Essays in energy economics and climate change adaptation

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Essays in Energy Economics and Climate Change

Adaptation

Thèse

Koffi Akakpo

Doctorat en économique

Philosophiæ doctor (Ph. D.)

Québec, Canada

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Essays in Energy Economics and Climate Change

Adaptation

Thèse

Koffi Akakpo

Sous la direction de:

Philippe Barla, directeur de recherche Vincent Boucher, codirecteur de recherche

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Résumé

Cette thèse, structurée en trois essais, porte sur l’économie de l’énergie et l’adaptation aux changement climatiques.

Le premier essai (Chapitre 1) étudie la présence de bris structurels dans la demande d’essence au Canada en réponse aux politiques sur l’efficacité énergétique visant à réduire les émissions de gaz à effet de serre (GES). L’analyse est effectuée sur des données trimestrielles de 1965 à 2012 en utilisant une approche bayésienne. Notre méthodologie est basée sur la méthode de Chib (1998) qui permet de multiples bris structurels et inconnus. Les résultats suggèrent l’existence de deux bris structurels définissant trois régimes. Dans le premier régime avant 1982, les élasticités prix et revenu se situent dans la plage de valeurs habituellement rapportée dans la littérature (-0,5 et 0,8 respectivement). Dans le troisième régime après 1986, le degré de réactivité du prix est quasiment nul alors que l’effet revenu est beaucoup plus faible (0,2). Le second régime fournit des résultats incohérents avec la théorie économique et correspond très probablement à une phase de transition perturbée par plusieurs chocs tels que la crise pétrolière, une récession et une réglementation de l’économie de carburant.

Dans le deuxième essai (Chapitre 2), nous examinons la réaction ou l’ajustement du prix de l’essence aux prix du pétrole brut. Certaines études indiquent que le processus d’ajustement est asymétrique c’est-à-dire que le prix de l’essence réagit beaucoup plus rapidement à la hausse des prix du pétrole brut qu’à sa baisse, ce qui est communément appelé en anglais “rockets and feathers” process. Cependant, d’autres études concluent à un processus d’ajustement symétrique. Dans cet essai, nous estimons un modèle dans lequel des périodes d’ajustement symétrique et asymétrique alternent. Nous estimons plus précisément un modèle à changement de régime markovien à deux états en utilisant des données hebdomadaires sur le prix de l’essence et du pétrole brut couvrant la période de 21 janvier 1991 au 29 octobre 2018. Nos

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résultats confirment l’idée selon laquelle la transmission des chocs du brut alternent entre processus d’ajustements symétrique et asymétrique. Nos résultats indiquent qu’en moyenne la durée du régime asymétrique est nettement plus longue que celle du régime symétrique soit 139 semaines contre 71. En outre, l’occurrence d’un régime asymétrique est bien plus importante après l’an 2000. Par ailleurs, nous observons que les variances des prix de l’essence et du brut sont beaucoup plus importantes en période d’ajustement asymétrique, ce qui laisse supposer que ce régime est marqué par l’instabilité des marchés. Nos résultats sont également robustes à l’analyse en sous-échantillons.

Les populations font face à des catastrophes naturelles (inondations, incendies et tornades par exemple) de plus en plus imprévisibles et dévastatrices. Dans le troisième essai (Chapitre 3), nous considérons un décideur responsable du lancement d’alertes d’inondation à l’intention de la population. La population est incertaine quant à la crédibilité des alertes et met à jour sa croyance à la suite de fausses alertes ou d’événements manqués. Nous montrons qu’une faible crédibilité amène le décideur à émettre des alertes pour de faible probabilités d’inondations. En pratique, ces probabilités sont fournies par des systèmes de prévisions hydrologiques. Dans un contexte réel, nous utilisons donc notre modèle pour comparer le bien-être social sous différentes prévisions hydrologiques. Nous observons que lorsque les prévisions incluent des scénarios extrêmes non réalistes, l’économie peut rester bloquée dans un état caractérisé par de nombreuses fausses alertes et une faible crédibilité.

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Abstract

This dissertation consists of three essays in energy economics and climate change adaptation. The first essay (Chapter I) focuses on the presence of structural breaks in the Canadian gasoline demand as a response to a fuel economy policy with a view to reducing greenhouse gas (GHG) emissions. Using a Bayesian approach, the analysis is carried out on quarterly data from 1965 to 2012. 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.

In the second essay (Chapter 2), we look at gasoline price response to crude oil prices by inves-tigating the relationship between gasoline prices and crude oil prices. Some studies indicate that the adjustment process is asymmetric with the price of gasoline responding much more quickly to crude oil price hikes than crude oil price drops what is referred as the “rockets and feathers” process. However, other studies find a symmetric adjustment process. In this essay, we estimate a model that allows switches between periods of symmetrical and asymmetrical adjustment processes. Specifically, we estimate a two-state Markov regime-switching model using weekly gasoline and crude oil prices in the United States from January 21, 1991 to October 29, 2018. My results support the idea that gasoline prices response to crude oil prices shocks sometimes with a symmetric pattern and sometimes with an asymmetric adjustment process. We find that the expected duration in the asymmetric regime is significantly longer

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than for the symmetric regime 139 weeks versus 71 weeks. In addition, the occurrence of a symmetric regime is greater up to 2000. Further, we find that the variances of gasoline and crude oil prices are much larger in periods of asymmetric regime suggesting that this regime is marked by market instability. My results are robust to subsamples analysis.

Populations are faced with more unpredictable and devastating natural disasters (e.g. floods, fires, tornadoes). In the third essay (Chapter 3), we consider a decision maker who is respon-sible for issuing flood warnings for the population. The population is uncertain about the credibility of the warnings and adjusts its beliefs following false alerts or missed events. We show that low credibility leads the decision maker to issue warnings for lower probabilities of flooding. In practice, those probabilities are provided by hydrological forecasts. We therefore use our model to compare welfare under alternative real-world hydrological forecasts. We find that when forecasts include non-realistic extreme scenarios, the economy may remain stuck in a state characterized by many false alerts and poor credibility.

