Day 3C Simulation: Maximum Simulated Likelihood

Transcription

1 Day 3C Simulation: Maximum Simulated Likelihood c A. Colin Cameron Univ. of Calif. - Davis... for Center of Labor Economics Norwegian School of Economics Advanced Microeconometrics Aug 28 - Sep 1, 2017 c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School ofaug Economics 28 - SepAdvanced 1, 2017 Microeconom 1 / 29

2 1. Introduction 1. Introduction Maximum simulated likelihood (MSL) I I for models where the density involves an integral with no closed form solution so replace the integral with a Monte Carlo integral. Leading applications I I random parameter models F random parameters multinomial logit random utility models F multinomial probit. These slides consider binary logit with a single random slope I Pr[y i = 1jx i, β 1, β 2i ] = Λ(β 1 + β 2i x i ), where β 2i jβ 2, σ 2 N[β 2, σ 2 2 ] c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School ofaug Economics 28 - SepAdvanced 1, 2017 Microeconom 2 / 29

3 1. Introduction Outline 1 Introduction 2 Binary logit model estimated using ml command 3 Random parameters binary logit MSL: Theory 4 Random parameters logit MSL by ml command 5 Random parameters logit MSL by mixlogit add-on 6 Random parameters logit MSL by Stata 15 asmixlogit 7 Random parameters logit model in general 8 MSL in General 9 References c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School ofaug Economics 28 - SepAdvanced 1, 2017 Microeconom 3 / 29

4 2. Binary logit Binary logit model using command ml 2. Binary logit model Logit example: individual choice between two cars I y = 1 if electric and y = 0 if regular I x is di erence in price, di erence in running cost per mile,... Binary logit model y i = 1 with probability Λ(x 0 i β) 0 with probability 1 Λ(x 0 i β) where Pr[y i = 1jx i, β] = Λ(x 0 i β) = ex0 i β 1 + e x0 i β. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School ofaug Economics 28 - SepAdvanced 1, 2017 Microeconom 4 / 29

5 2. Binary logit Binary logit MLE Binary logit MLE We can write the density (probability mass function) as f (y i jx i, β) = Λ(xi 0 β) y i (1 Λ(xi 0 β)) 1 y i. The MLE maximizes ln L = N i=1 ln f (y i jx i, β) = i lnfλ(xi 0 β) y i (1 Λ(xi 0 β)) 1 y i g. Some algebra yields the rst-order conditions: N i=1(y i exp(xi 0 β)) = 0. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School ofaug Economics 28 - SepAdvanced 1, 2017 Microeconom 5 / 29

6 2. Binary logit Binary logit example 2. Binary logit example Generated data example: logit with intercept plus single regressor Pr[y i = 1jx i, β 1, β 2 ] = Λ(β 1 + β 2 x i ) x i N[0, 2 2 ] Logit model can be generated as I y i = 1 if yi > 0 where yi = xi 0 β + u i where u i logistic I inverse transformation: logistic cdf F (w) = e w (1 + e w ) so setting u = e w (1 + e w ) gives w = ln u ln(1 u) Generate data as follows set obs 1000 set seed gen u = runiform() gen ulogistic = ln(u) - ln(1-u) // draw logistic gen x = rnormal(0,2) gen y = 1 + 1*x + ulogistic > 0 c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School ofaug Economics 28 - SepAdvanced 1, 2017 Microeconom 6 / 29

7 2. Binary logit Command logit Command logit Resulting estimates are close to β 1 = 1 and β 2 = 1.. summarize Variable Obs Mean Std. Dev. Min Max u ulogistic x y * Logit ml using logit command. logit y x, nolog Logistic regression Number of obs = 1000 LR chi2(1) = Prob > chi2 = Log likelihood = Pseudo R2 = y Coef. Std. Err. z P> z [95% Conf. Interval] x _cons c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School ofaug Economics 28 - SepAdvanced 1, 2017 Microeconom 7 / 29

