Day 1A Count Data Regression: Part 1

Transcription

1 Day 1A Count Data Regression: Part 1 A. Colin Cameron Univ. of Calif. - Davis... for Center of Labor Economics Norwegian School of Economics Advanced Microeconometrics Aug 28 - Sept 1, Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 1 / 65

2 1. ntroduction 1. ntroduction We consider nonlinear regression with cross-section data. The slides use count regression as a great illustrative example. Count data models are for dependent variable y = 0, 1, 2,... Example: y: Number of doctor visits (usually cross-section) x: health status, age, gender,... Many approaches and issues are general nonlinear model issues. Econometrics: F F Fully parametric: MLE Conditional mean: Quasi-MLE, generalized methods of moments (GMM) Statistics: generalized linear models (GLM). A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 2 / 65

3 1. ntroduction Analysis is straightforward in the usual case of model the conditional mean E[y jx]: in Stata replace command regress with poisson and for panel data (later) replace command xtreg with command xtpoisson nterpretation of marginal e ects, however, is more complicated: E[yjx] = exp(x 0 β) so ME j = E[yjx]/ x j = β j exp(x 0 β) 6= β j. Analysis is more complicated for better parametric models for prediction, censoring, selection autoregressive time series of counts (not covered here).n The session will focus on topics 2, 3 and 4. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 3 / 65

4 1. ntroduction Outline 1 ntroduction 2 Poisson (with cross-section data) Theory* 3 Poisson Application* 4 Summary of the remaining topics* 5 Generalized linear models 6 Diagnostics 7 Negative binomial model 8 Richer Parametric Model (censored, truncated, hurdle,..) 9 Summary 10 References A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 4 / 65

5 2. Poisson cross-section regression theory 2. Poisson cross section regression theory Count data example Many health surveys measure health use as counts as people have better recall of counts than of dollars spent U.S. Medical Expenditure Panel Survey (MEPS). Sample of Medicare population aged 65 and higher (N = 3,677) docvis = annual number of doctor visits. use mus17data.dta. summarize docvis Variable Obs Mean Std. Dev. Min Max docvis A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 5 / 65

6 2. Poisson cross-section regression theory Doctor visits: Histogram dropping observations with more than 40 visits Density # Doctor visits (for < 40 visits). Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 6 / 65

7 2. Poisson cross-section regression theory Poisson distribution Poisson distribution From stochastic process theory, natural model for counts is Probability mass function: Mean and variance: Equidispersion: variance = mean Restriction imposed by Poisson Overdispersion: variance > mean y Poisson[λ]. Pr[Y = yjλ] = e λ λ y y! E[y] = λ and V[y] = λ More common feature of count data Doctor visits data: ȳ = 6.82, sy 2 = ' 8.01ȳ. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 7 / 65

8 2. Poisson cross-section regression theory Poisson regression Poisson regression: summary Poisson regression is straightforward many packages do Poisson regression coe cients are easily interpreted as semi-elasticities. Do Poisson rather than OLS with dependent variable y ln p y (with adjustment for ln 0) y (a variance-stabilizing transformation). Poisson MLE is consistent provided only that E[yjx] = exp(x 0 β). But make sure standard errors etc. are robust to V[yjx] 6= E[yjx]. And generally don t use Poisson if need to predict probabilities. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 8 / 65

9 2. Poisson cross-section regression theory Poisson regression Poisson regression: Poisson MLE Let the Poisson rate parameter vary across individuals with x in way to ensure λ > 0. λ = E[yjx] = exp(x 0 β). MLE is straightforward given data independent over i. f (y) = e λ λ y /y! ) ln f (y) = exp(x 0 β) + yx 0 β ln y! ) ln L(β) = n i=1 f exp(x 0 i β) + y i x 0 i β ln y i!g ) ln L(β) β = n i=1 f exp(x 0 i β)x i + y i x i g A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School of Economics Aug 28 - Sept Advanced 1, 2017Microeconomet 9 / 65

10 2. Poisson cross-section regression theory Poisson regression Poisson regression: rst-order conditions The ML rst-order conditions are n i=1 (y i exp(xi 0 β))x b i = 0. No explicit solution for β. b nstead use Newton-Raphson iterative method. Fast as objective function is globally concave in β. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 10 / 65

11 2. Poisson cross-section regression theory Poisson regression Poisson regression: consistency of Poisson MLE ML rst-order conditions are n i=1 (y i exp(xi 0 β))x b i = 0. Consistency only requires (given independence over i) E[(y i exp(xi 0 β))x i ] = 0 So consistent if E[y i jx i ] = exp(xi 0 β) Poisson MLE is consistent if the conditional mean is correctly speci ed like MLE for linear model under normality (OLS) this robustness holds for only some likelihood based models. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 11 / 65

