Treatment-effect identification without parallel paths

Vandenberghe, Vincent Article Treatment-effect identification without parallel paths Economics: The Open-Access, Open-Assessment E-Journal Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Vandenberghe, Vincent (2018) : Treatment-effect identification without parallel paths, Economics: The Open-Access, Open-Assessment E-Journal, ISSN 1864-6042, Kiel Institute for the World Economy (IfW), Kiel, Vol. 12, Iss. 2018-9, pp. 1-19, https://doi.org/10.5018/economics-ejournal.ja.2018-9 This Version is available at: https://hdl.handle.net/10419/175360 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Licensed under the Creative Commons License - Attribution 4.0 International (CC BY 4.0) Vol. 12, 2018-9 | February 27, 2018 | http://dx.doi.org/10.5018/economics-ejournal.ja.2018-9 Treatment-effect identification without parallel paths Vincent Vandenberghe Abstract Imagine a region suffering from a widening income gap that becomes eligible for a generous transfer programme (the treatment). Imagine difference-in-differences analysis (DD) — a before-and-after comparison of the income-level difference — shows that the handicap has risen. Most observers would conclude to the policy's inefficiency. But second thoughts are needed, because DD rests heavily on the validity of a key assumption: parallel paths in the absence of treatment; an assumption that is often violated. To cope with this problem, economists traditionally include polynomial (linear, quadratic…) trends among the regressors, and estimate the treatment effect as a once-in-a-time trend shift. In practice that strategy does not work very well, because inter alia the estimation of the trend uses post-treatment data. What is needed is a method that i) uses pre-treatment observations to capture linear or non-linear trend differences, and ii) extrapolates these to compute the treatment effect. This paper shows how this can be achieved using a fully-flexible version of the canonical DD equation. It also contains an illustration using data on a 1994–2006 EU programme that was implemented in the Belgian province of Hainaut. JEL C21 R11 R15 O52 Keywords Treatment-effect analysis; difference-in-differences models; EU convergence policy Authors Vincent Vandenberghe, IRES-UCL, Economics School of Louvain, Louvain-laNeuve, Belgium, [email protected] Citation Vincent Vandenberghe (2018). Treatment-effect identification without parallel paths. Economics: The Open-Access, Open-Assessment E-Journal, 12 (2018-9): 1–19. http://dx.doi.org/10.5018/economics-ejournal.ja.2018-9 Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 2 1 Introduction This paper deals with how to properly evaluate the impact of convergence policies like Objective 1-Hainaut. At its core lies a methodological proposal. But, before turning to its full exposition, here are a few words about Objective 1 and the province of Hainaut in Belgium. Objective 1-Hainaut is an example of a European-Union (EU)-funded transfer policy aimed at helping European regions reduce their socio-economic handicap. The policies have a relatively old history. The underpinning idea was present in the preamble to the Treaty of Rome in 1957, and has been further emphasised in the 1980s with the entry of Greece, Portugal and Spain. In 1987, with the Single European Act, the EU received explicit competence for undertaking a regional policy aimed at ensuring convergence. Over the decades, a growing political concern for the so-called "regional problem" has meant that a considerable – and increasing – amount of resources has been spent to mitigate regional income disparities.1 Since the mid-1980s, the importance of EU development/convergence policies has not ceased to increase. In budgetary terms, the policies have grown from representing a mere 10% of the EU budget and 0.09% of the EU-15 GDP in 1980, to more than one third of the budget and around 0.37% of the EU GDP as an average of the period 1998–2001 (Rodríguez-Pose and Fratesi, 2003). The policies have become, after the Common Agricultural Policy (CAP), the second largest policy area in the EU. Also, every recent step towards greater economic integration at EU level has been accompanied by