this paper considers estimation and inference in linear and nonlinear panel data models with random coefficients and endogenous regressors .the quantities of interest are means , variances , and other moments of the distribution of the random coefficients . in a state level panel model of rational addiction , for example, we might be interested in the mean and variance of the distribution of the price effect on cigarette consumption across states , controlling for endogenous past and future consumptions .these models pose important challenges in estimation and inference if the relation between the regressors and individual coefficients is left unrestricted .fixed effects methods based on gmm estimators applied separately to the time series of each individual can be severely biased due to the incidental parameter problem .the source of the bias is the finite - sample bias of gmm if some of the regressors is endogenous or the model is nonlinear in parameters , or nonlinearities if the parameter of interest is the variance or other high order moment of the individual coefficients . neglecting the heterogeneity and imposing fixed coefficients does not solve the problem , because the resulting estimators are generally inconsistent for the mean of the random coefficients ( yitzhaki , 1996 , and angrist , graddy and imbens , 2000 ) .moreover , imposing fixed coefficients does not allow us to estimate other moments of the distribution of the random coefficients .we introduce a class of bias - corrected panel fixed effects gmm estimators .thus , instead of imposing fixed coefficients , we estimate different coefficients for each individual using the time series observations and correct for the resulting incidental parameter bias . for linear models , in addition to the bias correction ,these estimators differ from the standard fixed effects estimators in that both the intercept and the slopes are different for each individual .moreover , unlike for the classical random coefficient estimators , they do not rely on any restriction in the relationship between the regressors and random coefficients ; see hsiao and pesaran ( 2004 ) for a recent survey on random coefficient models .this flexibility allows us to account for roy ( 1951 ) type selection where the regressors are decision variables with levels determined by their returns .linear models with roy selection are commonly referred to as correlated random coefficient models in the panel data literature . in the presence of endogenous regressors , treating the random coefficients as fixed effects is also convenient to overcome the identification problems in these models pointed out by kelejian ( 1974 ) .the most general models we consider are semiparametric in the sense that the distribution of the individual coefficients is unspecified and the parameters are identified from moment conditions .these conditions can be nonlinear functions in parameters and variables , accommodating both linear and nonlinear random coefficient models , and allowing for the presence of time varying endogeneity in the regressors not captured by the individual coefficients .we use the moment conditions to estimate the model parameters and other quantities of interest via gmm methods applied separately to the time series of each individual .the resulting estimates can be severely biased in short panels due to the incidental parameters problem , which in this case is a consequence of the finite - sample bias of gmm ( newey and smith , 2004 ) and/or the nonlinearity of the quantities of interest in the individual coefficients .we develop analytical corrections to reduce the bias .  to derive the bias corrections , we use higher - order expansions of the gmm estimators , extending the analysis in newey and smith ( 2004 ) for cross sectional estimators to panel data estimators with fixed effects and serial dependence . if and denote the cross sectional and time series dimensions of the panel , the corrections remove the leading term of the bias of order , and center the asymptotic distribution at the true parameter value under sequences where and grow at the same rate .this approach is aimed to perform well in econometric applications that use moderately long panels , where the most important part of the bias is captured by the first term of the expansion .other previous studies that used a similar approach for the analysis of linear and nonlinear fixed effects estimators in panel data include , among others , kiviet ( 1995 ) , phillips and moon ( 1999 ) , alvarez and arellano ( 2003 ) , hahn and kuersteiner ( 2002 ) , lancaster ( 2002 ) , woutersen ( 2002 ) , hahn and newey ( 2004 ) , and hahn and kuersteiner ( 2011 ) .see arellano and hahn ( 2007 ) for a survey of this literature and additional references .a first distinctive feature of our corrections is that they can be used in overidentified models where the number of moment restrictions is greater than the dimension of the parameter vector .this situation is common in economic applications such as rational expectation models .overidentification complicates the analysis by introducing an initial stage for estimating optimal weighting matrices to combine the moment conditions , and precludes the use of the existing methods .for example , hahn and newey s ( 2004 ) and hahn and kuersteiner s ( 2011 ) general bias reduction methods for nonlinear panel data models do not cover optimal two - step gmm estimators .a second distinctive feature is that our results are specifically developed for models with multiple nonadditive heterogeneity , whereas the previous studies focused mostly on models with additive heterogeneity captured by an scalar individual effect .exceptions include arellano and hahn ( 2006 ) and bester and hansen ( 2008 ) , which also considered multiple heterogeneity , but they focus on parametric likelihood - based panel models with exogenous regressors .bai ( 2009 ) analyzed related linear panel models with exogenous regressors and multidimensional interactive individual effects .bai s nonadditive heterogeneity allows for interaction between individual effects and unobserved factors , whereas the nonadditive heterogeneity that we consider allows for interaction between individual effects and observed regressors .a third distinctive feature of our analysis is the focus on moments of the distribution of the individual effects as one of the main quantities of interest .we illustrate the applicability of our methods with empirical and numerical examples based on the cigarette demand application of becker , grossman and murphy ( 1994 ) . here , we estimate a linear rational addictive demand model with state - specific coefficients for price and common parameters for the other regressors using a panel data set of u.s . states .we find that standard estimators that do not account for non - additive heterogeneity by imposing a constant coefficient for price can have important biases for the common parameters , mean of the price coefficient and demand elasticities .the analytical bias corrections are effective in removing the bias of the estimates of the mean and standard deviation of the price coefficient .figure [ fig1 ] gives a preview of the empirical results .it plots a normal approximation to the distribution of the price effect based on uncorrected and bias corrected estimates of the mean and standard deviation of the distribution of the price coefficient .the figure shows that there is important heterogeneity in the price effect across states .the bias correction reduces by more than 15% the absolute value of the estimate of the mean effect and by 30% the estimate of the standard deviation .some of the results for the linear model are related to the recent literature on correlated random coefficient panel models with fixed .graham and powell ( 2008 ) gave identification and estimation results for average