Agricultural and Applied Economics 637 Applied Econometrics II. Assignment III Maximum Likelihood Estimation (Due: March 31, 2016)

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1 Agricultural and Applied Economics 637 Applied Econometrics II Assignment III Maximum Likelihood Estimation (Due: March 31, 2016) In this assignment I would like you to apply the theoretical Maximum Likelihood material to a number of empirical applications. Make sure you hand in your MATLAB code and output files. Total Points: 140 pts. 1. (65 pts) Let y1, y2,,yt be a random sample from a population with the following PDF: f y t yt t λ λ e where yt can take any non-negative integer value (e.g., 0, 1, 2, y! 3, ). A variable with this PDF is referred to as having a Poisson distribution. One can show that under this distribution, the mean and variance of this random variable is equal to the λ parameter. The file count_intro.ppt contains a brief overview of what is referred to as a Count Data model which is based on the Poisson distribution. For more detail refer to Greene p or Cameron and Trivedi, This modeling framework is often used when the dependent variable is the count of the number of times a particular event or activity occurs. Given the above distribution function, the sample log-likelihood function can be represented as: T t 1 t t t t L λ y -λ + y ln(λ ) ln y! (1.1) Let s extend the above model to where we allow the mean and variance of the Poisson random variable (λ) to be a function of a set of exogenous variables. That is, like the CRM we would like the mean value of the dependent variable to be conditional on a set of exogenous variables and a set of unknown parameters. The standard Poisson regression model assumption is to use the exponential mean parameterization: λt = exp(ztγ), where t=1,,t and Z is a matrix of exogenous variables and γ the vector of unknown coefficients to be estimated. Using the above likelihood function, one can 1

2 represent the sample log-likelihood function for the count model to be: T t1 t t t t L λ y Z γ ln y! (1.2) where λt=exp(ztγ). The Poisson Maximum Likelihood estimator, γ* is the solution to the K nonlinear equations corresponding to the 1 st order condition for maximum likelihood: y T t 1 t t t Kx1 exp Z γ Z 0. If Zt includes a constant term then the residuals yt exp(ztγ) sum to 0. Also, the likelihood function is globally concave, hence solving these equations via iterative algorithms yields unique parameter estimates. Using this modeling framework, I would like you to examine what determines the number of doctor visits undertaken by a sample of adults. The data file, randdata.xls contains data collected by the RAND Corporation from 1972 to The main goal of their study was to assess how a patient s use of health services is affected by types of randomly assigned health insurance, including both fee-for-service and health maintenance organizations (HMO s). In the experiment, data were collected from about 8,000 enrollees from 2,823 families and 6 sites across the U.S. Each family was enrolled in 1 of 14 different health insurance plans for either 3 or 5 years. The plans ranged from free care to 95% coinsurance below a maximum dollar expenditure and also included assignment in a prepaid group practice. 1 Data were collected from the enrollee s use of medical care services and health status throughout the randomly assigned term of enrollment for either 3 or 5 years. There are a total of 20,163 observations in the final dataset. Table 1 is used to show the frequency distribution of the number of doctor visits for those who had doctor visits. Approximately 31% of the sample had zero doctor visits during the study period. 1 NOTE: The fact that because the insurance plans are randomly assigned, not freely chosen by the participants, one does not face the problem of endogenous treatment effect. 2

3 Table 1: Distribution of the Number of Doctor Visits for Those With Visits Contacts % Contacts >15 Max % In Table 2 is provided a listing of variables included in the Rand Corporation s dataset. Table 2. Subset of Variables in the RAND Corporation Physician Visit Dataset Variable Definition mdvis Number of face-to-face doctor visits logc Ln(coinsurance + 1), 0 coinsurance 100 idp 1 if Individual Deductible Plan, 0 Otherwise fmde 0 if LDP=1, ln(max(1,mde/(.01*coinsurance))) otherwise linc Ln(Annual Income) lnum Ln(Family Size) xage Age That Year female 1 if Person is Female, 0 if male child 1 if XAGE < 18 fchild FEMALE*CHILD black 1 if Race of Household Head is Black educdec Education of Household Head in Years physlm 1 if the Person has a Physical Limitation disea Number of Chronic Diseases hlthg 1 if Self-Rated Health is Good hlthf 1 if Self-Rated Health is Fair hlthp 1 if Self-Rated Health is Poor Note: These variables are not stored in the dataset in this order. (a) (15 pts) Estimate the Poisson regression model using the likelihood function that allows for the count variable mean and variance to depend on a set of exogenous variables. Use mdvis as the dependent variable in this Poisson regression model 3