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List of Tables

1.1 Log of the Bayes Factor for comparing the basic specification model with

dif-ferent number of breakpoints . . . 18

1.2 Summary statistics on the posterior distribution of the elasticities . . . 20

1.3 Summary statistics on the posterior distribution of the elasticities-different spec-ifications. . . 21

1.4 Comparison of models with classical approach . . . 22

1.5 Classical approach - results for basic model for m = 2 . . . 22

1.6 Log of the Bayes Factor for comparing the basic specification model with dif-ferent number of breakpoints . . . 28

1.7 Summary statistics on the posterior distribution of the elasticities with prior from Basso and Oum (2007) . . . 28

1.8 Source of data, Statistics Canada . . . 31

2.1 Unit root and cointegration tests . . . 44

2.2 Error correction term specification . . . 46

2.3 Symmetric ECM and ECM with threshold cointegration . . . 48

2.4 Short run price dynamics with the Markov switching ECM . . . 49

2.5 Short run price dynamics with a Regime Switching for subsamples . . . 51

2.6 Comparison of different Error Correction Models . . . 59

3.1 Calibrated Parameters . . . 75

3.2 Probabilities of flooding. . . 83

3.3 Number of alerts, false alerts, and missed events for the three forecasts. . . 85

3.4 Welfare . . . 87

3.5 Ex-post utility for each system (maximal values are in bold). . . 89

3.6 Diagnostics of prediction errors. . . 91

3.7 Number of alerts, false alerts, and missed events for the three forecasting sys-tems. Baseline belief updating, b = 1.546. . . 91

3.8 Number of alerts, false alerts, and missed events for the three forecasting sys-tems. Less elastic belief updating, b = 1.546. . . 92

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List of Figures

1.1 Gasoline consumption, disposable income and average retail price . . . 16

1.2 Posterior density and probabilities M₁ . . . 19

1.3 Posterior density and probabilities M2 . . . 19

1.4 Posterior density of price and income elasticities . . . 29

1.5 Posterior density of price and income elasticities with prior from Basso and Oum (2007) . . . 30

2.1 Weekly U.S. gasoline retail prices and oil spot prices, Energy Information Administration (EIA) . . . 42

2.2 Estimates of Smoothed probabilities for regimes . . . 50

2.3 Long-run residuals from OLS regression of Equation (2.1) . . . 59

2.4 In-sample analysis: Estimates of Smoothed probabilities for regimes . . . 60

2.5 Out-of-sample analysis: Estimates of filtered probabilities for regimes . . . 61

3.1 Decision rule. . . 77

3.2 Value function V (x, p), as a function of the population’s confidence p, for fixed values of x. . . 78

3.3 Value function V (x, p), as a function of the flood probability x, for fixed values of p. . . 78

3.4 Schematic representation of the hydrological forecasting system. . . 80

3.5 Average of prediction error for each system. . . 83

3.6 Kernel density of prediction error for each system. . . 83

3.7 Probabilities of flooding. (x|x > 0) . . . 84

3.8 Evolution of the population’s confidence and the forecast probability of flooding for each forecast. . . 84

3.9 Kernel density of the population’s confidence for each system. . . 85

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A la mémoire de ma mère Dovi SAGBO et de mon père Awlou AKAKPO, qu’ils reposent en paix, A mon épouse Adjoa ADJOKPA,

A mes frères et soeurs Yves-Urbain, Martin, Eugène, Odette, Pierrette.

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An investment in knowledge pays the best interest.

Benjamin Franklin (January 17, 1706-April 17, 1790)

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Remerciements

Je me sens privilégié d’avoir pu mener à terme cette thèse grâce à de bienveillantes personnes à qui j’aimerais ici adresser en ces mots ma reconnaissance.

Je commencerai donc par exprimer mon immense gratitude à mon directeur de thèse Philippe Barla et mon codirecteur Vincent Boucher pour la supervision de ma thèse, leurs encadrements, leurs disponibilités, leurs rigueurs scientifiques et surtout leurs conseils qui m’ont aidé à per-sévérer dans la rédaction de la thèse. Les professeurs Philippe Barla et Vincent Boucher ont été d’un soutien inestimable pour moi pendant les périodes de doutes et d’hésitations. Je vous en suis reconnaissant.

Je voudrais également adresser mes remerciements à Marie-Amélie Boucher pour sa disponi-bilité lors de l’élaboration de cette thèse.

Je ne saurais remercier assez la Faculté des Sciences Sociales, le département d’Économique et le CREATE pour le soutien financier et matériel qui a permis d’avoir le cadre de travail propice à la réalisation et la vulgarisation de ma recherche doctorale.

Toute ma gratitude au Père Marian Schwark et au professeur Akuété Pédro Santos qui ont rendu possible cette thèse doctorale.

J’aimerais aussi adresser mes remerciements à tous mes collègues et amis Ibrahima Sarr, Sosthène-Blanchard Conombo, Jean-Louis Bago pour le soutien réciproque tout au long de ma thèse. Merci également à Isaora Zefania Dialahy, Daouda Belem, Kossi Thomas Golo, Kouamé Marius Sossou, Finagnon Antoine Dedewanou, Morvan Nongni Donfack, Rolande Kpekou Tossou, Guy Morel Kossivi Amouzou Agbe, Elysee Aristide Houndetoungan et Ibrahima Diallo pour toutes nos discussions dans le local 2241 du Pavillon De Sève, aux cours d’anglais et lors des séminaires de Jeudi Midi.

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Mes remerciements vont également à mes frères et mes soeurs au Togo et au Bénin pour leur présence permanente depuis toujours, à mes amis à Québec pour vos soutiens divers.

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Foreword

The three chapters of this thesis are separate articles submitted or in preparation for submis-sion to peer-reviewed scientific journals.

Chapter 1 was written jointly with my thesis director Philippe Barla, full professor in the Department of Economics at Université Laval.

Chapter 2 was written jointly with my co-supervisor Vincent Boucher, full professor in the Department of Economics at Université Laval, and Marie-Amélie Boucher1, full professor in the Department of Civil and Building Engineering at University of Sherbrooke.

I am the principal author of the three chapters presented in this thesis.

1_{Here is her address:} _{Department of Civil and Building Engineering, Université de Sherbrooke;}

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Introduction

In recent decades, the reliance on fossil fuel energy and climate change have been receiving a great deal of attention from economists. Many governments around the world are struggling to reduce greenhouse gas (GHG) emissions, especially from the transportation sector, but they also have to mitigate and adapt to current climate changes. This thesis focuses on energy and climate change economics. It contains three essays. The first two essays are about gasoline consumption and the determination of gasoline prices while the last essay is about climate change adaptation.

Fossil fuel energy sources are key components of modern life. The absence or lack of control over the use of transport and fossil fuel energy prices can have negative economic conse-quences. This was experienced by most industrialized countries, including the United States and Canada, in the aftermath of the oil crises of the 1970s. These oil crises led to unprece-dented disruptions and severe economic effects for several industrialized countries. Gasoline expenditures in the early 1980s for the average U.S. (Canadian) household reached just over 5% (just under 6%) of income before taxes, according to U.S. Energy Information Administration2 estimates (see Figure 1.1, panel C); whereas, it had been estimated at the beginning of the 1970s at around 3% (4.6%).3 For the first time in a long time, most industrialized countries faced economic uncertainty and concerned about the economic consequences of fossil fuel de-pendence. As result, governments the world over shared the sense of urgency expressed by experts from different fields about the need to design and implement appropriate policies to facilitate access to transportation and energy for different economic sectors.