8 2. Binary logit Command ml Command ml using user-written program Program lflogit de nes the logit log-likelihood I I I lnf is the rst argument and is the output - the log-density for observation i theta1 is the second argument and is the input - xi 0 β $Ml_y1 is a global macro for y i (it was not passed as a parameter).. * ML program lflogit to be called by command ml method lf. program lflogit 1. args lnf theta1 // theta1=x'b, lnf=lnf(y) 2. tempvar p // Will define p to make program more reada 3. local y "$ML_y1" // Define y so program more readable 4. generate double `p' = exp(`theta1')/(1+exp(`theta1')) 5. quietly replace `lnf' = `y'*ln(`p') + (1 `y')*ln(1 `p') 6. end c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School ofaug Economics 28 - SepAdvanced 1, 2017 Microeconom 8 / 29

9 2. Binary logit Command ml Command ml Tell command ml the program and data to be used I then maximize gives same results as logit. * Command ml model including defining y and x.. ml model lf lflogit (y = x).. ml maximize initial: log likelihood = alternative: log likelihood = rescale: log likelihood = Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = Number of obs = 1000 Wald chi2(1) = Log likelihood = Prob > chi2 = y Coef. Std. Err. z P> z [95% Conf. Interval] x _cons c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School ofaug Economics 28 - SepAdvanced 1, 2017 Microeconom 9 / 29

10 2. Binary logit Variation Variation For MSL below we need to treat intercept and slope di erently. And we need to explicitly compute the density f (y) And then take the log of this.. * The following program is a variation. * that will be extended to random parameters binary logit. * b1 and b2 are separate parameters, alternative way to defin lnf, orbust se's. program lflogitnew 1. args lnf b1 b2 // 1 is intercept and b2 is slope 2. tempvar p f 3. local y "$ML_y1" 4. gen double `p' = exp(`b1' + `b2') / (1 + exp(`b1' + `b2')) 5. quietly generate `f' = `p' if `y'==1 6. quietly replace `f' = 1 `p' if `y'==0 7. quietly replace `lnf' = ln(`f') 8. end c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 10 / 29

11 2. Binary logit Variation Variation (continued) Same results, except here have robust standard errors.. ml model lf lflogitnew (b1: y = ) (b2: x, nocons), vce(robust). ml init 1 1, copy. ml maximize initial: log pseudolikelihood = rescale: log pseudolikelihood = rescale eq: log pseudolikelihood = Iteration 0: log pseudolikelihood = (not concave) Iteration 1: log pseudolikelihood = Iteration 2: log pseudolikelihood = (not concave) Iteration 3: log pseudolikelihood = Iteration 4: log pseudolikelihood = (not concave) Iteration 5: log pseudolikelihood = Iteration 6: log pseudolikelihood = Iteration 7: log pseudolikelihood = Number of obs = 1000 Wald chi2(0) =. Log pseudolikelihood = Prob > chi2 =. Robust y Coef. Std. Err. z P> z [95% Conf. Interval] b1 b2 _cons x c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 11 / 29

12 3. Random parameters binary logit Random parameters binary logit model Random parameters binary logit Introduce a random slope parameter β 2i that is normally distributed Pr[y i = 1jx i, β 1, β 2i ] = Λ(β 1 + β 2i x i ) β 2i jβ 2, σ 2 N[β 2, σ 2 2 ]. Then β 2i = β 2 + w i where w i N[0, σ 2 2 ] so can rewrite as Then the density Pr[y i = 1jx i, β 1, w i ] Λ(β 1 + (β 2 + w i )x i ) w i jσ 2 N[0, σ 2 2 ]. f (y i jx i, β 1, β 2, w i ) = Λ(β 1 + (β 2 + w i )x i ) y i [1 Λ(β 1 + (β 2 + w i )x i )] 1 y i. We do not observe w i it needs to be integrated out. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 12 / 29

13 3. Random parameters binary logit Monte Carlo integration Monte Carlo integration We do not observe w i need to integrate it out = f (y i jx i, β 1, β 2, σ 2 ) Z Λ(β 1 + (β 2 + w i )x i ) y i [1 Λ(β 1 + (β 2 + w i )x i )] 1 y i g(w i jσ 2 )dw where g(w i jσ 2 ) is the N[0, σ 2 2 ] density. There is no closed form solution. So use Monte Carlo integration: bf (y i jx i, β 1, β 2, σ 2 ) = 1 S S s=1 f (y i jx i, β 1, β 2, w (s) i ) = 1 S S s=1 Λ(β 1 + (β 2 + w (s) i )x i ) y i [1 Λ(β 1 + (β 2 + w (s) i )x i )] 1 y i where w (s) i, s = 1,..., S are S draws from N[0, σ 2 2 ]. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 13 / 29