measures aimed at supporting financially the lagging countries or regions. For instance, the decision in the Maastricht reform to create the Single European Currency that was tied in with the establishment of the Cohesion Fund to alleviate the burdens that transition to EMU would impose on the less developed territories. After the reform, more than two thirds of all Structural Fund expenditure have been concentrated in the so-called Objective 1 regions. These are territories whose GDP per capita, measured in purchasing power standards (pps), is less than 75% of the EU average. In the 1990s, the list comprised 64 NUT2 regions2 (Tondl, 2007), one of them being Hainaut in Wallonia/Belgium (Figure 1). The 69 municipalities forming that province benefited from Objective 1 money between 1994 and 1999. And from 2000 to 2006 they also benefited from the "phasing out" programme. Yet, despite their rising macroeconomic importance, questions are being raised about the capacity of European development/convergence policies in general, and of policies targeted at Objective 1 regions, to achieve greater economic and social cohesion and to reduce income gaps. These questions are fundamentally based on rather mixed evidence about convergence following implementation (Magrini, 1999). In that context, it is a bit surprising that there are _________________________ 1 The European Commission’s focus on regional disparities has been paralleled by a renewed academic interest – both theoretical and empirical – in the economic analysis of growth and (non) convergence. From the work of Romer (1986), (1990) and Lucas (1988), a growing body of literature, known as ‘new growth theories’, has started to question the optimistic predictions of the traditional neoclassical model laid out by Solow (1956), which leaves little or no role to regional/convergence policy. 2 Nomenclature of Units for Territorial Statistics (NUTS) is a geocode standard for referencing the subdivisions of countries for statistical purposes. The standard is developed and regulated by the EU, and thus only covers the member states of the EU in detail. Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 3 Figure 1: Hainaut and its municipalities (+ the rest of Belgium) very few ex post economic evaluation studies3 of the monetary benefits of Objective 1. More precisely, there are very few papers answering questions such as “what would be the level of income per head in region X had it not benefited from Objective 1 money?” Along the same line, and in contrast with what economists and econometricians have done to evaluate other types of policy interventions (higher minimum wages, employment subsidies, active labourmarket or social policies…), very little work has been done using microdata, in a quasiexperimental setting, to evaluate the effectiveness of Objective 1 (or other EU policies aimed fostering convergence across regions or countries). In a sense, this paper aims at filling that void. This said, at its core, lies a methodological discussion of what can (or cannot) be achieved within the canonical differences-in-differences (DD) estimator, and how best to address its limitations. DD is a statistical technique commonly used in microeconometrics (Angrist and Krueger, 1999) that mimics an experimental research design using observational data, by studying the different evolution of 'treated' ' vs 'control' groups in a (quasi) natural experiment. It calculates _________________________ 3 There are several macroeconomic models that have been used to assess the potential impact of EU funds on economic growth (e.g. HERMIN model). All these models estimate growth effects from cohesion spending, but their size changes depending on the theoretical assumptions upon which the model is based. Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 4 the effect of a treatment (e.g. Objective 1) on an outcome (e.g. income per capita) by comparing i) the average change over time in the outcome variable for the treatment group (e.g. the municipalities of Hainaut), to ii) the average change over time for the control group (e.g. the municipalities forming the rest of Belgium). But the validity of that method rests heavily on the parallel-path assumption: in the absence of treatment (and in particular before its inception) the (average) outcome-level difference between the treated and the control entities (municipalities hereafter) must be time-invariant; so that the observation of a statistically significant change of the pre-treatment difference after the treatment's inception can be ascribed to the latter. Key to this paper is the idea that, whenever data permit, one should go beyond the canonical DD model and the parallel-path assumption underpinning its capacity to properly identify a treatment's effect. It is also that this should be done not simply via the addition of a polynomial (linear, quadratic…) time trend to the canonical DD model, as most authors do. A more promising avenue is to i) estimate the generalized, fully-flexible DD model proposed by Mora and Reggio (2012) ii) and so to account (and correct) for the absence of parallel paths, using only pretreatment observations. When data contain 2 or more pre-treatment periods it is easy to verify if parallel path holds. And quite often it does not. As said above, what most authors do when confronted to that problem, is to augment the canonical DD model (that contains a time dummy, a treatment dummy and the interaction between these two) with a polynomial (linear, quadratic…) time trend, and to estimate the treatment effect as a once-in-a-time shift of that trend (e.g. Friedberg, 1998; Autor, 2003; Besley and Burgess, 2004). In practice, that strategy does not work very well, because inter alia the estimation of the trend uses post treatment data.4 Wolfers (2006) for instance explains that Friedberg's (1998) work on the legalisation of divorce is a point in case. Friedberg controls for treated vs control US State diverging trends using a sample that covers only one year before treatment and many years after. Her estimates of the State trends rely almost completely on post-treatment developments, and absorbs most of the treatment's effect. Another – less commonly used – method to correct for the absence of parallelism consists of building a synthetic control group (Abadie et al., 2010, Bélot and Vandenberghe, 2009). The idea is that a combination of control entities is likely to provide a better (i.e. more "parallel") comparison for the treated one than any single entity alone. It might indeed. But there is no absolute guarantee that the synthetic entity will display parallelism prior to treatment. And the method is quite demanding in terms of data. Its use is conditional on the availability of relatively large set of control entities, for which the outcome variable of interest has been measured in the same way as for the treated entity. By comparison, the method exposed here below consists of taking whatever data is available as control, and to adapt the canonical DD identification method when the control entity does move parallel to the treated. How can the canonical DD model be adapted? It simply needs to be generalized. Mora and Reggio (2012) show that such a model – and the parallel-path assumption underpinning identification (DD[1]/Parallel[1] hereafter) – is a particular case of a more general one that can be used to estimate a whole family of DD models. The nicety of the Mora and Reggio framework _________________________ 4 Another reason has to do with what Cabras et al. (2017) call the" parametric straightjacket"; i.e. the fact that many regressions are not flexible enough to capture the true relationships as they tend to rely on arbitrary identification assumptions. Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 5 is to allow for all sorts of diverging trends verifying (or violating) Parallel[1], Parallel[2], Parallel[3], and even higher degrees of parallelism. Why still talk of “parallel-something” if there is divergence? Because, as will become clear in Section 2 hereafter, the identification idea remains that at the heart of the canonical DD[1]/Parallel[1]. model: in the absence of treatment, the differences between growth rates/Parallel[2], or acceleration rates/Parallel[3] …. should be constant. While DD[1] focuses on the evolution of outcome level differences, DD[2] tracks the evolution of outcome growth differences; and DD[3] that of