Hurdles and steps: Estimating demand for solar photovoltaics
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Gillingham, Kenneth; Tsvetanov, Tsvetan Article Hurdles and steps: Estimating demand for solar photovoltaics Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Gillingham, Kenneth; Tsvetanov, Tsvetan (2019) : Hurdles and steps: Estimating demand for solar photovoltaics, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 10, Iss. 1, pp. 275-310, https://doi.org/10.3982/QE919 This Version is available at: https://hdl.handle.net/10419/217143 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/
Quantitative Economics 10 (2019), 275–310 1759-7331/20190275 Hurdles and steps: Estimating demand for solar photovoltaics Kenneth Gillingham School of Forestry & Environmental Studies, Department of Economics, School of Management, Yale University and NBER Tsvetan Tsvetanov Department of Economics, University of Kansas This paper estimates demand for residential solar photovoltaic (PV) systems using a new approach to address three empirical challenges that often arise with count data: excess zeros, unobserved heterogeneity, and endogeneity of price. Our results imply a price elasticity of demand for solar PV systems of −065.Counterfactual policy simulations indicate that reducing state financial incentives in half would have led to 9% fewer new installations in Connecticut in 2014. Calculations suggest a subsidy program cost of $364/tCO2assuming solar displaces natural gas. Our Poisson hurdle approach holds promise for modeling the demand for many new technologies. Keywords. Count data, hurdle model, fixed effects, instrumental variables, Poisson, energy policy. JEL classification. C33, C36, Q42, Q48. 1. Introduction The market for rooftop solar photovoltaic (PV) systems has been growing rapidly around the world in the past decade. In the United States, there has been an increase in new installed capacity from under 500 MW in 2008 to over 4500 MW in 2013, along with a decrease in average (preincentive) PV system prices from over $8/W in 2008 to just above $4/W in 2013 (in 2014 dollars) (Barbose, Weaver, and Darghouth (2014)). These major changes in the market have come over a period of considerable government support at both the state and federal levels, ranging from state-level rebates to a 30% federal tax credit. For example, in Connecticut (CT), the 2008–2014 average combined state and federal incentives equaled 50% of the system cost for most PV system purchasers.1Yet Kenneth Gillingham: [email protected] Tsvetan Tsvetanov: [email protected] The authors would like to thank the Connecticut Green Bank for providing the data used in this analysis. We also thank Bryan Bollinger, CG Dong, Robert Mendelsohn, Corey Lang, Jim Sallee, Steve Sexton, Ted Juhl, Arthur van Benthem, Zhipeng Liu, and numerous seminar audiences for helpful discussions. Finally, we acknowledge funding from US Department of Energy award DE-EE0006128. 1This calculation is made based on an average system price in 2008–2014 in CT of $36,607, average state rebate amount of $10,288, and tax credit (taken post-incentive) of $7896 for consumers with at least this much tax burden (all in 2014 dollars). ©2019 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE919
276 Gillingham and Tsvetanov Quantitative Economics 10 (2019) such substantial government support is being slowly reduced in many markets throughout Europe and the United States. This paper develops a new approach to addressing key empirical challenges in estimating demand with count data and applies this approach to model the demand for residential PV systems in CT. We use rich installation-level data over the period 2008 to 2014 to shed new light on a small, but fast-growing market that is similar to many others around the world. We estimate a price elasticity of demand for a PV system of -0.65, a finding useful to both policymakers and firms. Policymakers are often interested in how changes in PV system prices—whether due to policy or other factors—influence the sales of PV systems. Such knowledge is essential for assessing the impacts of solar PV policies. Given the evidence that PV system installation markets are often imperfectly competitive (Bollinger and Gillingham (2016), Gillingham et al. (2016), Pless and van Benthem (2017)), firms may be able to use this elasticity to inform pricing decisions and market forecasts. The empirical challenges that motivate our approach are threefold: possible unobserved heterogeneity at a fine geographic level, excess zeros, and endogeneity of price (due to simultaneity). Existing studies have developed count data methods that can separately address the presence of excess zeros (e.g., Pohlmeier and Ulrich (1995), Santos Silva and Windmeijer (2001), Winkelmann (2004)) or endogeneity of covariates without unobserved heterogeneity (e.g., Mullahy (1997), Windmeijer and Santos Silva (1997), Terza (1998)). Windmeijer (2008) goes further in addressing endogeneity in the presence of unobserved heterogeneity, but does not address excess zeros. This paper is the first to address all three challenges in a count data setting with clear policy significance. We use panel data on the count of annual solar PV systems installed in a Census block group. Such panel data allow us to address likely unobserved heterogeneity in environmental preferences or other block group-specific factors with fixed effects. Aggregating at a higher level would leave this unobserved heterogeneity unaddressed. However, 74% of the observations have zero values for the number of installations in a block group in a year. Thus, classic models for count data are problematic, such as the Poisson and negative binomial, which likely misspecify the underlying data generating process. Moreover, the negative binomial model cannot readily accommodate block group fixed effects and endogeneity. To address this problem, we employ a hurdle model, which is recognized as an effective tool for dealing with the presence of excess zeros in count data settings (e.g., Cameron and Trivedi (2013)). We estimate a hurdle model based on two data generating processes: a standard logit for whether a block group has at least one adoption and a zero-truncated Poisson that models the rate of adoptions conditional on a block group having an adoption. This hurdle model has a clear behavioral interpretation in our setting: the first installation in an area is a rare event, but once there are multiple installations, peer effects may begin to influence adoption and installers can focus marketing on the area, so we have a count process. In order to tackle unobserved heterogeneity and endogeneity of the price variable, we extend the hurdle model to accommodate fixed effects and instrumental variables.