4 and the remaining variables shown in Table 1 as exogenous variables and a vector of 1 s to allow for an intercept in the mean generating function. (Hint: Make sure you include the intercept. Otherwise the model has a difficult time obtaining a solution.) To estimate these parameters, use the BHHH algorithm to estimate the ML model using numerical gradients of the log-likelihood function. Present the estimated coefficients, associated standard errors and total sample log-likelihood function value. Evaluate one of the measures of the degree of explanatory power of the regression model outlined in Greene p (NOTE: When implementing the above log-likelihood function you will need to evaluate the natural logarithm of the factorial number of doctor visits. Without having to actual evaluate the factorial of a large number and then taking the logarithm which could cause problems due to numerical accuracy use the return from the gammln(n+1) function. 2 ) (b) (5 pts) Undertake a single likelihood ratio test to examine the null hypothesis that self-rated health status has no effect on the expected number of doctor visits. (c) (10 pts) Using the results from the Poisson regression model, compare the estimated marginal effect of a change in LOGC for an individual in excellent health and an individual in poor health. Statistically test whether these two marginal impacts are equal. (Note: Except for the variables of concern, the remaining exogenous variables should be set at the overall sample mean values) (d) (10 pts) Using the results from the Poisson regression model, evaluate the elasticity of a change in family income, not the logarithm of income, on the expected number of doctor visits. Estimate this elasticity using two methods: (i) at the mean values of the mdvis and exogenous variables and (ii) the average value of the elasticity values calculated over all observations. What are the results obtained under both methods? Statistically test whether this elasticity different from 1.0 under both methods? 2 gammaln is the logarithm of the gamma function, which is the continuous version of the factorial 4

5 (e) (10 pts) For dummy variables, the use of marginal analysis is inappropriate as one evaluates the partial derivative of a variable that can only undertake discrete changes (i.e., 0 or 1). Develop a relative measure of the impact on the expected number of doctor visits of the respondent being female. What is the variance of this relative measure? (Hint: The term relative is important!! All other exogenous data should be set at the overall sample mean.) (f) (10 pts) Consider the Poisson regression model with conditional mean, λ=exp(zγ). Treat the estimation problem as an unweighted least squares problem in which y=e[y Z]+ε where E[y Z]=exp(Zγ) and ε~iid(0, σ 2 ). That is, I am not assuming the error terms are normally distributed. Estimate the above model via nonlinear least squares via the use of the numerical GN algorithm. Compare the ML versus least squares results. Using a single statistical test, test the null hypothesis that the results obtained under the two estimations are the statistically the same. (Hint: Using the MLE as a base, what would be the restricted model?) (g) (5 pts) At the mean of the data what is the expected number of doctor visits? Is this number of visits statistically different than the sample mean of office visits. 2. (25 pts) There are a variety of ways to control for the effect of heteroscedasticity in the estimation of a linear regression model. We reviewed in class the multiplicative heteroscedasticity approach. Under this approach we have yt = Xtβ + εt where E(εt 2 )= σt 2 = exp(dtα), Dt is a (1 x S) vector containing the t th observation on S nonstochastic explanatory variables and α is a (S x 1) parameter vector. Thus, the error variance changes across observation and the error covariances across observations are 0. The observation specific error variance depends on a set of exogenous variables. Assume you work for the Energy Information Agency of the U.S. Department of Energy. You are interested in examining trends in annual U.S. gasoline consumption. You collect data for 52 years (i.e., ) for your econometric analysis. For this assignment I would like you to use the BHHH maximum likelihood estimation algorithm to estimate the parameters of the multiplicative heteroscedastic model as discussed in Greene, p and by JHGLL, p You specify the dynamic model of U.S. per capita demand for gasoline via the following: 5

6 ln(q_gast/popt)=β0 + β1ln(gaspt) + β2ln(pc_inct) + β3ln(pnct) + β4ln(puct) (2.1) + β5shock73+ β6shock79 + β7recess + γln(q_gast-1/popt-1) + εt where εt~n(0,σt 2 ), σt 2 = exp(dtα) and Dt is composed of the variables: (i) an intercept, (ii) PC_Inct, (iii) Shock73, (iv) Shock79 and (v) Recess. Table 3 is used to provide definitions of the variables used in (2.1). This data can be obtained by accessing the US_gas_use.xlsx file located on the class website. The order of the variables may not be in the order listed above. (Hint: To simplify, you may want to use the pull_data.m MATLAB script file to grab the appropriate data to create specific matrices). Table 3. Description of Variables Used in Equation (2.1) Variable Description Q_Gas Total Quantity of Gasoline Consumed GasP Gasoline Price Index PC_Inc Per Capita Disposable Income PNC Price Index of New Cars PUC Price Index of Used Cars Shock73 Dummy variable identifying the effects of the 1973 Oil Embargo = 1 from , 0 otherwise Shock79 Dummy variable identifying the effects of the 1979 Price Increase = 1 from 1979-Present, 0 otherwise Dummy variable identifying years in which the U.S. Recess economy was in a recession = 1 for recession years, 0 otherwise Pop Population of the U.S. (in millions) (a) (10 pts) Estimate the parameters of (2.1) using maximum likelihood techniques. Present the usual parameter and regression-based statistics. (b) (10 pts) Using the information in Greene, p.556 and our class notes, undertake Wald, Likelihood Ratio and Lagrangian Multiplier tests of the null hypothesis that per capita gasoline demand is characterized as having a homoscedastic error structure. (b) (5 pts) Given the functional form shown in (2.1), we know that the short run price and income elasticities are β1 and β2, respectively. Due to the presence of lagged gasoline 6