effects .arellano and bonhomme ( 2010 ) studied identification of the distributional characteristics of the random coefficients in exogenous linear models .none of these papers considered the case where some of the regressors have time varying endogeneity not captured by the random coefficients or the model is nonlinear . for nonlinear models , chernozhukov , fernndez - val , hahn and newey ( 2010 ) considered identification and estimation of average and quantile treatment effects .their nonparametric and semiparametric bounds do not require large- , but they do not cover models with continuous regressors and time varying endogeneity .the rest of the paper is organized as follows .section [ s2 ] illustrates the type of models considered and discusses the nature of the bias in two examples .section [ s3 ] introduces the general model and fixed effects gmm estimators .section [ s4 ] derives the asymptotic properties of the estimators .the bias corrections and their asymptotic properties are given in section [ s5 ] .section [ s6 ] describes the empirical and numerical examples .section [ s9 ] concludes with a summary of the main results .additional numerical examples , proofs and other technical details are given in the online supplementary appendix fernndez - val and lee ( 2012 ) .in this section we describe in detail two simple examples to illustrate the nature of the bias problem .the first example is a linear correlated random coefficient model with endogenous regressors .we show that averaging iv estimators applied separately to the time series of each individual is biased for the mean of the random coefficients because of the finite - sample bias of iv .the second example considers estimation of the variance of the individual coefficients in a simple setting without endogeneity . here the sample variance of the estimators of the individual coefficients is biased because of the non - linearity of the variance operator in the individual coefficients .the discussion in this section is heuristic leaving to section [ s4 ] the specification of precise regularity conditions for the validity of the asymptotic expansions used .consider the following panel model : where is a response variable , is an observable regressor , is an unobservable error term , and and usually index individual and time period , respectively .denotes a group index and indexes the observations within the group . examples of groups include individuals , states , households , schools , or twins .] this is a linear random coefficient model where the effect of the regressor is heterogenous across individuals , but no restriction is imposed on the distribution of the individual effect vector .the regressor can be correlated with the error term and a valid instrument is available for , that is = 0 ] and \neq 0 ] , the mean of the random slope coefficient . we could neglect the heterogeneity andrun fixed effects ols and iv regressions in where in terms of the model ( [ livmodel ] ) . in this case , ols and iv estimate weighted means of the individual coefficients in the population ; see , for example , yitzhaki ( 1996 ) and angrist and krueger ( 1999 ) for ols , and angrist , graddy and imbens ( 2000 ) for iv .ols puts more weight on individuals with higher variances of the regressor because they give more information about the slope ; whereas iv weighs individuals in proportion to the variance of the first stage fitted values because these variances reflect the amount of information that the individuals convey about the part of the slope affected by the instrument .these weighted means are generally different from the mean effect because the weights can be correlated with the individual effects .  to see how these implicit ols and iv weighting schemes affect the estimand of the fixed - coefficient estimators , assume for simplicity that the relationship between and is linear , that is ( ) is normal conditional on ( ), is independent of ( ) , and ( ) is normal , for .then , the probability limits of the ols and iv estimators are where .when the first stage equation is different for each individual , the limit of the iv estimator is cov[\alpha_{1i } , \pi_{1i}]/\{e [ \pi_{1i}]^{2 } + var[\pi_{1i}]\}.\ ] ] see theorems 2 and 3 in angrist and imbens ( 1995 ) for a related discussion . ] + 2 e [ \pi_{1i } ] var[z_{it } ] cov[\alpha_{1i } , \pi_{1i}]\}/var[x_{it}],\\ \alpha_{1}^{iv } & = & \alpha_{1 } + cov[\alpha_{1i } , \pi_{1i}]/e [ \pi_{1i}].\end{aligned}\ ] ] these expressions show that the ols estimand differs from the average coefficient in presence of endogeneity , i.e. non zero correlation between the individual - time specific error terms , or whenever the individual coefficients are correlated ; while the iv estimand differs from the average coefficient only in the latter case . in the treatment - effects model ,there exists correlation between the error terms in presence of endogeneity bias and correlation between the individual effects arises under roy - type selection , i.e. , when individuals who experience a higher permanent effect of the treatment are relatively more prone to accept the offer of treatment .wooldridge ( 2005 ) and murtazashvile and wooldridge ( 2005 ) give sufficient conditions for consistency of standard ols and iv fixed effects estimators .these conditions amount to =0 ] .our proposal is to estimate the mean coefficient from separate time series estimators for each individual .this strategy consists of running ols or iv for each individual , and then estimating the population moment of interest by the corresponding sample moment of the individual estimators .for example , the mean of the random slope coefficient in the population is estimated by the sample average of the ols or iv slopes .these sample moments converge to the population moments of interest as number of individuals and time periods grow .however , since a different coefficient is estimated for each individual , the asymptotic distribution of the sample moments can have asymptotic bias due to the incidental parameter problem ( neyman and scott , 1948 ) .  to illustrate the nature of this bias ,consider the estimator of the mean coefficient constructed from individual time series iv estimators . in this casethe incidental parameter problem is caused by the finite - sample bias of iv .this can be explained using some expansions .thus , assuming independence across , standard higher - order asymptotics gives ( e.g. rilstone et .al . , 1996 ) , as  where ^{-1 } \tilde z_{it } \epsilon_{it} ] is the higher - order bias of iv ( see , e.g. , nagar , 1959 , and buse , 1992 ) , and the variables with tilde are in deviation from their individual means , e.g. ,  ]  let , the sample average of the iv estimators .the asymptotic distribution of is not centered around in short panels or more precisely under asymptotic sequences where . to see this , consider the expansion for  the first term is the standard influence function for a sample mean of known elements .the second term comes from the estimation of the individual elements inside the sample mean .assuming independence across and combining the previous expansions , this expression shows that the bias term dominates the asymptotic distribution of in short panels under sequences where .averaging reduces the order of the variance of , without affecting the order of its bias . in this casethe estimation of the individual coefficients has no first order effect in the asymptotic variance of because the second term is of smaller order than the first term .a potential drawback of the individual by individual time series estimation is that it might more be sensitive to weak identification problems than fixed coefficient pooled estimation . in the random coefficient model , for example, we require that = \pi_{1i } \neq 0 ] .the individual estimators are therefore more sensitive than