Dynamic panel data estimation of an integrated Grossman and Becker–Murphy model of health and addiction

Abstract

We propose a dynamic panel data approach to estimate a model that integrates the Becker–Murphy theory of rational addiction with the Grossman model of health investment. We define an individual’s lifetime smoking consumption and investments in health capital as simultaneous choices within a single optimisation problem. We show that this can be estimated using GMM system estimation of two stand-alone single fourth-order difference equations of health capital and smoking. These preserve roots and fundamental dynamics of the original system of four interrelated first-order equations. Monte Carlo simulations confirm that this reduced-form dynamic estimation also produces very similar estimates to the ones of the initial system of equations. We argue that, in the presence of long panel data, this approach may provide a feasible alternative for the estimation of a complex life-cycle model of human capital.

Appendices

Appendix A

Consider the system of two second-order difference equations (10) for $Y_1$ and $Y_2$ as follows:

$$\begin{aligned} Y_{1t}&= \phi _{10} + \phi _{11} Y_{1t-1} + \phi _{12} Y_{2t-1} + \theta _{11} X_{1t} + \theta _{12} X_{2t} + \varepsilon _{1t} \\ Y_{2t}&= \phi _{20} + \phi _{21} Y_{1t-1} + \phi _{22} Y_{2t-1} + \theta _{21} X_{1t} + \theta _{22} X_{2t} + \varepsilon _{2t} \end{aligned}$$

Abstracting from the constant terms without loss of generality, the above can be written in matrix form as,

$$\begin{aligned} \begin{bmatrix} Y_{1t} \\ Y_{2t} \end{bmatrix} =\begin{bmatrix} \phi _{11}&\phi _{12} \\ \phi _{21}&\phi _{22} \end{bmatrix} \begin{bmatrix} LY_{1t} \\ LY_{2t} \end{bmatrix} +\begin{bmatrix} \theta _{11}&\theta _{12} \\ \theta _{21}&\theta _{22} \end{bmatrix} \begin{bmatrix} X_{1t} \\ X_{2t} \end{bmatrix} +\begin{bmatrix} \epsilon _{1t} \\ \epsilon _{2t} \end{bmatrix} \end{aligned}$$

(18)

where L is the lag operator. Since L is multiplicative, Eq. (18) can be expressed as,

$$\begin{aligned} \begin{bmatrix} Y_{1t} \\ Y_{2t} \end{bmatrix} =\begin{bmatrix} L\phi _{11}&L\phi _{12} \\ L\phi _{21}&L\phi _{22} \end{bmatrix} \begin{bmatrix} Y_{1t} \\ Y_{2t} \end{bmatrix} +\begin{bmatrix} \theta _{11}&\theta _{12} \\ \theta _{21}&\theta _{22} \end{bmatrix} \begin{bmatrix} X_{1t} \\ X_{2t} \end{bmatrix} +\begin{bmatrix} \epsilon _{1t} \\ \epsilon _{2t} \end{bmatrix} \end{aligned}$$

Combining the vectors for $Y_t$ gives,

$$\begin{aligned} \begin{bmatrix} 1-L\phi _{11}&-L\phi _{12} \\ -L\phi _{21}&1-L\phi _{22} \end{bmatrix} \begin{bmatrix} Y_{1t} \\ Y_{2t} \end{bmatrix} =\begin{bmatrix} \theta _{11}&\theta _{12} \\ \theta _{21}&\theta _{22} \end{bmatrix} \begin{bmatrix} X_{1t} \\ X_{2t} \end{bmatrix} +\begin{bmatrix} \epsilon _{1t} \\ \epsilon _{2t} \end{bmatrix} \end{aligned}$$

This can be rearranged such that,

$$\begin{aligned} \begin{bmatrix} Y_{1t} \\ Y_{2t} \end{bmatrix} =\begin{bmatrix} 1-L\phi _{11}&-L\phi _{12} \\ -L\phi _{21}&1-L\phi _{22} \end{bmatrix}^{-1} \begin{bmatrix} \theta _{11}&\theta _{12} \\ \theta _{21}&\theta _{22} \end{bmatrix} \begin{bmatrix} X_{1t} \\ X_{2t} \end{bmatrix} +\begin{bmatrix} 1-L\phi _{11}&-L\phi _{12} \\ -L\phi _{21}&1-L\phi _{22} \end{bmatrix}^{-1} \begin{bmatrix} \epsilon _{1t} \\ \epsilon _{2t} \end{bmatrix} \end{aligned}$$

