Hurdles and Steps: Estimating Demand for Solar Photovoltaics

Transcription

1 Hurdles and Steps: Estimating Demand for Solar Photovoltaics Kenneth Gillingham Yale University, School of Forestry & Environmental Studies Tsvetan Tsvetanov University of Kansas, Department of Economics November 6, 2015 K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

2 Excess Zeros in Data Excess zeros: often the case with survey data examples: healthcare utilization consumption of a good (cigarettes, alcohol) What if we ignore the issue? treat non-negative response variables as if excess zeros were not a problem using simple ordinary least squares (OLS) regression leads to negative predicted outcomes cannot transform data due to zeros (log of zero does not exist) K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

3 Addressing Excess Zeros in Data One option: censor at zero (e.g., Tobit) y = x β + u y = y if y > 0 y = 0 if y 0 this is not appropriate if the zeros come from a different data-generating process: e.g., if participation decision is made separately from utilization decision (Humphreys, 2013) K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

4 Addressing Excess Zeros in Data Another option: model participation decision separately from use decision appropriate when these two decisions are chronologically sequential this is what two-part models do K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

5 Two-Part Model Basics of the two-part model (Jones, 2000) observations for which y = 0 are uninformative in estimating the determinants of the level of y y > 0 motivated by the conditional mean independence assumption E(y y > 0, x 2 ) = x 2 β 2 i.e., E(u 2 y > 0, x 2) = 0 in contrast to Heckman s (1979) sample selection model, where E(u 2 y > 0, x 2) > 0 estimation use a logit or probit model for the probability of observing a positive value of y use OLS for the sub-sample of positive observations (usually after a log transformation) K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

6 Count Data What if the dependent variable is a count variable? i.e., y {0, 1, 2, 3...} Use a version of the two-part model: estimate by logit or probit the probability of observing a positive value of y fit a truncated count data model for the sub-sample of positive observations This is known as a hurdle model originally formulated by Mullahy (1986) recognized as effective tool for dealing with excess zeros in count data settings (e.g., Cameron & Trivedi, 2013) K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

7 Background Motivation rebates, tax credits, etc.). As indicated previously, the data sample excludes systems for which the Data reported price was deemed Estimation likely to represent an appraised value, Results rather than a purchase price Conclusion paid to an installer (see Appendix A for further details). Year U.S. Solar Market Annual installed capacity (MW) New Installed Capacity (MW) Installed Prices Continued Their Precipitous Decline in 2013 and 2014 U.S.: a booming market for solar energy world s 3 rd largest solar photovoltaics (PV) market in 2013 persistent growth of solar installations particular note is that the national price trends Figure 6 are dominated by trends within pronounced decline of (pre-incentive) installation costs New PV Installations Installation Year Source: SEIA/GTM Research (2014) The present section begins by describing trends in installed price over time, decomposing those trends into underlying module and non-module costs, and presenting temporal trends related to cash incentives provided through state and utility programs. The section then compares installed prices between the United States and other international markets. It then examines the wide variability in installed prices across projects, describing trends by system size, among individual states, between third party-owned and customer-owned systems, across host customer sectors, and between various types of applications and technologies, including: microinverters vs. central inverters, systems with varying module efficiencies, Chinese-brand vs. non-chinese-brand modules, residential new construction vs. residential retrofit, BIPV vs. rack-mounted systems, rooftop vs. ground-mounted systems, and tracking vs. fixed-tilt systems. Figure 6 presents the median installed price of all residential and commercial projects within the sample, segmented into three system size groupings, from 1998 through Among the roughly 50,000 residential and commercial PV systems in the sample installed in 2013, the median installed price was $4.7/W for systems 10 kw, $4.3/W for systems kw in size, and $3.9/W for systems >100 kw. Importantly, though, these median values represent central tendencies, and considerable spread exists among the data, as will be summarized in subsequent figures. Also of California, which constitutes a large fraction of the total U.S. market and, as will be shown later, is relatively high-priced compared to other states. Installed Price (2013$/W DC) $12 $10 $8 $6 Residential & Commercial PV (Median Values) $4 10 kw $ kw >100 kw $ Installation Year Notes: See Table 1 and Table B-2 for residential and commercial PV sample sizes by installation year. Median Source: installed prices Barbose are shown only etif 15 al. or more (2014) observations are available for the individual size range. Figure 6. Installed Price of Residential & Commercial PV over Time 13 Tracking the Sun VII: The Installed Price of Photovoltaics in the United States from 1998 to 2013 K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