Quantitative Economics 10 (2019) Estimating demand for solar photovoltaics 277 At the basis of our approach is the conditional maximum likelihood (CMLE) estimator for fixed effects logit models introduced in a sequence of works by Rasch (1960,1961), Andersen (1972), and Chamberlain (1980). Majo and van Soest (2011) show that this conditional maximum likelihood approach can also be applied to the zero-truncated Poisson framework and present an application of this in a two-period setting. We generalize the model of Majo and van Soest (2011) to a setting with an arbitrary number of periods and possible endogeneity. In contrast to Majo and van Soest (2011), our approach uses a generalized method of moments (GMM) estimator that is based on the first-order conditions of the truncated Poisson CMLE. By using a GMM approach (as in Windmeijer (2008) and several other papers), we can draw upon the established procedures for addressing endogeneity using instrumental variables. To the best of our knowledge, this instrumental variables Poisson hurdle model with fixed effects is new to the literature, and we both prove the consistency of our estimator and verify it with a Monte Carlo simulation. This approach is particularly useful for solar PV markets, but we expect that it is more broadly applicable to many other settings with similar empirical challenges, such as the demand for many early-stage technologies. Identification in our setting is based on deviations from block group and year means in both the number of installations and PV system prices, after controlling for a variety of potential confounders and instrumenting for price. As instruments, we use a set of supply shifters: local roofing contractor wage rates and state incentives for PV systems. After controlling for income at a localized level, county-level roofer wages act as a valid contractor marginal cost shifter. Solar PV incentives in CT are given directly to the installing firm rather than the consumer, so the consumer sees the post-incentive price at the bottom of any contract to install a PV system. Thus, the incentives also act as a valid marginal cost shifter. Our preferred estimate of the price elasticity of PV system demand of −065 is the first estimate we are aware of for CT. This result is comparable to the existing literature, which uses data from California and very different empirical strategies. For example, Rogers and Sexton (2014) find a rebate elasticity of approximately −04, while Hughes and Podolefsky (2015) find an estimate of approximately −12.Incontrasttotheprevious papers, which both use reduced form approaches, Burr (2014) estimates a dynamic discrete choice model of demand, but does not specifically estimate a price elasticity of demand. The dynamic discrete choice approach may seem well-suited for our context, since subsidies may be changing over time and a household usually only installs a solar system once (although a household can always add capacity later if there is available roof space). However, we provide survey and descriptive evidence in Online Appendix A of the Supplemental Material (Gillingham and Tsvetanov (2019)) suggesting that solar PV demand in CT is more similar to the many other contexts where consumers do not appear to treat adoption as a dynamic “buy-or-wait” decision. Using our results, we perform policy counterfactual simulations to examine the impact of state financial incentives and permitting policies on PV system adoption. We first perform a simple analysis of the pass-through of the incentives to consumers, following Sallee (2011), and find the pass-through rate of 84%. Our simulation results suggest, under a 50% reduction of all financial incentives for purchasing PV systems, the number
278 Gillingham and Tsvetanov Quantitative Economics 10 (2019) Table 1. Timeline of pre-RSIP solar rebates. Rebate Structure Program Opens Change Change Closed Reopened Date July 1, 2004 Jan 29, 2007 Oct 27, 2008 Nov 19, 2008 May 18, 2009 ≤5kW $500/W $500/W $400/W - $175/W >5kW and ≤10 kW $000/W $430/W $250/W - $125/W Note: Source: The Connecticut Green Bank. of new installations in CT in 2014 would have been 9% less than observed. This would result in up to 13MW less added PV capacity in 2014. Simple calculations suggest a costeffectiveness of the program of $364 per avoided ton of CO2($594 if the federal tax credit is included), assuming that solar power displaces natural gas-fired generation. The remainder of this paper is structured as follows. Section 2provides a background on policies in CT targeted at stimulating solar demand. Section 3describes the data used in our analysis. Sections 4and 5outline the estimation methodology. Section 6presents our empirical results. Section 7presents the pass-through analysis, counterfactual policy simulations, and a discussion of cost-effectiveness and welfare. Finally, Section 8 concludes. 2. Background on solar PV policies in Connecticut Despite receiving fewer hours of sun than more southerly regions,2CT has a robust and growing market for solar PV systems, due to high electricity prices, many owneroccupied homes, and considerable state support for PV systems. A $5 per watt (W) rebate for residential solar PV systems (up to 5kW) was available in CT as early as 2004, as shown in Table 1. The structure of incentives changed on July 1, 2011, with the passage of Connecticut Public Act 11-802, directing the newly established CT Energy Finance and Investment Authority, which has since been renamed the Connecticut Green Bank (CGB), to develop a residential solar investment program that would result in at least 30 MW of new residential PV installations by the end of 2022 (Shaw, Fahey, and Solomon (2014)).3 Starting on March 2, 2012, two types of financial incentives were offered under the “Residential Solar Investment Program” (RSIP). For households that purchase a solar PV system, CT offers an upfront rebate, under an “expected performance based buy-down” (EPBB) program (replaced by the similar “homeowner performance based incentive” (HOPBI) rebate program after July 11, 2014). The rebate per W decreases based on the size of the system, with lower incentives per W for systems larger than 5kW and even lower for systems larger than 10 kW. Most systems in CT fall under this incentive program until near the end of our time period, when a higher percentage of systems are not purchased outright. For such third party-owned systems (e.g., installed under a solar lease or power purchase agreement), CT offers quarterly incentive payments based on 2See solar insolation maps at http://www.nrel.gov/gis/solar.html. 3This target had already been reached by the end of 2014. See http://www.ctcleanenergy.com.
Quantitative Economics 10 (2019) Estimating demand for solar photovoltaics 279 Table 2. Residential solar investment program timeline. Incentive Type Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 EPBB/HOPBI Start Date March 2, 2012 May 18, 2012 Jan 4, 2013 Jan 6, 2014 Sept 1, 2014 Jan 1, 2015 Incentive Design: ≤5kW $2450/W $2275/W $1750/W $1250/W $0800/W $0675/W >5kW and ≤10 kW $1250/W $1075/W $0550/W $0750/W $0800/W $0675/W >10 kW and ≤20 kW - - - - $0400/W $0400/W PBI Start Date March 2, 2012 May 18, 2012 April 1, 2013 Jan 6, 2014 Sept 1, 2014 Jan 1, 2015 Incentive Design: ≤10 kW $0300/kWh $0300/kWh $0225/kWh $0125/kWh $0180/kWh $0080/kWh >10 kW and ≤20 kW - - - - $0600/kWh $0060/kWh Note: Source: The Connecticut Green Bank. production (in kWh) over 6years, under a “performance-based incentive” (PBI) program (Shaw, Fahey, and Solomon (2014)). All incentives are being reduced over time in a series of steps as the solar PV market grows. Unlike other states, where learning-by-doing was an explicit policy motivation (Bollinger and Gillingham (2016)), in CT this declining step schedule is primarily motivated by budget constraints and a broad desire for a self-sustaining market. Table 2provides details of the programs and the step schedule. Online Appendix A gives more detail on the path of subsidies and consumer adoption of solar PV, providing evidence that the subsidy declines were unanticipated by consumers. Besides direct financial incentives, CT has several additional programs to promote solar PV systems. One program is “net metering,” which allows owners of solar PV systems to return excess generated electricity to the grid, offsetting electricity that is used during times of nongeneration, so the final electricity bill includes a charge for only the net electricity usage. PV system owners can also carry over credits for excess production for up to 1year. On March 31 of each year, the utility pays the PV system owners any net excess generation remaining at the avoided cost of wholesale electricity (a lower rate than the retail rate). Another program in CT is an effort starting in 2011 to streamline municipality and utility permitting, inspection, and interconnection processes. Several municipalities in CT have reduced their permitting costs and reduced the permitting time in response to this effort. Similar efforts have occurred in a number of other states, such as Arizona, California, and Colorado (CEFIA (2013)). Perhaps the most important nonrebate program for solar PV system adoption in CT is the CGB-sponsored “Solarize CT” grassroots marketing program in select municipalities.4Solarize CT involves local campaigns with a municipality-chosen installer, roughly 20-week time frame, and prenegotiated group discount pricing for all PV system installations in the town. According to Gillingham and Bollinger (2017), this program led to a substantial increase in demand in these municipalities: on average roughly 30 addi4Solarize CT is funded by foundations, ratepayers, and other grants.
