Decision-making by children

Abstract

In this paper, we examine the determinants of decision-making power by children and young adolescents. Moving beyond previous economic models that treat children as goods consumed by adults, we develop a noncooperative model of parental control of child behavior and child resistance. Using child reports of decision-making and psychological and cognitive measures from the NLSY79 Child Supplement, we examine the determinants of shared and sole decision-making based on indices created from seven domains of child activity. We find that the determinants of sole decision-making by the child and shared decision-making with parents are quite distinct: sharing decisions appears to be a form of parental investment in child development rather than a simple stage in the transfer of authority. In addition, we find that indicators of child capabilities and preferences affect reports of decision-making authority in ways that suggest child demand for autonomy as well as parental discretion in determining these outcomes.

Appendix

1.1 Height

The reference table from the Centers for Disease Control and Prevention, Department of Health and Human Services²¹ is used. It was released in 2000, based on children in the United States. A group is defined for each age-in-month and each gender (so there are 59 × 2 = 118 groups). For each group, three parameters (the median (M), the generalized coefficient of variation (S), and the power in the Box-Cox transformation (L)) are obtained from the table. With these parameters the z-score is calculated, as suggested by the Centers, for each child of height X as: $Z = \frac{{\left( {X/M} \right)^{L} - 1}}{L*S},\;L \ne 0$. We drop 46 observations with a z-score of absolute value more than 5.

1.2 Fixed-effects ordered logit model

Our three dependent variables (own, shared, and parent decisions) are all discrete dependent variables y _ij that can take K possible values 0, 1,…, K − 1. Since the number of decisions made is an ordinal measure of autonomy, we use the ordered logit model for estimation. Computational problems arise when we incorporate fixed effects in the model. This appendix describes a simple two-step procedure of estimating the fixed-effects ordered logit model first proposed by Das and van Soest (1999). See also Ejrnaes and Portner (2004) and Greene (2008).

We have a discrete dependent variable y _ij that can take K + 1 possible values. We do not assume cardinality (i.e., the difference between y _ij  = 1 and y _ij  = 2 is different from the difference between y _ij  = 2 and y _ij  = 3).

Consider the following model for a latent variable $y_{ij}^{*}$ for child j in family i, i = 1,…, I and j = 1,…, J:

$$ y_{ij}^{*} = \alpha_{i} + {\mathbf{x}}_{ij}^{\prime } {\varvec{\beta}} + \varepsilon_{ij} ,\;\varepsilon_{ij} \;{\text{has}}\;{\text{a}}\;{\text{logistic}}\;{\text{distribution}}\;{\text{conditional}}\;{\text{on}}\;{\mathbf{X}}_{i} \;{\text{and}}\;\alpha_{i} $$

The observed choice y _ij depends on the latent variable

$$ \begin{aligned} y_{ij} = 0&\quad{\text{ if }}y_{ij}^{*} \le 0 \hfill \\ y_{ij} = 1 &\quad{\text{ if }}0 < y_{ij}^{*} \le \mu_{1} \hfill \\ \ldots \hfill \\ y_{ij} = K & \quad{\text{ if }}\mu_{K - 1} < y_{ij}^{*} \hfill \\ \end{aligned} $$

Now derive the probability of observing a choice of y _ij  = k:

$$ \begin{aligned} \Pr \left( {y_{ij} = k|{\mathbf{X}}_{i} ,\alpha_{i} } \right) \hfill \\ = \Pr \left( {y_{ij}^{*} < \mu_{k} } \right) - \Pr \left( {y_{ij}^{*} < \mu_{k - 1} } \right) \hfill \\ = \Pr \left( {\varepsilon_{ij} < \mu_{k} - \alpha_{i} - {\mathbf{x}}_{ij}^{\prime } {\varvec{\beta}}} \right) - \Pr \left( {\varepsilon_{ij} < \mu_{k - 1} - \alpha_{i} - {\mathbf{x}}_{ij}^{\prime } {\varvec{\beta}}} \right) \hfill \\ = \Uplambda \left( {\mu_{k} - \alpha_{i} - {\mathbf{x}}_{ij}^{\prime } {\varvec{\beta}}} \right) - \Uplambda \left( {\mu_{k - 1} - \alpha_{i} - {\mathbf{x}}_{ij}^{\prime } {\varvec{\beta}}} \right) \hfill \\ \end{aligned} $$

where $\Uplambda ( \cdot )$ is the logistic CDF. Now we split the estimation problem into many small ones by defining a binary variable:

$$ s_{ij}^{k} = 1\;{\text{if}}\;y_{ij} > k,\quad k = 0, \ldots ,K - 1 $$

From this definition comes the following result:

$$\Pr \left( {s_{ij}^{k} = 1|\alpha_{i} ,{\mathbf{X}}_{i} } \right) = \Uplambda \left( { - \mu_{k} + \alpha_{i} + {\mathbf{x}}_{ij}^{\prime } {\varvec{\beta}}} \right) = \Uplambda \left( {\theta_{i} + {\mathbf{x}}_{ij}^{\prime } {\varvec{\beta}}} \right) $$

(4)

We lump the cutoff point μ _k , which is the same for all individuals, into the fixed family effects α _i and have a new family fixed effects θ _i . Obviously, we can fit Eq. 4 with the Chamberlain-type conditional fixed effects logit model (Chamberlain 1980). In total there are K such models, and from them we obtain K different sets of estimates for β, call it ${\varvec{\delta}} = \left( {{\varvec{\beta}}^{\left( 0 \right)} ,{\varvec{\beta}}^{\left( 1 \right)} , \ldots ,{\varvec{\beta}}^{{\left( {K - 1} \right)}} } \right)$.

The next step is to combine the K different sets of estimates. We use a minimum distance estimator with the following objection function:

$$ \left( {{\varvec{\delta}} - \iota_{K} \otimes {\varvec{\beta}}^*} \right)^{\prime } {\mathbf{W}}\left( {{\varvec{\delta}} - \iota_{K} \otimes {\varvec{\beta}}^*}\right) $$

where ι _K is a K vector of ones.

Our choice of the weighting matrix W makes use of the inverse of the covariance matrices from the K logit models:

$$ {\mathbf{W}} = \left( {\begin{array}{*{20}c} {\text{var} \left( {{\varvec{\beta}}^{\left( 0 \right)} } \right)} & \cdots & \cdots & 0 \\ \vdots & {\text{var} \left( {{\varvec{\beta}}^{\left( 1 \right)} } \right)} & {} & {} \\ \vdots & {} & \ddots & \vdots \\ 0 & {} & \cdots & {\text{var} \left( {{\varvec{\beta}}^{{\left( {K - 1} \right)}} } \right)} \\ \end{array} } \right)^{ - 1}$$

Intuitively, we put less weight on regressions that are less precisely estimated. The covariance matrix of the minimum distance estimator β* is given by

$$ \text{var} \left( {{\varvec{\beta}}^{*} } \right) = \left[ {\left( {\iota_{K} \otimes {\text{I}}} \right)^{\prime } {\mathbf{W}}\left( {\iota_{K} \otimes {\text{I}}} \right)} \right]^{ - 1} \left( {\iota_{K} \otimes {\text{I}}} \right)^{\prime } {\mathbf{W}}\left( {\iota_{K} \otimes {\text{I}}} \right)\left[ {\left( {\iota_{K} \otimes {\text{I}}} \right)^{\prime } {\mathbf{W}}\left( {\iota_{K} \otimes {\text{I}}} \right)} \right]^{ - 1} \cdot $$