8 Solar Incentives Financial incentives for PV installations federal tax credit + state financial incentives tax credit given to customer state incentives realized by installer can substantially reduce system prices in 2013, median PV incentives 33% of median system cost (Barbose et al. 2014) federal tax credit has remained unchanged since 2005 median state incentives have decreased by 95% since 2003 equivalent to 80% of the drop in pre-incentive system costs K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

9 Motivation Question: How would the reduction in state financial support for solar installations impact PV adoption? K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

10 Case Study Connecticut (CT): small, but rapidly growing solar market Number of New Installations 0 1,000 2,000 3,000 4,000 5,000 New Residential PV Installations in CT $/W Average Price of Residential PV Systems in CT Pre-incentive price Post-incentive price spread of solar technology favored by significant government support for residential PV installations 30% federal tax credit, state incentives : average state + federal incentive 50% of PV system cost financial support is slowly being reduced CT has already started a gradual phase-out of solar incentives K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

11 Setting Study area: Connecticut Study period: Research objectives: 1 estimate a model of residential solar demand 2 conduct policy simulations Data setting panel data: unit of observation: Census block group-year include block group fixed effects + quadratic time trend use within-block group variation in prices over time count data: demand is quantified by number of installations in each unit of observation K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

12 Empirical Challenge #1: Excess of Zero Outcomes Lack of installations in multiple time periods excess of zero values for the outcome count variable Implications for the data distribution: skewed count data distribution with high probability of zeros Frequency Number of installations If not addressed, results in model misspecification and inaccurate conclusions (e.g., Cameron & Trivedi, 2013) K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

13 Empirical Challenge #2: Unobserved Heterogeneity Local product and environmental preferences potentially important determinants of solar demand e.g., Bollinger & Gillingham (2012), Millard-Ball (2012) unobserved to the econometrician likely correlated with other observed demand determinants (including price) If not addressed, results in inconsistent parameter estimates K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

14 Empirical Challenge #3: Simultaneity Bias Estimating demand from market equilibrium data price and number of installations determined simultaneously in equilibrium classic simultaneity bias problem price is likely to be endogenous in the model If not addressed, results in inconsistent parameter estimates K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

15 Addressing Challenges 1 Set up and estimate a hurdle model use logit to estimate first stage use zero-truncated Poisson to estimate second stage 2 Include block group-specific fixed effects (FE) exploit panel structure of the data need to accommodate FE both in logit and zero-truncated Poisson portion of hurdle model 3 Use instrumental variable (IV) procedure use supply shifters as instruments (incentive levels, labor costs) need to accommodate IV procedure both in logit and zero-truncated Poisson portion of hurdle model K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

16 Methodological Contribution Develop a method for estimating count data with excess zero outcomes in the presence of unobserved heterogeneity and endogeneity of one or more of the regressors first study to derive a consistent estimator of hurdle models with individual-specific fixed effects and endogenous regressors this method can be applied to other solar markets or similar contexts of early-stage technology adoption K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

17 Substantive Contribution Obtain estimate of price elasticity of residential solar demand in CT of important information for forecasting and pricing decisions offers guidance to policymakers and firms in the industry Policy implications eliminating state financial incentives leads to a 47% reduction in new PV installations in CT relative to 2014 switching to a lower level of state financial incentives leads to a 13% reduction in new PV installations in CT relative to 2014 K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

18 Data Sources Study period: Sample: all purchased residential PV systems in CT Connecticut Green Bank (CGB): costs, technical characteristics, financing mechanisms, customer address, and project status of each residential solar installation in CT U.S. Census: demographic and socioeconomic data at Census block group level U.S. BLS: county-level wage data K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