During the 1970s and 1980s, the problem was to decrease energy sales and consumers

every-2

Statistics on U.S. are available at the following URL:https://www.eia.gov/todayinenergy/detail.php?id=9831

3_{Statistics on Canada were computed by the authors based on the data reported in the Statistics Canada}

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where were aware of the need for adjustment in energy consumption patterns. Surveys (Dahl and Sterner,1991;Espey,1998;Basso and Oum,2007;Dahl,2012) highlight that researchers have been unanimous about the importance of understanding the sensitivity of fuel demand to changes in prices and real disposable income of households. Indeed, this plays a key role in shaping policies related to climate change, optimal taxation in the transportation sector, national security or urbanization. However, recent works suggest that the values of these pa-rameters may have changed in recent decades (Hughes et al.,2006;Small and Dender,2007;

Park and Zhao,2010;Liu,2014). Indeed, several behavioral and structural factors such as the share of passenger-kilometers traveled in transit compared to other modes of transportation, urban sprawl, the growth of multiple income households as well as the dramatic fluctuations of gasoline prices have changed making consumers less sensitive to price increases (Polzin and Chu,2005;Hughes et al.,2006). This indicates the scale of the problem of energy dependence from the road transportation sector and thereby its GHG emissions. Overall, there is some ev-idence of changes in price and reactiveness of gasoline demand overtime. However, none of the existing studies proceeds to a formal structural break analysis. Yet, a break triggers a change in the relevant parameters of fuel demand, mainly price and income elasticities. Moreover, each new structural break involves an estimation of a new set of parameters. This estimation with only recent data after a break is more accurate than in the case of an estimation based on the entire dataset.

We address this issue in the first essay by focusing on the existence of structural breaks, and their timing, in the Canadian demand for gasoline from 1965 to 2012. Given the current eco-nomic and political context in Canada, this analysis is particularly relevant for two reasons. First, the transportation sector is the second most important source of GHGs with a share of 19% with gasoline accounting for over 68% of total emissions (WWF-Canada,2012). In addi-tion, the Company Average Fuel Consumption (CAFC) standards have been mandatory since 2011 (McCauley, 2011) to ensure Canada meets its global climate change obligations under the Kyoto Protocol. Second, since the trend in gasoline consumption per capita is changing4_,

we considered it advisable to explore the data in order to identify relevant information for the estimation of the appropriate price and income elasticities.

Another significant development in recent decades has been the debate over the determination

4

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of the price of gasoline, one of the key determinants of gasoline consumption. A common perception among consumers is that gasoline prices exhibit asymmetric behavior with the price of gasoline responding much more quickly to crude oil price hikes than crude oil price drops, which is referred to as the “rockets and feathers” process. This suggests that the gasoline market is non-competitive according to some economists. Bacon (1991) was the first to provide empirical evidence. Some studies confirm the consumers’ perception of the rockets and feathers process (Bremmer and Kesselring, 2016; Grasso and Manera, 2007; Borenstein et al., 1997). However, many other studies find that the marketing of gasoline distribution is competitive (Douglas, 2010; Oladunjoye, 2008; Bachmeier and Griffin,2003;Karrenbrock,

1991). It is also possible that both processes (asymmetric and symmetric) are played under different circumstances in the gasoline market, which is not covered in this literature. In the second essay, we analyze how the price of gasoline responds to crude oil price hikes and drops. To do so, we exploit weekly gasoline and crude oil prices in the United States from January 21, 1991 to October 29, 2018. We contribute by developing an empirical framework to test for this possibility. Specifically, we estimate a two-state Markov-switching model that allows switches between periods of symmetrical and asymmetrical adjustment processes. The second essay is along the same line as the first essay. Indeed, its methodological approach identifies relevant information, or data for each type of gasoline response.

The third essay of this dissertation is related to climate change and presents an independent study by investigating whether information about a population’s beliefs is relevant for a de-cision maker who is responsible for issuing flood warnings. Since the 1990s, there has been a growing awareness at the international level about the resurgence of natural disasters such as floods, fires and tornadoes. For example, today, in several countries, floods are more frequent, more unpredictable, and more devastating. In most countries, government authorities have been responsible for managing natural disasters. Part of this management consists of issuing warnings in the event of natural disasters based on decision models. Traditional (Murphy,

1977; Murphy, 1985; Katz and Murphy, 1997; Richardson, 2000) and more recent realistic (Zhu et al., 2002;Matte et al., 2017) decision models only consider natural disaster risk as a state variable. Basically, in a static setting, the decision maker considers only the probability of an adverse event (e.g. floods) and disregards the potential impact of the flood warning on the population’s beliefs and in turn on its optimal decision-making rule. However, these

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warnings are only as effective as their credibility to the population; too many false alerts or missed events are likely to affect this credibility. In this essay, we study the optimal flood warning rule under endogenous credibility. Specifically, we develop a dynamic (versus static in the literature) model of a decision maker. In a theoretical dynamic setting, the population’s beliefs evolve through time as a result of an updating process and consequently change the decision maker’s optimal rule. We use our decision model to compare the performance of different hydrological forecasts from the province of Quebec, Canada.

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Bibliography

Bachmeier, L. J. and J. M. Griffin (2003): “New Evidence on Asymmetric Gasoline Price Responses.” Review of Economics and Statistics, 85, 772–776.

Bacon, R. W. (1991): “Rockets and feathers: the asymmetric speed of adjustment of UK retail gasoline prices to cost changes,” Energy Economics, 13, 211–218.

Basso, L. J. and T. H. Oum (2007): “Automobile Fuel Demand: A Critical Assessment of Empirical Methodologies,” Transport Reviews, 27, 449–484.

Borenstein, S., A. Cameron, and R. Gilbert (1997): “Do gasoline prices respond asym-metrically to crude oil price changes?” Quarterly Journal of Economics, 112, 305–339. Bremmer, D. S. and R. G. Kesselring (2016): “The relationship between U.S. retail

gaso-line and crude oil prices during the Great Recession: "Rockets and feathers" or "balloons and rocks" behavior?” Energy Economics, 55, 200–210.

Dahl, C. and T. Sterner (1991): “Analysing gasoline demand elasticities: a survey,” Energy Economics, 13, 203–210.

Dahl, C. A. (2012): “Measuring global gasoline and diesel price and income elasticities,” Energy Policy, 41, 2–13.

