthly data from 1965:1 price to 2012:6 Sales by reporting companies, exclusive of exports and sales to other reporting companies, and adjusted for exports and imports by non-reporting companies. All gasoline type fuels for internal combustion engines other than aircraft. (Harmonized System code ,.14 and.15) Unadjusted disposable income for Canada from sectors accounts, persons and unincorporated business. For regular unleaded self-serve gasoline, average city prices are weighted by provincial volume supplied and cities population to calculate the Canada average retail price. Treasury bills at the last Wednesday unless otherwise stated, Bank of Canada. Government of Canada marketable bonds, average yield, over 10 years. Estimates of population, Canada, provinces and territories. Consumer Price Index for Canada, base(2002=100) Gasoline consumer price index for Canada, base(2002=100) References Bai J and Perron P Estimating and testing linear models with multiple structural changes. Econometrica, Vol. 66, pp Bai J and Perron P Computation and Analysis of Multiple Structural Change Models. Journal of Applied Econometrics, Vol. 18 (1), pp Basso LJ and Oum TH Automobile Fuel Demand: A Critical Assessment of Empirical Methodologies. Transport Reviews, Vol. 27(4), pp Chernoff H and Zacks S Estimating the current mean of a Normal distribution which is subject to changes in time. Annals of Mathematical Statistics, Vol. 35, pp Chib Siddhartha Marginal likelihood from the Gibbs output. Journal of the American Statistical Association, Vol. 90, pp

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