traditional pooled estimators to weak instruments problems . on the other hand , individual by individual estimationrelaxes the exogeneity condition by conditioning on additive and non - additive time invariant heterogeneity , i.e , = 0 ] and its fixed effects estimator is where and .let and .assuming independence across and , a standard asymptotic expansion gives , as , the first term corresponds to the influence function of the sample variance if the s were known .the second term comes from the estimation of the s .the third term is a bias term that comes from the nonlinearity of the variance in .the bias term dominates the expansion in short panels under sequences where . as in the previous example, the estimation of the s has no first order affect in the asymptotic variance since the second term is of smaller order than the first term .we consider a general model with a finite number of moment conditions . to describe it ,let the data be denoted by  .we assume that is independent over and stationary and strongly mixing over .also , let be a of common parameters , be a sequence of with the realizations of the individual effects , and be an of functions , where . has to be less than the time dimension . ]the model has true parameters and , satisfying the moment conditions = 0 ,  \  ( t = 1 , ... , t ; i = 1 , ... , n),\ ] ] where  ] denote the expectation taken with respect to the distribution of the individual effects .in the previous model , the ultimate quantities of interest are smooth functions of parameters and observations , which in some cases could be the parameters themselves , ,\ ] ] if , or moments or other smooth functions of the individual effects ,\ ] ] if . in the correlated random coefficient example , , , , , and . in the variance of the random coefficients example , , , , , and )^2 ] .  for a matrix or vector , let denote the euclidean norm , that is  ] for some ; and ( v ) there exists a deterministic sequence of symmetric finite positive definite matrices such that and , for each  > 0,\ ] ] where .\ ] ]  conditions [ cond1](i)-(ii ) impose cross sectional independence , but allow for weak time series dependence as in hahn and kuersteiner ( 2011 ) .conditions [ cond1](iii)-(iv ) describe the asymptotic sequences that we consider where and grow at the same rate with the sample size , whereas the number of moments is fixed .condition [ cond2 ] adapts standard assumptions of the gmm literature to guarantee the identification of the parameters based on time series variation for all the individuals , see newey and mcfadden ( 1994 ) . the dominance and moment conditions in [ cond2](iv ) are used to establish uniform consistency of the estimators of the individual effects .[ th1 ] suppose that conditions [ cond1 ] and [ cond2 ] hold .then , for any  where and . also , for any  where , and , where .let , , , and  ] ; and ( ii ) is finite positive definite , and is a sequence of finite positive definite matrices , where is the sequence of matrices of condition [ cond2](v ) .[ cond5 ] ( i ) there exists a function such that , for  and < \infty, ] and < \infty, ] and where  ]  ] for some and ( ii ) is a sequence of finite positive definite matrices ; and ( iii ) for each , is a martingale difference sequence with respect to .  conditions [ cond4](i)-(ii ) are used to establish the uniform consistency of the estimators of the individual weighting matrices .condition [ cond4](iii ) is convenient to simplify the expressions of the optimal weighting matrices .it holds , for example , in rational expectation models that commonly arise in economic applications .[ uniform consistency of two - step estimators ] [ th4]suppose that conditions [ cond1 ] , [ cond2 ] , [ cond3 ] and [ cond4 ] hold .then , for any  where and . also , for any  where , and , where .we replace condition [ cond5 ] by the following condition to obtain the limit distribution of the two - step estimator :  [ cond7 ] there exists some such that , for  and < \infty, ] , , with , for ,  \right ) , \\b_{\gamma_{i}}^{g } & = & \left(\begin{array}{c }  b_{\alpha_{i}}^{g } \\b_{\lambda_{i}}^{g } \\\end{array}\right ) =  \left(\begin{array}{c }  -\sigma_{\alpha_{i } } \\h_{\alpha_{i } } '  \\ \end{array}\right ) \sum_{j=0}^{\infty }  e \left [ g_{\alpha_{i}}(z_{it})'p_{\alpha_{i}}g(z_{i , t - j } ) \right ] , \\b_{\gamma_{i}}^{\omega } & = & \left(\begin{array}{c }  b_{\alpha_{i}}^{\omega } \\b_{\lambda_{i}}^{\omega } \\\end{array}\right ) = \left(\begin{array}{c }  h_{\alpha_{i } }  \\p_{\alpha_{i } }  \\\end{array}\right )  \sum_{j=0}^{\infty } e[g(z_{it } ) g(z_{it})'p_{\alpha_{i } } g(z_{i , t - j } ) ] , \\b_{\gamma_{i}}^{w } & = & \left(\begin{array}{c }  b_{\alpha_{i}}^{w } \\b_{\lambda_{i}}^{w } \\\end{array}\right ) = \left(\begin{array}{c }   h_{\alpha_{i } }  \\   p_{\alpha_{i } }  \\\end{array}\right ) \sum_{j=1}^{d_{\alpha } } \omega_{\alpha_{i , j } } \left ( h_{\alpha_{i , j}}^{w ' } - h'_{\alpha_{i , j } } \right).\end{aligned}\ ] ]  [ limit distribution for two - step estimators of common parameters ] [ th6 ] under the conditions [ cond1 ] , [ cond2 ] , [ cond3 ] , [ cond5 ] , [ cond4 ] and [ cond7 ] , where , ]  ] .the expressions for , , and are given in lemma [ lemma2 ] .theorem [ th6 ] establishes that one iteration of the gmm procedure not only improves asymptotic efficiency by reducing the variance of the influence function , but also removes the variance and non - optimal weighting matrices components from the bias .the higher - order bias of the estimator of the individual effects , , now has four components , as in newey and smith ( 2004 ) .these components correspond to the asymptotic bias for a gmm estimator with the optimal score , ; the bias arising from estimation of , ; the bias arising from estimation of , ; and the bias arising from the choice of the preliminary first step estimator , .an additional iteration of the gmm estimator removes the term .the general procedure for deriving the asymptotic distribution of the fe - gmm estimators consists of several expansions .first , we derive higher - order asymptotic expansions for the estimators of the individual effects , with the common parameter fixed at its true value .next , we obtain the asymptotic distribution for the profile score of the common parameter at using the expansions of the estimators of the individual effects . finally , we derive the asymptotic distribution of estimator for the common parameter multiplying the asymptotic distribution of the score by the limit profile jacobian matrix .this procedure is detailed in the online appendix fernndez - val and lee ( 2012 ) . herewe characterize the asymptotic bias in a linear correlated random coefficient model with endogenous regressors .motivated by the numerical and empirical examples that follow , we consider a model where only the variables with common parameter are endogenous and allow for the moment conditions not to be martingale difference sequences .* example : correlated random coefficient model with endogenous regressors . *we consider a simplified version of the models in the empirical and numerical examples .the notation is the same as in the theorems discussed above .the moment condition is where and .that is , only the regressors with common coefficients are endogenous .let .to simplify the expressions for the bias , we assume that and = e[x_{2it } \epsilon_{i , t - j}], ] which do not depend on and .we can therefore obtain the optimal gmm estimator in one step using the sample averages to estimate the optimal weighting matrices .  in this model, it is straightforward to see that the estimators of the individual effects have no bias , that is . by linearity of the first order conditions in and  the only source of bias is the correlation between the estimators of and after some straightforward but tedious algebra , this bias simplifies to .