12 2. Poisson cross-section regression theory Poisson regression Poisson regression: distribution of Poisson MLE f distribution is Poisson then β b a N [β, V MLE [ β]] b where bv MLE [ β] b 1 = i bµ i x i xi 0 using 2 ln L(β)/ β β 0 = i exp(xi 0 β)) f distribution is not Poisson but E[y i jx i ] = exp(xi 0 β) and V[y i jx i ] = σ 2 i then β b a N [β, V ROB [ β]] b and we use the robust sandwich estimate of variance (White (1982), Huber (1967)) bv ROB [ β] b 1 = i bµ i x i xi 0 i (y i bµ i ) 2 x i x 0 i i bµ i x i x 0 i 1 V ROB [ β] b =V MLE [ β] b if σ 2 i = µ i (imposed by Poisson) V ROB [ β] b = αv MLE [ β] b if σ 2 i = αµ i (used in GLM literature) Robust se s are much larger than default ML se s if α >> 1. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 12 / 65

13 2. Poisson cross-section regression theory Poisson regression ASDE: derivation of robust sandwich Take a rst-order Taylor series expansion of i (y i about β. exp(x 0 i b β))x i i (y i exp(x 0 i b β))x i = i (y i exp(x 0 i β))x i i exp(x 0 i β)x i x 0 i ( b β β) + F.o.c. set this to zero and can show that R disappears asymptotically i (y i µ i )x i + ( i µ i x i xi 0) (b β β) = 0 ) ( β b β) = ( i µ i x i xi 0) 1 i (y i µ i )x i a ( i µ i x i xi 0) 1 N 0, i σ 2 i x i xi 0 h a N 0, ( i µ i x i xi 0) 1 i σ 2 i x i xi 0 ( i µ i x i xi 0) 1i where µ i = exp(x 0 i β), bµ i = exp(x 0 i b β) and σ 2 i = E[(y i µ i ) 2 ]. Asymptotically can estimate i σ 2 i x i x 0 i by i (y i bµ i ) 2 x i x 0 i. f density is Poisson then simpli es to ( i µ i x i x 0 i ) 1 as σ 2 i = µ i. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 13 / 65

14 3. Poisson regression example 3. Poisson regression example 2003 MEPS data for over 65 in Medicare Dependent variable: docvis Regressors grouped into three categories: Health insurance status indicators F F private medicaid Socioeconomic F F F age age2 educyr Health status measures F F actlim totchr global xlist private medicaid age age2 educyr actlim totchr in commands refer to as $xlist A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 14 / 65

15 3. Poisson regression example Summary statistics. describe docvis $xlist storage display value variable name type format label variable label docvis float %9.0g # doctor visits private byte %8.0g =1 if has private supplementary insuran medicaid byte %8.0g =1 if has Medicaid public insurance age byte %8.0g Age age2 float %9.0g Age squared educyr byte %8.0g Years of education actlim byte %8.0g =1 if activity limitation totchr byte %8.0g # chronic conditions.. summarize docvis $xlist, sep(10) Variable Obs Mean Std. Dev. Min Max docvis private medicaid age age educyr actlim totchr A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 15 / 65

16 3. Poisson regression example Poisson MLE with robust sandwich standard errors - preferred. * Poisson with robust standard errors. poisson docvis $xlist, vce(robust) nolog // Poisson robust SEs Poisson regression Number of obs = 3677 Wald chi2(7) = Prob > chi2 = Log pseudolikelihood = Pseudo R2 = Robust docvis Coef. Std. Err. z P> z [95% Conf. nterval] private medicaid age age educyr actlim totchr _cons A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 16 / 65

17 3. Poisson regression example Poisson MLE with default ML standard errors - do not use - These are misleadingly small due to overdispersion!!. * Poisson with default ML standard errors. poisson docvis $xlist // Poisson default ML standard errors teration 0: log likelihood = teration 1: log likelihood = teration 2: log likelihood = Poisson regression Number of obs = 3677 LR chi2(7) = Prob > chi2 = Log likelihood = Pseudo R2 = docvis Coef. Std. Err. z P> z [95% Conf. nterval] private medicaid age age educyr actlim totchr _cons Robustq se s are times larger Note: sy 2 /ȳ = p /6.82 = p 8.01 = A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 17 / 65

18 3. Poisson regression example Coe cient interpretation Poisson regression: coe cient interpretation For the exponential conditional mean the marginal e ect ME j = E[yjx] x j = exp(x 0 β) β j = E[yjx] β j 1 Conditional mean is strictly monotonic increasing (or decreasing) in x ij according to the sign of β j. 2 Coe cients are semi-elasticities: β j is proportionate change in conditional mean when x ij changes by one unit. 3 More precisely (exp(β j ) 1) is proportionate change. Programs have options to report exponentiated coe cients (incidence-rate ratios). 4 Like all single-index models, if β j = 2β k, then the e ect of one-unit change in x j is twice that of x k. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 18 / 65