14 3. Random parameters binary logit Maximum simulated likelihood Maximum simulated likelihood The maximum simulated likelihood estimator maximizes ln L(β 1, β 2, σ 2 ) = N i=1 ln bf (y i jx i, β 1, β 2, σ 2 ) 1 = N i=1 ln S S s=1 Λ(β 1 + (β 2 + w (s) i )x i ) y i To implement in Stata [1 Λ(β 1 + (β 2 + w (s) i )x i )] 1 y i I Generate data to test program I Generate uniform draws that are held constant throughout I write a program lflogitmsl that calculates ln bf (y i jx i, β 1, β 2, σ 2 ) I call this program from ml maximize. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 14 / 29

15 3. Random parameters binary logit Generate data with random coe cient Generate data with random coe cient Parameters β 1 = 1, β 2 = 1, σ β2 = 1.. * Generate the data Pr[y=1] = LAMDA(1 + (1+e)*x). clear all. set obs 1000 number of observations (_N) was 0, now 1,000. set seed gen u = runiform(). gen ulogistic = ln(u) ln(1 u). gen x = rnormal(0,2). gen e = rnormal(0,1). gen y = 1 + (1+e)*x + ulogistic > 0. summarize Variable Obs Mean Std. Dev. Min Max u 1, ulogistic 1, x 1, e 1, y 1, c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 15 / 29

16 4. Random parameters logit by ml command Generate uniform draws 4. Random parameters logit by ml command We code up this random parameters logit example using ml command. The inverse transformation method is used for normal draws I I they are from the same underlying uniform draws but vary with each iteration as sd (σ 2 ) changes. To avoid chatter we will use the same underlying random uniform draws.. * Create 100 draws (S=100) from the uniform for each observation (n=1000). * These will be used to in turn get draws from the normal distribution. set seed forvalues i = 1/100 { 2. gen draws`i' = runiform() 3. } c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 16 / 29

17 4. Random parameters logit by ml command Program for log simulated density Program for log simulated density The following code builds up the log-density for one observation. * Program to calculate the log density using Monte Carlo integration. program lflogitmsl 1. args lnf b1 b2 ln_sd // if use sd then problems if sd < 0 2. tempvar p sim_f sim_avef 3. local y "$ML_y1" 4. local sd = exp(`ln_sd') // convert back to sd 5. qui gen `sim_avef' = 0 6. set seed forvalues d = 1/100 { 8. gen double `p' = exp(`b1' + `b2' + `sd'*invnormal(draws`d')*x) /// > / (1 + exp(`b1' + `b2' + `sd'*invnormal(draws`d')*x)) 9. qui gen `sim_f' = `p' if `y'==1 10. qui replace `sim_f' = 1 `p' if `y'==0 11. qui replace `sim_avef' = `sim_avef' + `sim_f'/ drop `p' `sim_f' 13. } 14. qui replace `lnf' = ln(`sim_avef') 15. end c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 17 / 29

18 4. Random parameters logit by ml command nl maximize ml maximize. * Now calculate the maximum simulated likelihood estimator. ml model lf lflogitmsl (b1: y = ) (b2: x, nocons) (ln_sd:), vce(robust). ml init 1 1 0, copy. ml maximize, difficult initial: log pseudolikelihood = rescale: log pseudolikelihood = rescale eq: log pseudolikelihood = Iteration 0: log pseudolikelihood = (not concave) Iteration 1: log pseudolikelihood = (not concave) Iteration 2: log pseudolikelihood = (not concave) Iteration 3: log pseudolikelihood = (not concave) Iteration 4: log pseudolikelihood = (not concave) Iteration 5: log pseudolikelihood = (not concave) Iteration 6: log pseudolikelihood = (not concave) Iteration 7: log pseudolikelihood = (not concave) Iteration 8: log pseudolikelihood = (not concave) Iteration 9: log pseudolikelihood = Iteration 10: log pseudolikelihood = (not concave) Iteration 11: log pseudolikelihood = (not concave) Iteration 12: log pseudolikelihood = (not concave) Iteration 13: log pseudolikelihood = (not concave) Iteration 14: log pseudolikelihood = (not concave) Iteration 15: log pseudolikelihood = Iteration 16: log pseudolikelihood = c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 18 / 29

