Rational Addiction and Time Consistency: An Empirical Test

This paper deals with one of the main empirical problems associated with the rational addiction theory, namely that its derived demand equation is not empirically distinguishable from models with forward looking behavior, but with time inconsistent preferences. The implication is that, even when forward looking behavior is supported by data, the standard rational addiction equation cannot distinguish between time consistency and inconsistency in preferences. We show that an encompassing general specification of the rational addiction model embeds the possibility of testing for time consistent versus time inconsistent naïve agents.We use a panel of Russian individuals to estimate a rational addiction equation for tobacco with time inconsistent preferences, where GMM estimators deal with errors in variables and unobserved heterogeneity. The results conform to the theoretical predictions and the proposed test for time consistency does not reject the hypothesis that Russian cigarettes consumers discount future utility exponentially. We further show that the proposed empirical specification of the Euler equation, whilst being indistinguishable from the general empirical specification of the rational addiction model, it allows to identify more structural parameters, such as an upper-bound for the parameter capturing present bias in time preferences.

1 Introduction Becker & Murphy (1988) explored the dynamic behavior of consumption of addictive goods, showing how many phenomena previously thought to be irrational can be consistent with utility optimization according to stable preferences. In their model, individuals recognize both the current and future consequences of consuming addictive goods. This model has subsequently become the standard approach to modeling consumption of goods such as cigarettes. A sizable empirical literature has emerged since then, beginning with Becker et al. (1994), which has tested and generally supported the empirical predictions of the Becker and Murphy model. These past contributions, however, run into a number of critical drawbacks. This paper is concerned with one of these problems, namely that forward looking behavior, implied by the model, does not imply time consistent preferences. Indeed even when evidence of forward looking behavior is found, the standard rational addiction demand equation does not allow to separately identify the short-run and long-run discount factor applied to future consumption periods (Picone, 2005). This is a crucial issue, because dynamic inconsistency can deliver radically dierent implications for government policy. In particular, while time consistency implies that the optimal tax on addictive goods should depend only on the externalities imposed on society, time inconsistency suggests a much higher tax depending also on the internalities that drugs' use imposes on consumers future selves (Gruber & Köszegi, 2001;O'Donoghue & Rabin, 2006). This paper oers the following contributions to the literature on addiction and time preferences. First, it provides the solution to a generalization of the rational addiction model that embeds time inconsistency through quasi-hyperbolic discounting, similar to Gruber & Köszegi (2001). We show that while the empirical specication is indistinguishable from the general specication of the rational addiction model (Becker et al., 1994) 1 , the testable implications are richer.
Second, it provides an estimate, using panel data at the individual level, of the general specication of the rational addiction demand equation that includes current, past and future prices. As far as we know this general specication has been estimated before only by Becker 1 We use the expression general specication of the demand equation as opposed to the most popular standard specication which is usually estimated in the empirical literature on rational addiction. While the general specication includes current, past and future prices of the addictive good as explanatory variables, the standard version only includes current prices. We shall return on this. et al. (1994); Chaloupka (1991); Waters & Sloan (1995).
Third, we implement a simple test of time consistency based on the additional structural information that can be extracted from the empirical general specication of the rational addiction demand equation. The purpose of the proposed test is to disentangle time consistent versus naïve agents, and it is possible because the proposed theoretical framework encompasses the rational addiction theory with time consistency as a special case involving no present bias in time preferences. Fourth, we compare our results with a time consistency test proposed by Gruber & Köszegi (2000), the working paper version of Gruber & Köszegi (2001).
Fifth, we show that while point estimates of the present bias parameter cannot be obtained under the proposed theoretical framework, it is still possible to recover an upper-bound, which provides further insights on the degree of time inconsistency.
In our framework we can only discriminate between time consistent and naïve agents. This is because the equilibrium of both naïve and time consistent individuals can be solved as an optimization problem and leads to the same empirical demand equation. We recognize, however, that time inconsistent agents could also be sophisticated. However, the equilibrium of sophisticated agents can only be analyzed as the equilibrium of a dynamic game (see O'Donoghue & Rabin, 1999b, 2002, for more details).