19 3. Poisson regression example Coe cient interpretation Poisson regression: coe cient interpretation (continued) Example: bβ Private = Private insurance is associated with an increase in mean doctor visits of 14.2%. More precisely the increase is 100 (e ) = 100 ( ) = 15.3%. Alternatively the exponentiated coe cient is e = 1.153, so the multiplicative e ect is Example: bβ Private = and bβ totchr = Since 0.142/0.248 = 0.57, private insurance has the same impact on mean doctor visits as 0.57 more chronic conditions. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 19 / 65

20 3. Poisson regression example Marginal e ects Marginal e ects: Three types 1. Average marginal e ect (AME): Evaluate at each x i and average AME = i E[y i jx i ] x ij = i exp(x 0 i b β) bβ j. 2. Marginal e ect at mean (MEM): Evaluate at x = x MEM = E[yjx] x j = exp(x 0 β) b bβ j x=x 3. Marginal e ect at representative value (MER): Evaluate at x = x AME is nest For population AME use population weights in computing AME. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 20 / 65

21 3. Poisson regression example Marginal e ects For Poisson with intercept in model AME = ȳ bβ j Reason: f.o.c. i (y i exp(xi 0 β)) b = 0 imply i exp(xi 0 β) b = ȳ For Poisson can show that AME > MEM. Computation of marginal e ects in Stata after poisson (or other regression command) give command margins, dydx(*) for AME margins, dydx(*) atmean for MEM margins, dydx(*) at(age=30 educyr=12) for MER Old Stata 10: add-ons mfx for MEM and margeff for AME. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 21 / 65

22 3. Poisson regression example Marginal e ects Marginal e ects: AME (This page) versus MEM (next page). * AME and MEM for Poisson. quietly poisson docvis $xlist, vce(robust). margins, dydx(*) // AME: Average marginal effect for Poisson Average marginal effects Number of obs = 3,677 Model VCE : Robust Expression : Predicted number of events, predict() dy/dx w.r.t. : private medicaid age age2 educyr actlim totchr Delta method dy/dx Std. Err. z P> z [95% Conf. nterval] private medicaid age age educyr actlim totchr A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 22 / 65

23 3. Poisson regression example Marginal e ects Marginal e ects: MEM. margins, dydx(*) atmean // MEM: ME for Poisson evaluated at average of x Conditional marginal effects Number of obs = 3,677 Model VCE : Robust Expression : Predicted number of events, predict() dy/dx w.r.t. : private medicaid age age2 educyr actlim totchr at : private = (mean) medicaid = (mean) age = (mean) age2 = (mean) educyr = (mean) actlim = (mean) totchr = (mean) Delta method dy/dx Std. Err. z P> z [95% Conf. nterval] private medicaid age age educyr actlim totchr A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 23 / 65

24 3. Poisson regression example Factor Variables Factor variables Problem: obtaining ME in models with interactions and polynomials Solution: Factor variable operators # and ## also i. for discrete variables uses nite di erence not derivatives Now can get ME with respect to age allowing for age 2 in model. Also now the i. variables have a somewhat di erent ME.. * Also factor variables and noncaculus methods. * The i. are discrete and will calculate ME of one unit change (not derivative). * The c.age##age means age and age sauared appear and ME is w.r.t. age. quietly poisson docvis i.private i.medicaid c.age##c.age educyr i.actlim totchr, vce(robust). margins, dydx(*) // MEM: ME for Poisson evaluated at average of x Average marginal effects Number of obs = 3,677 Model VCE : Robust Expression : Predicted number of events, predict() dy/dx w.r.t. : 1.private 1.medicaid age educyr 1.actlim totchr Delta method dy/dx Std. Err. z P> z [95% Conf. nterval] 1.private medicaid age educyr actlim totchr Note: dy/dx for factor levels is the discrete change from the base level. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 24 / 65

25 4. Summary of remaining topics 4. Summary of remaining topics The Poisson is a generalized linear model (GLM) the framework used in the statistics literature for nonlinear regression leading examples are OLS, logit, probit, gamma regression in Stata use command glm with family member poisson and log ling function. The Poisson MLE is generally ine cient but generally not great e ciency loss (like OLS versus GLS) But it is usually the wrong model if we want to predict probabilities intuitively Poisson has only one parameter µ whereas e.g. the normal has µ and σ 2 The standard better ML model is the negative binomial distribution F then Var (y ) = µ + αµ 2 (overdispersion) versus Poisson Var (y ) = µ. F in Stata use command nbreg. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 25 / 65

26 4. Summary of remaining topics 4. Summary of remaining topics (continued) Count data may be record no zeroes e.g. record number of doctor visits only for those who visited the clinic at least once then use truncated MLE. Count data may be topcoded e.g. record number of doctor visits as 0, 1, 2, 3, 4 0r more. then use censored (from above) MLE. Count data may be interval recorded. Count data even if fully observed often have too few zeroes e.g. fewer zeroes than predicted by a negative binomial model. then two approaches F hurdle model - one model for = 0 or > 0 and separate model given > 0 F with zeroes model Most natural extension of R 2 is R 2 Cor = ccor 2 [y i, by i ]. Compare nonnested parametric models using AC or BC. Quantile regression has been extended to counts. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 26 / 65