280 Gillingham and Tsvetanov Quantitative Economics 10 (2019) tional installations per municipality over the length of the program. The program is also associated with a roughly $040 to $050 per W decrease in installation prices. 3. Data Our primary dataset contains nearly all residential solar PV system installations in CT from the period 2008–2014. Each installation that receives a rebate in the two major investor-owned electric utility regions in CT, United Illuminating and Eversource Energy (formerly Connecticut Light & Power) is entered into a database by CGB, with information on price, rebates granted, system size (in kW), technical characteristics, financing arrangements, address, date of application processing, and date of installation.5Until late 2014, there were few third party-owned (TPO) systems in CT (i.e., solar leases or power-purchase agreements) and the price data for these TPO systems are well known to be less reliable than owned-system price data. Thus, we exclude these TPO systems from our primary analysis.6Our raw dataset contains 5070 residential PV installations approved for the rebate by the CGB between January 1, 2008 and December 31, 2014. We geocode the installations and match each to the U.S. Census block group they reside in. We then collect U.S. Census data at the block group level from the 2006–2010, 2007–2011, 2008–2012, and 2009–2013 waves of the American Community Survey (ACS). We include data on total population, median household income, median age, and education level. We obtain a measure of block group population density by dividing population by land area. To generate panel data for each variable, we use the average value for each variable across all waves available for that year. So, for example, the value for population in 2010 would be the average of the 2006–2010, 2007–2011, 2008–2012, and 2009–2013 values for this variable. We use the 2009–2013 values for 2014.7In addition to Census data, we also draw town-level voting registration data from the Office of CT’s Secretary of the State (http://portal.ct.gov/sots) for each year in our study period. Finally, we bring in annual county-level roofing contractor wage data from the U.S. Bureau of Labor Statistics (http://www.bls.gov). 3.1 Preparation of the panel dataset Due to the possibility of unobserved heterogeneity at the localized level, we convert our data to a panel dataset, where an observation is at the Census block group-year level. This leads to a balanced panel of 10,738 observations in 1534 block groups. For each 5The only other utilities in CT are small municipal utilities in Bozrah, Norwich, and Wallingford (Graziano and Gillingham (2015)). 6As a robustness check, we also analyze the full dataset of purchased and third-party-owned (TPO) systems, controlling for third party ownership in the regressions. The results, presented in Online Appendix E in the Supplemental Material (Gillingham and Tsvetanov (2019)), should be interpreted with great caution due to the known issues with over-reporting of TPO system prices. They are suggestive of a different consumer decision-process for TPO systems. 7We also employed an alternative approach, using only the mid-year of each ACS wave and interpolating for all missing years, but this made no practical difference to our results. More generally, as shown in Section 6.2, our results are also robust to the exclusion of all demographic variables.
Quantitative Economics 10 (2019) Estimating demand for solar photovoltaics 281 block group-year, we take the average of the system price, system size, and incentive level granted. We also create an indicator variable for a Solarize campaign occurring in the given block group-year. Using this panel dataset for our empirical analysis necessitates one further step. Many observations refer to block group-year combinations where there are no installations. In fact, our hurdle model approach is motivated by these excess zeros in our dataset. However, we still have to determine the relevant installation price, system size, and incentive for observations with no recorded contracts. This is a common issue in the empirical literature and we take a conservative stance by examining two different approaches. In our primary approach, we fill in the missing price and other variables with the average annual value for the same municipality, and if this is not possible, we use the average annual value within the county of the installation. Given that our adoption data in these block groups is censored at zero, this approach may underestimate the price. Thus, as shown in Section 6.2, we perform several robustness checks and find that our results are robust to this choice. Our post-incentive PV system price per W variable is based on the ratio of the average block group-year PV system prices and the average system size. 3.2 Trends and summary statistics Figure 1displays the overall trend in installations and PV system prices (in 2014 dollars) during our study period. Between 2008 and 2014, the real average (post-incentive) price falls by more than 40%, with a brief spike in 2009. This spike in the post-incentive price is commonly attributed to a roughly 50% drop in the incentive between 2008 and 2009. Consistent with larger trends in the global PV market (Barbose, Darghouth, Weaver and Wiser (2013)), real installation costs have been steadily decreasing during the entire study period, while the efficiency of panels and quality of installations remained relatively constant. Figure 1also shows a substantial increase in PV system installations after 2011. Some of this increase after 2011 consists of installations under Solarize programs, which involve both lower prices and additional solar marketing. Table 3presents summary statistics for our panel dataset of 10,738 block group-year observations. The number of PV system installations is a count variable, with a mean of 048 and a variance of 119. Hence, these data exhibit clear overdispersion relative to standard count models, such as the Poisson model, in which an underlying assumption is the equality of mean and variance of the count variable (see Section 4.2). Figure 2reveals why this might be the case: the distribution of installations in a block group in a given year is very strongly skewed, with most of the mass (over 74%) centered at zero. This visually illustrates the issue of excess zeros in our dataset. Table 4presents summary statistics for the truncated dataset with positive installations. Comparing Tables 3 and 4reveals some differences. Most notably, there is a much stronger presence of Solarize campaigns in the truncated sample (12% versus 6%). The truncated sample has lower average population density, consistent with Graziano and Gillingham (2015), who show that most installations in CT occur in suburban or rural areas. This sample also
282 Gillingham and Tsvetanov Quantitative Economics 10 (2019) Figure 1. Trends in installations and average post-incentive system price. Table 3. Summary statistics for the full sample. Variable Mean St. Dev. Min Max Number of PV installations 04781 10891 0 20 System capacity (kW) 6972 18417 07221 Post-incentive system price ($/W) 37645 14369 00171 205824 Solarize campaign 00595 02366 0 1 Population density (per km2)853651 1249742 0 21,784 Median household income (in 1000$/year) 887752 417775 2499 250001 Median age 426938 70762 148806 %population above 25 with some college or college degree 467736 99282 0 81203 %population above 25 with graduate or professional degree 183589 121701 0 807786 %Republican voters 219084 78536 369 5114 %Democrat voters 340913 105664 1692 723 Incentive level ($/W) 35288 20203 06824 64879 Roofing contractor wage ($/week) 1031203 171964 548894 1312658 Note: All variables have 10,738 observations. All dollars in 2014 dollars. features slightly larger systems, likely because it more heavily samples block groups that are slightly wealthier and have a greater unobserved preference for PV systems. 4. Empirical specification:Preliminaries Let Yit denote the number of PV installations in block group ipurchased in year t.We use Wit to denote a vector containing the installation price pit and demand shifters. Demand for solar is represented by the following general function: Yit =Dit(WitΘ),(1)
