Econometric Models of Expenditure

Transcription

1 Econometric Models of Expenditure Benjamin M. Craig University of Arizona ISPOR Educational Teleconference October 28,

2 Outline Overview of Expenditure Estimator Selection Two problems Two mistakes Example with Stata code Summary 2

3 Econometrics of Expenditure Econometrics includes: The measurement of economics outcomes; e.g., primary data collection The estimation of empirical relationships; and e.g. statistics The analysis, interpretation, and communication of results e.g., Theory In this talk, I will address the measurement, estimation, and analysis of medical expenditures. 3

4 Medical Expenditure Medical expenditure is a key economic outcome in pharmacoeconomics as it may represent: the economic burden of poor health e.g., a cost-of of-illness study e.g., how much does end-stage renal disease cost? the direct cost of an intervention e.g., a technology assessment e.g., how much does gastric bypass (i.e., stomach stapling) surgery cost? 4

5 Prices and Quantity Expenditure = price k quantity k k K Expenditure is composed of two interrelated elements Prices Quantity If price is marginal cost, expenditure is the same as cost. Obviously, price is related to quantity (i.e., law of demand); however we will ignore this nontrivial relationship for now. 5

6 Quantity of Medical Care Quantity represents the consumption of resources (a.k.a. opportunity cost), and bundled by time or by episode. Interval Expenditure Amount spent over a period of time e.g., medical services consumed over five years. Episode Expenditure Amount spent for an episode of care, regardless of time e.g., medication consumed while hospitalized 6

7 Price of Medical Care Price is monetary value of the medical care While consumption of goods and services is easily measured, value depends on perspective. A physician visit may be worth $10 from the perspective of a patient (out-of of-pocket) $100 from the perspective of the insurer (reimbursement) $80 from the perspective of a physician (cost) $110 from the perspective of the open market (market price) Therefore, the measurement of expenditure is highly dependent on perspective. 7

8 Suppose You have some expenditure data, and You want to estimate the relationship between expenditure and some set of variables, and predict expenditures What estimator should you use? 8

9 Distribution of Expenditure P e r c e n t expendi t ur e Zero-mass Too many zeros to fit nicely under a single distribution Skewed Asymmetric with mostly low values and a thick tail on the right. 9

10 Problem #1: Zero-mass First, what to do about the zero-mass? Are people with zero expenditure distinctly different from those with positive expenditure? The mechanism that determines zero or nonzero expenditures may not be the same as the mechanism that determines the amount of positive expenditure? What does theory tell you? 10

11 Two-Part Model E(Y X) ) = P(Y>0 >0 X)* )*E(Y X,Y>0) (First Part)*( )*(Second Part) For example, allergy drug expenditure may be dependent on the presence of allergies in the first part and the severity of allergies in the second part. The first part is a binary model and the second part is a continuous model on only positive expenditures. 11

12 Two-Part Model E(Y X) ) = P(Y>0 >0 X)* )*E(Y X,Y>0) (First Part)*( )*(Second Part) This model has two error terms, one for each part. Under this specification, we assume that these error terms are independent. Under independence, the two parts can be estimated separately e.g., First part = Logit, and second part = OLS 12

13 Problems with the Two-Part Model Interdependence between first and second parts Cake Wars : Joseph Newhouse and Joel Hay Do unobservable characteristic in the binary model change the profile of those with positive expenditures? Average treatment effect must incorporate the average effect of X on Part 1 and on Part 2. This is doable, but complicated. 13

14 Skewness P e r c e n t expendi t ur e Whether or not the model is one or two parts, the expenditure distribution is still skewed. First I will describe two potentially biased estimators, followed by a viable alternative. 14

15 Mistake #1: Ordinary Least Squares OLS linear regression models may yield Badly biased estimates as they are very sensitive to outliers in the right tail Less precise estimates of means and marginal effects Erroneous predictions such as negative values OLS may be justifiable is if There are few values near zero Dataset is large 15

16 Mistake #2: log(expenditure) Running OLS on log(expenditure) ) will force the distribution to be less skewed; however the estimates will be biased, regardless of how well they fit the data. Other reasons for not logging include: You can not log zeros and by replacing the zeros with positive numbers, you will further bias your results. You can not predict zero (the modal response). Predicted values will be downwardly biased (Proof available upon request), and proposed correction for this bias (i.e., Duan s estimate) makes unreasonable assumptions. Logging is not necessary; better solutions are available. 16

17 Expected Conditional Models Instead, estimate an expected conditional model E[ y x] = exp( β + β x β hx 1 h ) This model specifies the expected conditional value, not the dependent variable. This particular model uses a log link. Prediction of expected outcomes does not require re- transformation (i.e., Duan s smearing estimate). These generalized linear models (GLM) allow for heteroskedasticity through choice of family. 17

18 [ y x ] exp( x β ) E = Link Selection There are other link functions that address skewness (e.g., square root and Box-Cox) E[ y x] = β 0 + β 1 x β h x h E[ y x] = [( ) ) ] λ β + β x β x 1 / λ h h where λ 0 Notice that the log link model has the same expected value form as common discrete models (i.e., Poisson, negative binomial, and complementary log-log). log). 18

19 Variables, Link and Family The GLM procedure in STATA facilitate the estimation expected conditional models (ECM). To estimate these models requires the specification of the variables, link function, and an error distribution family. Choose variables and link based on theory, and family using Park s test (Manning and Mullahy, Journal of Health Economics 2001). 19

20 2 λ [ y x] = σ v( x) κ exp( xβ ) V = Distribution Family Selection Using a GLM framework under heteroskedasticity, we can determine family using a Park s test By estimating lambda, the distribution may be: Gaussian, Poisson, Gamma, Log-normal, Weibell,, or Chi- squared. V If If [ ] 2 y x = σ v( x) λ = 0 λ = 1 = κ exp( xβ ) then Family = Gaussian then Family = Poisson If λ = 2 then Family = Gamma, Homoskedastic Log Normal, Weibell, or Chi squared If λ > 2 then Family =? λ 20

21 2 λ [ y x] = σ v( x) κ exp( xβ ) V = Park s Test V [ ] y x = σ 2 v( x) = κ exp( x β ) λ ln ln ( [ ]) V y x = ln( κ ) + λ ln( exp( xβ )) ) ( y yˆ ) 2 = α + λ ln( yˆ ) + v i i where yˆ i is predicted y Step 1. Estimate the model three times using the Gaussian, Poisson, and Gamma families. Step 2. Predict y (a.k.a( yhat) ) using the three models. Step 3. Estimate lambda using OLS on the transformed equation. Step 4. Choose the family that corresponds to lambda. i i 21

22 Example P e r c e n t expendi t ur e MEPS 2003 Hospital Event File (N=2898) Third-party Expenditure Notice: Zero-mass and skewed distribution 22

23 Econometrics is built on Science not Mathematics. Theory Data Results Feedback 23

24 Summary Statistics and common sense may dictate what not to do (e.g., OLS on log(expenditure)). Estimator selection and interpretation depends on economic theory. Variable selection? One-part or Two-part? If two-part, correlated errors or not? 24

25 Questions? 25