19 4. Random parameters logit by ml command MSL results using ml method MSL results using ml method bβ 1 = 1.17, bβ 2 = 1.08, bσ β2 = Number of obs = 1,000 Wald chi2(0) =. Log pseudolikelihood = Prob > chi2 =. Robust y Coef. Std. Err. z P> z [95% Conf. Interval] b1 b2 ln_sd _cons x _cons * And convert back to sd = exp(ln_sd). nlcom exp(_b[ln_sd:_cons]) _nl_1: exp(_b[ln_sd:_cons]) y Coef. Std. Err. z P> z [95% Conf. Interval] _nl_ c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 19 / 29

20 4. Random Parameters Binary Logit by mixlogit 4. Random Parameters Binary Logit by mixlogit Can instead use user-written addon mixlogit This requires converting data to a data set with one line for each alternative I similar format to that used by command clogit. * Data before conversion. list y x in 1/5, clean y x * Now convert to dataset with data for each alternative. gen id = _n. gen x1 = 0. rename x x2. rename y y2. gen y1 = 1 y2. reshape long y x, i(id) j(alt) c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 20 / 29

21 4. Random Parameters Binary Logit by mixlogit We now have two lines per initial observation I x 1i = 0 and x 2i = x i so the di erence (x 2i x 1i ) = x i. * See what expanded data set looks like. sum id alt y x Variable Obs Mean Std. Dev. Min Max id 2, alt 2, y 2, x 2, list id alt y x in 1/10, clean id alt y x c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 21 / 29

22 4. Random Parameters Binary Logit by mixlogit MSL results using mixlogit MSL results using mixlogit This uses Hammersley draws as default so no need for seed. bβ 1 = 1.217, bβ 2 = 1.15, bσ β2 = * Now do mixlogit which has similar command structure to clogit. mixlogit y d2, group(id) rand(x) nrep(50) Iteration 0: log likelihood = (not concave) Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = Iteration 5: log likelihood = Mixed logit model Number of obs = 2,000 LR chi2(1) = Log likelihood = Prob > chi2 = y Coef. Std. Err. z P> z [95% Conf. Interval] Mean SD d x x The sign of the estimated standard deviations is irrelevant: interpret them as being positive c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 22 / 29

23 5. Random Parameter Logit by Stata 15 asmixlogit 5. Random Parameter Logit by Stata 15 asmixlogit Stata 15 introduced a command for the mix logit model I this uses Hammersley draws as default so no need for seed. bβ 1 = 1.20, bβ 2 = 1.15, bσ β2 = * asmixlogit has similar command structure to asclogit. asmixlogit y, case(id) alternatives(alt) random(x) nolog Alternative specific mixed logit Number of obs = 2,000 Case variable: id Number of cases = 1,000 Alternative variable: alt Alts per case: min = 2 avg = 2.0 max = 2 Integration sequence: Hammersley Integration points: 50 Wald chi2(1) = Log simulated likelihood = Prob > chi2 = y Coef. Std. Err. z P> z [95% Conf. Interval] alt Normal 1 x sd(x) _cons (base alternative) LR test vs. fixed parameters: chibar2(01) = Prob >= chibar2 = c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 23 / 29

24 Comparison Comparison d.g.p. values: β 1 = 1, β 2 = 1, σ β2 = 1. ml code: bβ 1 = 1.17, bβ 2 = 1.08, bσ β2 = lnl = mixlogit: bβ 1 = 1.21, bβ 2 = 1.15, bσ β2 = lnl = asmixlogit: bβ 1 = 1.20, bβ 2 = 1.15, bσ β2 = lnl = Results will get closer as N increases and number of draws or evaluation points increases. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 24 / 29