Quantitative Economics 10 (2019) Estimating demand for solar photovoltaics 289 Truncated Poisson The truncated Poisson model is estimated using only observations for which Yit >0. Allowing for unobserved block group heterogeneity, the truncated Poisson parameter λit for each one of these observations can be expressed as follows: λit =expαP i+w itθ2+μP t≡ciexpX itδ2≡ciβit(13) with year fixed effects represented by μP t,ci≡exp(αP i),δ2, as before, denoting a vector of parameters for the time variables and all remaining observed regressors, and βit ≡ exp(X itδ2). Unlike the fixed effects Poisson model, estimating a zero-truncated Poisson with individual fixed effects through maximum likelihood does not allow for δ2to be estimated independently of the fixed effects coefficients. Therefore, MLE is inconsistent in this specification. As shown by Majo and van Soest (2011), conditional maximum likelihood can be employed in this setting, in a similar fashion to the fixed effects logit model, which eliminates the fixed effects parameters from the estimation. Majo and van Soest (2011) demonstrated this method for a two-period panel dataset. In what follows, we generalize their procedure to any panel with an arbitrary number of longitudinal observations. Let Ti⊆{1T}be the subset of periods with Yit >0and let Y∗ i={Yit :t∈Ti}.Also, let Xi={Xit :t∈Ti}denote the matrix of regressors and λi={λit :t∈Ti}be the vector of corresponding truncated Poisson parameters. Define the statistic ni=t∈TiYit = T t=1Yit.Finally,letHi(ni)be the set of all possible histories of strictly positive natural numbers di={dit ∈N+:t∈Ti}such that t∈Tidit =ni. Then we have that PrY∗ i=yi|Xiniciδ2=PrY∗ i=yi|Xiciδ2 Pr t∈Ti Yit =ni|Xiciδ2 = t∈Ti λyit it yit!exp(λit)−1 di∈Hi(ni) t∈Ti λdit it dit!exp(λit)−1 =1 hi(λi) t∈Ti ni! yit!λyit it (14) where hi(λi)≡di∈Hi(ni)t∈Tiλdit it . Since λit =ciβit,itiseasytoshowthathi(λi)= hi(ciβi)=cni ihi(βi),whereβi={βit :t∈Ti}. In other words, the function hi(·)is homogeneous of degree niin βi. So, from (14), 1 hi(λi) t∈Ti ni! yit!λyit it =1 cni ihi(βi) t∈Ti ni! yit!cyit iβyit it =1 hi(βi) t∈Ti ni! yit!βyit it rameters, and is therefore not comparable to the unrestricted likelihood of alternative model specifications. This prevents the implementation of model selection tests based on likelihood ratio statistics, including selection tests among nonnested models, such as the Vuong test (Vuong (1989)) and its extensions (e.g., Chen, Hong, and Shum (2007)).
290 Gillingham and Tsvetanov Quantitative Economics 10 (2019) that is, upon conditioning on T t=1Yit, the truncated Poisson distribution no longer depends on the nuisance parameter cibut still depends on the parameters of interest δ2, as long as Ticontains at least two periods and the explanatory variables Xit are not constant over Ti. The conditional log-likelihood then takes the following form: L(δ2)= N i=1 P iδ2 T t=1 Yit = N i=1log t∈Ti Yit!− t∈Ti log[Yit!] + t∈Ti Yit log(βit)−log(hi) (15) where P iis the conditional log-likelihood for all observations from block group iwith Yit >0. This procedure is rather general and can be applied to an arbitrarily large number of longitudinal panel observations and any number of model parameters. The parameter estimator of this fixed effects truncated Poisson model is obtained as ˆ δ2CMLE = argmaxδ2L(δ2). Under strict exogeneity of the regressors X, ˆ δ2CMLE can be shown to be a consistent estimator of δ2(See Appendix A). 5.3 Endogeneity Both the logit and truncated Poisson conditional likelihood estimators, discussed in Section 5.2, are consistent, provided that the respective models are specified appropriately. However, this would no longer be the case with endogeneity of price. We thus extend the hurdle model further in order to accommodate the implementation of a suitable instrumental variable procedure. Once again, we address the problem separately in the logit and truncated Poisson portions of the model. Logit In the discrete choice literature, a number of methods have been developed to tackle endogeneity in demand settings. These include the product-market control (and instrumental variable) approach of Berry, Levinsohn, and Pakes (1995), a simulated maximum likelihood approach developed by Gupta and Park (2009), and Bayesian methods employed by Yang, Chen, and Allenby (2003)andJiang, Manchanda, and Rossi (2009). More recently, Petrin and Train (2010) proposed a control function approach that is arguably easier to estimate and more flexible than the above methods, as it does not require invoking equilibrium or imposing strict distributional assumptions for the identification of demand parameters. As in Section 4.2, dropping subscripts for simplicity, let W=(p Z1)and Z1⊂Z, where Zis a vector of exogenous variables. Also, let Θ1=(γ11). Suppose the purchase of any positive number of PV systems in a given block group generates utility u,givenby u=γ1p+Z 11+ζ1(16) where ζ1denotes an idiosyncratic term that is correlated with price p.AsinSection4.2, let p=ZΠ+r (17)
Quantitative Economics 10 (2019) Estimating demand for solar photovoltaics 291 Since ζ1is correlated with pbut not with Z, there exists some function CF(rρ1),where ρ1is a parameter, such that ζ1=CF(rρ1)+˜ ζ1and ˜ ζ1is uncorrelated with p.Wecan then rewrite (16)as u=γ1p+Z 11+CF(rρ1)+˜ ζ1=WΘ1+CF(rρ1)+˜ ζ1(18) Therefore, if we estimate our model including the control function CF(rρ1)with valid instruments, we can consistently estimate Θ1. Petrin and Train (2010) suggested several simplifying assumptions. First, as an alternative to specifying a joint distribution for both error terms in (16)and(17), one could enter rflexibly in the utility and then choose a distributional assumption for ˜ ζ1.Second, the control function can be approximated as a linear function. Accordingly, we specify CF(rρ1)as ρ1rin (18) and assume that ˜ ζ1is distributed i.i.d. type I extreme value.11 The probability of Y>0is now given by Pr[Yit >0|WitΘ1]= biexpX itδ1+ρ1r 1+biexpX itδ1+ρ1r As in our earlier discussion from Section 4.2, this CF specification implies a twostage estimation procedure. In the first stage, we run a linear regression of price on all excluded and included instruments. The residuals from this stage are then included as a covariate in the second-stage estimation, which, following the discussion from Section 5.2, is carried out using conditional maximum likelihood in order to eliminate the nuisance parameters. Finally, we bootstrap the standard errors to ensure that our inference accounts for both stages. Truncated Poisson To our knowledge, we are the first to address endogeneity of one or more of the regressors in a fixed effects truncated Poisson model. In what follows, we develop a generalized method of moments (GMM) procedure for the consistent estimation of the vector of truncated Poisson slope parameters δ2in cases where Xis no longer strictly exogenous.12 Recall that the function hi(βi), described in Section 5.2, is homogeneous of degree ni in βi. This implies that t∈Ti ∂hi(·) ∂βit βit =nihi(·),ort∈Ti ∂hi(·) ∂βit βit hi(·)=ni. For convenience, we now define a nonlinear function φit of all regressors and slope parameters in the truncated Poisson model: φit(Xiδ2)≡ ∂hi(βi) ∂βit βit hi(βi) 11Alternatively, CF(rρ1)can be specified as a quadratic function (e.g., Olley and Pakes (1996)). We also reestimated our model using a quadratic control function and found the results to be quite robust. 12Alternatively, endogeneity in the truncated Poisson model could be addressed through a control function approach. However, unlike the GMM approach, this would require further assumptions about the functional form and distribution of the model’s error term, which would be complicated by the nonlinearity in λ.