7 consumption as an exogenous variable, the long run price (ξp) and income (ξi ) elasticities are given by the following: β1 β 2 P I (2.2) 1γ 1γ Test the null hypothesis that ξp -1. Then test the null hypothesis that ξi = 1. Finally, test the joint hypothesis that ξp -1 and ξi = (50 pts) In the PPT file describing the maximum likelihood estimation (MLE) of the General Variance linear model there contained a discussion of the MLE of the AR(1) specification. Although we did not review in detail in class the MLE of the AR1 model, we provided the foundations of how to apply MLE to an AR(1) model. For more detail refer to Judge et al, Ch 9, p Let s assume we have the following linear regression with AR(1) errors: Yt = Xtβ + εt (t=1,,t) εt = ρεt-1 + μt ; ρ <1 E[μt X] = 0 E[μtμs] = σ 2 μ if t = s and 0 otherwise, μt ~N(0, σ 2 μ) Prais and Winsten (1954) shows how one can include the 1 st observation in a 2 2 1ρ y 1 1ρ x 1 y2 ρy1 x2 ρx1 * * transformed model: Y y X 3 ρy 2 x3 ρx. Given the above 2 yt ρyt1 xt ρxt1 structure the total sample log-likelihood function can be shown to be: * * 1 1 1ρ ε 1 T y 2 t xtβ L ln 2π ln σμ ln 1 ρ ln 2π ln σ 2 μ σ μ t σ μ One can use the 2-step feasible generalized least squares (FGLS) algorithm to obtain parameter estimates and to examine the presence of autocorrelation in such a model. The 2-step estimation algorithm is described on page in Judge, et.al., Chapter 9. In the following I would like you to compare the FGLS estimates with those obtained from the use of the above total sample log-likelihood function. 7

8 For this question I would like you to use the same dataset as you used in question #2 above. In this application you are not concerned with the presence of heteroscedasticity but with the presence of autocorrelated errors. You decide that you would like to estimate the following econometric model assuming AR(1) errors: ln(q_gast/popt)=β0 + β1ln(gaspt) + β2ln(pc_inct) + β3ln(pnct) + β4ln(puct) + β5shock73+ β6shock79 + β7recess + εt (3.1) (a) (10 pts) Estimate the above model using the CRM with the traditional CRM parameter standard errors using the correct but inefficient standard errors. In general when we have error terms that are characterized as being autocorrelated (or heteroscedastic) the correct formulas to use to estimate the CRM parameter covariance matrix is not σ 2 (X X) -1 but rather the following: Σβ = (X X) -1 X ΦX(X X) -1 = σ 2 (X X) -1 X ΨX(X X) -1 where Φ is the error variance matrix, Φ=σ 2 Ψ, σ is a scalar, Ψ is a (T x T) symmetric matrix and T the number of observations. In addition to the typical regression summary statistics, coefficient estimates, the above 2 sets of coefficient standard errors, equation F-statistic, etc., modify your CRM estimation code so that it automatically calculates and displays (i) the Durbin- Watson (DW) statistic, (ii) the ρ value assuming AR(1) and (iii) the asymptotic standard normal test statistic for testing for AR(1) based on your estimate of ρ. 3 What is the DW statistic for this model? Does this statistic indicate the presence of an AR(1) error structure? [Hint: Remember the relationship between consecutive errors under the AR(1) error structure.] (b) (10 pts) Given the above CRM estimates, I would like you to estimate the 2-Step FGLS estimates of the coefficients and coefficient standard errors. Remember, 3 Under suitable conditions, it can be shown that ˆ will be approximately normally distributed with mean ρ and variance (1-ρ 2 )/T. If the null hypothesis that ρ=0 is true the variance becomes 1/T. You can therefore use a Z-statistic to test the above hypothesis. 8

9 that in general, the FGLS estimates are obtained via the following: ˆ β ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 β X Ω X X Ω Y= X X X Y, X Ω X where Ω=σ ˆ Show the parameter estimates, estimated standard errors, p-values etc. Remember the error covariance matrix under the AR(1) assumption, Ω, has the following structure: μ 2 2 E ee σ,ρ σμ 2 T-1 1 ρ ρ ρ T-2 ρ 1 ρ ρ 1 2 T-3 2 ρ ρ 1 ρ 1-ρ T-1 T-2 T-3 ρ ρ ρ (c) (15 pts) Finally, I would like you to use a maximum likelihood procedure to obtain your parameter estimates. Remember that you need to insure that -1 < ρ < 1. Use the tanh function to guarantee this. Display your estimation results. Make sure you evaluate the variance of ρ where ρ = tanh(γ) where γ is a coefficient that appears in the list of estimated parameters. That is, ρ is not a parameter directly estimated. (d) (5 pts) Using the results in (a) (c) create a single table showing the coefficients and correctly calculated parameter standard errors and Z-values for the CRM, FGLS, ML parameter estimates of the AR(1) model. Are there any differences in 9

10 the standard normal Z-statistics across the four methods used to obtain parameter estimates? (e) (10 pts) Using the above 3 sets of results test the null hypotheses that the income elasticity equals the new car price elasticity and equals the used car elasticity. Are there any differences in the results? 10