27 5. Generalized linear models 5. Generalized linear models Generalized linear models (GLM) is the framework used in the statistics literature for nonlinear regression a brief introduction that may be skipped depending on time. Leading examples are OLS regression for y 2 (, ) Logit and probit regression for y 2 f0, 1g Poisson regression for y 2 f0, 1, 2, 3,...g Gamma regression including exponential for y 2 (0, ) n Stata use command glm specify the GLM family member: here poisson specify the link function (inverse of the conditional mean function): here log get robust standard errors: vce(robust) A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 27 / 65

28 Exactly same as poisson, vce(robust) A. Colin Cameron Univ. of Calif. - Davis... for Center of Labor Economics Norwegian School of Economics Advanced Microeconomet Count Regression: Part 1 Aug 28 - Sept 1, / Generalized linear models Poisson GLM with robust sandwich standard errors. glm docvis $xlist, family(poisson) link(log) vce(robust) nolog Generalized linear models No. of obs = 3677 Optimization : ML Residual df = 3669 Scale parameter = 1 Deviance = (1/df) Deviance = Pearson = (1/df) Pearson = Variance function: V(u) = u [Poisson] Link function : g(u) = ln(u) [Log] AC = Log pseudolikelihood = BC = Robust docvis Coef. Std. Err. z P> z [95% Conf. nterval] private medicaid age age educyr actlim totchr _cons

29 5. Generalized linear models ASDE: What is a generalized linear model? Class of models based on linear exponential family (LEF): normal, binomial, Bernoulli, gamma, exponential, Poisson. Speci cally for the LEF f (y i jµ i ) = expfa(µ i ) + b(y i ) + c(µ i )y i g E[y i ] = µ i = a 0 (µ i )/c(µ i ) V[y i ] = 1/c(µ i ) For regression specify a model of the mean µ i = µ i (β) = µ i (x i, β). Poisson is a member with a(µ) = µ; c(µ) = ln µ a 0 (µ) = 1 and c 0 (µ) = 1/µ E[y] = ( 1)/(1/µ) = µ and V[y] = 1/(1/µ) = µ. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 29 / 65

30 5. Generalized linear models Quasi-MLE maximizes the log-likelihood ln L(β) = i ln f (y i jµ i (β)) = i fa(µ i (β)) + b(y i ) + c(µ i (β))y i g. The rst-order conditions are i fa 0 (µ i (β)) + c 0 (µ i (β))y i g µ i (β) β = 0 ) i c 0 (µ i (β) fy i a 0 (µ i (β))/c 0 (µ i (β))g µ i (β) β = 0 ) i 1 V[y i ] fy i µ i (β)g µ i (β) β = 0 MLE based on LEF with µ i = g(xi 0 β) shares the robustness properties of normal and Poisson MLE consistency requires correct speci cation of the mean (so E[fy i µ i (β)g] = 0). But correct standard errors should use a robust estimate of variance Robust sandwich s.e. s or Default ML s.e. s multiplied by p bα where V[y i jx i ] = α h(e[y i jx i ]). A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 30 / 65

31 5. Generalized linear models ASDE: Nonlinear least squares estimator Alternative estimator for counts that is not used, because Poisson is simpler and is usually more e cient. Specify same conditional mean as Poisson: E[y i jx i ] = exp(x 0 i β). Minimize sum of squared residuals: N i=1(y i exp(x 0 i β))2. NLS is consistent provided E[y i jx i ] = exp(xi 0β) a N [β, VMLE [ β]] b and use robust sandwich variance estimate: bβ NLS bv ROB [ β b 1 NLS ] = i bµ 2 i x i xi 0 i (y i bµ i ) 2 bµ 2 i x i x 0 i i bµ 2 i x i x 0 i 1. For doctor visits data nl (docvis = exp({xb: $xlist}+{b0})), vce(robust) NLS robust standard errors are 5-20% larger than those for Poisson A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 31 / 65

32 6. Diagnostics 6. Diagnostics: residuals and in uence measures Some diagnostics come out of the GLM literature. Residuals (for Poisson) Raw: r i = (y i bµ i ) Pearson: p i = (y i bµ i )/ p bµ i Deviance: d i = sign(y i bµ i ) p 2fy i ln(y i /bµ i ) (y i bµ i )g Anscombe: a i = 1.5(yi 2/3 µ 2/3 i )/µ 1/6 i Last three will be standardized if V[y i ] = µ i. Small-sample corrections (for Poisson) Hat matrix: H = W 1/2 X(X 0 WX) 1 X 0 W 1/2 ; W = Diag[bµ i ]. Studentized residual: p i = p i / p 1 h ii and d i = d i / p 1 h ii. n uential observations: Rule of thumb: h ii > 2K /N Cook s distance: C i = (pi )2 h i /K (1 h ii ) measures change in β b when observation i is omitted. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 32 / 65