Douglas, C. C. (2010): “Do gasoline prices exhibit asymmetry? Not usually!” Energy Economics, 32, 918–925.

Espey, M. (1998): “Gasoline demand revisited: an international meta-analysis of elasticities,” Energy Economics, 20, 273–295.

Grasso, M. and M. Manera (2007): “Asymmetric error correction models for the oil-gasoline price relationship,” Energy Policy, 35, 156–177.

Hughes, J., C. Knittel, and D. Sperling (2006): “Evidence of a Shift in the Short-Run Price Elasticity of Gasoline Demand,” .

Karrenbrock, J. D. (1991): “The behavior of retail gasoline prices: symmetric or not?” Federal Reserve Bank of St. Louis, 73, 19–29.

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Katz, R. and A. Murphy (1997): Economic value of weather and climate forecasts, New York: Cambridge University Press.

Liu, W. (2014): “Modeling gasoline demand in the United States: A flexible semiparametric approach,” Energy Economics, 45, 244–253.

Matte, S., M.-A. Boucher, V. Boucher, and F. F. Thomas-Charles (2017): “Moving beyond the cost–loss ratio: economic assessment of streamflow forecasts for a risk-averse decision maker,” Hydrology and Earth System Sciences, 21, 2967–2986.

McCauley, S. (2011): “Canada’s Passenger Automobile and Light Truck Greenhouse Gas Emission Regulations for Model Years 2011-2016,” Tech. rep., Environment Canada. Murphy, A. (1977): “The value of climatological, categorical and probabilistic forecasts in

the cost-loss ratio situation,” Monthly Weather Review, 105, 803–816.

——— (1985): “Decision Making and the Value of forecasts in a Generalized Model of the Cost-Loss Ratio Situation,” Monthly Weather Review, 113, 362–369.

Oladunjoye, O. (2008): “Market structure and price adjustment in the US wholesale gasoline markets,” Energy Economics, 30, 937–961.

Park, S. Y. and G. Zhao (2010): “An estimation of U.S. gasoline demand: A smooth time-varying cointegration approach,” Energy Economics, 32, 110–120.

Polzin, S. E. and X. Chu (2005): “A closer look at Public Transportation Mode Share Trends,” Journal of Transportation and Statistics, 8, 41–53.

Richardson, D. (2000): “Skill and relative economic value of the ECMWF ensemble predic-tion system,” Quarterly Journal of the Royal Meteorological Society, 126, 649–667.

Small, K. A. and K. V. Dender (2007): “Fuel Efficiency and Motor Vehicle Travel: The Declining Rebound Effect,” The Energy Journal, 28, 25–52.

WWF-Canada (2012): “Road Transportation Emissions Reduction Strategies,” Tech. rep., World Wildlife Fund - Canada.

Zhu, Y., Z. Toth, R. Wobus, and K. Mylne (2002): “The economic value of ensemble-based weather forecasts,” Bulletin of the American Meteorological Society, 83, 73–83.

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Chapter 1

The Analysis of Structural Breaks in

Gasoline Demand Using a Bayesian

Approach

Résumé

Cet éssai porte sur la présence de bris structurels dans la demande d’essence au Canada en réponse aux politiques sur l’efficacité énergétique visant à réduire les émissions de gaz à effet de serre (GES). L’analyse est effectuée sur données trimestrielles de 1965 à 2012 en utilisant une approche bayésienne. Notre méthodoloie est basée sur la méthode de Chib (1998) qui permet de multiples bris structurels et inconnus. Les résultats suggèrent l’existence de deux bris structurels définissant trois régimes. Dans le premier régime avant 1982, les élasticités prix et revenu se situent dans la plage de valeurs habituellement rapportée dans la littérature (-0,5 et 0,8 respectivement). Dans le troisième régime après 1986, le degré de réactivité du prix est proche de zéro alors que l’effet revenu est beaucoup plus faible (0,2). Le second régime fournit des résultats incohérents avec la théorie économique et correspond très probablement à une phase de transition perturbée par plusieurs chocs tels que la crise pétrolière, une récession et une réglementation de l’économie de carburant.

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Abstract

This essay focuses on the presence of structural breaks in the Canadian gasoline demand using a Bayesian approach. The analysis is carried out on quarterly data from 1965 to 2012. 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.

JEL Codes: C11, Q41, Q54

Keywords: Gasoline demand, price elasticity, income elasticity, structural breaks, Bayesian approach

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1.1 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 -0.6. 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 et al. (2006) 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 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 -0.30 to -0.05 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 2008. 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 address this gap by testing for the existence of structural breaks in the Canadian demand for gasoline from 1965 to 2012.

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

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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, 2004andSpirling,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 1985. During the first regime from 1965 to 1982, the expected value of the posterior distribution of the price elasticity is -0.493 (standard deviation at 0.077). For the income elasticity, the expectation is 0.831 (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 0.226 (0.121). The period covering the second regime between 1982 and 1986 is characterized by a major recession caused

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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 1.2 discusses the econometric issues to model structural breaks and presents the Bayesian approach with its particularities. Section

1.3 describes the data and provides some basic descriptive statistics. We outline the results and the robustness analysis in Section1.4. Finally, Section1.5discusses the main conclusions and policy implications.

1.2 Methodology

Most empirical studies of gasoline demand adopt a log-log linear functional of the following form:

lnGt= β0+ β1lnPt+ β2lnYt+ β 0

3Xt+ εt (1.1)

where Gtis gasoline consumption per capita at time t, Ptis the retail price of gasoline per liter,

Ytis the real disposable income per capita, Xt represents others potential covariates and εt is

the error term. β1 and β2 measure respectively the price and income elasticities. Specification

1.1 does however assume that the parameters are constant over time. Thus, we use instead specification 1.2which allows for an undetermined m number of shifts or breakpoints in the parameters values.

lnGt= β0,r+ β1,rlnPt+ β2,rlnYt+ β 0

3,rXt+ εt (1.2)

with: r = 1, 2, · · · , m + 1, τ_r−1< t ≤ τr and εt∼ N (0, σr2).

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 τr that also need to be estimated.2 For example, if the data revealed one

breakpoint this means two regimes with different gasoline demand parameters.

1

For the US, the fuel economy rose by up to 25% during this period (BST, 2017). While comparable statistics are unavailable for Canada, the same phenomenon occurred in Canada (Beauregard-Tellier,2004).

2

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The number and timing of structural breaks is treated as a problem of model selection based on some information criterion (Kass and Raftery, 1995; Raftery, 1995; Chib, 1995; Bai and Perron,1998; Chib and Jeliazkov, 2001). The optimal number of breaks may be determined either through a sequential (Yao, 1988; Liu et al., 1997; Girón 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 and Perron,1998).