292 Gillingham and Tsvetanov Quantitative Economics 10 (2019) By definition, it follows that t∈Tiφit =t∈TiYit since both are equal to ni. Using this equivalence and the Weak Law of Large Numbers, 1 NN i=11 T∗ it∈Tiφit =1 NN i=11 T∗ i × t∈TiYit p −→ EP[Yit],whereT∗ i=max{t:Yit >0}.Hence,inourcontextφit can be interpreted asymptotically as the predicted number of installations in block group iat time t, conditional on the occurrence of a positive number of installations. Note that this interpretation follows directly from the functional form of φit and does not require any assumptions about the exogeneity of Xi. Based on this interpretation, we proceed to construct the following model of the zero-truncated demand for solar PV: Yit =φit(Xiδ2)+ξit(19) where ξit is the econometrician’s error, which represents block group and year-specific idiosyncratic shocks that influence the number of adoptions. We now use this model to derive moment conditions for the estimation of δ2, both in the absence and in the presence of endogenous regressors. Let ξi=(ξi1ξiT∗).IfXis strictly exogenous in the demand model, the resultant orthogonality with the error term ξiprovides a moment condition E[X iξi]=0,withthe sample analog of this condition leading to a GMM estimator of δ2.Asdemonstratedin Appendix B, this GMM estimator turns out to be equivalent to ˆ δ2CMLE, which was shown in Appendix Ato be a consistent estimator of δ2under strict exogeneity of all regressors. However, with one or more endogenous regressors in the model, E[X iξi]=0no longer holds. In that case, a vector Z, comprised only of variables that are exogenous in the model, would be orthogonal to the error term in (19), that is, EZ iξi=0(20) where Zi=(Zi1ZiT ∗). Then, if δ2is a P-dimensional vector, we can use the sample analog of (20) in order to estimate δ2through a GMM estimator, as long as there are a total of at least Pexogenous variables in Z. Suppose Zit ∈Z⊂RQ,whereQ>P,andlet ψ(Ziδ2)=Z iξi.Then ˆ δ2GMM =argmax δ21 N N i=1 ψ(Ziδ2) ˆ Ξ1 N N i=1 ψ(Ziδ2)(21) where ˆ Ξis an optimal weighting matrix. In Appendix A, we demonstrate that under the standard GMM assumptions ˆ δ2GMM is a consistent estimator of δ2. 5.4 Deriving the price elasticity The estimators derived in Sections 5.2 and 5.3 do not allow us to estimate the fixed effects parameters in the model. As a result, we are not able to recover the predicted conditional expectations or calculate the exact marginal effects, needed for the derivation
Quantitative Economics 10 (2019) Estimating demand for solar photovoltaics 293 of price elasticity.13 Instead, we develop, in the spirit of Kitazawa (2012), a procedure for obtaining average elasticity estimates that are shown to converge to the true average elasticity values as N→∞. Logit The average price elasticity in the logit model is given by ηL=(1−Pr[Yit > 0])γ1E[pit],whereγ1is the coefficient on price. The following proposition derives an expression for the consistent estimator of ηL. Proposition 1. Let ˆηL=(1−¯ι) ˆγ1¯ p,where ¯ι=1 NT N i=1T t=1ιit,¯ p=1 NT N i=1T t=1pit, and ˆγ1 p −→ γ1.Then ˆηLp −→ ηL. Proof. By the Weak Law of Large Numbers (WLLN), ¯ιp −→ Pr[Yit >0]and ¯ pp −→ E[pit]. Then, since ˆγ1 p −→ γ1, by the Continuous Mapping Theorem (CMT), ˆηL=(1−¯ι) ˆγ1¯ pp −→ (1−Pr[Yit >0])γ1E[pit]=ηL. This is intuitive. Note that Pr[Yit >0|WitΘ1]= biexp(X itδ1) 1+biexp(X itδ1).Sincewedonothave an estimate of bi, we cannot use the post-estimation predicted probabilities to calculate ηL. However, asymptotically, the sample averages ¯ιand ¯ pconverge in probability to Pr[Yit >0]and E[pit], respectively. Hence, we can use these averages, together with the consistent logit CF estimate of γ1, to obtain a consistent estimator of ηL. Truncated Poisson Similarly, we derive an estimator for the average elasticity in the zero-truncated Poisson portion of the hurdle model. First, since λit =ciexp(X itδ2)and we do not estimate ci, we proceed to express λit as a function of EP[Yit]. Note that EP[Yit]= λit 1−exp(−λit)≡m(λit) Then λit =m−1(EP[Yit]). As shown in Appendix C,m(·)is a monotonic function over the relevant range of λit values in this model, implying that m−1(·)is a one-to-one mapping from EP[Yit]to λit. Next, it can be shown that the average price elasticity is given by ηP=1+λit −EP[Yit]γ2EP[pit]=1+m−1EP[Yit]−EP[Yit]γ2EP[pit] where γ2is the coefficient on price in the truncated Poisson model. This suggests a straightforward estimator of ηP, presented in the following proposition. Proposition 2. Let ˆηP=(1+m−1(¯ YP)−¯ YP)ˆγ2¯ pP,where ¯ YP=1 NN i=11 T∗ it∈TiYit, ¯ pP=1 NN i=11 T∗ it∈Tipit,and ˆγ2 p −→ γ2.Then ˆηPp −→ ηP. Proof. By WLLN, ¯ YPp −→ EP[Yit]and ¯ pPp −→ EP[pit].ByCMT,m−1(¯ YP)p −→ m−1(EP[Yit]). Since ˆγ2 p −→ γ2,ˆηP=(1+m−1(¯ YP)−¯ YP)ˆγ2¯ pPp −→ (1+m−1(EP[Yit])−EP[Yit])γ2EP[pit]= ηP,byCMT. 13Furthermore, this implies that model selection tests based on conditional expectations (e.g., Silva, Tenreyro, and Windmeijer (2015)) cannot be implemented.