The implications of our ndings are the following. First, the rational addiction model and its derived general demand equation (Becker & Murphy, 1988;Becker et al., 1994) can be easily extended to discriminate between time consistent and naïve time inconsistent consumers, with the advantage of producing exactly the same empirical specication. Second, the possibility of distinguishing time consistent from naïve agents is nestled within the same general specication. Stated dierently, the information extracted from the general rational addiction demand equation is sucient to test for both forward looking behavior and time consistency.
The possibility of testing for time consistency using eld data opens up the opportunity of using the rational addiction demand equation to predict the impact of price measures on consumption of addictive goods in a more general way. This has relevant policy implications as time inconsistent preferences generally imply larger optimal taxes on the addictive goods.
The paper proceeds as follows. Section 2 summarizes the literature on time consistency and addiction. Section 3 reviews the general formulation of the rational addiction model while Section 4 introduces time inconsistent preferences and our test strategy. Section 5 discusses the data. Section 6 details the estimation methods and instruments choice. Section 7 presents and discusses the results and Section 8 concludes.

Prior Research
The early literature on dynamic consumption behavior modeled impatience in decision making by assuming that agents discount future streams of utility or prots exponentially over time.
Exponential discounting is pivotal. Without this assumption, inter-temporal marginal rates of substitution will change as time passes, and preferences will be time inconsistent (Strotz, 1956). Behavioral economics has built on the work of Strotz (1956) to explore the consequences of relaxing the standard assumption of exponential discounting. Ainslie (1992) and Loewenstein & Elster (1992) indicate that some basic features of inter-temporal decision-making that are inconsistent with simple models of exponential discounting, namely that many individuals value consumption in the present more than any delayed consumption, may be explained by a particular type of time inconsistency: hyperbolic discounting. In the formulation of quasihyperbolic discounting adopted by Laibson (1997) the degree of present bias is captured by an extra discount parameter β ∈ (0, 1) which accounts for instant gratication. Accordingly, the consumption path planned at each time period for the future time periods may never be realized, because the inter-temporal trade-o changes over time. The implications of such selfcontrol problems depend on individuals' awareness of their future preferences (O'Donoghue & Rabin, 1999a, 2002. Extreme assumptions about such awareness, e.g. full awareness and full unawareness, identify two types of individuals usually considered in the literature (Strotz, 1956;Pollak, 1968;O'Donoghue & Rabin, 1999a): naïve and sophisticated. A sophisticated person is fully aware of what her future selves preferences will be. A naïve person believes her future selves' preferences will be identical to her current self 's, not realizing that as she gets closer to executing decisions her tastes will have changed. To analyze equilibrium behavior of individuals with dierent time preferences, researchers have formally modeled a consumer as a sequence of temporal selves making choices in a dynamic game (Laibson, 1997;O'Donoghue & Rabin, 1999a, 1999b, 2002. Hence, a T −period consumption problem translates into a tion decision. In their analysis of consumption behavior of time consistent (TC henceforth), sophisticated and naïve agents, O'Donoghue & Rabin (1999a, 1999b, 2002 assume individual behavior to be described by perception-perfect strategies, i.e. solution concepts describing the individual's optimal action in all periods given her current preferences and her perception of future behavior. Naïfs have present biased preferences, but believe that they are time consistent. Therefore, the decision process for naïfs is identical to that for TCs even though naïfs have dierent time preferences. For naïfs and TCs this amounts to just choosing an optimal future consumption path. Thus, both naïfs and TCs equilibrium can be solved as an optimization problem.
Since naïfs' optimization problem is encompassing the TC case, its solution and the resulting demand equation oer the opportunity to develop an empirical test of time consistency.