outcome acceleration differences. The other key feature of the Mora and Reggio framework is that it solves the problem identified by Wolfers (2006) whith polynomial time-trend corrected DD.5 We show hereafter that this is because – unlike what is done by authors resorting to polynomial time-trend corrections – only pre-treatment observations are used to capture trend differences, and because the estimation of the treatment effect rests on a simple extrapolation of these pre-treatment trends.6 This said, the reader should be aware that the economic efficiency criteria associated with the different DD estimators vary dramatically. In the context of a deprived region receiving financial aid, using DD[1] as a treatment-evaluation method means a focus on the reduction and the initial income-level difference of that region. Under DD[2] the requirements are intrinsically milder. Efficiency exists as soon as one detects a reduction of the pre-treatment income-growth rate difference. And there is no paradox in DD[1] results being negative, while those delivered by DD[2] are positive. That simply means that the initial income-level difference has risen, but less than it would had the growth rate difference not been reduced (see Figure 2 for an illustration). By contrast, if even DD[2] shows no significant gains, then it means that the policy has not been very effective at all; as it has not even been able to reduce the pre-treatment growth rate difference. In the case of Objective 1-Hainaut and its 69 treated municipalities, using per head income data and the rest of Wallonia or Belgium as a control group, our DD[1] results suggest a negative impact. But the analysis of pre-treatment data clearly shows that Parallel[1] does not hold before inception. We rather find statistically significant evidence of Parallel[2] (constant growth rate difference before 1994). This is thus the assumption we retain for identifying Objective 1’s causal impact. And when doing so, results change considerably, as our DD[2] estimates are positive and statistically significant. This is supportive of the idea that Objective 1 reduced the growth rate difference that affected Hainaut before 1994. In the absence of this correction, the income-level difference increment – the one typically measured by DD[1] – would have been larger. Over the year 2010 horizon, we find that Hainaut experienced a rise of its income-level difference compared to the rest of Belgium of 426 euros. But we find a statistically significant DD[2] of 491 euros. This means is that in the absence of the growth rate (positive) correction; the income-level difference rise would have been of 426 + 491 euros. _________________________ 5 The (linear) trend-augmented version of the canonical DD model writes Yit= α + ∑𝛼𝜏𝐼𝜏,𝑡 𝑇 𝜏=𝑡2 + αDDi + ηAFTERt.Di + θ t.Di+εit with Iτ,t=1 if t=τ and 0 otherwise, and where Yit is entity i's outcome in time t, D the treatment dummy, AFTER the after treatment dummy, and here t is a continuous variable. Coefficient θ captures the linear trend characterizing the treated entities. And η – a trend shift around t=0 – measures the treatment effect. 6 For an illustration of how the generalized DD method we put forth here compares with polynomial time-trend corrected DD, see Vandenberghe (2018). Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 6 Figure 2: The inadequacy of traditional difference-in-(level) differences estimator (DD[1]) in the presence of non-parallel paths$ The rest of the paper is divided into five sections. Section 2 exposes analytically the DD[1]/Parallel[1], DD[2]/Parallel[2]… DD[q]/Parallel[q] sequence, and how it can be implemented using OLS estimation. Section 3 briefly discusses Objective 1-Hainaut ; its particularities and the calendar of its implementation. Section 4 presents the dataset used in this paper and some descriptive statistics. Section 5 contains and comments the main estimation results. Section 6 concludes. 2 Beyond parallel 1 To generalise the idea of treatment effect identification via DD, Mora and Reggio (2012) suggest estimating a fully-flexible equation, where the right-hand part only consists of time, treatment and timeXtreatment dummies7 Yit= γ + ∑𝛾𝜏𝐼𝜏,𝑡 𝑇 𝜏=𝑡2 + γDDi + ∑𝛾𝜏 𝐷𝐼𝜏,𝑡 𝑇 𝜏=𝑡2𝐷𝑖 +εit [1.] with t=t1,…. T and Iτ,t=1 if t=τ and 0 otherwise, covering before and after treatment periods. _________________________ 7 That equation can also be augmented by including time-varying continuous variables Xit. Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 7 The advantages of such a specification are manifold. First, conditional on the availability of many pre-treatment periods in the data, the OLS-estimated coefficients can be used to compute a whole family of difference-in-difference estimators DD[p], where p=1, 2...q is the degree of parallelism underpinning identification. The canonical differences-in-differences model is DD[1], and rests on parallelism of degree 1 (Parallel[1]), meaning that outcome levels must stay parallel in the absence of treatment.8 Without Parallel[1] – as depicted on Figure 2 – one can estimate DD[2] that rests on Parallel[2], i.e. outcome growth-rate parallelism.9 If Parallel[2] fails, one should turn to DD[3] with requires Parallel[3] or outcome-acceleration10 parallelism… and so on up to degree p=q, if data permit. Second, Eq. [1], can capture dynamic (i.e. lagged) responses to treatment.11 Finally – and this is something we particularly stress in the context to this paper – corrections for the violation of Parallel[p] rests solely on pre-treatment observations. Consider the canonical DD[1]/Parallel[1] estimator, with just before-and-after observations t* and t*+1.12 Treatment effect corresponds to13;14 DD[p=1]t*+1;t*=(γDt*+1+γD)- (γDt* +γD)=γDt*+1 - γDt*. [2.] Eq. [2] can be used to assess Parallel[1] prior to treatment. Using 2 pre-treatment periods (say t*-2, t*-1), one can compute 'placebo' DD[1] capturing the deviation from Parallel[1] prior to the start of the treatment. For instance, DD[1]t*;t*-1=γDt* - γDt*-1. should not be statistically different from zero. If not, then treated and control trends diverge before treatment (as illustrated on Figure 2). And the identification of the treatment effect should rest on Parallel[2] The point is that this can be easily achieved by computing15 DD[p=2]t*+1; t*-1=DD[1]t*+1; t*-DD[1]t*;t*-1=(γDt*+1γDt*)-(γDt*-γDt*-1) = γDt*+12γDt* +γDt*-1 [3.] _________________________ 8 If outcome level change by unit of time is "speed" (i.e. 1st-order derivate of outcome vis-à-vis time), then Parallel[1] means stable outcome level differences due to identical speeds 9 If outcome growth rate change by unit of time is "acceleration" (2nd-order derivate), then Parallel[2] means stable outcome growth rate differences due to same accelerations. 10 If outcome acceleration change by unit of time is "surge" (3nd-order derivate), then Parallel[3] corresponds to a situation where outcome acceleration differences remain stable due to identical surges. 11 The pattern of lagged effects is usually of substantive interest. We might, for example, believe that treatment effect should grow or fade as time passes. 12 Hereafter the range of periods used by the estimator appears as superscript in DD[p=1]t*+1;t* 13 When estimating eq. [1] with only 2 periods (T=2), γDt* is subsumed into the constant γD and DD[1] is directly captured by the timeXtreatment interaction term coefficient. 14 Treatment effect' standard error must account for the fact that it is a linear combination of estimated coefficients. Variance/standard error must account for the covariance between corresponding variables. That is done e.g. by STATA test or lincom commands, that use the variance-covariance matrix of the estimated coefficients. 15 Again, when estimating eq. [1] with only 3 periods, γDt*-1 is subsumed into γD and DD[2] is computed using only 2 coefficients. Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 8 or, said differently, the difference between the OLS-estimated observed post-treatment t*+1 outcome level difference i.e. γDt*+1 and the predicted one (γDt*+DD[1]t*;t*-1) given the level difference in t* and its expected rise due to growth rate difference between t* and t*-1 (see Figure 3 for the link between the algebra and the graphical representation). This prediction uses only regression coefficients driven by pre-treatment observations; a major difference with the traditional polynomial time-trend corrected method mentioned in the introduction. The DD[2]t*+1;t*-1 estimator can be generalised to the case where one wants/has de possibility to calibrate Parallel[2] using more than 2 adjacent pre-treatment periods. Imagine one has v>2 pre-treatment and 1 post-treatment observations. The estimator becomes DD[p=2]t*+1/t*-v= DD[1]t*+1; t* - 1/v DD[1]t*;t*-v =γDt*+1- (1+1/v)γDt* +1/vγDt* -v [4.] with v≥p-1 One can also account for the possibility that treatment lasts more than one period or, alternatively, that its effects are lagged (i.e. it takes several periods for the treatement to deliver Figure 3: How difference-in-[growth-rate] differences (DD[2]/Parallel[2]) can cope with non-parallel paths$ $ On this figure, t*-1 is the first period observed in the data. Hence, γDt*-1 is subsumed into γD and, in contrast with [3], DD[2] is computed using only 2 coefficients Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 15 Table 3: Estimates of Objective 1’s impact on the level of taxable income per head (in 2010 euros), s periods ahead of t*=1993; DD[2]t*+s;t*-1/Parallel[2] vs DD[1]t*+s;t*/Parallel[1], t*+s=1994 to 2013 t*+s s In 2010 euros In 2010 euros relative to Belgium average income Control=Liège Control=rest of Wallonia Control=rest of Belgium Control=Liège Control=rest of Wallonia Control=rest of Belgium DD[1] DD[2] DD[1] DD[2] DD[1] DD[2] DD[1] DD[2] DD[1] DD[2] DD[1] DD[2] 1994 1 72.2 113.8 -130.8 -64.2 -118.2 13.0 0.60% 0.95% -0.99% 0.11% -1.09% -0.54% 1995 2 -3.8 79.6 -198.7 -65.5 -171.6 90.7 -0.03% 0.64% -1.39% 0.73% -1.60% -0.53% 1996 3 -21.1 103.8 -215.4 -15.7 -227.4 166.1 -0.17% 0.83% -1.81% 1.33% -1.72% -0.13% 1997 4 -14.5 152.1 -217.3 49.0 -253.2 271.4 -0.11% 1.20% -1.99% 2.13% -1.71% 0.38% 1998 5 -30.0 178.3 -254.0 78.9 -335.2 320.5 -0.23% 1.36% -2.56% 2.45% -1.94% 0.60% 1999 6 -75.0 174.9 -325.9 73.6 -473.2 313.7 -0.56% 1.30% -3.52% 2.34% -2.43% 0.55% 2000 a 7 177.0 468.6 -131.4 334.6 -426.1 491.9 1.30% 3.45% -3.14% 3.62% -0.97% 2.46% 2001 8 153.6 486.8 -97.8 434.8 -547.9 501.3 1.10% 3.50% -3.94% 3.60% -0.70% 3.13% 2002 9 215.1 590.0 -106.3 492.9 -519.7 660.7 1.51% 4.15% -3.66% 4.65% -0.75% 3.47% 2003 10 235.0 651.5 -97.9 567.9 -528.3 783.2 1.62% 4.50% -3.65% 5.41% -0.68% 3.93% 2004 11 51.2 509.4 -361.9 370.4 -452.5 990.2 0.34% 3.39% -3.01% 6.59% -2.41% 2.46% 2005 12 94.0 593.9 -334.6 464.3 -371.8 1202.0 0.62% 3.94% -2.47% 7.98% -2.22% 3.08% 2006 13 94.9 636.4 -395.9 469.6 -433.3 1271.6 0.62% 4.15% -2.82% 8.29% -2.58% 3.06% 2007 b 14 91.4 674.6 -442.7 489.4 -506.8 1329.3 0.59% 4.32% -3.24% 8.51% -2.83% 3.13% 2008 15 -4.8 620.0 -520.9 477.8 -482.0 1485.3 -0.03% 3.98% -3.09% 9.53% -3.34% 3.06% 2009 16 -41.2 625.2 -556.1 509.1 -481.1 1617.3 -0.26% 3.94% -3.03% 10.19% -3.50% 3.21% 2010 17 14.8 722.9 -529.6 602.3 -405.0 1824.5 0.09% 4.63% -2.60% 11.70% -3.39% 3.86% 2011 18 -17.2 732.6 -547.6 650.8 -378.0 1982.7 -0.11% 4.71% -2.43% 12.75% -3.52% 4.18% 2012 19 -28.4 763.0 -620.9 644.1 -497.0 1994.8 -0.18% 4.88% -3.18% 12.76% -3.97% 4.12% 2013 20 -17.1 816.0 -612.5 719.1 -480.8 2142.2 -0.11% 5.16% -3.04% 13.55% -3.87% 4.55% Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 16 Figure 7: Estimates of Objective 1’s impact on the level of taxable income per head (in 2010 euros), s periods ahead of t*=1993; DD[2]t*+s;t*-1/Parallel[2] vs DD[1]t*+s;t*/Parallel[1], t*+s=1994 to 2013 Obj1 Obj1Phasing out -500 0500 1000 1500 2000 Euros 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Year Estimates obtained using Statistics Belgium municipal-level taxable income data weighted by population sizes & deflated by CPI (1=2010) Liege =ref Obj1 Obj1Phasing out -500 0500 1000 1500 2000 Euros 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Year Estimates obtained using Statistics Belgium municipal-level taxable income data weighted by population sizes & deflated by CPI (1=2010) r. of Wal. =ref Obj1 Obj1Phasing out -500 0500 1000 1500 2000 Euros 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Year Estimates obtained using Statistics Belgium municipal-level taxable income data weighted by population sizes & deflated by CPI (1=2010) r. of Belgium=ref DD[1] vs DD[2] estimates; t=1993 vs t+s=2000/2013 Change in income level handicap DD[1] DD[2] Obj1 Obj1Phasing out -.05 0.05 .1 .15 % 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Year Estimates obtained using Statistics Belgium municipal-level taxable income data weighted by population sizes & deflated by CPI (1=2010) Liege =ref Obj1 Obj1Phasing out -.05 0.05 .1 .15 % 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Year Estimates obtained using Statistics Belgium municipal-level taxable income data weighted by population sizes & deflated by CPI (1=2010) r. of Wal. =ref Obj1 Obj1Phasing out -.05 0.05 .1 .15 % 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Year Estimates obtained using Statistics Belgium municipal-level taxable income data weighted by population sizes & deflated by CPI (1=2010) r. of Belgium=ref DD[1] vs DD[2] estimates; t=1993 vs t+s=2000/2013 Change in income level handicap in % of Belgian average DD[1] DD[2] Economics: The Open-Access, Open-Assessment E-Journal 12 (2018–9) www.economics-ejournal.org 17 comparing Hainaut to the rest of Wallonia and the rest of Belgium. On the lower part of Figure 7, results are normalized by the average taxable income per head of the whole of Belgium. Qualitatively, the results are unaffected. In particular DD[2] estimates suggest that Objective 1 has had a positive impact on the growth-rate difference that Hainaut was suffering from before 1994. That positive effect is particularly visible beyond 1999, in comparison with Liège and the rest of Wallonia. In the absence of this correction, the rise of the income-level difference (captured by DD[1]) would have been larger. Over the year 2000 horizon (Table 3), Hainaut experienced a rise of its income level difference compared to the rest of Belgium of 426.1 euros. What DD[2]t*+s = 491.9 euros means is that in the absence of a growth rate difference positive correction; that rise would have been of 426.1 + 491.9 euros. 6 Concluding remarks The traditional difference-in-differences DD[1] model – and the parallel-paths Parallel[1] assumption on which it rests – seems to be particularly irrelevant in the case of Objective1Hainaut; and perhaps also for other EU rust-belt regions that became eligible to Ojective1. Remember that Hainaut got selected by the EU expressly because "it was suffering from a substantial deterioration of its economic and social situation". This statement hints at a development path that was not parallel to that of other EU or Belgian regions. We show in this paper that this was indeed the case before the introduction of Objective 1. And this is something that disqualifies DD[1] to be a proper treatment-effect identification strategy. From a methodological point of view, we also show that if data contain more than one point of observation before treatment, it is very easy to drop Parallel[1] – i.e. the parallel-paths assumption on which DD[1] is based – and implement DD[2]/Parallel[2] ; or even models allowing for higher degree of parallelism. In a nutshell, Parallel[2] i) allows for (time-invariant) growth-rate differences in the absence of treatment and ii) ascribes to the treatment (the outcome effect of) any change of the ex ante growth difference. The paper also shows that the estimation of treatment outcome under Parallel[2], or higher degree of parallelism, can be achieved via OLS applied to a generalized version the canonical linear DD equation. Last, and not least, the correction for trend divergences between treated and control only rests on pretreatment observations; and the estimation of the treatment effect is based on a simple extrapolation of pre-treatment trends: an improvement in comparision with can be achieved by resorting to polynomial time-trend corrected DD. This being said, as our Hainaut-Objective 1 results clearly show, DD[2]/Parallel[2] [or higher order] estimates are much more likely to lead to the conclusion that the treatment has been effective: all it takes is a small reduction of the pre-treatment growth rate difference to conclude that treatment has generated economic gains. And in the case of Hainaut, we show that this has happened against a background of a steadily rising income level difference; i.e. something that most people would probably interpret as an absence of convergence. 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