19 Summary Statistics of Full Sample Variable Mean St. Dev. Min Max Number of PV installations Post-incentive system price ($/W) Solarize campaign Population density (per km 2 ) Median household income (in 1,000$/year) Median age % population above 25 with some college or college degree % population above 25 with graduate or professional degree % Republican voters % Democrat voters Incentive level ($/W) Electrical/wiring contractor wage ($/week) Note: All variables have 10,150 observations. All dollars in 2014 dollars. Subsample with positive installations K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

20 Trends in Price and Installations installations price (2014$/watt) year Solarize installations post-incentive price non-solarize installations K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

21 Hurdle Model Count variable is modeled as outcome of a two-stage data-generating process first stage determines whether the count variable has zero or positive realization if positive realization, the hurdle is crossed and the exact outcome is determined by a zero-truncated count distribution Distribution given by: { f1 (0) if y = 0, Pr(Y = y) = (1 f 1 (0)) f 2 (y) if y = 1, 2,...,. f 1 : first-stage distribution (usually logistic) f 2 : second-stage distribution (usually zero-truncated Poisson) K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

22 Hurdle Model Pros: count data model effectively addresses the presence of excess zeros in the data depending on the parameters of the logit distribution, can allow for any number of zero outcomes some flexibility in the estimation: log-likelihood function is a sum of the logit and truncated Poisson log-likelihood functions logit and truncated Poisson components of the model can be estimated separately from each other Log-likelihood function for the hurdle model Challenges: need to accommodate fixed effects and an IV procedure K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

23 Hurdle Model: Fixed Effects Fixed effects in logit model Chamberlain (1980): logit with fixed effects can be estimated consistently using conditional maximum likelihood (CMLE) Details Fixed effects in truncated Poisson model Majo & van Soest (2011): CMLE is also consistent in this setting (similar to the logit model) they only demonstrate this for T = 2 we extend their proof to multiple periods Details K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

24 Hurdle Model: Instrumenting with Fixed Effects IV in FE logit model Petrin & Train (2010) propose a flexible control function (CF) approach for logit models first stage: linear regression of price on all excluded and included instruments second stage: logit CMLE with first-stage residual included among the regressors use bootstrapped standard errors K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

25 Hurdle Model: Instrumenting with Fixed Effects IV in FE truncated Poisson model conditional maximum likelihood estimator (CMLE) can be mapped directly to a generalized method of moments (GMM) estimator start from first-order condition of CMLE (the score): S i (δ 2 ) = J J j=1 t=1 j=1 t=1 T [Y ijt φ ijt (X ijt, δ 2 )] X ijt T ξ ijt X ijt = X i ξ i S i (δ 2) = score vector φ ijt = a function of regressors and parameters ξ ijt = Y ijt φ ijt for exogenous X i, E[X i ξ i ] = 0 by the properties of the score function K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

26 Hurdle Model: Instrumenting with Fixed Effects IV in FE truncated Poisson model (cont d) reduced-form representation of the model: Y ijt = φ ijt (X ijt, δ 2 ) + ξ ijt (1) E[X i ξ i ] = 0 if X i is exogenous (2) use ψ(x i, δ 2 ) = X i ξ i to construct a GMM estimator: if X i is exogenous, then E [ψ(x i, δ 2)] = 0 and ˆδ 2,GMM = ˆδ 2,CMLE if X i is endogenous, but there exists an exogenous vector Z i, such that E [ψ(z i, δ 2)] = 0, then: [ [ 1 N ˆδ 2,GMM = argmax N i=1 ψ(z ] i, δ 2 )] ˆΞ 1 N N i=1 ψ(z i, δ 2 ) δ 2 Monte Carlo simulations K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

27 Choice of Instrumental Variables Due to endogeneity of price, we need instrumental variable(s) Two conditions for a valid instrument correlated with price relevant uncorrelated with the regression error term exogenous Use supply shifters as instrumental variables 1 solar subsidy levels Details 2 installation labor costs Details K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

28 Regression Results Linear Poisson Hurdle Variable Logit Trun. Poisson Price ** *** *** *** (0.0227) (0.0617) (0.0632) (0.182) Controls yes yes yes yes Block group FE yes yes yes yes Time trend yes yes yes yes Instruments yes yes yes yes Price elasticity Observations 10,150 10,150 10,150 2,636 Note: (*) p < 0.1; (**) p < 0.05; (***) p < Controls used: Solarize indicator, population density, median income, median age, proportion with some college edutation, proportion of African American population, proportion of Asian population, proportion of Republican voters, proportion of Democrat voters. Robust standard errors clustered at the town level. Deriving elasticity in a hurdle model Robustness checks K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