33 6. Diagnostics. summarize rraw rpearson rdeviance ranscombe hat cooksd, sep(10) Variable Obs Mean Std. Dev. Min Max rraw e rpearson rdeviance ranscombe hat cooksd e correlate rraw rpearson rdeviance ranscombe (obs=3677) rraw rpearson rdevia~e ransco~e rraw rpearson rdeviance ranscombe The various residuals are highly correlated. The raw residuals sum to zero due to f.o.c.. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 33 / 65

34 6. Diagnostics Diagnostics: R-squared measures Di erent interpretations of R 2 in linear model lead to di erent R 2 in nonlinear model. Most are di cult to interpret in nonlinear models. Simplest: squared correlation coe cient between y i and by i = bµ i R 2 Cor = ccor 2 [y i, by i ] Sums of squares measures di er in nonlinear models R 2 Res = 1 ResSS/TotalSS 6= R 2 Exp = ExpSS/TotalSS Relative gain in log-likelihood (L 0 is intercept model only) R 2 RG = ln L t ln L 0 ln L max ln L 0 = 1 ln L max ln L t. ln L max ln L 0 Works for Poisson as ln L max occurs when µ i = y i. Unlike others 0 RRG 2 < 1 and R2 RG always increases as add regressors. Stata measure is only applicable to binary and multinomial models R 2 Pseudo = 1 ln L t / ln L 0. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 34 / 65

35 6. Diagnostics ASDE: Diagnostics: overdispersion test H 0 :V[y i jx i ] = E[y i jx i ] versus H 1 :V[y i jx i ] = E[y i jx i ]+ α(e[y i jx i ]) 2. Test H 0 : α = 0 against H 1 : α > 0. mplement by auxiliary regression ((y i bµ i ) 2 y i )/bµ i = αbµ i + error and do t test of whether the coe cient of bµ i is zero. n practice can skip this test and just do Poisson with robust s.e. s. Test is useful as can also use this test for test of underdispersion, whereas other tests (such as LM of Poisson vs. negative binomial) only test overdispersion. f we are just modelling the conditional mean, overdispersion is okay provided robust standard errors are calculated. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 35 / 65

36 6. Diagnostics ASDE: Example of overdispersion test.. * Overdispersion test against V[y x] = E[y x] + a*(e[y x]^2). quietly poisson docvis $xlist, vce(robust). predict muhat, n. quietly generate ystar = ((docvis muhat)^2 docvis)/muhat. regress ystar muhat, noconstant noheader ystar Coef. Std. Err. t P> t [95% Conf. nterval] muhat Very strongly reject H 0. Data here are overdispersed.. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 36 / 65

37 6. Diagnostics Diagnostics: predicted probabilities Now suppose we want to predict probability of 0 doctor visits, 1 doctor visits,... Observed frequency p j (fraction of observations with y i = j). Fitted frequency bp j = N 1 N i=1 bp ij predicted probability bp ij = Pr[y i = j] = e bµ i bµ j i /j! for Poisson. Expect bp j close to p j, j = 0, 1, 2,... nformal statistic is Pearson s chi-square test j (np j nbp j ) 2 nbp j but this is not χ 2 distributed due to estimation to get bp j. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 37 / 65

38 6. Diagnostics nstead do a formal chi-square goodness of t test. Assuming that the density is correctly speci ed (so more applicable to models more general than Poisson) this can be computed as NR 2 u (uncentered R 2 ) from the arti cial regression where 1 = s i (y i, x i, bθ) 0 γ + j (d ij (y i ) bp ij ) 0 δ j + error j denotes cells (e.g. values 0, 1, 2, 3, and 4 or more) d ij (y i ) equals 1 if y i is in cell j and 0 otherwise bp ij equals predicted probability for that cell s i (y i, x i, θ) = ln f (y i jx i, θ)/ θ (= (y i exp(xi 0 β)) for Poisson). Reject at level α if NR 2 u > χ 2 α (J 1) where J is number of cells. Use Stata add-on chi2gof: e.g. chi2gof, cells(11) table A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 38 / 65

39 6. Diagnostics Stata add-on chi2gof, cells(11) table yields χ 2 α (10) statistic equals Clearly problem as Poisson greatly underpredicts low counts e.g. for y = 0.. chi2gof, cells(11) table Chi square Goodness of Fit Test for Poisson Model: Chi square chi2(10) = Prob>chi2 = 0.00 Fitted Cells Abs. Freq. Rel. Freq. Rel. Freq. Abs. Dif or more (supersedes addon countfit: compares actual and tted frequencies but no test). A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 39 / 65

40 7. Negative binomial regression Motivation 7. Negative binomial regression: motivation Count data are often overdispersed with more zeros and more high values than a Poisson distribution predicts. Doctor visits: Frequencies with and grouped. tabulate dvrange dvrange Freq. Percent Cum Total 3, A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 40 / 65