25 6. Random Parameters Binary Logit in General 6. Random Parameters Binary Logit in General The previous example had just one regressor. Now generalize. Random parameters allow di erent individuals even with same x i to respond di erently (big in marketing studies) Binary logit but replace β with β i N [β, Σ] with density φ(β i jβ, Σ) Pr[y i = 1jx i, β i ] = Λ(x 0 i β i ) = e x0 i β i /(1 e x 0 i β i ) Conditional on β i density (or p.m.f.) of y i is f (y i jx i, β i ) = Λ(x 0 i β i ) y i (1 Λ(x 0 i β i )) 1 y i Unconditional analysis requires integrate out β i : f (y i jx i, β, Σ) = R R Λ(x 0 i β i ) y i (1 Λ(x 0 i β i )) 1 y i φ(β i jβ, Σ)d β i c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 25 / 29

26 6. Random Parameters Binary Logit in General Random parameters binary logit (continued) Compute f (y i jx i, β, Σ) by Monte Carlo integration bf (y i jx i, β, Σ) = 1 S S s=1 Λ(x 0 i β (s) i ) y i (1 Λ(x 0 i β (s) i )) 1 y i I uses s draws β (s) i, s = 1,..., S from φ(β i jβ, Σ) I note: at r th round of gradient method draw is from φ(β i jβ r, Σ r ) The ML estimator for binary outcome model maximizes ln L(β, Σ) = N i=1 ln f (y i jx i, β, Σ). The simulated maximum likelihood (SML) estimator maximizes ln L(β, Σ) = N i=1 ln bf (y i jx i, β, Σ) = N i=1 ln 1 S S s=1 Λ(xi 0 β (s) i ) y i (1 Λ(xi 0 β (s) i )) 1 y i = i fy i ln bp i + (1 y i ) ln(1 bp i )g; bp i = 1 S s Λ(xi 0 β (s) i ) Especially popular for multinomial data I then random parameters logit overcomes independence of irrelevant alternatives limitation of regular conditional logit. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 26 / 29

27 7. MSL in General 7. MSL in general Problem: MLE (with independent data over i) maximizes ln L(θ) = N i=1 ln f (y i jx i, θ). I but f (y i jx i, θ) does not have a closed form solution. I e.g. f (y i jx i, θ) = R g(y i jx i, θ 1, α)h(αjθ 2 )dα =? Solution: Maximum simulated likelihood (MSL) estimator maximizes ln bl(θ) = N i=1 ln bf (y i jx i, θ) I I bf (y i jx i, θ) is a simulated approxn. to f (y i jx i, θ) based on S draws e.g. f (y i jx i, θ) = S 1 S s=1 g(y i jx i, θ, α (s) ), α (s) are draws from h(α). MSLE consistent with the usual MLE asymptotic distribution if I bf () is an unbiased simulator and satis es other conditions given below I S!, N! and p N/S! 0 where S is number of simulations. I note that many draws S (to compute bf ()) are required. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 27 / 29

28 7. MSL in General MSL details MSL details Assumed properties of the simulator: I bf () is an unbiased simulator with: E[bf (y i jx i, θ)] = f (y i jx i, θ) I bf () is di erentiable in θ (or smooth simulator) so gradient methods can be used I the underlying draws to compute bf () are unchanged so no "chatter". MSL needs S! because simulator is nonetheless biased for ln f () E[bf ()] = f () ; E[ln bf ()] 6= ln f (). Variation: Method of simulated (MSM) estimator instead works with moment conditions that allow an unbiased simulator I bθ is a method of moments estimator that solves N i =1 m(y i jx i, θ) = 0. I Assume unbiased simulator such that E[ bm(y i jx i, θ)] = m(y i jx i, θ) I The MSM solves N i =1 bm(y i jx i, θ) = 0 F Consistent even for small S, though there is then e ciency loss. F When bm() is the frequency simulator V[bθ MSM ] = (1 + S 1 )V[ bθ MSL ]. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 28 / 29

29 8. References 8. Some References The general principles of MSL are covered in I CT(2005) MMA chapter 13. c A. Colin Cameron Univ. of Calif. - Davis... for Maximum Center of Labor Simulated Economics Likelihood Norwegian School of Aug Economics 28 - SepAdvanced 1, 2017 Microeconom 29 / 29