(19)

The inverse matrix which appears twice on the RHS of Eq. (19) can be written:

$$\begin{aligned} \begin{bmatrix} 1-L\phi _{11}&-L\phi _{12} \\ -L\phi _{21}&1-L\phi _{22} \end{bmatrix}^{-1} =\frac{1}{DetA} \begin{bmatrix} 1-L\phi _{22}&L\phi _{12} \\ L\phi _{21}&1-L\phi _{11} \end{bmatrix} \end{aligned}$$

(20)

where DetA is the determinant of the matrix being inverted:

$$\begin{aligned} DetA = [1-L\phi _{11}] [1-L\phi _{22}] - L^2 \phi _{12} \phi _{21} = 1 - [\phi _{11} + \phi _{22}]L + [\phi _{11}\phi _{22} - \phi _{12}\phi _{21}]L^2 \end{aligned}$$

(21)

Using Eq. (20) allows us to write,

$$\begin{aligned} DetA\begin{bmatrix} Y_{1t} \\ Y_{2t} \end{bmatrix} =\begin{bmatrix} 1-L\phi _{22}&L\phi _{12} \\ L\phi _{21}&1-L\phi _{11} \end{bmatrix} \begin{bmatrix} \theta _{11}&\theta _{12} \\ \theta _{21}&\theta _{22} \end{bmatrix} \begin{bmatrix} X_{1t} \\ X_{2t} \end{bmatrix} +\begin{bmatrix} 1-L\phi _{22}&L\phi _{12} \\ L\phi _{21}&1-L\phi _{11} \end{bmatrix} \begin{bmatrix} \epsilon _{1t} \\ \epsilon _{2t} \end{bmatrix} \end{aligned}$$

(22)

recalling that DetA is a scalar—the determinant of the matrix to be inverted.

Multiplying out the first two matrices on the RHS, such that

$$\begin{aligned} \begin{bmatrix} 1-L\phi _{22}&\quad L\phi _{12} \\ L\phi _{21}&\quad 1-L\phi _{11} \end{bmatrix} \begin{bmatrix} \theta _{11}&\quad \theta _{12} \\ \theta _{21}&\quad \theta _{22} \end{bmatrix} =\begin{bmatrix} [1-L\phi _{22}] \theta _{11} + L\phi _{12}\theta _{21}&\quad [1-L\phi _{22}] \theta _{12} + L\phi _{12}\theta _{22} \\ \theta _{11} L\phi _{21} + [1-L\phi _{11}] \theta _{21}&\quad L\phi _{21} \theta _{12} + [1-L\phi _{11} ]\theta _{22} \end{bmatrix} \end{aligned}$$

(23)

Substituting Eq. (23) into Eq. (22) gives,

$$\begin{aligned} DetA\begin{bmatrix} Y_{1t} \\ Y_{2t} \end{bmatrix}= & {} \begin{bmatrix} [1-L\phi _{22}] \theta _{11} + L\phi _{12}\theta _{21}&\quad [1-L\phi _{22}] \theta _{12} + L\phi _{12}\theta _{22} \\ \theta _{11} L\phi _{21} + [1-L\phi _{11}] \theta _{21}&\quad L\phi _{21} \theta _{12} + [1-L\phi _{11} ]\theta _{22} \end{bmatrix} \begin{bmatrix} X_{1t} \\ X_{2t} \end{bmatrix}\nonumber \\&+\begin{bmatrix} 1-L\phi _{22}&L\phi _{12} \\ L\phi _{21}&1-L\phi _{11} \end{bmatrix} \begin{bmatrix} \epsilon _{1t} \\ \epsilon _{2t} \end{bmatrix} \end{aligned}$$

(24)

Now take the top row of Eq. (24), which is the expression for $Y_{1t}$ where we are assuming that $Y_{1}$ is the observable variable. That will be

$$\begin{aligned} DetA Y_{1t}= & {} \theta _{11} X_{1t} + [\phi _{12} \theta _{21} - \phi _{22}\theta _{11}] LX_{1t} + \theta _{12}X_{2t} + [\phi _{12}\theta _{22} - \phi _{22} \theta _{12}] LX_{2t} \\&+\,[\epsilon _{1t} - \phi _{22} L \epsilon _{1t} + \phi _{12} L \epsilon _{2t}] \end{aligned}$$