Such distinction is very important from a policy perspective. Because quasi-hyperbolic individuals tend to over-consume the addictive good, the optimal value of a Pigouvian tax on addictive goods' consumption, for example, increases drastically when present biased (instead of time consistent) consumers are considered. This is because the internal costs of impatience add to the external costs caused by consumption of the addictive goods when calculating the optimal level of the Pigouvian tax. 2 The implications of present biased preferences, and their associated problems of selfcontrol, have been studied under a variety of economic choices and environments: Laibson Rabin (1999aRabin ( , 1999bRabin ( , 2002, and Angeletos et al. (2001) applied this formulation to consumption and saving behavior; Diamond & Köszegi (2003) explored retirement decisions; Barro (1999) applied it to growth; Gruber & Köszegi (2001) and Levy (2010) to smoking behavior; Shapiro (2005) to caloric intake; Fang & Silverman (2009) to welfare program participation and labor supply of single mothers with dependent children; Della Vigna & Paserman (2005) to job search (see Della Vigna, 2009, for a review); Acland & Levy (2015) to gym attendance.
Few works have attempted to use a parametric approach to estimate structural dynamic 2 Since the equilibrium of sophisticated cannot be solved as an optimization problem, our empirical analysis and our test of time consistency do not apply to sophisticated agents. O'Donoghue & Rabin (1999a) stress that even though sophistication is closer to standard economic assumptions than naïveté, it may have departures from conventional predictions that are even more extreme than those implied by naïveté. Focusing on naïfs only produces results which arise from present biased preferences only, rather than from present biased preferences in conjunction with sophistication. In addition, naïveté is more empirically relevant than sophistication as some studies demonstrate (Ariely & Wertenbroch, 2002;Della Vigna & Malmendier, 2003). models with hyperbolic time preferences (Fang & Silverman, 2009;Laibson et al., 2007;Paserman, 2008). Focusing on addictive goods, Levy (2010)  To our best knowledge no research has to date tested the assumption of time consistency within the structural demand equation derived from the rational addiction model of Becker & Murphy (1988). As pointed out by Picone (2005), the standard version of the rational addiction demand equation does not allow identication of the short and long run discount factor thus making it impossible to empirically test for time consistency of the agents. However, as we shall explain in the next section, the less popular general formulation of the rational addiction demand equation opens the possibility of testing for time consistency.

3
The General Rational Addiction Demand Equation Following Boyer (1978, 1983 and Becker et al. (1994) (BGM henceforth), considering time as discrete, the individual is assumed to maximize over time the following concave instantaneous utility function where C t is the quantity of a single addictive good consumed in period t, A t is the stock of past consumption in period t, Y t is the consumption of a composite commodity in period t and e t reects the impact of unmeasured life-cycle variables on utility. δ = 1 (1+r) is the long run discount factor and r is the individual rate of time preference. Preferences are stationary the sense that the instantaneous utility function in (1) does not change over time. This means that a person's instantaneous utility function depends on his current consumption level but not on the specic period t. This utility function has the following properties ∂Ut ∂Ct > 0 ; ∂Ut ∂Yt > 0 ; and ∂Ut ∂At < 0. Utility maximization is subject to the lifetime budget constraint where W 0 is the present value of wealth and P t is the relative price of the addictive good at time t. The evolution of the addictive stock A t is described by of the stock over time and represents the exogenous rate of disappearance of the eects of the physical and mental eects of past consumption (Becker & Murphy, 1988). When the stock depreciates completely in one time period, the depreciation rate is γ = 1, the depreciation factor becomes 0 and A t = C t−1 . Assuming this restriction holds, considering a quadratic instantaneous utility function in the three arguments 3 subject to the inter-temporal budget constraint, and solving the rst-order conditions for C t and A t BGM obtain the following second-order dierence demand equation: Equation (2) gives current consumption as a function of past and future consumption, the current price P t and the unobservable shift variables e t and e t+1 . This is the restricted or standard formulation of the rational addiction demand equation usually estimated in the empirical literature (Baltagi & Grin, 2001, 2002Baltagi & Geishecker, 2006;Gruber & Köszegi, 2001;Jones & Labeaga, 2003;Labeaga, 1993Labeaga, , 1999Liu et al., 1999;Olekalns & Bardsley, 1996;Saer & Chaloupka, 1999;Ziliak, 1997).