\ ] ] for the limit jacobian , we find e[\tilde{w}_{2it } \tilde{w}_{2it}']^{-1 } e[\tilde{w}_{2it } \tilde{x}_{2it } ' ] \right\},\ ] ] where variables with tilde indicate residuals of population linear projections of the corresponding variable on for example e[x_{1it } x_{1it}']^{-1 } x_{1it} ] < \infty, ] ,\ ] ] for and .\ ] ]  [ cor2 ] let be a twice differentiable function such that < \infty ] ,\ ] ] for and . ] and = n^{-1 } \sum_{i=1}^n  { \widehat}f_i(\theta) ] and to simplify the presentation , we only give explicit formulas for fe - gmm three - step estimators in the main text .we give the expressions for one and two - step estimators in the supplementary appendix .let , \ \ { \widehat}j_s(\theta ) = \widehat{\bar{e}}[{\widehat}g_{\theta_i}(\theta ) ' { \widehat}p_{\alpha_i } ( \theta ) { \widehat}g_{\theta_i}(\theta)],\ ] ] where  ] and variables with tilde now indicate residuals of sample linear projections of the corresponding variable on for example .  we can estimate the bias of from the analytic formula in expression ( [ bias : lineariv ] ) replacing population by sample moments and by and trimming the number of terms in the spectral expectation , the one - step bias corrected estimates of the common parameter and the average of the individual parameter  ] .  the convergence condition for the estimators of and holds for sample analogs evaluated at the initial fe - gmm one - step or two - step estimators if the trimming sequence is chosen such that and as .theorem [ th7 ] also shows that all the bias - corrected estimators considered are first - order asymptotically efficient , since their variances achieve the semiparametric efficiency bound for the common parameters in this model , see chamberlain ( 1992 ) .the following corollaries give bias corrected estimators for averages of the data and individual effects and for moments of the individual effects , together with the limit distributions of these estimators and consistent estimators of their asymptotic variances .to construct the corrections , we use bias corrected estimators of the common parameter .the corollaries then follow from lemma [ lemma2 ] and theorem [ th7 ] by the delta method .we use the same notation as in the estimation of the bias of the common parameters above to denote the estimators of the components of the bias and variance .[ c1th5 ] let be a twice continuously differentiable function in its second and third argument , such that > 0, ] < \infty, ] and .for , let -  { \widehat}{b}_{\mu}({\widehat}{\theta}^{c})/t, ] then , under the conditions of theorem [ th7 ] where  ] also , for any and , ,\ ] ] is a consistent estimator for .the second term in ( [ eq : std_errors ] ) is included to improve the finite sample properties of the estimator in short panels .we illustrate the new estimators with an empirical example based on the classical cigarette demand study of becker , grossman and murphy ( 1994 ) ( bgm hereafter ) .cigarettes are addictive goods . to account for this addictive nature , early cigarette demand studiesincluded lagged consumption as explanatory variables ( e.g. , baltagi and levin , 1986 ) .this approach , however , ignores that rational or forward - looking consumers take into account the effect of today s consumption decision on future consumption decisions .becker and murphy ( 1988 ) developed a model of rational addiction where expected changes in future prices affect the current consumption .bgm empirically tested this model using a linear structural demand function based on quadratic utility assumptions .the demand function includes both future and past consumptions as determinants of current demand , and the future price affects the current demand only through the future consumption .they found that the effect of future consumption on current consumption is significant , what they took as evidence in favor of the rational model .most of the empirical studies in this literature use yearly state - level panel data sets .they include fixed effects to control for additive heterogeneity at the state - level and use leads and lags of cigarette prices and taxes as instruments for leads and lags of consumption .these studies , however , do not consider possible non - additive heterogeneity in price elasticities or sensitivities across states .there are multiple reasons why there may be heterogeneity in the price effects across states correlated with the price level .first , the considerable differences in income , industrial , ethnic and religious composition at inter - state level can translate into different tastes and policies toward cigarettes .second , from the perspective of the theoretical model developed by becker and murphy ( 1988 ) , the price effect is a function of the marginal utility of wealth that varies across states and depends on cigarette prices . if the price effect is heterogenous and correlated with the price level , a fixed coefficient specification may produce substantial bias in estimating the average elasticity of cigarette consumption because the between variation of price is much larger than the within variation .wangen ( 2004 ) gives additional theoretical reasons against a fixed coefficient specification for the demand function in this application .we consider the following linear specification for the demand function where is cigarette consumption in state at time measured by per capita sales in packs ; is an additive state effect ; is a state specific price coefficient ; is the price in 1982 - 1984 dollars ; and is a vector of covariates which includes income , various measures of incentive for smuggling across states , and year dummies .we estimate the model parameters using ols and iv methods with both fixed coefficient for price and random coefficient for price .the data set , consisting of an unbalanced panel of 51 u.s .states over the years 1957 to 1994 , is the same as in fenn , antonovitz and schroeter ( 2001 ) .the set of instruments for and in the iv estimators is the same as in specification 3 of bgm and includes , , , , , , and , where is the state excise tax for cigarettes in 1982 - 1984 dollars .table 1 reports estimates of coefficients and demand elasticities .we focus on the coefficients of the key variables , namely , and . throughout the table, fc refers to the fixed coefficient specification with and rc refers to the random coefficient specification in equation ( [ eq : rc ] ) .bc and ibc refer to estimates after bias correction and iterated bias correction , respectively .demand elasticities are calculated using the expressions in appendix a of bgm .they are functions of , , , and , linear in . for random coefficient estimators, we report the mean of individual elasticities , i.e. where are price elasticities at different time horizons .standard errors for the elasticities are obtained by the delta method as described in corollaries [ c1th5 ] and [ c2th5 ] . forbias - 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( 1996 ) on using linear regressions in welfare economics , _ journal of business and economic statistics _ 14 , 478 - 486 .