294 Gillingham and Tsvetanov Quantitative Economics 10 (2019) Table 5. Monte Carlo simulation results. Parameters Elasticity Estimated Implied True Value True Value Specification Mean Bias MSE Value Bias Logit CF δ1=−01−00961 00039 00003 η1=−00767 −00780 −00013 Tr. Poisson GMM δ2=−01−00999 00001 00001 η2=−01716 −01712 00004 Poisson hurdle n/a n/a n/a n/a η=−02483 −02493 −00010 Linear 2SLS n/a −02257 n/a n/a η=−02483 −02654 −00171 Poisson CF n/a −01103 n/a n/a η=−02483 −02659 −00176 Note: See Online Appendix B.2 in the Supplemental Material (Gillingham and Tsvetanov (2019)) for details about construction of the simulated data. Output is based on 5000 replications. Sample means and parameter values, averaged over the 5000 replications, are used to compute elasticity under each specification. Elasticity values in the hurdle model are calculated using the formulas in Section 5.4. The elasticity in the linear model is derived from ˆη=ˆ δ¯ w ¯ y. The elasticity in the Poisson model is derived from ˆη=ˆ δ¯ w. Thus, ηPcan be consistently estimated using the mean values for installations and prices in the truncated sample, along with the GMM estimate ˆγ2. 5.5 Monte Carlo simulations While we have already proven the consistency of our estimator, we conduct a set of Monte Carlo simulations to evaluate its performance. In short, we simulate data from our data generating process and then apply our estimator and alternative estimators to the simulated data. We use a simple panel data framework, in which the outcome count variable is determined by a set of unobserved time-invariant and time-varying crosssection-specific factors and a single endogenous regressor. The model, data generation process, and parameter values used in these simulations are described in Online Appendix B.2 in the Supplemental Material (Gillingham and Tsvetanov (2019)). Our results are encouraging. Table 5shows the results from one representative simulation. The bias on each of the parameters of interest using the logit CF and truncated Poisson GMM is small: 3% for the logit CF and 02% for the truncated Poisson. Converting this to an elasticity, we find that our estimated (combined) elasticity is within 12% of true elasticity value. Compared to the results obtained using linear two-stage least squares (2SLS) and Poisson CF approaches, we find that our estimated elasticity is much closer to the true value. We repeat this procedure with several different sets of parameter values and find similar results.14 14We also run another set of Monte Carlo simulations (described in Online Appendix B.3 in the Supplemental Material (Gillingham and Tsvetanov (2019))), which shows that even in the case where the true data generation process is Poisson with fixed effects and an endogenous regressor (i.e., it does not have excess zeros), the hurdle model still performs very well in recovering the true values.
Quantitative Economics 10 (2019) Estimating demand for solar photovoltaics 295 6. Results 6.1 Primary results Our identification strategy relies on the validity of our instruments. As described earlier, we use two marginal cost shifters as instruments for the post-incentive price: average incentive levels in $/W for the first 5kW of installed capacity in each block group-year and county-year average roofer wages. Incentives are given directly to installers in CT, so they act as a shifter of the firm marginal cost. Due to the declining incentives and time lag between the submission and approval of contract applications, there is both temporal and cross-sectional variation in incentive levels.15 County-year average roofer wages are used as a proxy for PV system installer labor costs, and are a valid shifter after controlling for income.16 Online Appendix C contains the results of the first-stage regression, demonstrating that we do not have to worry about weak instruments. Table 6presents the results from estimating a linear model, Poisson model, and Poisson hurdle model with instruments for price.17 Our preferred specification is the instrumental variables hurdle model, consisting of a logit regression estimated with a control function approach in column (3) and a trucated Poisson estimated by GMM in column (4). All columns include block group fixed effects and year dummies. Standard errors are clustered at the town level. At the bottom of the table, we present estimates of the price elasticity of demand for each specification. We are most interested in the statistical and economic significance of the price coefficient. We find negative and statistically significant coefficients on the price variable in all model specifications. Rather than interpreting the coefficient directly, we find it more instructive to consider the price elasticity implied by the coefficient taken at the mean of our sample.18 Estimating the model using a Poisson specification implies a considerably higher (in absolute value) price elasticity relative to the linear specification. A linear 2SLS regression yields a price elasticity of −062. Fitting a Poisson model generates an elasticity coefficient that is almost twice as high. However, both of these empirical specifications are unsuited for our data with excess zeros, as discussed in Section 4. As shown earlier, the total price elasticity of the hurdle model, which is our preferred specification, is the sum of the elasticities from the logit and truncated Poisson. Summing the price elasticity estimates in columns (3) and (4) shows that the hurdle model 15The time lag provides cross-sectional variation because the incentives granted are based on the approval date of the contract. Consumers are assumed to have rational expectations about what the awarded incentive will be at the time of approval. 16While over-identification tests are never definitive, we use a Hansen’s J overidentification test to examine the validity of the instruments in the truncated Poisson GMM specification of our hurdle model. Comfortingly, we find that we fail to reject the null of valid instruments, with a Chi-squared test statistic of 047 and a p-value of 049. We find similar results for other specifications. 17For completeness, Online Appendix D in the Supplemental Material (Gillingham and Tsvetanov (2019)) also presents output from the same model specification without the use of instrumental variables. We view these results with great caution as the coefficients are most likely biased due to the endogeneity of price. 18The price elasticity in the linear model is estimated as η=ˆγ¯ p ¯ Y,where ˆγis the two-stage least squares (2SLS) estimate of the price coefficient and ¯ pand ¯ Yare the sample means for price and number of installations. The price elasticity in the Poisson model is estimated as η=ˆγ¯ p,where ˆγis the Poisson CF estimate of the price coefficient.