In sum, conventional approaches to the point problem fail to address the change-point problem appropriately either by requiring excessive prior knowledge, by ignoring state-dependent

The estimation can be carried out either within the classical (frequentist) or Bayesian frame-work. 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, interest centers on the posterior distribution, which is proportional to the likelihood times the prior distribution. Note that the estimation is based on Markov chain Monte Carlo (MCMC) simulation methods which depend 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 of the change-point model based on a Markov process. A discrete random variable s_tis 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 constrained:

P =            p11 p12 0 . . . 0 0 p22 p23 . . . 0 .. . ... ... . .. ... 0 0 . . . pmm pm,m+1 0 0 . . . 0 1           

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where p_ij = P r(st= j|st−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.

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. Basically, given state i at time t, states i and i + 1 are the only accessible states at time t + 1. 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). Given data and m the number of break points, Chib’s changepoint model applies a Markov chain Monte Carlo scheme for sampling the posterior distribution, which is proportional to the likelihood times the prior distribution:3

π (S, θθθ, P|data) ∝ π(θθθ, P) × f (data|θθθ, S, P) (1.3) where data = (G_1:T, Z1:T), Z1:T = (11:T, P1:T, Y1:T, X1:T) contains all the explanatory

vari-ables, θθθ = β0_r, σ_r2 0 , r = 1, · · · , m + 1

the parameters of the model (1.2), S = (s1, s2, · · · , sT)

the unobserved states. TheChib(1998)’s approach is completed by setting prior distributions to θθθ and P. The sampling algorithm consists of three steps: (i) sampling the states S condi-tioned on the data and the other parameters; (ii) simulating the parameters θθθ conditioned on the data, S and the other parameters; (iii) updating the non-zero elements in the matrix P given the value of S. See Appendix 1.6.1,1.6.2and 1.6.3 for more details.

S | data, θθθ, P θ

θθ | data, S, P

P | S

(1.4)

The model selection (i.e. the optimal number of breaks) is based on the Bayes factors method which compares the marginal likelihoods (ML) across models (Kass and Raftery,1995;Raftery,

1995;Chib,1995;Chib and Jeliazkov,2001). Chib(1995) has developed a simple computation technique for the estimation of the marginal likelihood. For a model with m breaks (Mm),

the marginal likelihood is calculated as follows: M L(lnG1:T|Z1:T, Mm) =

Z

f (lnG1:T|Z1:T, θ, P, Mm)π(θ, P |Mm)d(θ, P ) (1.5) 3

In this case, Chib applies Monte Carlo scheme to the posterior density π (S, θθθ, P|data) rather than π (θθθ, P|data) after the parameter space is augmented to include the unobserved states S.

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where f (·) is the likelihood function, π(θ, P |M_m) is a prior density on the parameters. Chib

(1995) re-expressed Equation (1.5) as a combination of the likelihood function, the prior den-sity and the posterior denden-sity π(θ, P |lnG_1:T, Z1:T, Mm) at any point (θ, P ) in the parameter

space (see Appendix 1.6.4 for details).

The Bayes factor is then defined as the ratio between the marginal likelihood of two models M_m and M_n:

Bm,n =

M L(lnG1:T|Z1:T, Mm)

M L(lnG1:T|Z1:T, Mn)

. (1.6)

The choice of model is then determined by a set of rules proposed byJeffreys(1961). Essentially, a value of Bm,nlarger than one means that model Mmbetter fits the data than Mn. Moreover,

the larger is this ratio the stronger is the conclusion. The evidence is considered decisive if Log10(Bm,n) is greater than 2.

The log-likelihood function is

Logf (lnG1:T|Z1:T, θ, P, Mm) = T

X

t=1

Log [f (lnGt|Zt, θ, P, Mm)] . (1.7)

Now given Chib (1998) transition matrix, we have:

f (lnGt|Zt, θ, P, Mm) = m

X

k=1

f (lnGt|Zt, θ, P, st= k)P r(st= k|G1:t−1, Z1:t−1, θ, P ) (1.8)

where f (lnGt|Zt, θ, P, st = k) is the density of observation at time t conditional to the state

st= k, P r(st= k|G1:t−1, Z1:t−1, θ, P, ) is the marginal probability to be in regime k which is

defined through Bayes’ rule:

P r(st= k|G1:t−1, Z1:t−1, θ, P ) = k X i=k−1 P r(st= k|st−1= i)P r(st−1= i|G1:t−1, Z1:t−1, θ, P ). (1.9) For comparison purposes, we also estimate the model using a frequentist approach. Specifically, we useBai 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 (Bai and Perron, 1998). In each regime, a minimal number h of observations is imposed such that τk− τk−1 ≥ h. The optimal number

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and timing of breaks as well as the parameter values are derived by minimizing the following function: LS(β, τ ) = (lnG1:T − β 0 Z1:T) 0 (lnG1:T − β 0 Z1:T) = m+1 X k=1 τk X t=τk−1+1 (lnGt− β 0 Zt)2. (1.10)

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(β, Γ) given m. The estimation is basically based on a dynamic programming and the use of the Bellman principle, unlike a standard grid search procedure. The constraint on the minimum duration of a regime considerably decreases the number of potential regimes to be tested (see Appendix 1.6.5 for more details). 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.4

1.3 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 1965. 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 2009.5 Table1.8in Appendix1.6.6 details the sources of the variables used in the empirical analysis.

Panel A of Figure 1.1shows per capita gasoline consumption and real gasoline price over the analysis period. Panel B shows per capita gasoline consumption and real disposable income.

4_{The Bayesian results are obtained with Matlab while the classical results are computed in R.} 5

A Canadian dollar was worth 0.87 U.S. dollar in 2009. Published on

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Panel A: Gasoline consumption and average retail price Years 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 200 250 300 350 400 3000 4000 5000 6000 7000

Gasoline Consumption in liters per capita

Real Disposab

le Income

,$ of 2009

Real.Dis.Inc Gas.Cons.Cap

Panel B: Gasoline consumption and disposable

Years 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 200 250 300 350 400 60 80 100 120

Gasoline Consumption in liters per capita

Retail A ver age Pr ice ,$ of 2009 Retail.Price Gas.Cons.Cap

Panel C: Share of gasoline expenses in relation to income (%)

Years 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 3.5 4.0 4.5 5.0 5.5

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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 to 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.

Panel C of Figure 1.1 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 2011. 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.