* table 1 : estimates of rational addiction model for cigarette demand *  ccccccccc & ols - fc & iv - fc & & ( rl)4 - 6(rl)7 - 9 & & & nbc & bc & ibc & nbc & bc & ibc & & & & & & & & & & & & & & & & ( mean ) & -9.58 & -34.10 & -13.49 & -13.58 & -13.26 & -36.39 & -31.26 & -31.26 & ( 1.86 ) & ( 4.10 ) & ( 3.55 ) & ( 3.55 ) & ( 3.55 ) & ( 4.85 ) & ( 4.62 ) & ( 4.64 ) & & & & & & & & ( std . & & & 4.35 & 4.22 & 4.07 & 12.86 & 10.45 & 10.60 & & & ( 0.98 ) & ( 1.02 ) & ( 1.03 ) & ( 2.35 ) & ( 2.13 ) & ( 2.15 ) & & & & & & & & & 0.49 & 0.45 & 0.48 & 0.48 & 0.48 & 0.44 & 0.44 & 0.45 & ( 0.01 ) & ( 0.06 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & & & & & & & & & 0.44 & 0.17 & 0.44 & 0.43 & 0.44 & 0.23 & 0.29 & 0.27 & ( 0.01 ) & ( 0.07 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & ( 0.05 ) & ( 0.05 ) & ( 0.05 ) & & & & & & & & & & & & & & & & & & & & & & & & long - run & -1.05 & -0.70 & -1.30 & -1.31 & -1.28 & -0.88 & -0.91 & -0.90 & ( 0.24 ) & ( 0.12 ) & ( 0.28 ) & ( 0.28 ) & ( 0.28 ) & ( 0.09 ) & ( 0.10 ) & ( 0.10 ) & & & & & & & & own price & -0.20 & -0.32 & -0.27 & -0.27 & -0.27 & -0.38 & -0.35 & -0.35 ( anticipated ) & ( 0.04 ) & ( 0.04 ) & ( 0.06 ) & ( 0.06 ) & ( 0.06 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & & & & & & & & own price & -0.11 & -0.29 & -0.15 & -0.16 & -0.15 & -0.33 & -0.29 & -0.29 ( unanticipated ) & ( 0.02 ) & ( 0.03 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & ( 0.04 ) & & & & & & & & future price & -0.07 & -0.05 & -0.10 & -0.10 & -0.09 & -0.09 & -0.10 & -0.09 ( unanticipated ) & ( 0.01 ) & ( 0.03 ) & ( 0.02 ) & ( 0.02 ) & ( 0.02 ) & ( 0.02 ) & ( 0.02 ) & ( 0.02 ) & & & & & & & & past price & -0.08 & -0.14 & -0.11 & -0.11 & -0.10 & -0.16 & -0.15 & -0.15 ( unanticipated ) & ( 0.01 ) & ( 0.02 ) & ( 0.03 ) & ( 0.02 ) & ( 0.03 ) & ( 0.02 ) & ( 0.02 ) & ( 0.02 ) & & & & & & & & short - run & -0.30 & -0.35 & -0.41 & -0.41 & -0.40 & -0.44 & -0.44 & -0.43 & ( 0.05 ) & ( 0.06 ) & ( 0.12 ) & ( 0.12 ) & ( 0.12 ) & ( 0.06 ) & ( 0.06 ) & ( 0.06 )  rc / fc refers to random / fixed coefficient model .nbc / bc / ibc refers to no bias - correction / bias correction / iterated bias correction estimates .+ note : _ standard errors are in parenthesis ._  * supplementary appendix to panel data models with nonadditive unobserved heterogeneity : estimation and inference * +  +  this supplement to the paper `` panel data models with nonadditive unobserved heterogeneity : estimation and inference '' provides additional numerical examples and the proofs of the main results .the appendix is organized in seven sections .the first section contains a monte carlo simulation calibrated to the empirical example of the paper .the second section gives the proofs of the consistency of the one - step and two - step fe - gmm estimators .the third section includes the derivations of the asymptotic distribution of one - step and two - step fe - gmm estimators .the forth section provides the derivations of the asymptotic distribution of bias corrected fe - gmm estimators .the fifth and sixth sections contain the characterization of the stochastic expansions for the estimators of the individual effects and the scores .the last section includes the expressions for the scores and their derivatives .  throughout the appendices and will denote uniform orders in probability .for example , for a sequence of random variables , means , and means .it can be shown that the usual algebraic properties for and orders also apply to the uniform orders and .let denote a unitary vector with a one in position .for a matrix , denotes euclidean norm , that is  ] .we find substantial biases for ols and iv - fc estimators .rc - iv displays some bias , which is removed by the corrections in some configurations .the bias corrections provide significant improvements in the estimation of standard errors .iv - rc standard errors overestimate the dispersion by more than when is greater or equal than , whereas iv - bc or iv - ibc estimators have se / sd ratios close to . as a result bias corrected estimatorsshow smaller size distortions .this improvement comes from the bias correction in the estimates of the dispersion of that we use to construct the standard errors .the bias of the estimator of the dispersion is generally large , and is effectively removed by the correction .we can see more evidence on this phenomenon in table a3 .table a3 shows the results for the estimators of the standard deviation of the individual specific coefficient ^{1/2} ] as defined in condition [ cond2 ] . using the standard argument for consistency of extremum estimator , as in newey and mcfadden ( 1994 ) , with probability  by definition of and lemma [ la8 ] .thus , by continuity of and the definition of the lefthand side above , we conclude that = o\left ( t^{-1 } \right) ] for some and .define , and  ] for some and .define , and  ] where and .let for . then ,   note that \right| \\ \leqd_g^{2 } \max_{1 \leq k \leq l \leq d_g } \left| g_{k}(z_{it};\overline{\theta } , \overline{\alpha}_{i } ) g_{l}(z_{it};\overline{\theta } , \overline{\alpha}_{i } ) ' - e \left [ g_{k}(z_{it};\overline{\theta } , \overline{\alpha}_{i } ) g_{l}(z_{it};\overline{\theta } , \overline{\alpha}_{i } ) ' \right ] \right|.\end{gathered}\ ] ] then we can apply lemma [ ld0 ] to and with .a similar argument applies to the derivatives , since they are sums of products of elements that satisfy the assumption of lemma [ ld0 ] .[ le5 ] assume that conditions [ cond1 ] , [ cond2 ] , [ cond3 ] , [ cond5 ] , [ cond4 ] , and [ cond7 ] hold , and such that as . for any such that between and let ^{-1}, ]  and  j\le d_{\alpha} j > d_{\alpha} j \leq d_{\alpha} j > d_{\alpha} ]  and .  the results follow by lemmas [ lg1 ] and [ lf5 ] , noting that & = & m_{i}^{w } \left(\begin{array}{cc }  h_{\alpha_{i}}^{w } \omega_{i } h_{\alpha_{i}}^{w ' } & h_{\alpha_{i}}^{w } \omega_{i } p_{\alpha_{i}}^{w } \\p_{\alpha_{i}}^{w } \omega_{i } h_{\alpha_{i}}^{w ' } & p_{\alpha_{i}}^{w } \omega_{i } p_{\alpha_{i}}^{w } \\\end{array}\right ) m_{i}^{w ' } , \\ e \left [ \tilde{c}_{i}^{w } \tilde{\psi}_{i}^{w } \right ] & = & \sum_{j=-\infty}^{\infty}e \left [ g_{\theta_{i}}(z_{it } ) ' p_{\alpha_{i}}^{w } g(z_{i , t - j } ) \right ] , \\e \left [ \tilde{\psi}_{i , j}^{w } m_{i , j}^{w } \tilde{\psi}_{i}^{w } \right ] & = & \left\{\begin{array}{ll }   - g_{\theta \alpha _ { i , j } } ' p_{\alpha_{i}}^{w } \omega_{i } h_{\alpha_{i}}^{w ' } , & \hbox{if ; } \\   - g_{\theta \alpha _ { i } } ' ( i_{d_{\alpha } } \otimes e_{j - d_{\alpha } } ) h_{\alpha_{i}}^{w } \omega_{i } p_{\alpha_{i},j}^{w } , & \hbox{if . } \\\end{array}\right.\end{aligned}\ ] ]  [ lg3 ] suppose that conditions [ cond1 ] , [ cond2 ] , [ cond3 ] , and [ cond5 ] hold .then , for  where and are defined in lemma [ lg2 ] .  by lemma [ lg1 ] , then , the result follows by lemma [ lg2 ] .[ li1 ] suppose that conditions [ cond1 ] , [ cond2 ] , [ cond3 ] , [ cond5 ] , [ cond4 ] , and [ cond7 ] hold .then , where all the terms are identical to that of lemma [ lg1 ] after replacing by . also , the properties of all the terms of the expansion are the analogous to those of lemma [ lg1 ] .the proof is similar to the proof of lemma [ lg1 ] .[ li2 ] suppose that conditions [ cond1 ] , [ cond2 ] , [ cond3 ] , [ cond5 ] , [ cond4 ] , and [ cond7 ] hold .then , \\ \frac{1}{n } \sum_{i=1}^{n }  q_{1si }  & { \overset{p}{\to } } &  \bar{e } e \left [ q_{1si } \right ]  = \bar{e}[b_{si}^{b } + b_{si}^{c } ] = : b_{s } , \end{aligned}\ ] ] where , j\le d_{\alpha} j > d_{\alpha} j \le d_{\alpha } , k\le d_{\alpha} j \le d_{\alpha } , k > d_{\alpha} j > d_{\alpha } , k\le d_{\alpha} j > d_{\alpha } , k > d_{\alpha} j \le d_{\alpha} j > d_{\alpha} j\le d_{\alpha } , k\le d_{\alpha} j \le d_{\alpha } , k > d_{\alpha} j > d_{\alpha } , k\le d_{\alpha} j > d_{\alpha } , k > d_{\alpha}$. } \\ \end{array}\right .\\\end{aligned}\ ] ]   * + first derivatives *   =  0.\end{aligned}\ ] ]      note that the formulae for the derivatives of appendix [ appendixk ] apply for , replacing by .hence , we only need to derive the derivatives for .   * + first derivatives *   = - \left(\begin{array}{cc }  0 & 0 \\  0 & e \left [ { \widehat}{\omega}_{i } - \omega_{i } \right ] \\\end{array}\right).\end{aligned}\ ] ]  * second and third derivatives *  since does not depend on , the derivatives ( and its expectation ) of order greater than one are zero .* + first derivatives *        since this score does not depend explicitly on , the formulae for the derivatives are the same as in appendix [ appendixl ] .andrews , d. w. k. ( 1991 ) , `` heteroskedasticity and autocorrelation consistent covariance matrix estimation , '' _ econometrica _ 59 , 817 - 858 .