29 Implied Price Elasticity of Residential Solar Demand Preferred specification implies price elasticity of households are relatively sensitive to changes in PV system prices 1% increase in system prices results in a 1.76% drop in number of installations demanded in an average block group-year $1 reduction in average system price 1,102 additional systems demanded statewide in that year K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

30 Implied Public Cost of Solar Subsidies in CT Policy cost: state dollars per additional kwh of solar capacity Compare to environmental benefits from solar (reduced emissions and local pollution) assume solar fully displaces natural-gas fired generation coal provides a very small fraction of electricity in CT results in lower greenhouse gas emissions policy cost of $135/tCO 2 higher than social cost of carbon of $40/tCO 2 reduces damages from local pollutants benefits = $0.021/kWh lower than policy cost of $0.035/kWh K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

31 Policy Simulations Use estimates to simulate the impact of a number of hypothetical policy changes in CT compare number of installations under each hypothetical scenario to actual number observed in 2014 Results: 1 complete phase-out of incentives: 47% drop in new installations in CT in cutting incentives in half: 24% drop in new installations in CT in reducing incentives to Step 6: 13% drop in new installations in CT in 2014 K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

32 Conclusion Methodological contribution first to incorporate both fixed effects and IV procedure in a Poisson hurdle model develop a flexible approach that can be used in modeling demand in a variety of count data settings Substantive contribution: guidance for policymakers and firms Findings: estimate demand for residential solar PV systems in CT analysis suggests a price elasticity of assess the impact of solar policies in CT reducing or eliminating state incentives can have a substantial near-term impact on new installations K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

33 Additional Slides Summary Statistics of Subsample with Positive Installations Variable Mean St. Dev. Min Max Number of PV installations Post-incentive system price ($/W) Solarize campaign Population density (per km 2 ) Median household income (in 1,000$/year) Median age % population above 25 with some college or college degree % population above 25 with graduate or professional degree % Republican voters % Democrat voters Incentive level ($/W) Electrical/wiring contractor wage ($/week) Note: All variables have 2,636 observations. All dollars in 2014 dollars. Back K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

34 Additional Slides Log-likelihood Function for Hurdle Model Let ι ijt = I (Y ijt > 0). Then: L(Θ 1, Θ 2 ) = { [ ] N J T 1 (1 ι ijt )log i=1 j=1 t=1 1 + exp(w ijt Θ + ι ijt log 1) + N J T i=1 j=1 t=1 L l (Θ 1 ) + L t(θ 2 ) [ exp(w ijt Θ 1) 1 + exp(w ijt Θ 1) ]} [ ( )]} {ι ijt Y ijt W ijt Θ 2 log(y ijt!) log exp(exp(w ijt Θ 2)) 1 Back K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

35 Additional Slides Fixed Effects in Logit Model Back Pr[Y ijt > 0] = exp(α i +X ijt δ 1) 1+exp(α i +X ijt δ 1) b i exp(x ijt δ 1) 1+b i exp(x ijt δ 1) X ijt : vector of regressors b i : block group specific parameter δ 1 : slope parameter need to derive ˆδ 1 independently of ˆb i to avoid bias due to incidental parameters problem (Neyman & Scott, 1948) Chamberlain (1980): logit with fixed effects can be estimated consistently using conditional maximum likelihood (CMLE) l i (b i, δ 1 J j=1 t=1 T J Y ijt ) = l i (δ 1 j=1 t=1 T Y ijt ) sum of outcomes in group i: minimal sufficient statistic for b i conditional log-likelihood does not depend on b i K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

36 Additional Slides Fixed Effects in Truncated Poisson Model Back Pr[Y ijt = y λ ijt ] = λ y ijt y!(e λ ijt 1) λ ijt = exp(β i + X ijt δ 2) c i exp(x ijt δ 2) c i : block group specific parameter δ 2 : slope parameter again, need to derive ˆδ 2 independently of ĉ i to avoid bias Majo & van Soest (2011): conditional maximum likelihood estimator is consistent (similar to the logit model) they only demonstrate this for T = 2 we extend their proof to multiple periods K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