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1.4 Results

First, we analyze the results of the most basic specification with gasoline consumption depend-ing upon price, disposable income and quarterly dummies to account for seasonable effects. Table 1.1 shows the Log₁₀(Bm,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 1.1allow comparing the likelihood of models with m breaks (Mm) with respect to models with n breaks (Mn). A positive value means that Mm

is more likely than Mn while the opposite holds for negative values. Thus, it appears that

M2 fits best the data. However, Jeffreys’ rule indicates that the Log10(Bm,n) should be higher

than 2 in order to decisively conclude that Mmis better than Mn. In our setting, this means

that we cannot clearly select M2 over M1 as Log10(B2,1) is well below two.

Table 1.1: Log of the Bayes Factor for comparing the basic specification model with different number of breakpoints Models n break(s) 0 1 2 3 4 m break(s) 0 0 -139.4915 -139.6303 -138.2065 -132.1109 1 139.4915 0 -0.1387 1.2815 7.3807 2 139.6303 0.1387 0 1.4238 7.5194 3 138.2065 -1.2851 -1.4238 0 6.0956 4 132.1109 -7.3807 -7.5194 -6.0956 0

Thus, we analyze the results of both models. Figure 1.2and 1.3plot the posterior density of regime change (Panel A) and the posterior probabilities of states (Panel B) for M1 and M2

respectively.

The posterior density of regime change for M₁is 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 1.3 shows a first break around the last quarter of 1982 and a second break at the end of 1985. 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 1.4 illustrates the posterior density of the price and income elasticities for M₁ and M₂. We also show the results for M₀ as it is the model most often estimated in the literature. Table 1.2 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

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Panel A Panel B

19600 1970 1980 1990 2000 2010 2020 0.05

0.1 0.15

Posterior Density of a Regime Change Probabilities

Years 19650 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Years Pr(S t = k | Y t , X t )

posterior probability of regime

Regime1 Regime2

Figure 1.2: Posterior density and probabilities M1

Panel A Panel B 19600 1970 1980 1990 2000 2010 2020 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Posterior Density of a Regime Change Probabilities

Years 19650 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Years Pr(S t = k | Y t , X t )

posterior probability of regime

Regime1 Regime2 Regime3

Figure 1.3: Posterior density and probabilities M2

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.

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Table 1.2: Summary statistics on the posterior distribution of the elasticities

Bayesian results - Basic Model

Type of regime Statistics Price Income Intercept Q2 Q3 Q4

No break Regime 1 Mean -0.3338 0.3396 4.2058 0.1047 0.1658 0.0904 St.dev 0.0521 0.0479 0.2718 0.0184 0.0189 0.0185 Lower -0.4356 0.2440 3.6677 0.0688 0.1286 0.0542 Upper -0.2312 0.4331 4.7548 0.1411 0.2027 0.1264 One break Regime 1 Mean -0.4983 0.8305 0.8413 0.0948 0.1279 0.0931 St.dev 0.0735 0.0480 0.3514 0.0215 0.0224 0.0213 Lower -0.6367 0.7366 0.1322 0.0521 0.0841 0.0516 Upper -0.3450 0.9259 1.5119 0.1369 0.1722 0.1342 Regime 2 Mean 0.0084 0.2150 3.7548 0.0819 0.1337 0.0724 St.dev 0.0637 0.1198 0.8200 0.0157 0.0167 0.0167 Lower -0.1270 -0.0063 2.0420 0.0512 0.1010 0.0394 Upper 0.1281 0.4714 5.2750 0.1131 0.1665 0.1046 Two breaks Regime 1 Mean -0.4930 0.8309 0.8165 0.0954 0.1290 0.0936 St.dev 0.0766 0.0482 0.3537 0.0214 0.0228 0.0216 Lower -0.6335 0.7333 0.1234 0.0526 0.0842 0.0509 Upper -0.3302 0.9223 1.5203 0.1385 0.1739 0.1361 Regime 2 Mean 0.3149 0.4861 0.1193 0.0323 0.0152 0.0286 St.dev 1.9847 1.0686 2.2159 2.2104 2.2063 2.2128 Lower -3.6108 -1.5987 -4.2919 -4.3265 -4.2870 -4.3843 Upper 4.1456 2.6121 4.4928 4.4861 4.4038 4.4370 Regime 3 Mean 0.0032 0.2264 3.6778 0.0819 0.1333 0.0715 St.dev 0.0649 0.1210 0.8274 0.0157 0.0170 0.0168 Lower -0.1320 0.0080 1.9405 0.0511 0.1000 0.0385 Upper 0.1231 0.4819 5.1796 0.1127 0.1662 0.1041

St.dev indicates standard deviation, and Lower and Upper indicate 95 Bayesian credible intervals.

1.4.1 Robustness analysis

In order to access the robustness of our results, we examine several alternative specifications.

Covariates and price-income models

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

1.3) 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

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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 1.3 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.

Table 1.3: Summary statistics on the posterior distribution of the elasticities-different speci-fications

Model Basic Model Model with covariates Price-Income

Interaction Model

Type of regime Statistics Price Income Price Income Price Income

Regime 1 Mean -0.4930 0.8309 -0.5943 0.7493 -0.4766 0.8205 St.dev 0.0766 0.0482 0.0977 0.1037 0.0798 0.0547 Lower -0.6335 0.7333 -0.7671 0.5505 -0.6224 0.7142 Upper -0.3302 0.9223 -0.3858 0.9572 -0.3096 0.9278 Regime 2 Mean 0.3149 0.4861 0.1289 0.2488 1.221 0.6657 St.dev 1.9847 1.0686 2.1292 1.7588 4.239 1.2088 Lower -3.6108 -1.5987 -4.0147 -3.1931 -7.1011 -1.7181 Upper 4.1456 2.6121 4.4059 3.7298 9.5761 3.0138 Regime 3 Mean 0.0032 0.2264 -0.0568 0.295 0.0334 0.1979 St.dev 0.0649 0.1210 0.0754 0.1463 0.0622 0.1106 Lower -0.1320 0.0080 -0.2083 0.0098 -0.0955 0.0002 Upper 0.1231 0.4819 0.0865 0.5861 0.1483 0.432

Classical approach based on Bai and Perron (2003)

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

1.4). 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. 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 -0.05 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

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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 1.4: Comparison of models with classical approach

m breaks 0 1 2 3

SSR 1.4018 0.1618 0.1310 0.1128

BIC -356.8253 -730.3234 -733.6644 -725.4757

LogLik 196.7772 383.5262 393.0673 388.9728

Table 1.5: Classical approach - results for basic model for m = 2

Price Income Intercept Q2 Q3 Q4

Regime 1 -0.3939*** 0.8177*** 0.5148*** 0.1024*** 0.1348*** 0.0977*** (0.0446) (0.0221) (0.1817) (0.0101) (0.0106) (0.0101) Regime 2 -0.0370 -0.3996*** 9.2478*** 01367*** 0.2416*** 0.1459*** (0.0705) (0.1360) (1.3776) (0.0145) (0.0196) (0.0165) Regime 3 -0.0488* 0.3696*** 2.6597*** 0.0742*** 01122*** 0.0539*** (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. Break dates are {1982-Q1; 1991-Q3}

.