Abstract: this paper considers fixed effects estimation and inference in linear and nonlinear panel data models with random coefficients and endogenous regressors . the quantities of interest means , variances , and other moments of the random coefficients are estimated by cross sectional sample moments of gmm estimators applied separately to the time series of each individual . to deal with the incidental parameter problem introduced by the noise of the within - individual estimators in short panels , we develop bias corrections . these corrections are based on higher - order asymptotic expansions of the gmm estimators and produce improved point and interval estimates in moderately long panels . under asymptotic sequences where the cross sectional and time series dimensions of the panel pass to infinity at the same rate , the uncorrected estimator has an asymptotic bias of the same order as the asymptotic variance . the bias corrections remove the bias without increasing variance . an empirical example on cigarette demand based on becker , grossman and murphy ( 1994 ) shows significant heterogeneity in the price effect across u.s . states .  * jel classification * : c23 ; j31 ; j51 .  * keywords * : correlated random coefficient model ; panel data ; instrumental variables ; gmm ; fixed effects ; bias ; incidental parameter problem ; cigarette demand .   boston university , department of economics , 270 bay state road , boston , ma 02215 , ivanf.edu . ] department of economics , mit , 50 memorial drive , cambridge , ma 02142 , jhlee82.edu . ]


at `` punctual '' space scale ( cm^2^ ) rain can be described by a stochastic sequence of couples : being the diameter of the -th drops , and the interval of time between the arrival of the -th drop and the -th drop ( ) .partitioning the time axis in sampling time intervals of equal duration it is possible to build from the sequence the sequence , with the probability density function of drop diameter , and the number of drops observed at the ground during the -th sampling time interval ( ) .these two last quantities are usually measured by disdrometers with a sampling time in the range 10s-5min .observational evidences suggest that rain is a non - homogeneous process as the variability observed in the sequence can not simply ascribed to the variability expected when sampling from a single stochastic process .the sequence , and thus , is non - stationary : the `` statistical rules '' to which the couples obey are not invariant under time translation .a proper description of the distribution of drop sizes , which is fundamental for understanding the microphysics involved in the mechanisms of precipitation formation and retrieving rainfall from radar sensors , has been the main research goal since the introduction of the impact disdrometer has made reliable and prolonged measurements of drop diameters feasible .the main tool the meteorological community uses to describe the variability of the rainfall phenomenon is the drop size distribution , namely the concentration per unit volume and diameter . in their seminal work marshall and palmer proposed the exponential as functional dependence for with a decaying constant depending on the rainfall rate .later , joss and gori made clear that the exponential form is the results of sampling long time intervals ( minutes ) , and at smaller time intervals ( minute ) the drop size distribution does not have an exponential functional dependence .ulbrich proposed the gamma distribution as the functional form for the drop size distributions sampled at short time intervals .the rationale for this choice is that while an exponential decay is observed for the right tail of the distribution ( recently have shown that break - up due to air viscosity is the main mechanism underlying the occurrence of exponential right tail ) , the density at smaller diameters is smaller than that of an exponential distribution .the gamma distribution has been the only choice adopted in literature ( if one excludes sporadic attempts with the log - normal distribution , see e.g. , , and ) mainly because of two properties : 1 ) it is defined by two parameters , and 2 ) its moments can easily calculated so that any rain bulk variable can easily expressed in terms of the parameters of the gamma distribution .the gamma distribution has been widely accepted by radar meteorology and cloud physics communities , even if measurements show that the gamma distribution is not general enough to represent adequately the full range of the sample variability , .the fit accuracy of the gamma distribution to disdrometer data has not been properly addressed ( aside from subjective statements : `` a good fit '' or `` a reasonable fit '' not supported by any objective measurement ) or mistakenly addressed ( see section [ sec : results ] ) as in . in the present work , we consider data from 10 different sites , and use the kolmogorov - smirnov goodness of fit test ( e.g. ) to show that the gamma distribution is a poor fit to 1 minute sampled drop size distribution .  to obviate to this inadequacy , we propose a parametrization of the drop size distribution based on a standard procedure in statistical science : the description of an unknown probability density function in terms of its mean , standard deviation , skewness , and kurtosis .in particular , we show that mean , standard deviation and skewness are the minimum number of parameters , which effectively describe the drop size distributions .these parameters , together with the drop count , are the variables necessary to describe the rainfall phenomenon at a punctual scale in space and at short time scales .all bulk variables of interest can be derived by the 4-tuple of parameters . adopting the jargon of the scientific community studying dynamical systems, we refer to the four dimensional cartesian product spanned by the 4-tuple as the rainfall phase space .the work is organized as follows . in section [ sec : data ] , we discuss the data used in our analysis , section [ sec : methods ] we illustrate all the methods of analysis adopted .section [ sec : results ] reports our results , and section [ sec : conclusions ] our conclusions .we consider joss - waldvogel disdrometer data sampled at 1 minute time intervals from ten different locations on earth s surface .table 1 gives the list of the locations with a three letter code used to reference the site in the figures .in addition , the kppen - geiger climate classification ( ) of each site is provided .table 2 completes the description of the data sets providing for each site the latitude , longitude , altitude , together with number of minutes and drops considered in the analysis .all data are processed as follows .we consider for each data base only minutes for which the drop count is ( drop arrival rate per second ) .the rationale for this choice is twofold : a ) it guarantees a minimum reasonable accuracy for the estimation of the probability density function , and the other statistical parameters ( mean , standard deviation , ... ) .b ) it excludes time intervals of observation which are quiescent ( sparse precipitation ) whose contribution to the total cumulated precipitated volume is negligible ( see discussion in ) .after this first filtering , we eliminate from the remaining minutes of observation any outlier drop count . for each minute, we find the disdrometer class with the maximum probability density , and we calculate the central continuous non - zero span of the probability density : the set of contiguous disdrometer classes with non zero counts and which includes the class with maximum density .the counts in all disdrometer classes which do not belong to the central continuous non - zero span are considered as outliers and are discarded : e.g. the disdrometer count ( 3,11,18,31,30,35,80,52,41,39,44,21,5,0,1,0,0,0,0,0 ) has an outlier in the 15-th class which is disregarded leading to the count ( 3,11,18,31,30,35,80,52,41,39,44,21 , + 5,0,0,0,0,0,0,0 ) .the removal of outliers drop counts improves the estimate of higher moments of the probability density function .a detailed discussion on outliers and their effects on estimated statistical parameters can be found in .we now describe the methods used to quantify the variability of the rainfall phenomenon .two equivalent descriptions are possible for the variability of drop diameters .1 ) the drop size distribution defined as the concentration per unit volume and unit diameter     where is the number of drops per unit volume and is the probability density function of drop diameter in the unit volume .2 ) the flux - equivalent of eq.([eq : nd_proper ] ) : and . is the number of drops observed at the ground , and is the probability density function of drop diameter at the ground .the two descriptions are equivalent as    and    in the above equations , is the time interval of observation ( in seconds ) , the capture area of the instrument ( in m^2^ ) , and the drop velocity of diameter ( in m / s ) .usually it is assumed that the arrival velocity of drops is equal to their limit velocity and , where m/(s mm^0.67^ ) and ( ) . with these definitions    by use of eq.([eq : cloud - ground-2 ] ) , we can connect the moments of the probability density function observed at the ground with those of the concentration per unit volume :    using the above equation one is able to derive expressions for any rain bulk variable .e.g. the rainfall rate ( in mm / h ) is    the concentration per unit volume ,eq.([eq : nd_proper ] ) is by far the most common quantitiy in literature , even if measurements of drop sizes at the ground are by large the only available data .hereby , we adopt the probability density of drop diameter at the ground , eq.([eq : ground - cloud ] ) , as measured by disdrometer counts .   the most common functional form adopted for the probability density function in eq.([eq : nd_proper ] ) is the gamma distribution    where is the shape, the inverse scale parameter , and is gamma function .consequently the statistical moment of order of the drop size distribution is    however , it is common practice , in literature , to write eqs.