In the more general case, i.e. with δ < 1, we have past and future prices entering equation (2) (Becker et al., 1990;Chaloupka, 1990;Picone, 2005): (3) Equation (3) is a generalization of equation (2). A serious problem in estimating this general specication is the likely high collinearity between prices, possibly resulting in low statistical signicance of the relevant eects. To overcome this problem Becker et al. (1990) impose the restrictions implied by theory. In particular, since the ratio of future-to-past price eects is equal to the ratio of future-to-past consumption eects, i.e.
the coecients of P t+1 and C t+1 are equal to the respective past eects multiplied by the discount factor. Becker et al. (1994) and Chaloupka (1990) nd that this restriction is valid and improves the statistical signicance of the price and consumption coecients. The dicult identication of price eects in the general specication explains why the vast empirical literature on rational addiction has focused on estimating the restricted equation (2). However, its great advantage is that it provides deeper insights into inter-temporal preferences, since we can estimate consumption responses to price changes at three dierent time periods. Gruber & Köszegi (2000) proposed an alternative model which is also consistent with forward looking behavior, but embeds quasi-hyperbolic preferences (Laibson, 1997) and showed that tests of forward looking behavior only cannot distinguish the rational addiction model from their own. They also proposed a test of time consistency based on the idea that the responses to changes in prices at three dierent time periods could be used to distinguish time consistent from present biased consumers. The ratio of the responses to a two-periods-ahead price change and to a one-period-ahead price change should be the same if the individual is time consistent, but the response to a one-period-ahead price change should be smaller than the response to a two-periods-ahead price change if the individual is present biased.
Unfortunately, Gruber & Köszegi (2000) could not implement this test with their data and the test disappeared from the published version of the paper (Gruber & Köszegi, 2001).
Building on this previous contribution, we show that, by introducing quasi-hyperbolic discounting into the general version of the rational addiction model, it is possible to develop an easy test allowing to distinguish TCs from naïfs consumers. In addition, we borrow the idea behind the test proposed by Gruber & Köszegi (2000) and adapt it to our theoretical framework, giving it a slightly dierent interpretation.

Quasi-Hyperbolic Discounting and the Test of Time Consistency
In what follows we embed quasi-hyperbolic discounting (Laibson, 1997) in the previous model.
We solve the maximization problem step by step reproducing passages from Chaloupka (1990) mathematical appendix.
O 'Donoghue & Rabin (1999b, 2002 show that, under stationarity of preferences, for both TCs and naïfs the innite-horizon perception-perfect strategy is unique and corresponds to the unique nite-horizon perception-perfect strategy as the horizon, T , becomes long. Therefore, in what follows we will keep assuming an innite time-horizon T = ∞ in part for expositional ease and in part because an innite time horizon is the typical assumption in rational addiction models. Individuals are assumed to maximize the sum of lifetime discounted utility where δ = 1 (1+r) is the long-run discount factor, r is the discount rate, and the extra discount parameter β ∈ (0, 1] is intended to capture the essence of hyperbolic discounting, namely, that the discount factor between consecutive future periods (δ) is larger than between the current period and the next one (βδ). If β = 1 preferences in equation (4) are dynamically inconsistent, in the sense that preferences at date t are inconsistent with preferences at date t+ 1. 4 To analyze equilibrium behavior when preferences are dynamically inconsistent researchers usually model a consumer as a sequence of temporal selves making choices in a dynamic game (Laibson, 1997;O'Donoghue & Rabin, 1999a, 1999b, 2002, as explained in Section 2. However, (2002) show that the equilibrium of both naïfs and TCs solves the same optimization problem. Therefore, the demand equation that solves (4) applies to both TCs and naïfs consumers. As before, interaction between past and future consumption is modeled where C 0 measures the level of addictive consumption in the period prior to that under consideration. The consumer's problem becomes: Taking a quadratic function in the three arguments, the resulting instantaneous utility is: The maximized value of utility becomes: which can be re-written as: where λ is the marginal utility of wealth. Maximizing (5) with respect to Y t and subject to the budget constraint results in the following rst order condition for Y t : Plugging this result into (6) results in the maximization problem being a function of only consumption of the addictive good and the stock of the addictive good: and (9) with respect to C t implies the following rst-order condition: Noting that: and dene the rst term on the right hand side of (12) as U C,t = [α C + α CC C t + α CA A t ] and the right hand side of (13) as V A,t = α A + α AA A t + α CA C t . Substituting these denitions in equation (11) the rst order condition can be rewritten as: The consumption demand equation can be obtained starting from equation (14), as similar equations can be derived for each time period. Consider the dierences δ(1 − γ)U C,t − U C,t−1 and δ(1 − γ)U C,t+1 − U C,t . Using equation (14) they can be written as Now multiply both sides of equation (15) by (1 − γ) and subtract it from equation (16): Substituting U C,i and V A,i with their denitions, using A t = (1 − γ)A t−1 + C t−1 to eliminate the stock of habits, and solving for C t produces the demand equation: (18) where: Equation (18) is very similar to equation (3) except that the coecient θ that multiplies C t+1 and C t−1 is not exactly the same. The dierence between θ − and θ + is that the α CA parameter in equation (22) is multiplied by β.

