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

Andrew M. Jones; Audrey Laporte; Nigel Rice; Eugenio Zucchelli

doi:10.1007/s00181-017-1367-6

Empirical Economics

February 2019, Volume 56, Issue 2, pp 703–733 | Cite as

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

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Authors and affiliations

Andrew M. Jones
Audrey Laporte
Nigel Rice
Eugenio Zucchelli

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First Online: 18 January 2018

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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.

Keywords

Addiction Health capital Smoking Dynamic panel data models

JEL classification

C1 C3 C5 C6 I1

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Notes

Acknowledgements

The authors gratefully acknowledge funding from the Economic and Social Research Council (ESRC) under Grant Reference RES-060-25-0045. We are grateful to Victor Aguirregabiria, Jonneke Bohlaar, Brenda Gannon, Eleonora Fichera, Aloysius Siow, Matt Sutton, and participants of the 8th International Health Economics Association (iHEA) Congress in Toronto, the 9th European Conference on Health Economics (ECHE) in Zurich, the 2012 Applied Health Econometrics Symposium at the University of Leeds, the 2012 Netspar Conference on Health and Inequality in the Life-cycle in Rotterdam as well as participants of seminars at Universities of Barcelona, Leeds, Newcastle, Toronto and York for discussions and helpful comments. Data from the British Household Panel Survey were supplied by the Economic and Social Research Council Data Archive. Neither the original collectors of the data nor the Archive bear any responsibility for the analysis or interpretations presented in this manuscript.

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

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

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Authors and Affiliations

Andrew M. Jones
- 1
Audrey Laporte
- 2
- 3
Nigel Rice
- 1
- 4
Eugenio Zucchelli
- 5
Email author

1.Department of Economics and Related StudiesUniversity of YorkYorkUK
2.Institute for Health Policy, Management and EvaluationUniversity of TorontoTorontoCanada
3.Canadian Centre for Health EconomicsUniversity of TorontoTorontoCanada
4.Centre for Health EconomicsUniversity of YorkYorkUK
5.Division of Health ResearchLancaster UniversityLancashireUK

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

Abstract

Keywords

JEL classification

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Acknowledgements

References

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