Democracy, Electoral Systems, and Judicial Empowerment in Developing Countries

Appendix

277 appendix Chapter 3 The Sample Selection Ordered Probit (S-OP) Model Recall from the text that the S-OP statistical model consists of two stages and is defined (after dropping subscript t for time for notational convenience ) as (A3.1) (A3.2) Since yi,xi is observed when di is equal to 1, the underlying latent variable yi * for yi is thus also given by the following ordered probit (i.e., outcome) equation in the second stage of the S-OP model: (A3.3) d u i i i * = ′ + α z d if d otherwise i i = >      1 0 0 * yi i i * = ′ + β ε x y j if y i j i j = < ≤ −µ µ 1 * ( , )~ ( , , , , ) ε ρ i i u N2 0 0 1 1 y d i i i i * = ′ + + β θ ε x y j if y i j i j = < ≤ −µ µ 1 * 278 appendix where (εi,ui) ∼ N2(0,0,1,1,r) and di ∈ (0,1) is a dichotomous variable with di = 1 indicating a new democratic regime that is realized from the nonrandom occurrence of a democratic transition. Conditional on di = 1, yi is related to the latent variable yi * in (A3.1) and a boundary or cut-off parameter m as follows: The log likelihood function of the S-OP statistical is, according to Greene and Hensher (2010), defined as (A3.5) where Φ (.) is the standard normal multivariate distribution. We estimate the log likelihood function in (A3.5) with random effects. Note that maximizing the log likelihood function in (A3.5) on a time-series cross section data set (which is precisely the kind of data that we are using) requires calculating normal multidimensional distribution integrals, the dimension of which grows with T where T denotes time. Since evaluating multidimensional integrals of the likelihood presented above is computationally intensive , we use simulation methods—specifically, the Geweke-HajivassiliouKeane (GHK) smooth recursive conditioning simulator—to maximize the log likelihood function in (A3.5). The GHK maximum simulated likelihood (MSL) method is often used by econometricians (see, e.g., Hajivassiliou and McFadden 1990; Train 2003) to evaluate the likelihood in (A3.5) as it is used to calculate multivariate normal probabilities, which is required for MSL estimation. The GHK method leads to a simulated log likelihood y if y and d if y and d if i i i i i = − ≤ = ≤ = 1 0 1 0 0 1 1 * * 0 1 2 1 1 1 2 < ≤ = < ≤ = y and d if y and d J if i i i it * * ... ...µµ µµ µ J i J i y and d − < ≤ =              1 1 * log log ( ) log[ ( , L m i d d ij j i i i = − ′ + − ′ ′ = = ∑ ∑ Φ Φ αµ β α z x z 0 0 2 i i d J j i i i , ) ( , , )] ρµ β α ρ − − ′ ′ = − ∑ 0 2 1 Φ × x z (A3.4) appendix 279 function, which we maximize with respect to the parameter vectors xi, zi, mJ and the covariance matrix Ω by using the Broyden-Fletcher-GoldfarbShanno (BFGS) numerical optimization method (see Train 2003). Once we obtain the estimates, the variance-covariance matrix is directly derived by inverting the Hessian evaluated in the obtained maximum likelihood estimators . The estimation of the ordered probit and the S-OP model has been done by using R. Calculation of Interaction Effect: Recall that the ordered probit model (and also the ordered outcome equation of the S-OP model) is given by yi * = ′ + β ε x where yi is related to yi * as (A3.6) where j = 0,1,2 . . . J is the discrete ordered outcome and the m’s are the (J − 1) unknown parameters known as the boundary or cut-off parameters. Assume without loss of generality that there are only three covariates (x1, x2, and x3) in the x vector of the ordered probit equation in (A3.2) where x2and x3are interacted while x1 is not. This implies that b′x = b1x1 + b2x2 + b3x3 + b23(x2 * x3). Suppose further that x2 is the continuous independent variable (legislative) concentration in new democracies and x3 is the continuous independent variable (judiciary) trust in new democracies, and the interaction of these two variables is x2x3 = x2 * x3. The m’s and b′ = (b1,b2,b3,b23) are jointly estimated by the GHK MSL method (mentioned above). Suppose that ε ∼ N (0,1); then the probability for the jth outcome is given by (A3.7) where Φ is the cumulative standard normal distribution function, which is continuous and twice differentiable. The marginal effect of the continuous variable x2 (concentration) on the probability of the jth outcome in the ordered probit model, according to Mallick (2009, 4), is given by (A3.8) where φ (.) is the standard normal density function, φj−1(.) = φ(mJ−1 − b′x), and φj(.) = φ(mJ − b′x). The marginal effect of x3 is similar to (A3.8) and will hence not be repeated here. Note that the formula in (A3.8) accounts for the fact that y J if y i J i J = < ≤ −µ µ 1 * prob y j i J J ( ) ( ) ( ) = = − ′ − − ′ − Φ Φµ β µ β x x 1 δ ϕ β β ϕ β β 2 2 1 2 23 3 2 23 3 , [ | ] (.)[ ] (.)