Testing Time Consistency
Noting that β = 1 implies time consistency and the model reduces to the standard BGM rational addition model, equation (18) can thus be used to test whether consumers are time consistent or not by testing the equality θ − = θ + .
Recalling that the empirical reduced form of the demand equation (18) is identical to the general formulation of the rational addiction demand equation, i.e. 5 it is possible to identify all needed coecients.
and the test of time consistency reduces to a non linear hypothesis test on the estimated parameters, i.e. that φ 1 φ 5 = φ 2 φ 4 . If the test rejects the null, then β = 1 and the data do not support time consistent preferences as implied by the BGM rational addiction model in favor of quasi-hyperbolic discounting for naïve agents.
Given the parametric specication of equation (18) and the corresponding reduced form equation (24), it is not possible to directly identify the value of present bias parameter β.
It is however possible to extract further information about it, and in particular an upper limit compatible with the estimated coecients. From equations (21) and (22), the ratio of consumption coecients can be written as from which an expression for β can be obtained, i.e.
Thus, β is a linear function of the unknown ratio α CC /α CA . If, as suggested by the theory, α CA is positive, α CC is negative, and β ≤ 1, then by equations (21) and (22) the function is increasing in the α CC /α CA ratio, and a natural upper bound for equation (26) is hit when In the unpublished version of their paper, Gruber & Köszegi (2000) also developed a model that embeds the hyperbolic discounting preferences by Laibson (1997) and proposed a test of time consistency. Their test is based on the idea that the responses to changes in prices at three dierent time periods could be used to distinguish time consistent from hyperbolic discounters. We implement this test using model (24). This general specication allows calculation of the eects on consumption at time t of price changes at three dierent time periods (t-1, t and t+1). The consumption responses to price changes at dierent points in the future can be used to assess whether consumers discount exponentially or not. According to the authors, the ratio of the response to a current price change over a lagged price change should be the same as the ratio of the response to a one-period-ahead price change over the response to a current price change if consumers discount exponentially. The rst ratio will be smaller than the second if in reality the underlying consumer is a hyperbolic discounter. The ratio of future-to-current and current-to-past price eects from equation (24) are Under the null hypothesis of time consistency, and conditional on an exogenous depreciation rate, the two ratios (28) and (29)  To each individual aged 13 and above, the survey asks whether she/he smokes and if so the number of cigarettes smoked per day. This is the main consumption measure used in our study. The household questionnaire also asks about family tobacco expenditure and quantity, but that is at the household level and is not suitable for individual consumption analysis.