Substituting for DetA from Eq. (21),

$$\begin{aligned}&\left[ 1 - \left[ \phi _{11} + \phi _{22} \right] L + \left[ \phi _{11}\phi _{22} - \phi _{12} \phi _{21} \right] L^2 \right] Y_{1t} = \phi _{11} X_{1t} + \left[ \phi _{12}\theta _{21} - \phi _{22} \theta _{11} \right] LX_{1t} \nonumber \\&\quad +\, \theta _{12} X_{2t} + \left[ \phi _{12} \theta _{22} - \phi _{22} \theta _{12} \right] LX_{2t} + \left[ \epsilon _{1t} - \phi _{22} L \epsilon _{1t} + \phi _{12} L \epsilon _{2t} \right] \end{aligned}$$

(25)

Applying the lag operator gives,

$$\begin{aligned}&Y_{1t} - \left[ \phi _{11} + \phi _{22} \right] Y_{1t-1} + \left[ \phi _{11}\phi _{22} - \phi _{12} \phi _{21} \right] Y_{1t-2} = \phi _{11} X_{1t}\\&\quad +\, \left[ \phi _{12}\theta _{21} - \phi _{22} \theta _{11} \right] X_{1t-1} \\&\quad +\, \theta _{12} X_{2t} + \left[ \phi _{12} \theta _{22} - \phi _{22} \theta _{12} \right] X_{2t-1} + \left[ \epsilon _{1t} - \phi _{22} \epsilon _{1t-1} + \phi _{12} \epsilon _{2t-1} \right] \end{aligned}$$

Isolating $Y_{1t}$ on the left-hand side gives,

$$\begin{aligned} Y_{1t}= & {} \left[ \phi _{11} + \phi _{22} \right] Y_{1t-1} - \left[ \phi _{11}\phi _{22} - \phi _{12} \phi _{21} \right] Y_{1t-2}\nonumber \\&+\, \phi _{11} X_{1t} + \left[ \phi _{12}\theta _{21} - \phi _{22} \theta _{11} \right] X_{1t-1} \nonumber \\&+\, \theta _{12} X_{2t} + \left[ \phi _{12} \theta _{22} - \phi _{22} \theta _{12} \right] X_{2t-1} + \left[ \epsilon _{1t} - \phi _{22} \epsilon _{1t-1} + \phi _{12} \epsilon _{2t-1} \right] \nonumber \\ \end{aligned}$$

(26)

The purpose of the transformation using the lag operators is to leave us with, on the RHS, lagged values of the observable Y, values of the exogenous variables, and a disturbance term which involves (in the 2 equation case) both $\epsilon _{1}$ and $\epsilon _{2}$ values and, by its involvement of both current and lagged $\epsilon$’s, requiring us to use an estimating method which allows for a serially correlated residual term.

The structure in Eq. (26) also has the appeal that we can see how the roots of the system are derived from the coefficients on the two lags of $Y_1$ and, by comparing it with the original matrix form, we can see that the roots of Eq. (26) will be the same as the roots of the original system of two FODEs, justifying our argument that the dynamics of the system are retained by the transformation.

Appendix B

Instrument set for single fourth-order equation in first-differenced form

$$\begin{aligned} \left( \begin{array}{ccccccccccccc} S_{i1} &{} 0 &{} 0 &{} \ldots &{} \ldots &{} 0 &{} \ldots &{} \ldots &{} 0 &{} \Delta X_{i6} &{} \Delta X_{i5} &{} \Delta X_{i4} &{} \Delta X_{i3}\\ 0 &{} S_{i1} &{} S_{i2} &{} \ldots &{} \ldots &{} 0 &{} \ldots &{} \ldots &{} 0 &{} \Delta X_{i7} &{} \Delta X_{i6} &{} \Delta X_{i5} &{} \Delta X_{i4} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} &{} \vdots &{} \ddots &{} &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ \vdots &{} \vdots &{} \vdots &{} &{} \ddots &{} \vdots &{} &{} \ddots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \ldots &{} \ldots &{} S_{i1} &{} \ldots &{} \dots &{} S_{iT-5} &{} \Delta X_{iT} &{} \Delta X_{iT-1} &{} \Delta X_{iT-2} &{} \Delta X_{iT-3} \\ \end{array} \right) \end{aligned}$$