Importance of Prior Knowledge

Finally, considering the results obtained from the two approaches (Bayesian and classical), we redid the basic model estimation exercise with the Bayesian approach by setting out informative priors on some parameters, especially price and income elasticities. For this we used the values of the Basso and Oum (2007)’s survey. The numerical values used to calibrate these distributions are: (β1,r, σβ1,r) = (−0.335, 0.024) for the price elasticity and

(β2,r, σβ2,r) = (0.467, 0.096) for the income elasticity. At first, we analyzed how these priors

can be a determinant for the best model. Our results clearly suggest that the model M1 with

one break is the best model. Indeed, the Bayes factor Log₁₀(B1,2) = 4.6507 is higher than 2.

Both models are better than the model without break M0. Second, we analyzed the effect

on posterior distributions of price and income elasticities. We highlighted three main points, the first of these being structural breaks which are almost identical to the previous results of

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the basic model M₁ and M₂. The second main point is the statement that the price and income reactiveness of gasoline demand has declined after the second oil crisis according to the posterior distributions (see Table 1.7 and Figure 1.5 in Appendix 1.6.6). However, the average values are a little higher than those found previously. The third point is the coherence of results with economic theory in all regimes, particularly in the second of the model with two breaks.

1.5 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 1986. 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.

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1.6 Appendix

1.6.1 Sampling of Latent State Variables

The algorithm of sampling the states is based on Chib (1996). Let S_t = (s1, s2, · · · , st) and

St+1 = (st+1, st+2, · · · , sT) indicates the state history up to time t and the history of state

from t + 1 to T , respectively. Basically, the joint density of states and observations is: p(s1, s2, · · · , sT|data, θθθ, P) = p(sT|data, θθθ, P) × p(sT −1, · · · , s2, s1|sT, data, θθθ, P)

= p(sT|data, θθθ, P) × · · · × p(st|St+1, data, θθθ, P)×

· · · × p(s1|S2, data, θθθ, P).

(1.11)

where data = (G_1:T, Z1:T), Z1:T = (11:T, P1:T, Y1:T, X1:T) contains all the explanatory

vari-ables, θθθ = β_r0, σ_r2 0 , r = 1, · · · , m + 1

the parameters of the model (1.2). By using Backward algorithm, a simple decomposition of1.11 yields

This expression depends on two terms. The first term p(s_t+1|s_t, data, θθθ, P) should be obtained from the transition probabilities while the second term p(st|data, θθθ, P) required a recursive

calculation defined in Equation 1.9.

1.6.2 Sampling of transition probabilities

As proposed by Chib (1998), we use Beta prior,

pii ∼ Beta(a, b), i = 1, · · · , m p(pii|a, b) = Γ(a + b) Γ(a)Γ(b)p a−1 ii (1 − pii)b−1. (1.13)

Only the non-zero elements p_ii of the diagonal and p_ij of the upper diagonal in the matrix P need to be revised. Let niibe the number of one-step transitions from state i to i and nij be the

number of one-step transitions from state i to j = i + 1 in the sequence S_n= (s1, s2, · · · , sn),

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The numerical values used to calibrate this prior are: a = 10 and b = 1. Thus, the prior mean of pii is equal to

a

a + b ' 0.91. It yields a higher probability of staying in the initial regime i than to switch out of it.

1.6.3 Sampling algorithm of posterior distributions of elasticities

Consider the following linear model equation and assume m structural breaks (τ₁, τ2, · · · , τm):

lnGt= β0,r+ β1,rlnPt+ β2,rlnYt+ β 0

3,rXt+ εt (1.15)

with: r = 1, 2, · · · , m + 1, τr−1< t ≤ τr and εt∼ N (0, σr2).

Below, we summarize the steps of the computation of posterior distribution of β_r and σ_r2. Using normal and gamma priori distribution for respectively βr and σr2

βr ∼ N b0, B0−1 , σ−2_r ∼ G µ₀ 2 , ν0 2 , (1.16)

then the posterior distribution of βr given Gτr−1:τr, Zτr−1:τr, σ 2

r, S and the posterior

distri-bution of σ2_r given G_τ_r−1_:τ_r, Zτr−1:τr, βr, S are also normal and gamma distribution:

βr| Gτr−1:τr, Zτr−1:τr, σ 2 r, S ∼ N ¯βr, Ω−1r , σ−2_r | Gτr−1:τr, Zτr−1:τr, βr, S ∼ G µ_r 2 , νr 2 , (1.17) where: ¯ βr = Z_τ0_r−1_:τ_rZτr−1:τr + B −1 0 −1 Z_τ0_r−1_:τ_rGτr−1:τr + B −1 0 b0 Ωr = Z_τ0_r−1_:τ_rZτr−1:τr + B −1 0 −1 µr = µ0+ τr− τr−1+ 1 νr = ν0+ G 0 τr−1:τrGτr−1:τr + b 0 0B −1 0 b0− ¯β 0 rΩ−1r β¯r

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In this essay, we use two sets of parameters to calibrate these distributions. First, we consider absolutely flat priors on all βr, r = 1, · · · , m. So, these priors are noninformative and that make

this Bayesian analysis equivalent to the classical distribution theory. Second, the Bayesian estimation was redone for the basic model in equation 1.2 by setting out informative priors only on price β_1,r, r = 1, · · · , m and income β2,r, r = 1, · · · , m elasticities using values of

the Basso and Oum (2007) survey. In the literature, the price and income elasticities are in the range of -0.335 and 0.467 respectively with a variance in the range of 0.024 and 0.096 respectively. Given that we do not have any information about others variables, we consider flat priors for them.

Furthermore, the hyper-parameters (µ₀, ν0) of variance σ2r, r = 1, · · · , m of the error term in

the context of model (1.15) are chosen in Maheu and Gordon (2008). The numerical values used to calibrate the hyper-parameters (µ₀, ν0) are: µ0 = 25 and ν0 = 10. In this case, the

prior mean of σ_r2 is equal to ν0 µ0− 2

' 0.43.