The price variable is computed from the community questionnaire, where interviewers go to local stores in the community and check minimum and maximum prices of a large sample of commodities, including domestic and foreign branded cigarettes. Because several missing values are recorded at community level (if, for instance, no store had a particular item or if the store was closed), the price was averaged across communities within the same primary sample units to reduce the impact of measurement errors. Because the prices are at current level, and the survey does not provide consumer price indices do deate prices, we compute a consumer price index at PSU level following the Törnqvist procedure (Törnqvist, 1936). The reference price is that of the Moscow PSU in 1998, and the index is computed on a wide set of food commodities, excluding tobacco and alcohol items. Cigarettes prices together with other monetary variables described below are then deated using this consumer price index.   This information would be sucient to estimate the model presented in equation (24), but the introduction of control variables can improve the precision of estimates. 7 The list of 7 In addition, as for most panel data surveys, also the RLMS suers a certain level of attrition. According to Gerry & Papadopoulos (2015), who analyze years 2001-2010, the average yearly attrition rate is below 10%, but attrition is signicantly correlated with some individual characteristics that makes a missing completely at

Estimating the General Rational Addiction Model
Our empirical demand equation is a variant of equation (24): where C it is the number of smoked cigarettes by individual i in period t, P it is cigarettes real price, X it is a vector of exogenous economic and socio-demographic variables that aect cigarettes consumption, v i are individual xed eects capturing time invariant preferences that are correlated with lead and lagged consumption and probably with other determinants of consumption, d t are time xed eects, and u it = φ 7 e t + φ 8 e t+1 is the idiosyncratic error term. OLS estimates of the dynamic panel data equation (30) can suer from an omitted variable bias due to unaccounted demand shifters that may also be serially correlated (Becker et al., 1994). To correct for the endogeneity bias we follow Arellano & Bond (1991) in using a GMM procedure to obtain the vector of parameters. The GMM estimators exploit a set of moment conditions between instrumental variables and time-varying disturbances. The basic idea is to take rst-dierences to deal with the unobserved xed eects and then use the suitably lagged levels of the endogenous and predetermined variables as instruments for the rst-dierenced series, under the assumption that the error term in levels is spherical and taking into account the serial correlation induced by the rst-dierence transformation. This idea extends to the case of lags and leads of the dependent variable and to the case where serial correlation already exists in the error term of the original model, as in equation (30).
After rst dierencing equation (30) becomes: the strategy is to nd a set of instruments Z it that are uncorrelated with the rst-dierenced error term ∆u it and correlated with the regressors. By denition for i = 1, ..., N and t = 3, ..., T − 1. Given the error term u it in (32), the following moment conditions are available: E(C it−s ∆u it ) = 0 for t = 4, ..., T − 1 and s ≥ 3. This allows the use of lagged levels of observed consumption series dated t − 3 and earlier as instruments for the rst-dierenced equation (31). The moment restrictions can be written in matrix form as , whose i th block is given as where the block diagonal structure at each time period exploits all of the instruments available, concatenated to one-column rst dierenced exogenous regressors ∆W it = (∆P it , ∆P it−1 , ∆P it+1 , ∆X it ) that act as instruments for themselves (Arellano & Bond, 1991).
Ever since the work of BGM on US cigarette consumption, past and future prices have been considered as natural instruments for lagged and lead consumption, as well. We maintain this pivotal option here.
The rst-dierenced GMM estimator is poorly behaved in terms of nite sample properties (bias and imprecision) when instruments are weak. This can occur here given that the lagged levels of consumption are usually only weakly correlated with subsequent rst-dierences.
More plausible results and better nite sample properties can be obtained using a system-GMM estimator (Arellano & Bover, 1995;Blundell & Bond, 1998). This augmented version exploits additional moment conditions, which are valid under mean stationarity of the initial condition. This assumption yields (T − 4) further linear moment conditions which allow the use of equations in levels with suitably lagged rst-dierences of the series as instruments E(∆C it−2 u it ) = 0 for t = 4, ..., T − 1 The complete system of moment conditions available can be expressed as E(Z + i u + i ) = 0, where u + i = (∆u i4 , ..., ∆u iT −1 , u i4 , ..., u iT −1 ) . The instrument matrix for this system is Whether we actually need all these moment conditions is debatable, since in nite samples there is a bias/eciency trade-o (Biørn & Klette, 1998). A large instruments collection, like that generated by the system GMM, over-ts the endogenous explanatory variables and weakens the power of the over-identication tests (Roodman, 2009b). Ziliak (1997) showed that GMM may perform better with suboptimal instruments and argued against using all available moments. We use only a subset of them, testing their validity with the Sargan over-identication test (Baltagi & Grin, 2001). After some experimentation, we report in the next section estimates using the following parsimonious matrix of instruments Z i , which represents a compromise between theory, previous applied work and the characteristics of the data.
where the least informative lagged levels of consumption have been dropped and where the instruments X is the Törnqvist consumer price index 10 .