296 Gillingham and Tsvetanov Quantitative Economics 10 (2019) Table 6. Primary estimation results. Hurdle Linear Poisson Logit Trun. Poisson 2SLSiCFii CFii GMMi Variable (1)(2)(3)(4) Price −0079*** −0286*** −0201** −0292** (00292)(00982)(00999)(0132) Solarize 0929*** 0913*** 0863*** 0877*** (0242)(015)(0204)(0161) Pop. density −56×10−6−00002*** −00002** 00004 (94×10−6)(000007)(000009)(00005) Income 00008 00004 −00014 00013 (000077)(000154)(000152)(000267) Age −0003 −0012** −00096*−0008 (00026)(00047)(00053)(0012) %(some) college 00003 00002 0002 0005 (0001)(0003)(0004)(0007) %grad/prof degree 0001 −00003 0006 −0005 −0002 −0004 −0005 (0008) %Republican 0063*** 0077*0036 0153*** (0024)(0040)(0041)(0059) %Democrat 0027** 0041*0011 0102*** (00126)(00222)(00248)( 00384) BG FE yes yes yes yes Year Dummies yes yes yes yes Instruments yes yes yes yes Price elasticityiii −0621*** −1076*** −0528** −0123** (02301)(03698)(02626)(00554) Observations 10,738 10,738 10,738 3238 Note: Dependent variable is number of residential PV installations. Unit of observation is block group-year. BG FE refers to block group fixed effects. Year FE refers to year fixed effects. The instruments used for all IV specifications are the EPBB/HOPBI state financial incentives given to the installers and the county roofing contractor wage rate. p<01(*), p<005 (**), p<001 (***). iClustered standard errors at the town level in parentheses. ii Block bootstrapped standard errors (100 replications), clustered at the town level, in parentheses. iii Standard errors for the price elasticity obtained by the delta method. implies a price elasticity of solar PV system demand of −065. At the average system price and number of installations in our sample, this elasticity estimate suggests that a $1/W decrease in the installation price (well within the variation in our data) would lead to an increase in demand of approximately 0083 additional PV systems in each block group during the respective year. In our data, there are 1534 block groups in any given year. Thus, a $1/W decrease in system price translates into 127 additional installations demanded statewide in that year. The other coefficients are of less interest to us, but we are reassured to see that the signs are generally consistent and make sense. One of the more interesting of these is the coefficient on the indicator for whether a block group has a Solarize campaign, which
Quantitative Economics 10 (2019) Estimating demand for solar photovoltaics 297 is positive and highly statistically significant in all specifications. This is consistent with results in Gillingham and Bollinger (2017), which uses quasi-experimental and experimental approaches to find a large treatment effect of the Solarize program. The other statistically significant coefficients largely make sense in sign. Solar demand is higher in less densely populated areas and in areas with a younger population. Political views also appear to have a positive effect on the decision to adopt in the linear and truncated Poisson specifications. The general lack of statistical significance for the demographic and voting variables is likely attributable to the lack of sufficient time series variation in these variables. 6.2 Robustness checks We perform several robustness checks, which are both reassuring and provide insight into the variation behind our results. Table 7shows the estimated elasticities from each of these robustness checks and includes the elasticity estimates from our preferred specifications in columns (3) and (4) of Table 6for comparison. First, we are concerned that our results in the logit regression may be driven by the method we use to fill in missing price data in block group-years where no contracts were signed. We therefore reestimate the model using the highest, rather than average, recorded prices as proxies, preserving the order of our approach outlined in Section 3.1 (column (I)). We find that our logit CF estimates are only modestly affected by this change in the interpolation approach, a very reassuring result. As an alternative, we also estimate a hedonic equilibrium price equation for each year using the subsample with positive installation counts at the block group level and Table 7. Elasticity values under different specifications. Robustness Checks Model Specification Baseline I II III IV V Logit CFi−0528** −0478*−0661*−0530** −0521** −0543** (02626)(02778)(03498)(02635)(02698)(02622) Tr. Poisson GMMii −0123** −0123** −0123** −0122** −0118** −0067 (00554)(00554)(00554)(00553)(00522)(00426) Combined elasticityiii −0651** −0601** −0784** −0652** −0639** −061** (02684)(02833)(03542)(02692)(02748)(02656) BG FE yes yes yes yes yes yes Year Dummies yes yes yes yes yes yes Market size no no no yes no no Demographics/voting yes yes yes yes no yes Solarize included yes yes yes yes yes no Missing price proxy average price highest price hedonic price average price average price average price Observations 10,738 10,738 10,738 10,738 10,738 10,099 Note: Dependent variable is number of residential PV installations. Unit of observation is block group-year. All other variables are the same as in Table 6. Standard errors are obtained by the delta method. p<01(*), p<005 (**), p<001 (***). iBlock bootstrapped standard errors (100 replications), clustered by town. ii Robust standard errors, clustered by town. iii Standard errors of combined elasticity coefficients obtained assuming independence of the data generating processes.
298 Gillingham and Tsvetanov Quantitative Economics 10 (2019) including county fixed effects. The regressors are a Solarize dummy and the set of demographic and voting variables from the baseline specification. We then use the estimated coefficients to predict prices in block groups with zero installations. We are careful in interpreting the estimates in this robustness check (shown in column (II)) due to the likely selection bias that we cannot control for in the hedonic equation. It is nonetheless reassuring to note that the elasticity we obtain of −078 is relatively close to our baseline result of −065. Next, we explore the extent to which the changing market size for solar may affect our results. In particular, the pool of potential buyers continuously declines as more households adopt solar, resulting in a sequential truncation in the distribution of willingness to pay for the consumers remaining in the market.19 To test the effects of this endogenous exit of adopters, we estimate an alternative specification in which we control for the market size. We scale the number of Census occupied housing units in each block group-year by the percentage of solar-viable homes in the area calculated from satellite data by GeoStellar, and then subtract the cumulative number of installations in the block group up to that year. This provides a time-varying measure of market size. While the results from this specification should be viewed with caution, as the market size regressor may not be strictly exogenous but only predetermined, they are quite supportive of the robustness of our main model. As shown in column (III) of Table 7,our elasticity estimate is almost identical to the baseline result. Furthermore, the market size variable is not statistically significant effect in either the logit or truncated Poission, with p-values of 023 and 091, respectively. In addition, we test the sensitivity of our results to an alternative specification without demographic and voting variables among the regressors (column (IV)). Excluding these variables, we obtain almost identical elasticity estimates as in our baseline logit and truncated Poisson runs. This result is not surprising, given the lack of substantial temporal variation in these variables, as noted earlier. Lastly, we examine the effect of excluding observations from Solarize campaigns from the analysis (column (V)). This robustness check also provides insight into the origins of our results. We find an elasticity estimate from the truncated Poisson GMM that is of smaller magnitude than the baseline value, while the logit CF is roughly unchanged. The overall implied elasticity from the hurdle model is quite close to our baseline elasticity. These results underscore the importance of the Solarize campaigns for our observations with multiple installations in a block group-year. Figure 3shows a histogram of observations with at least one installation. Non-Solarize observations with positive outcomes are largely centered at low counts, while higher-count outcomes are mostly Solarize observations. Hence, when we drop all Solarize observations, our outcome variable is effectively reduced to a binary variable, enabling us to capture most of the variation in the data through the logit component of the hurdle model. This suggests that the hurdle model, as a mix of components that can exploit both binary and zero-truncated count data variation, offers a flexible approach for estimating solar demand in various settings: from emerging markets with few installations to booming higher-demand markets. 19We thank an anonymous referee for suggesting this robustness check.