1.6.4 Marginal likelihood from the Gibbs sampler

The method of Chib (1995) consists on re-expressing the marginal likelihood which is simple to implement. For a model Mm, the marginal likelihood (M L) function is

M L(lnG1:T|Z1:T, Mm) =

Z

f (lnG1:T|Z1:T, θ, P, Mm)π(θ, P |Mm)d(θ, P )

where Z1:T = (11:T, P1:T, Y1:T, X1:T) contains all the explanatory variables, f (·) is the

likeli-hood function, π(θ, P |M_m) is a prior density on the parameters. It may be re-expressed as ˆ M L(lnG1:T|Z1:T, Mm) = f (lnG1:T|Z1:T, θ∗, P∗, Mm)π(θ∗, P∗|Mm) π(θ∗_{, P}∗_|G 1:T, Z1:T, Mm) (1.18) where π(θ∗, P∗|G1:T, Z1:T, Mm) is the posterior density. (θ∗, P∗) is any point in the parameter

space. The latter expression follows from Bayes theorem. The choice of the point (θ∗, P∗) is based on a high posterior density point such as the maximum likelihood estimate or the posterior mean.

The estimate of the marginal likelihood in Chib (1995) exploits the output of the Gibbs sampler. Below, we summarize the calculation in the following:

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1. find the mean or mode of the posterior distribution θ∗; 2. decompose the posterior density into two parts:

π(θ∗, P∗|data) = π(θ∗_{|data) × π(P}∗_{|data, θ}∗_). _(1.19)

Then, appropriate Monte Carlo estimates of these two densities at θ∗ can be written as: π(θ∗|data) = R π(θ∗_{|data, P, S)π(P, S|data)d(P, S)}

ˆ

π(θ∗|data) = PG

g=1π θ

∗_{|data, P}(g)_{, S}(g)

(1.20)

π(P∗|data, θ∗) = R π(P∗_{|data, θ}∗_{, S)p(S|data, θ}∗_)dS

ˆ

π(P∗|data, θ∗) = PG

g=1π P

∗_{|data, θ}∗_{, S}(g) (1.21) where S(g) is a draw from the distribution S(g)|data, θ∗_{, P and P |data, θ}∗_{, S.}

1.6.5 The dynamic programming algorithm and the principle of Bellman

In this section, we describe briefly the principle of dynamic programming 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(Tr,n)

be the sum of squared residuals associated with the optimal partition containing r structural breaks using the first n observations. The recursive problem is defined as follows:

SSR(Tm,T) = M inmh≤j≤T −h{SSR(Tm−1,j) + SSR(j + 1, T )} (1.22)

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.

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Table 1.6: Log of the Bayes Factor for comparing the basic specification model with different number of breakpoints Models n break(s) 0 1 2 3 4 m break(s) 0 0 -124.0765 -119.4258 -122.4580 -122.1278 1 124.0765 0 4.6507 1.6185 1.9487 2 119.4258 -4.6507 0 -3.0322 -2.7021 3 122.4580 -1.6185 3.0322 0 0.3301 4 122.1278 -1.9487 2.7021 -0.3301 0

Table 1.7: Summary statistics on the posterior distribution of the elasticities with prior from

Basso and Oum (2007)

Bayesian results - Basic Model

Type of regime Statistics Price Income Intercept Q2 Q3 Q4

No break Regime 1 Mean -0.3529 0.3803 3.9399 0.1047 0.1626 0.0892 St.dev 0.0506 0.0467 0.2723 0.0195 0.0202 0.0195 Lower -0.4529 0.2916 3.4036 0.0663 0.1231 0.0504 Upper -0.2545 0.4721 4.4651 0.1419 0.2006 0.1265 One break Regime 1 Mean -0.5018 0.8178 0.9603 0.0941 0.1287 0.0933 St.dev 0.0646 0.0483 0.3493 0.0221 0.0235 0.0231 Lower -0.6224 0.7216 0.2761 0.0498 0.0804 0.0458 Upper -0.3725 0.9118 1.6660 0.1355 0.1743 0.1385 Regime 2 Mean -0.1157 0.4814 1.9802 0.0840 0.1205 0.0607 St.dev 0.0607 0.1039 0.7073 0.0166 0.0180 0.0171 Lower -0.2354 0.2882 0.5811 0.0524 0.0841 0.0265 Upper 0.0027 0.6968 3.3622 0.1152 0.1555 0.0936 Two breaks Regime 1 Mean -0.4401 0.8183 0.6996 0.0992 0.1328 0.0959 St.dev 0.0723 0.0482 0.3455 0.0237 0.0252 0.0240 Lower -0.5729 0.7191 0.0149 0.0526 0.0845 0.0510 Upper -0.2978 0.9122 1.3797 0.1462 0.1820 0.1417 Regime 2 Mean -0.2819 0.7598 0.4223 0.0547 0.0932 0.0623 St.dev 0.1514 0.1293 0.9436 0.1607 0.1810 0.1762 Lower -0.5799 0.5081 -1.4119 -0.1787 -0.1476 -0.1816 Upper 0.0145 1.0112 2.2548 0.2764 0.3222 0.3019 Regime 3 Mean -0.1572 0.5605 1.4711 0.0847 0.1161 0.0570 St.dev 0.0571 0.0956 0.6598 0.0179 0.0182 0.0184 Lower -0.2659 0.3681 0.2095 0.0499 0.0801 0.0222 Upper -0.0417 0.7445 2.8066 0.1202 0.1528 0.0938

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No break −0.60 −0.5 −0.4 −0.3 −0.2 −0.1 1 2 3 4 5 6 7 8 Price Elasticity Regime1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0 1 2 3 4 5 6 7 8 9 Income Elasticity Regime1 One break: 1984-Q1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0 1 2 3 4 5 6 Price Elasticity Regime1 Regime2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 1 2 3 4 5 6 7 8 Income Elasticity Regime1 Regime2

Two breaks: 1982-Q4 and 1986-Q1

−4 −3 −2 −1 0 1 2 3 4 5 0 1 2 3 4 5 6 7 Price Elasticity Regime1 Regime2 Regime3 −20 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 Income Elasticity Regime1 Regime2 Regime3

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No break −0.60 −0.5 −0.4 −0.3 −0.2 −0.1 1 2 3 4 5 6 7 8 Price Elasticity Regime1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0 1 2 3 4 5 6 7 8 9 Income Elasticity Regime1 One break −0.80 −0.6 −0.4 −0.2 0 0.2 1 2 3 4 5 6 7 Price Elasticity Regime1 Regime2 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 Income Elasticity Regime1 Regime2 Two breaks −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0 1 2 3 4 5 6 7 Price Elasticity Regime1 Regime2 Regime3 0 0.5 1 1.5 0 1 2 3 4 5 6 7 8 9 10 Income Elasticity Regime1 Regime2 Regime3

Figure 1.5: Posterior density of price and income elasticities with prior fromBasso and Oum