41 7. Negative binomial regression Motivation Poisson (white) with λ = ȳ compared to actual data (grey) Density # Doctor visits versus Poisson Poisson clearly inappropriate: ȳ = 6.82, s y = 7.39, s 2 y = ' 8.01ȳ. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 41 / 65

42 7. Negative binomial regression Negative binomial distribution Negative binomial distribution Negative binomial is a Poisson-gamma mixture. then y Poisson[λυ] υ Gamma[µ = 1, σ 2 = α] y Negative Binomial[µ = λ, σ 2 = λ + αλ 2 ]. Probability mass function: Pr[Y = yjλ, α] = Γ(α 1 α + y) α 1 Γ(α 1 )Γ(y + 1) α 1 + λ Mean and variance: Overdispersion: variance > mean. E[y] = λ V[y] = λ + αλ 2 1 λ λ + α 1 y. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 42 / 65

43 7. Negative binomial regression Negative binomial distribution Negative binomial for λ = ȳ and α = compared to actual data Density # Doctor visits versus negative binomial Negative binomial much more appropriate than Poisson for these data. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 43 / 65

44 7. Negative binomial regression Negative binomial distribution Negative binomial regression Negative binomial (Negbin 2) permits overdispersion. f (yjλ, α) = Γ(y + α 1 ) α 1 α 1 λ y Γ(y + 1)Γ(α 1 ) α 1 + λ α 1. + λ Same conditional mean but di erent conditional variance to Poisson E[yjx] = λ = exp(x 0 β) V[yjx] = λ + αλ 2 = exp(x 0 β) + α(exp(x 0 β)) 2. The ML rst-order conditions w.r.t. β and α are (with µ i = exp(x 0 i β)) N i=1 N i=1 1 α 2 ln(1 + αµ i ) y i 1 j=0 y i exp(xi 0β) 1 + α exp(xi 0β)x i = 0 1 (j + α 1 ) + y i µ i α (1 + αµ i ) = 0. Can additionally allow α = exp(x 0 γ) (generalized negative binomial). Can instead use Negbin 1: V[yjx] = (1 + α)λ = (1 + α) exp(x 0 β). Often little e ciency gain (if any) over Poisson with robust s.e s. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 44 / 65

45 7. Negative binomial regression Negative binomial distribution Negative binomial MLE with ML default standard errors. nbreg docvis $xlist, nolog Negative binomial regression Number of obs = 3677 LR chi2(7) = Dispersion = mean Prob > chi2 = Log likelihood = Pseudo R2 = docvis Coef. Std. Err. z P> z [95% Conf. nterval] private medicaid age age educyr actlim totchr _cons /lnalpha alpha Likelihood ratio test of alpha=0: chibar2(01) = Prob>=chibar2 = Likelihood ratio test of α = 0 prefers NB to Poisson (p < 0.05) - where critical values use half χ 2 (1) as α = 0 is on boundary of NB. Fitted frequencies close to observed frequencies (from output not given) chi2gof, cells(11) table yields χ 2 α (10) statistic equals A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 45 / 65

46 7. Negative binomial regression Negative binomial distribution Poisson and negative binomial MLE with di erent standard error estimates docvis private *** *** *** *** *** (0.0143) (0.0364) (0.0360) (0.0332) (0.0369) medicaid *** * * (0.0189) (0.0568) (0.0475) (0.0454) (0.0567) age *** *** *** *** *** (0.0260) (0.0630) (0.0652) (0.0602) (0.0646) age *** *** *** *** *** (0.0002) (0.0004) (0.0004) (0.0004) (0.0004) educyr *** *** *** *** *** (0.0019) (0.0048) (0.0047) (0.0042) (0.0049) actlim *** *** *** *** *** (0.0146) (0.0397) (0.0366) (0.0348) (0.0394) totchr *** *** *** *** *** (0.0046) (0.0126) (0.0117) (0.0121) (0.0132) _cons *** *** *** *** *** (0.9720) (2.3692) (2.4415) (2.2474) (2.4241) lnalpha _cons *** *** (0.0307) (0.0378) N pseudo R sq Standard errors in parentheses * p<0.05, ** p<0.01, *** p<0.001 (1) (2) (3) (4) (5) PDEFAULT PROBUST PPEARSON NBDEFAULT NBROBUST A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 46 / 65

47 8. Richer parametric models 8. Richer parametric models Data frequently exhibit non-poisson features: Overdispersion: conditional variance exceeds conditional mean whereas Poisson imposes equality. Excess zeros: higher frequency of zeros than predicted by Poisson. This provides motivation for richer parametric models than basic Poisson. Some models still have E[yjx] = exp(x 0 β) Then richer model can provide more e cient estimates. Other models imply E[yjx] 6= exp(x 0 β) Then Poisson QMLE is inconsistent And marginal e ects and coe cient interpretation more di cult. Many of these models are fully parametric and require correct speci cation for consistency. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 47 / 65