Quantitative Economics 10 (2019) Estimating demand for solar photovoltaics 305 Lemma A.2. Suppose Zit ∈Z⊂RQ,where Q>P.Let ψ(Ziδ2)=Z iξiand E[ψ(Zi δ0 2)]=0.Furthermore,let G(δ2)=[E[ψ(Ziδ2)]Ξ[E[ψ(Ziδ2)]and GN(δ2)= [1 NN i=1[ψ(Ziδ2)]ˆ Ξ[1 NN i=1[ψ(Ziδ2)],where ˆ Ξis a symmetric positive semidefinite weight matrix and Ξis a symmetric and positive definite matrix.Under the following assumptions: A1. is a compact set. A2. For each z∈Z,ψ(z·)is a continuous function on . A3. For all z∈Z,ψ(zδ2)= ψ(zδ0 2)if δ2= δ0 2. A4. ˆ Ξp −→ Ξ. A5. The uniform weak law of large numbers holds, ˆ δ2GMM p −→ δ0 2,where ˆ δ2GMM = argminδ2∈GN(δ2). Proof. A1 and A2 imply that the problem minδ2∈GN(δ2)always has a solution. Let ˆ δ2GMM =argminδ2∈GN(δ2). By definition, we know that δ0 2∈solves the problem minδ2∈G(δ2).ByA3andbecauseΞis a positive definite matrix, δ0 2is a unique solution to minδ2∈G(δ2). Then, using A4 and A5, |GN( ˆ δ2GMM)−G(δ0 2)|p −→ 0. Therefore, ˆ δ2GMM p −→ δ0 2. Appendix B: Equivalence of CMLE and GMM estimators under strict exogeneity In what follows, we review the equivalence of CMLE and GMM estimators in our setting that only holds under strict exogeneity of the vector of regressors. Suppose Xit ∈X⊂RP. Then, starting from (15), we can express the P-dimensional score vector of derivatives of the log-likelihood corresponding to block group ias Si(δ2)=∇ δ2P iδ2 T t=1 Yit= t∈TiYit −φit(Xiδ2)Xit where φit ≡ ∂hi ∂βit βit hi.LetY∗ i∈Ydenote the T∗-dimensional vector of outcomes in block group i,andletPr(Y∗ i=y|Xiniδ2)≡g(y|Xiniδ2). Assuming that the model has been correctly specified, y∈Yg(y|Xiniδ2)=1for all Xi,ni,andδ2, and it can be shown that the score of the log-likelihood function, evaluated at the true parameter vector δ0 2, has a zero conditional mean. Let Si(δ0 2)(p) denote the pth element of the score vector, corresponding to ∂ ∂δp 2 P i(δ2)|δ2=δ0 2.Then ESiδ0 2(p) |Xini= y∈Y ∂ ∂δp 2 P i(δ2)|δ2=δ0 2gy|Xiniδ0 2 = y∈Y ∂ ∂δp 2 g(y|Xiniδ2)|δ2=δ0 2
306 Gillingham and Tsvetanov Quantitative Economics 10 (2019) =∂ ∂δp 2 y∈Y g(y|Xiniδ2)δ2=δ0 2 =0 Since the above is true for any element of the score, it follows that ESiδ0 2|Xini=0(B.1) Note that the log-likelihood function is derived after conditioning on the vector of regressors. Therefore, this result would not hold if one or more of the variables in Xare endogenous in the model. Let ξit ≡Yit −φit(Xiδ0 2),andletξi=(ξi1ξiT ∗). By the law of iterated expectations, (B.1) implies that EX iξi=0 that is, the expected value of the score leads to the same moment condition as the orthogonality of the regressors to the error term in (19) under strict exogeneity of X.Thus, with no endogenous regressors, the sample analog of E[X iξi]=0is identical to the firstorder conditions of the conditional likelihood from (15) and yields a GMM estimator that is equivalent to ˆ δ2CMLE. Appendix C: Monotonicity of m(λ) In order to ensure that m−1(·)is a one-to-one function, we need to show that m(λ) is monotonic over the relevant range of λvalues. In what follows, we prove that, as long as λis positive, the function m(λ) is strictly increasing. Lemma C.1. For any λ>0,m(λ) > 0. Proof. Dropping all subscripts for simplicity, m(λ) =[1 λ−e−λ 1−e−λ]m(λ) =[eλ−1−λ λ(eλ−1)]m(λ). Note that, by the properties of the truncated Poisson model, m(λ) =Et(Y) > 0for all λ. Furthermore, λ>0, which implies that λ(eλ−1)>0. Hence, we need to ensure that eλ−1−λis either positive or negative over the relevant range of λ. Let h1(λ) =eλand h2(λ) =λ+1and note that h1(0)=h2(0). Also note that h 1(0)= h 2(0)=1, while, for any λ>0,h 1(λ) > 1=h 2(λ).Hence,h1(λ) > h2(λ) for all λ>0, implying that m(λ) > 0for all λ>0, which is the relevant range of λin a truncated Poisson model. References Andersen, E. B. (1970), “Asymptotic properties of conditional maximum-likelihood estimators.” Journal of the Royal Statistical Society, Series B, 32, 283–301. [288] Andersen, E. B. (1972), “The numerical solution of a set of conditional estimation equations.” Journal of the Royal Statistical Society, Series B, 34, 42–54. [277,288] Barbose, G., N. Darghouth, S. Weaver, and R. Wiser (2013), Tracking the Sun VI: An Historical Summary of the Installed Price of Photovoltaics in the United States From 1998 to 2012. Lawrence Berkeley National Laboratory, Berkeley, CA. [281]
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