([eq : nd_proper ] ) and ( [ eq : defgamma ] ) as    where is defined as    with the above notation    finally , if the probability density function is a gamma distribution , then also the probability density at the ground is a gamma distribution     we briefly review the two main methods which have been adopted in literature to obtain the three parameters in eqs.([eq : nd_proper ] ) and ( [ eq : defgamma ] ) .the method of moments ( mm ) uses the moments of the observed distribution to derive the parameters of the desired fitting function . for a drop size distribution with a gamma distribution for diameter density ,the mm method finds the number of drops per unit volume , the scale and shape such that the resulting distribution exactly matches the moments of order n1 , n2 and n3 of the observed distribution .hereby , we use the mm and mm procedures as they are the ones usually adopted in literature ( e.g. and ) , together with the mm adopted by .this last procedure finds the concentration , the scale and shape of the drop size distribution matching the observed number of drops at the ground ( moment 0.67 ) , the observed average drop diameter at the ground ( moment 1.67 ) , and the observed second moment of drop diameter at the ground ( moment 2.67 ) .note that the last two conditions imply that the resulting drop size distribution matches the standard deviation of drop diameter observed at the ground .therefore , we will use for brevity the notation mm instead of mm .  using eqs .( [ eq : nd_proper ] ) and ( [ eq : momentsgamma ] ) , the parameters , , and are         the method of maximum likelihood is the main alternative to the method of moments .let be the probability density function of the variable given the vector of parameters , and is a sample of size .the likelihood that the sample is drawn from the distribution is defined as . in practice it is more convenient to deal with the logarithm of the likelihood , denominated the log - likelihood , or the average log - likelihood .the ml method makes an estimation of maximizing the average log - likelihood , i.e.  ] and ^{4} ] , which will denote with the letter ^{5/2}}.\label{eq:5th - central - moment}\ ] ] mean , standard deviation , skewness , and kurtosis are sufficient to achieve a satisfactory ( albeit not full ) description of any probability density function of interest . in many cases they are redundant as some of the statistical descriptors are shown to be function of the others .for example if then only two elements of the 4-tuple are independent .this redundancy also occurs for rainfall as we will show in section [ sec : results ] .   for a dynamical system ,the term `` phase space '' indicates the cartesian product of the variables necessary to describe the system .note that the cardinality of a dynamical system is always larger or equal to its degrees of freedom . to describe the motion a point particle of mass in one dimensionwe need its position and its quantity of motion ( , with ) . in this casethe phase space is the plane ( ) , and at each time the state of the particle is associated to a point in the plane .the time evolution of a dynamical system is described by the time evolution ( trajectory ) of its associated point in the phase space .  in the case of the rainfall phenomenon ,the variables of interest are bulk variables ( e.g rainfall rate , reflectivity ) , which implies a `` summation '' over the number of drops in a given interval of time .all the bulk variables are functions of either the couple , if one uses the concentration per unit volume , or the couple , if one uses the flux - equivalent description . if the probability density function at the ground ( or that in a unitary volume ) has a parametric description : in term of parameters , then we define the phase space of rainfall as the cartesian product spanned by ( + 1)-tuple .e.g. , if we consider the concentration and assume a gamma distribution for the probability density function in eq.([eq : nd_proper ] ) , the rainfall phase space is spanned by the 3-tuple . hereby , we do not impose any particular functional form on the function ( and thus ) .we show ( section [ sec : results ] ) that the parameters , , and are sufficient to describe the variability of the probability density function at the ground therefore the cartesian product spanned by the 4-tuple can be considered as the rainfall phase space .any bulk variable can be written as function of the phase space parameters . in the case of the rainfall rate ( expressed in mm / h ).\label{eq : r_mitica}\ ] ]  this equation states that the set of observation time intervals ( 1 minute in our case ) with equal rainfall rate is a three dimensional manifold in the phase space .   the kolmogorov - smirnov ( k - s ) goodness - of - fit test is a non - parametric test used to check if a sequence of random samples can be considered as a realization of a stochastic process with a given cumulative distribution function . the test compares the hypothetical with the cumulative frequency , where for , is the i - th order statistics , and .the k - s uses as test statistic the maximum difference .if no parameter in is determined from data , then has a distribution which is independent by .thus the critical value of for a significance level of and for large samples , , is , and reported in all statistical textbooks ( see e.g. ) .contrary , if the parameters of are estimated , then the distribution of is dependent on , and the critical value of must be re - calculated , e.g. via montecarlo simulations ( ) .the critical value of re - calculated is always smaller than the value corresponding to the canonical case where it is assumed that `` no parameter in is determined from data '' .disdrometer data report the occurrence of a drop in a given range of diameter values ( diameter class ) and not an `` exact '' diameter value which is needed to perform the kolmogrov - smirnov test . to bypass this limitationwe assign to a drop in the j - th diameter class a random value selected uniformly in the range defined by the class itself .all the results reported in this section refer to probability density function observed at the ground .similar results can be obtained for the concentration per unit volume and diameter since the two distribution are connected via eq.([eq : ground - cloud ] ) .the method of the moments mm for the gamma distribution finds the number of drops per unit volume , the scale and shape such that the fitting concentration per unit volume and unit diameter exactly matches the moments of order n1 , n2 and n3 of the observed concentration per unit volume and unit diameter . none of the mm , mm , and mm methods exactly match the moment 3.67 , so that the `` reproducibility '' of the rainfall rate ( observed versus the one derived from the fitted parameters ) has been considered as a `` measure '' of fit goodness : e.g. ( ) and ) .hereby , we show that accuracy with which the mm method reproduces the -th moment of the concentration can not be taken as a measure of fit goodness as the accuracy depends on the separation between and the orders ,, and not just on the particular functional form chosen .let us consider the mm method .the fitting distribution by construction matches the third , fourth and sixth moment of .therefore it is not surprising that the 3.67-th moment of the fitting ( the rainfall rate ) is close to the observed value : the middle - left panel of figure 1 indicates the relative error is bounded in the range -0.5%,0.5% .however , instead of a gamma distribution for the functional form of , one could use any other distribution with two parameters ( e.g. gaussian , lognormal , beta ) and obtain similar accuracies .what about the number of drops observed at the ground ( the 0.67-th moment of ? as shown in the middle - right panel of figure 1 , the agreement is not so good as relative error of the order of % are possible .next , we consider the mm method .it reproduces the rainfall rate reasonably well , relative error bounded in the -5%,5% range ( upper - left panel of figure 1 ) , but poorly reproduces the reflectivity ( the sixth moment of ) as relative errors larger than % are not so uncommon ( upper - right panel of figure 1 ) .finally , the mm method reproduces with the same accuracy of the mm method ( extremely well ) the rainfall rate , the relative error is bounded in the range -0.5%,0.5% ( bottom left panel of figure 1 ) , but is better than the mm method with respect the drop count ( 2 is closer to 0.67 than 3 ) , relative error in the range -15%,15% ( bottom right panel of figure 1 ) .figure 1 suggests that the gamma distribution , as convenient and as parsimonious it may be , is not a satisfactory functional form for the drop size distribution . to prove this point we use a proper measure of goodness - of - fit , such as the kolmogorov - smirnov test , and what is considered to be the best fitting procedure , the mml method . in table 2 , we report the percentage of acceptance ( acp ) and rejection ( rjc ) of the lower truncated gamma distribution with parameters estimated using the mml , using the kolmogorov - smirnov goodness - of - fit test with a 5% level of significance to each minute of the drw data ( 6863 ) . in columns 2 and 3 , the percentages are calculated using as 5% critical value , the classical value reported in all statistical textbooks ( e.g. ) assuming that no parameters of the gamma distribution are estimated . in columns 4 and 5the critical value is determined via montecarlo simulations taking into account the fact that the parameters are estimated via the mml from data . the first row ( all )reports the fraction of the total number of minutes in the drw data sets for which the gamma distribution can be considered a good fit .the percentage of acceptance passes from 71% to 45% when one takes in proper consideration that the parameters of the distribution are obtained from the sample ( ) .the remaining rows report the percentage of acceptance and rejection for subsets obtained using as thresholds the percentiles of the distribution of number of drops per minute : 5%- , 25%- , 50%- , 75%- , 95%-percentile .e.g. the second row reports the results for the subsets with number of drops smaller than the 5%-percentile ( in our case ) : minutes with a small sample size . on the other hand ,the last row reports the results for the subsets with number of drops larger than or equal to the 95%-percentile ( in our case ) : minutes with a large sample size .we see that when we move to subsets of minutes with large drop counts the percentage of acceptance ( rejection ) diminishes ( increases ) .  