48 8. Richer parametric models Truncated data Counts left-truncated at zero Sampling rule is such that observe only y and x for y 1 i.e. only those who participate at least once are in sample. Truncated density (given untruncated density f (y jx, θ)) is f (yjx, θ, y 1) = f (yjx, θ) f (yjx, θ) = Pr[y 0jx, θ] [1 f (0jx, θ)]. MLE is inconsistent if any aspect of the parametric model is misspeci ed. Need to assume that the process for nonzeroes is the same as zeroes. e.g. f data are on annual number of hunting trips for only those who hunted this year, then a missing 0 is interpreted as being for a hunter who did not hunt this year (rather than for all people). Stata commands ztp (poisson) and ztnb (negative binomial) ztnb docvis $xlist if docvis>0, nolog A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 48 / 65

49 8. Richer parametric models Censored data Counts right-censored Sampling rule is that observe only 0, 1, 2,..., c 1, c or more i.e. Only record counts up to c and then any value above c. Censored density (given uncensored density f (yjx, θ) and cdf is F (yjx, θ)) f (yjx, θ) y c 1 1 F (c 1jx, θ) = 1 c j=0 1 f (jjx, θ) y = c Log-likelihood (where d i = 1 if uncensored and d i = 0 if censored) L(θ) = N i=1 fd i ln f (y i jx i, θ) + (1 d i ) ln(1 c 1 j=0 f (jjx i, θ))g MLE is inconsistent if any aspect of the parametric model is misspeci ed So pick a good density - at least negative binomial. Stata code up yourself using command ml (for user-de ned likelihood). A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 49 / 65

50 8. Richer parametric models Counts recorded in intervals Counts recorded in intervals Sampling rule is that observe only counts in ranges. e.g. 0, 1-4, 5-9, 10 and above. nterval density is simply Pr[a y b] = b f (jjx, θ). j=a Let interval ranges by [a 0, a 1 1], [a 1, a 2 1],..., [a m, a m+1 ), where a 0 = 0, a m+1 =. Let d k be binary indicators for whether in interval k (k = 0,..., m). Then ln L(θ) = N i=1 h i m k=0 d a ij ln k+1 1 f (jjx, θ). k=a k MLE is inconsistent if any aspect of the parametric model is misspeci ed. Stata has no command so need to code up. For convenience could instead use ordered logit or probit here. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 50 / 65

51 8. Richer parametric models Hurdle model or two-part model Hurdle model or two-part model Suppose zero counts are determined by a di erent process to positive counts. Zeros: density f 1 (yjx 1, θ 1 ) so Pr[y = 0] = f 1 (0) and Pr[y > 0] = 1 f 1 (0). Positives: density f 2 (yjx 2, θ 2 ) so truncated density f 2 (y)/(1 f 2 (0)). e.g. First - do hunt this year or not? Second - given chose to hunt, how many times ( 1)? Combined density is 8 < f (yjx 1, x 1, θ 1, θ 2 ) = : f 1 (yjx 1, θ 1 ) y = 0 1 f 1 (0jx 1, θ 1 ) 1 f 2 (0jx 2, θ 2 ) f 2(yjx 2, θ 2 ) y 1 MLE is inconsistent if any aspect of model misspeci ed. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 51 / 65

52 8. Richer parametric models Hurdle model or two-part model Conditional mean is now E[yjx] = Pr[y 1 > 0jx 1 ] E y2 >0[y 2 jy 2 > 0, x 2 ]. This makes marginal e ects more complicated. Example: f 1 () is logit and f 2 () is negative binomial. Then E[yjx] = Λ(x1β) 0 exp(x2β)/[1 0 (1 + α 2 exp(x2β)) 0 1/α 2 ], where Λ(z) = e z /(1 + e z ). A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 52 / 65

53 8. Richer parametric models Hurdle model or two-part model Hurdle model - logit and negative binomial: Stata addon hnblogit. hnblogit docvis $xlist, nolog Negative Binomial Logit Hurdle Regression Number of obs = 3677 Wald chi2(7) = Log likelihood = Prob > chi2 = logit private medicaid age age educyr actlim totchr _cons negbinomial private medicaid age age educyr actlim totchr _cons /lnalpha AC Statistic = Coef. Std. Err. z P> z [95% Conf. nterval] A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 53 / 65

54 8. Richer parametric models Zero-in ated model Zero-in ated model (or with-zeroes model) Suppose there is an additional reason for zero counts Extra model for 0: density f 1 (yjx 1, θ 1 ) Usual model for 0: realization of 0 from density f 2 (yjx 2, θ 2 ). e.g. Some zeroes are mismeasurement and some are true zeros. Zero-in ated model has density = f (yjx 1, x 1, θ 1, θ 2 ) f1 (0jx 1, θ 1 ) + [1 f 1 (0jx 1, θ 1 )] f 2 (0jx 2, θ 2 ) y = 0 [1 f 1 (0jx 1, θ 1 )] f 2 (yjx 2, θ 2 ) y 1 MLE is inconsistent if any aspect of model misspeci ed. Not used much in econometrics - hurdle model more popular. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 54 / 65

