3.3.1.1 Defining the Model
In line with the reduced form equation (13) in section 3.2 describing the child health production function, consider the following standard model under probit specification-
H* = X′ β + ε, where
X is a vector that includes an intercept term and other variables of interest, as will be discussed shortly. β denotes the vector of the relevant coefficients on each of the
14
The ordered probit model differs from probit model in having more than two (three in this study)
categories of a given dependent variable instead of two, and is largely similar to probit, which I skip in this section.
regressors included in X and ε denotes the error terms, assumed to be normally distributed with mean zero and variance 1.
i.e. ε ~N (0, 1)
H* is an unobserved latent variable between -∞ and +∞, capturing either the perceived physical health status or perceived mental health status of the child. H (without the star) represents the corresponding observed physical health status (HLTH) or observed mental health status (MLTH), each of which is a an indicator variable taking the values-
0 (Poor/Fair/Good); and 1 (Very Good/Excellent)
The observed value of H depends on where the unobserved value H* stands in the scale from -∞ and +∞, as marked by the cut point (0)15
. Specifically, following rules apply as regards the observed values of H:
H = 0 if H* ≤ 0; and H = 1 if H* > 0
Pertaining to the objective of exploring the channels that may transmit the beneficial effects of parental schooling on child health, X may represent one of the following four vectors, thereby giving rise to four different equations under this model-
Equation (I): X = (1 MoSch FaSch Z)
Equation (II): X = (1 MoSch FaSch Pinc Z) Equation (III): X = (1 MoSch FaSch Insur Z) Equation (IV): X = (1 MoSch FaSch Pinc Insur Z)
15Considering a non-zero cut-point (μ) is equally valid, but this can be normalized to make 0 as the cut point, if the model contains a constant term (Green, 2003, p. 669).
where MoSch and FaSch denote the mother‟s and father‟s schooling, each observed in three ordinal categories - 0 (No Degree); 1 (HSD/GED); and 2 (College Degree), as defined earlier. Pinc denotes total parental income (during their two years in MEPS, adjusted for inflation and expressed in log). Insur denotes the health insurance type of the child observed in three categories - 1 (Any Private); 2 (Only Public); and 3 (None). Z is the vector of other regressors (demographic, geographic and others including parental health). Inclusion of parental income and health insurance type of child successively in the regression equations inform us of the importance of income and insurance themselves, and to what extent the effects of income and insurance stem from parental schooling.
3.3.1.2 Response Probabilities
The estimated response probabilities are distributed as: Prob (H = 0 | X)
= Prob (H* ≤ 0 | X) = Prob (X′β + ε ≤ 0 | X) = Prob (ε ≤ - X′β | X)
= ∫ ( ) , which is denoted as Φ(-X′β) [or 1- Φ(X′β) because of the symmetry], where φ (z) is known as standard normal density, defined as-
φ (z) = exp(-0.5z2) / (2π)0.5
, and „Φ‟ is the standard normal cumulative distribution. Similarly,
Prob (H = 1 | X)
= Prob (H* > 0 | X) = Prob (X′β + ε > 0 | X)
= Prob (ε > - X′β | X) = Prob (ε < X′β | X) = Φ(X′β)
The model is then estimated by the method of maximum likelihood. For this, a joint probability (or likelihood) function is first defined using these response probabilities, and the values of the parameters that maximize this function are then obtained.
As discussed earlier, the siblings within the household share many common family characteristics - demographic, economic, genetic (which may determine their initial health endowment), environment, family background, parental preferences, parental health, parental behavior, and the perception of child health status by the parents. In order to address potential autocorrelation on the error terms arising from these sibling effects, all the regressions are clustered at the household level. Also, I obtain the robust standard errors instead of the regular ones to address the inherently heteroskedastic nature of the non-liner form of model specifications and the maximum likelihood estimation technique. Under the validity of the model assumptions, the estimates thus obtained are asymptotically efficient.
3.3.1.3 Marginal Effects
It is noteworthy that the parameter vector β in these models, unlike in the linear regression models, does not give the marginal effects of the predictors on the observed value of H, but merely gives the contributions of the predictors on the unobserved H*. More precisely, the estimated regression coefficients give the change in the z-score or probit index for a one unit change in a given predictor, other things same. Therefore, these parameters, at the time they are estimated, are less insightful and only indicative
of the direction of their contributions. However these estimated coefficients form the raw materials for the marginal effects of their respective regressors on the probability of observing a particular value of the dependent variable H. The expressions for the estimation of marginal effects are as below-
The marginal effects, which measures the ratio of the change in the probability of observing H = 1 to a small change in X, are generally computed using the following expression-
( )
= φ (X′β)β
Clearly, the marginal effects are functions of X, and therefore their interpretations are not straightforward. This is made easier either by evaluating these expressions at sample means of the data (the method used in this study), or the marginal effects are evaluated at every observation and their sample average is computed. In large samples like the current one, both approaches will give similar results.
The above expression is typically appropriate for a continuous variable (say x), where a „small change‟ in x is conceivable. But in many cases such as in this study, X may include dummy (or categorical) variables. The appropriate marginal effect associated to a dummy variable, say d, which now measures the change in the probability of observing H = 1 when d changes the values from 0 to 1, is expressed as- Marginal effect = Δ Prob [H = 1| ̅(d)] = Prob [H = 1| ̅(d), d = 1] - Prob [H = 1| ̅(d), d = 0],
where ̅(d) denotes the means of all the other variables except for d in the model.
Similar marginal effects for a variable, say MoSch in this study, which can take three values 0, 1 or 2, can be expressed as-
Marginal effect (1) = Prob [H = 1| ̅(MoSch), MoSch = 1]
- Prob [H = 1| ̅(MoSch), MoSch = 0], and
Marginal effect (2) = Prob [H = 1| ̅(MoSch), MoSch = 2]
- Prob [H = 1| ̅(MoSch), MoSch = 0],
where the first marginal effect (1) corresponds to the change in mother‟s schooling from level 0 to level 1, and the second corresponds to change in the schooling from level 0 to level 2.
3.3.2 Bivariate Probit Model