in summary , if we consider the entire drw data set , we are confident ( at the 95% level ) that the gamma distribution can be a proper fit for probability density function only for 45% of 1 minute sampling time intervals .more disturbingly the percentage of rejection increases as the sample size increases .note that test like the kolmogorov - smirnov should be administered to sample with a size of at least ( ) to have of any significance ( in other words if the sample size is very small the effect of random fluctuations is large enough that almost any tested distribution will pass the test ) .results for the other nine data sets ( no reported here for brevity ) are similar to that of darwin . on the ground of these results , we reject the gamma distribution as a proper fit for drop size distributions .we show that the value of standardized central moments of order are strictly dependent from than the third one ( skewness ) .in particular we study the dependence on the skewness of the fourth standardized central moment ( kurtosis ) and the fifth one denoted by the symbol . as a consequence ,mean ( ) , standard deviation ( ) , and skewness ( ) provide a satisfactory description of the variability of the probability density function and therefore the 4-tuple can be considered as the rainfall phase space .   to examine the dependence of the parameters and on , we calculate for each data set the median , 5%- , and 95%-percentile of the observed values of and for a given value of ( in practice this is accomplished dividing each data sets in subsets with `` equal '' ( ) value of skewness ) .the results are reported in figure 2 .the median values are depicted with solid lines of different colors one for each dataset .we also calculate , merging all the datasets , the 5% and 95% percentile for any given range value of the skewness if at least 100 samples ( 1 minute time interval of observation ) are present .the range between the 5% and 95% percentile is shaded in gray in the figure .we see how in both cases ( vs , central panel and vs lower panel ) the median lines are independent from the site chosen .the discrepancies observed for values of skewness larger than .56 are mostly due to lack of statistics as shown in the upper panel of figure 2 , where we plot the number of sample for each range value of the skewness .figure 2 shows that mean standard deviation and skewness are effective statistical descriptors for the probability density function of drop diameters at the ground .next , we show that these three parameters are not independent .we divide the range of value of the skewness in intervals of length 0.64 ( [ -1.60,-0.96], ... ,[2.88,3.52 ] as in ) , and the values of the mean diameter as follows : [ 0.3,0.4 ] , [ 0.4,0.5 ] , [ 0.5,0.6 ] , [ 0.7,0.8 ] , [ 0.9,1 ] , [ 1,1.2 ] , [ 1.2,1.4 ] , [ 1.4,1.6 ] , [ 1.6,1.8 ] , [ 1.8,2.0 ] , [ 2.0,2.5 ] , and [ 2.5,3 ] ( indicated by the vertical dashed lines in figure 3 ) . the rationale for these choices is to have inside each range of skewness and mean a `` reasonable '' number ( > 10 ) of 1 minute interval observations to calculate the median value of the parameter .the results are reported in figure 3 .we see how for value of the mean diameter less than 1 mm the median curves are approximatively linear and do not depend on the site of observation . in this case , the site average angular coefficient and average intercept are reported at the bottom right of each panel , if at least the median from 5 sites was available .we see how the slope ( intercept ) increases ( decreases ) with increasingly larger value of the skewness until a sort of plateau is reached for the skewness ranges  ] after which the slope decreases ( intercept increases ) again . for values of the mean diameter larger than 1the median curves depend on the particular site of observation .however , the median curve is also estimated from a smaller number of samples ( in the range10 - 100 ) .thus , with the present data , we can not judge if the discrepancies between sites for the median curve ( in the range ) are real properties of the rainfall phenomenon or merely artifacts due to the poor statistical sampling available .  in summary ,if is in the interval [ 0.3,1 ] then where is the intercept and the slope .note that if the gamma distribution was indeed an extremely accurate fit to probability density function of drop diameter at the ground p(d ) then which is not supported by the experimental evidences depicted in figure 3 .eq.([eq : musigmagamma ] ) suggests that only two parameters of the triplets are necessary to describe the probability density of drop diameters at the ground .thus one could define the rainfall phase space as a tridimensional space : e.g. the cartesian product of the 3-tuple .however , longer data sets are necessary to effectively estimate the functions and , and to explore the relationship in the range mm .therefore , for the purpose of this manuscript , we conservatively consider defined by the 4-tuple as the rainfall phase space .to each 1 minute time interval of observation is associated the point of coordinates ( ) in the phase space .the entire data set occupies a volume in . to visualize this volume, we need to consider the six 2d projections : , , , , , and ( we use instead of for better visualization ) . given a data set ,we calculate the density of points in the phase space for all six 2d projections .ten separate figures ( one for each data set ) would be necessary to illustrate the results .to obviate to this difficulty and give an idea to the reader of the differences / similarities between data sets we adopted the concept of average bounding perimeter .for all 2d projections we calculate the center of mass of the phase space points ( each point has the same mass ) . with the center of mass as fixed point we span with an 10^o^ angle step the plane of the 2d projection .for each 10^o^ cone we calculate the average distance from the center of mass of the points within the cone .the connection with a continuous line of all average distance creates the average bounding perimeter which is a `` measurement '' of the volume of the phase space occupied by the database .the results are shown in figure 4 .the plots on the projection plane show how the average bounding perimeters reflect the linear relationship between mean drop diameter and standard deviation of drop diameter depicted in figure 3 .the plots on the plane projection indicate that large value of counts ( ) are reached ( on average ) only for values of the mean drop diameter which are small ( in the range 0.4 - 0.7 mm ) or large ( . the first case , many drops with small diameter , is a common feature of orographic precipitation as shown ( e.g. , , , , and ) . the bby and czc data sets are the same ones used in ( , , and ) while bao data set come from an instrument located at 1,577 m asml .the second case , many drops with possibly large diameters , is typical of strong convective events .the drw data sets ( rain of monsoonic origin ) and the mik ( small island at the equator ) and bkt ( another equatorial site ) are the data sets , among those considered , where the combination of large number of drops and large drop diameters occur more frequently on average .the average bounding perimeter on the plane projection show that a decrease in value of the mean drop diameter is linked to a raise of the skewness value , although this may be in part an effect of the limitation of the instrument ( joss - waldvogel impact disdrometer ) which is not capable of detecting drop diameters smaller than 0.3 mm ( reducing the contribution of left tails of to the skewness ) . due to the approximate linear relationship between and , results of projection on planesfor which one of the axis is the standard deviation are similar to those for which axis is substituted by the mean . if we consider the result on the plane projection , we see how these two variable are quite uncorrelated as the shape of the average bounding perimeters do not suggest any particular relation .the rainfall rate aside from a multiplicative constant is the sum of three factors ( eq.[eq : r_mitica ] ) : , , and .each factor accounts for a fraction  ] .different colors indicate different databases . ]ignaccolo , m. and c. de michele , 2012 : skewness as measure of the invariance of instantaneous renormalized drop diameter distributions part 1 : convective vs. stratiform precipitation ._ hydrology and earth system sciences _ , * 16 ( 2 ) * , 319327 .  ignaccolo , m. and c. de michele , 2012 : skewness as measure of the invariance of instantaneous renormalized drop diameter distributions part 2 : orographic precipitation . _ hydrology and earth system sciences _ , * 16 ( 2 ) * , 329343 .martner , b. e. , s. e. yuter , b. w. allen , s. y. matrosov , d. e. kingsmill , and f. m. ralph , 2008 : raindrop size distributions and rain characteristics in california coastal rainfall for periods with and without a radar bright band ._ j. hydrometeor ._ , * 9 * , 408425 .testud , j. , s. oury , r. a. black , p. amayenc , and x. dou , 2001 : the concept of normalized distribution to descrbie raindrop spectra : a tool for cloud physics and cloud remote sensing ._ j. appl ._ , * 40 ( 6 ) * , 11181140 .

Abstract: we show that the gamma distribution is not an adequate fit for the probability density function of drop diameters using the kolmogorov - smirnov goodness of fit test . we propose a different parametrization of drop size distributions , which not depending by any particular functional form , is based on the adoption of standardized central moments . the first three standardized central moments are sufficient to characterize the distribution of drop diamters at the ground . these parameters together with the drop count form a 4-tuple which fully describe the variability of the drop size distributions . the cartesian product of this 4-tuple of parameters is the rainfall phase space . using disdrometer data from 10 different locations we identify invariant , not depending on location , properties of the rainfall phenomenon .