[ ] j i j j prob y j x x x = ∂ = ∂ = + − + − x 280 appendix the impact of x2 is also dependent on its combined effect of x2 and x3 on the dependent variable. Following Mallick (2009, 4), the magnitude of the interaction effect of x2x3 on the probability of the jth outcome is obtained by computing the partial derivative of (A3.8) with respect to x3, which leads to (A3.9) where ′ ϕj (.) is the first derivative of the density function with respect to its argument. Observe that the expression in (A3.9) is different from the marginal effect formula of interaction terms in ordered probit models in standard software packages, where it is simply calculated as . To understand the asymptotic properties of the interaction effect in (A3.12) and calculate its standard error, we first need to rewrite equation (A3.7) as prob(yi = j) = Fj(x,b). Then the estimated value of the marginal effect of x2 and x3can be computed as (A3.10) where β̂ is the consistent estimator of b that is estimated via the GHK MSL method. The consistency of 23, ˆ j δ is ensured by the continuity of Fj and the consistency of β̂ . We can compute the standard error of the above interaction effect by applying the delta method, which is given by (A3.11) In the expressions for σ32,j and 32, ˆ j σ where ˆ β Ω is the consistent covariance estimator of β̂ and 2 23, 23, 23, ˆ ~ ( , ) j j j N δ δ σ for all j = 0, 1, 2 . . . J. Note that the delta method estimates the variance using a first-order Taylor approximation . Since a first-order Taylor approximation may provide a poor approximation in nonlinear functions (such as the ordered probit function), we follow Spanos (1999, 493–94) and use a second-order approximation by δ ϕ ϕ β β β 23 2 2 3 1 23 2 23 3 , [ | ] [ (.) (.)] [ ] [ j i j j prob y j x x x = ∂ = ∂ ∂ = − − + − x × β β β ϕ ϕ 3 23 2 1 + ′ − ′ − x j j ][ (.) (.)] ∂ = ∂ = − − prob y j x x i j j [ | ] ( * ) [ (.) (.)] x 2 3 1 23 ϕ ϕ β 2 23, 2 3 ˆ (x, ) ˆ j j F x x β δ ∂ = ∂ ∂ 2 2 23, 2 3 2 3 ˆ ˆ (x, ) (x, ) ˆ ˆ ' j F F x x x x β ∂ β ∂ β σ ∂β ∂ ∂β ∂ ⎧ ⎫ ⎧ ⎫ ∂ ∂ ⎪ ⎪ ⎪ ⎪ = Ω ⎨ ⎬ ⎨ ⎬ ∂ ∂ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ appendix 281 replacing the normal distribution with the chi-square distribution. Finally, observe that the t-statistic that tests the null that the interaction effect is zero is 23, 23, ˆ ˆ / j j t σ δ = . Chapter 5 Log-Likelihood of Bivariate Ordered Probit Model The bivariate ordered probit model (described in equations (5.4) and (5.5) in the text) can be written more comprehensively (after dropping the subscript t for time for notational convenience) as (A5.1) (A5.2) Thus the joint probability for yi,1 = j and yi,2 = k is according to Greene and Hensher (2010, 223) given by (A5.3) Using the information from (A5.1)–(A5.3), the log likelihood function of the bivariate ordered probit model can be defined as yi i i , * , , 1 1 1 1 = ′ + β ε x y j if y j J i j i j , , * , ,..., 1 1 1 1 0 = < ≤ = −µ µ yi i i , * , , 2 2 2 2 = ′ + β ε x y j if y j J i j i j , , * , ,..., 2 1 2 2 0 = < ≤ = − δ δ ε ε ρ ρ i i N , , ~ , 1 2 0 0 1 1                                 prob y j y k i i i i j i k i ( , | , ) [( ),( ) , , , , , , 1 2 1 2 2 1 1 1 2 = = = − ′ − ′ x x x x Φ µ β δ β , , ] [( ),( ), ] [( , , ρµ β δ β ρµ β − − ′ − ′         − − ′ − Φ Φ 2 1 1 1 1 2 2 1 j i k i j x x xi i k i j i k i , , , , ),( ), ] [( ),( ), 1 1 1 2 2 1 1 1 1 1 2 δ β ρµ β δ β ρ − − − − ′ − − ′ − ′ x x x Φ ] ]         282 appendix (A5.4) where mi,j = 1 if yi,1 = j and is 0 otherwise and likewise, ni,k = 1 if yi,2 = k and is 0 otherwise. We estimate the log likelihood function in (A5.4) with random effects. Maximizing the log likelihood function in (A5.4) on a TSCS data set requires calculating normal multidimensional distribution integrals , the dimension of which grows with T where T denotes time. Evaluating multidimensional integrals of the likelihood presented above is computationally intensive. Thus we use simulation methods—specifically, the GHK smooth recursive conditioning simulator—to maximize the log likelihood function in (A5.4). The GHK MSL method is often used by econometricians (e.g., Train 2003) to evaluate the likelihood in (A5.4) as it is used to calculate multivariate normal probabilities, which is required for MSL estimation . The GHK method leads to a simulated log likelihood function, which we maximize with respect to the parameter vectors xi, dk, zi, mJ, and the covariance matrix Ω by using the BFGS numerical optimization method. After obtaining the estimates, the variance-covariance matrix is directly derived by inverting the Hessian evaluated in the obtained maximum likelihood estimators. LL m n i N j J i j i k k J j i k i = − ′ − ′ = = = ∏ ∑ ∑ 1 0 0 2 1 1 1 2 1 2 , , , , [( ),( ), Φ µ β δ β ρ x x ] ] [( ),( ), ] [( , , , − − ′ − ′         − − ′ − Φ Φ 2 1 1 1 1 2 2 1µ β δ β ρµ β j i k i j i x x x 1 1 1 1 2 2 1 1 1 1 1 2 ),( ), ] [( ),( ), ] , , , δ β ρµ β δ β ρ k i j i k i − − − − ′ − − ′ − ′  x x x Φ         ...