37 Additional Slides CMLE in Truncated Poisson Model Pr(Y i = y i X i = x i, J c n i i J T 1 h(x i,n i,δ 2 ) j=1 t=1 j=1 t=1 n i! y ijt! cy ijt i = Pr(Y i = y i X i = x i, J T Y ijt = n i, c i, δ 2) = [ exp(x ijtδ 2)] yijt = j=1 t=1 T Y ijt = n i, δ 2) 1 h(x i,n i,c i,δ 2 ) 1 h(x i,n i,δ 2 ) J J j=1 t=1 T j=1 t=1 T n i! y ijt! n i! y ijt! λy ijt ijt = [ exp(x ijtδ 2) ] yijt Therefore: l i (c i, δ J T 2 Y ijt ) = l i (δ J T 2 Y ijt ) j=1 t=1 j=1 t=1 Back K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

38 Additional Slides Monte Carlo Simulations Monte Carlo simulations based on a randomly generated panel dataset with i = 1,..., 20 and t = 1,..., 5. Parameters Mean(ˆγ) Bias(ˆγ) Var(ˆγ) MSE(ˆγ) Baseline Longer Panel (T = 8) Weak Instruments (π = 0.5) Notes: True parameter value is γ = Estimated output is based on 5,000 replications. Back K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

39 Additional Slides IV #1: Incentive Level (in $/W) Incentive for purchased systems: a subsidy to the installer incentive given directly to the installer consumer sees post-incentive price at the bottom of the contract changes in incentive level should shift marginal cost of an installation without impacting the demand curve Incentive level determined in advance by state authorities based on budget restrictions rather than demand Back Incentive levels over time K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

40 Additional Slides IV #2: Labor Costs Use average county wages of electrical and wiring contractors labor from this industry typically involved in a PV installation changes in wages expected shift marginal costs of installers Exploit cross-sectional and temporal variation in wages variation is plausibly exogenous after controlling for income Back County wages over time K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

41 Additional Slides Incentive Levels Incentive(2014$/W) Year Back K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

42 Additional Slides County Wages Average monthly wage ($) Fairfield Hartford Litchfield Middlesex New Haven New London Tolland Windham Year Back Note: Wages are in 2013 USD. K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

43 Additional Slides Price Elasticity in a Hurdle Model Mean of the outcome variable is given by: E[Y ijt W ijt, Θ] = Pr[Y ijt > 0 W ijt, Θ 1 ]E t [Y ijt W ijt, Θ 2 ] = exp(w ijt Θ 1) λ ijt 1+exp(W ijt Θ 1) 1 exp( λ ijt ) Price elasticity η at the mean values of Y and p is then: η = [E[Y ijt W ijt,θ]] E[p ijt ] p ijt E[Y ijt W ijt,θ] = [Pr[Y ijt >0 W ijt,θ 1 ]] E[p ijt ] p ijt η l + η t Back Pr[Y ijt >0 W ijt,θ 1 ] + [Et[Y ijt W ijt,θ 1 ]] E t[p ijt ] p ijt E t[y ijt W ijt,θ 1 ] K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

44 Additional Slides Robustness Checks Baseline Robustness Checks Model Specification I II III IV Logit CF *** *** *** *** (0.1982) (0.1972) (0.6451) (0.1950) (0.2658) Tr. Poisson GMM *** *** *** *** (0.0819) (0.0819) (0.0891) (0.0797) (0.0398) Combined elasticity *** *** *** *** (0.2145) (0.2135) (0.6512) (0.2106) (0.2687) Missing price proxy average price highest price average price average price average price BG FE yes yes yes yes yes Time trend yes yes no yes yes Year FE no no yes no no Demographics/voting yes yes yes no yes Solarize included yes yes yes yes no Back K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32

45 Additional Slides Histogram of the Count of Installations Frequency Number of installations Solarize installations non-solarize installations Back K. Gillingham, T. Tsvetanov Demand for Solar Photovoltaics November 6, / 32