Results and discussion
The empirical specication ( The right-hand side variables are the lead and lag consumption, current, lead and lag real price of cigarettes at PSU level from the community questionnaire, and a number of sociodemographic characteristics described in Section 6. In all specications we use time dummies to make the assumption of no correlation across individuals in the idiosyncratic disturbances more likely to hold (Roodman, 2009a).
We use a subset of all available instruments of prices. Our chosen estimator is the system-GMM (Blundell & Bond, 1998). This unifying GMM framework incorporates orthogonality conditions of both types of equations, transformed and in levels, and performs signicantly better in terms of eciency as compared to other IV estimators of dynamic panel data models.
We estimate both one-step and two-step system-GMM estimators, but we only report two-step estimates with a robust covariance matrix using the Windmeijer (2005) correction.
In terms of empirical studies and nite sample properties of the GMM estimator, the choice of transformation used to remove individual eects is important. First dierencing (FD) is one option, but Arellano & Bover (1995) propose an alternative transformation for models with predetermined instruments: forward orthogonal deviations (FOD). This transformation involves subtracting the mean of all future observations for each individual. The key dierence between FD and FOD is that the latter does not introduce a moving average process in the disturbance, i.e. orthogonality among errors is preserved. Another practical dierence is that the FOD transformation preserves the sample size in panels with gaps, where FD would reduce the number of observations (Roodman, 2009a). 11 Results under the FOD transformation method are reported in Table 3 for model (30)  12 We also estimated model (30) using OLS and Fixed Eects (FE) (columns 1 and 2 in Table 3). As Bond (2002) points out, while in the OLS regression the lagged dependent variable is positively correlated with the error, biasing its coecient estimate upward, the 11 Notice that precisely the same instruments set would be used to estimate the model in orthogonal deviations.
12 As reported by Roodman (2009a), in general we check for serial correlation of order l in levels by looking for correlation of order l + 1 in dierences. Because in our model current consumption depends on both past and future consumption, this is an autoregressive process of order 2 (AR2) and we have second-order serial correlation by construction. So, for the validity of our instrument set, we need to detect no serial correlation of order 3 in the residuals.
opposite is the case in the FE model. Good estimates of the true parameters should therefore lie in the range between the FE and OLS estimates. These OLS and FE estimates provide bounds that can be used as useful checks on results from theoretically superior estimators.
Estimates reported for our preferred specication, System GMM with FOD and covariates (column 4 in Table 3), are consistent with the rational addiction framework. First, past consumption has a signicant positive eect. Second, future consumption also has a signicant positive eect, supporting the idea that smokers' behavior is forward looking. Third, the coecient of lagged consumption is greater than the coecient of lead consumption, giving rise to a positive discount rate. Fourth, we obtain a negative coecient on the current price and a positive coecient on both past and future prices. So, the signs on the two consumption variables and the three price variables conform to theoretical predictions. As to the socio- 13 Slightly dierent specications using a dierent set of covariates and of instruments for the strictly exogenous variables produce qualitatively similar results. They are available from the authors upon request. We implemented the tests of time consistency on the estimated parameters from our preferred specication (model 4 in Table 3). As explained in section 5 our test boils down to testing the null hypothesis φ 1 φ 5 = φ 2 φ 4 . Our test has a χ 2 distribution with 1 degree of freedom.
As a further test of time preference we also tested the null hypothesis that the ratio of current-to-past responses to a price change is equal to the ratio of future-to-current responses to a price change, as suggested by Gruber & Köszegi (2000). The test is again distributed as a χ 2 (1). We obtain χ 2 (1) = 2.35 with P rob > χ 2 = 0.125, supporting again the hypothesis of time consistency.