55 8. Richer parametric models Zero-in ated model Zero-in ated negative binomial: Stata command zinb and zip. zinb docvis $xlist, inflate($xlist) vuong nolog Zero inflated negative binomial regression Number of obs = 3677 Nonzero obs = 3276 Zero obs = 401 nflation model = logit LR chi2(7) = Log likelihood = Prob > chi2 = Coef. Std. Err. z P> z [95% Conf. nterval] docvis private medicaid age age educyr actlim totchr _cons inflate private medicaid age age educyr actlim totchr _cons /lnalpha alpha Vuong test of zinb vs. standard negative binomial: z = 6.48 Pr>z = A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 55 / 65

56 8. Richer parametric models Continuous mixture models Continuous mixture models Mixture motivation for negative binomial assumes y jθ Poisson (θ) where θ = λυ is the product of two components: observed individual heterogeneity λ = exp(x 0 β) unobserved individual heterogeneity υ Gamma[1, α]. ntegrating out Z Z h(yjλ) = f (yjλ, υ)g(υ)dυ = [e λυ (λυ) y /y!] g(υ)dυ gives yjλ NB [λ, λ + αλ 2 ] if υ Gamma[1, α]. Di erent distributions of υ lead to di erent models e.g. Poisson-lognormal mixture (random e ects model) e.g. Poisson-nverse Gaussian. Even if no closed form solution can estimate using numerical integration (one-dimensional) e.g. Gaussian quadrature. Monte Carlo integration e.g. maximum simulated likelihood. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 56 / 65

57 8. Richer parametric models Hierarchical models Hierarchical models For multi-level surveys cross-section data individuals i may be in cluster j e.g. patient i in hospital j e.g. individual i in household j or village j Hierarchical model or generalized linear mixed model example y i Poisson[µ ij = exp(x 0 ij β j + ε ij )] β j = W j γ + v j ε ij N [0, σ 2 ε ] v j N [0, Diag[σ 2 jk ]] Estimate by MLE or by Bayesian methods Stata command mepoisson A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 57 / 65

58 8. Richer parametric models Model comparison Model comparison for fully parametric models Choice between nested models using likelihood ratio tests e.g. Poisson versus negative binomial. Choice between non-nested models using Vuong s (1989) likelihood ratio test e.g. Zero-in ated NB versus NB Choice between non-nested mixture models using penalized log-likelihood Akaike s information criterion (AC) and extensions (q = # parameters) AC = 2 ln L + 2q BC = 2 ln L + qk ln N CAC = 2 ln L + q(1 + ln)n Prefer model with small AC or BC. AC penalty for larger model too small. Bayesian C (BC) better. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 58 / 65

59 8. Richer parametric models Model comparison Compare predicted means: E[y jx, bθ]. Compare observed frequencies p j to average predicted frequencies where bp ij = bpr[y i = j]. bp j = N 1 N bp ij, i=1 A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 59 / 65

60 8. Richer parametric models Model comparison Compare AC, BC for regular NB, hurdle logit/nb and zero-in ated NB. Statistics NBREG HURDLENB ZNB N ll aic bic Hurdle NB and ZNB are big improvement on regular NB - lnl is approximately 100 higher than for NB - AC and BC is much smaller (with only 9 extra parameters) Little di erence between Hurdle NB and ZNB. A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 60 / 65

61 8. Richer parametric models Model comparison The conditional means from the three models are similar.. summarize docvis dvnbreg dvhurdle dvzinb Variable Obs Mean Std. Dev. Min Max docvis dvnbreg dvhurdle dvzinb correlate docvis dvnbreg dvhurdle dvzinb (obs=3677) docvis dvnbreg dvhurdle dvzinb docvis dvnbreg dvhurdle dvzinb A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 61 / 65

62 8. Richer parametric models Quantile regression Quantile regression The q th quantile regression estimator b β q minimizes over β q N N Q(β q ) = qjy i xi 0 β q j + (1 q)jy i xi 0 β q j, 0 < q < 1. i :y i xi 0 β i :y i <xi 0 β Example: median regression with q = 0.5. For count y adapt standard methods for continuous y by: Replace count y by continuous variable z = y + u where u Uniform[0, 1]. Then reconvert predicted z-quantile to y-quantile using ceiling function. Machado and Santos Silva (2005). Stata example qcount docvis $xlist, q(0.5) rep(50) qcount_mfx // MEM A. Colin Cameron Univ. of Calif. - Davis... for Center Count of Labor Regression: Economics PartNorwegian 1 School ofaug Economics 28 - Sept Advanced 1, 2017 Microeconomet 62 / 65