The estimated discount factor δ = φ 5 /φ 4 = 0.917, which corresponds to a long run discount rate of 8.9%. This set of results seems to suggest a pretty standard time preference structure, with no evidence of time inconsistency and with a reasonably small discount rate.

Dynamics of Consumption
Our demand equation is a second-order dierence equation in current consumption. The roots of this dierence equation are useful for describing the dynamics of consumption and are positive if and only if consumption is addictive (Chaloupka, 1990). For equation (30) these roots are λ 1,2 = 1± √ 1−4φ 1 φ 2 2φ 1 with 4φ 1 φ 2 < 1 from the assumption of concavity. BGM note that both roots are real and depend on the sign of φ 1 and φ 2 . Both roots are positive if and only if consumption is addictive (φ 1 > 0); both roots will be zero or negative otherwise.
The smaller root, λ 1 , gives the change in current consumption resulting from a shock to future consumption. The inverse of the larger root λ 2 indicates the impact of a shock to past consumption on current consumption. These shocks may be the result of a change in any of the factors aecting demand for cigarettes. In our preferred specication (system GMM with FOD) φ 1 = 0.393 and both roots are positive (λ 2 = 2.112; λ 1 = 0.432), so cigarettes consumption is actually addictive.
Besides the restrictions on the values of the two roots, the conditions necessary for stability include that the sum of the coecients on past and future consumption is less than unity and that the sum of the coecients on prices is negative (Chaloupka, 1990). Our results fulll both these stability conditions as the sum of coecients on past and future consumption is less than unity (0.751), and the sum of price coecients is negative (−0.0116), as required by theory. 14 8 Conclusion This paper addresses one of the main theoretical and empirical shortcomings of the rational addiction model, namely that forward looking behavior, implied by theory, does not necessarily imply time consistency. Then, even when forward looking behavior is supported by data, the dynamic consumption equation derived from the rational addiction theory does not provide evidence in favor of time consistent preferences against a model with dynamic inconsistency (Gruber & Köszegi, 2001).
We show that the possibility of testing for time consistency is nested within the rational addiction demand equation. Rather than relying on additional assumptions or on a dierent theoretical or empirical framework, we use price eects and the rarely estimated general formulation of the rational addiction demand equation to implement a test of time consistency.
The test's purpose is to check whether consumers behind our data reveal time consistent or naïve time inconsistent preferences. Our estimates of the general rational addiction demand equation conform to theory and our test of time consistency does not reject the hypothesis that consumers discount exponentially.
The value added of our contribution is to show that the possibility of distinguishing time consistent from time inconsistent naïve preferences is nestled within a rational addiction model with quasi-hyperbolic discounting. The information extracted from a general rational addiction demand equation is then sucient to test for both forward looking behavior and time consistency of the underlying consumers.
This has relevant policy implications for the optimal taxation of addictive goods. As Gruber & Köszegi (2002, 2004) point out, when agents are time inconsistent positive taxation 14 In a recent paper Laporte et al. (2017) investigate whether the unstable root (λ2) complicates the estimation of the rational addiction model even when the true data generating process possesses the true characteristics of rational addiction. In our case, however, all the restrictions implied by theory are satised by the data without imposing constraints on the estimated parameters.
is optimal even in the absence of externalities, as time inconsistency will imply self-control problems and the optimal future consumption path planned at time t will not be realized by the agent. Hence, in case of time inconsistent agents taxes on addictive goods are substantially larger than for time consistent consumers. O'Donoghue & Rabin (2006) considering nonaddictive unhealthy goods with quasi-hyperbolic time preferences, shows that the optimal tax is proportional to (1 − β) times the marginal health cost of consumption and that even very small levels of present bias (β close to 1) produce signicantly large optimal taxes.
In this context, having the possibility of testing for time consistency can be of great value to the policy maker. In addition, the possibility of estimating at least an upper bound for the present bias parameter (β max ), would enable the policy maker to compute the lower bound of an optimal tax on addictive goods by extending O'Donoghue & Rabin (2006) optimal taxation model to addictive consumption. This would be an interesting avenue for future research.