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Empirical Economics

Erschienen in:

18.01.2018

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

verfasst von: Andrew M. Jones, Audrey Laporte, Nigel Rice, Eugenio Zucchelli

Erschienen in: Empirical Economics | Ausgabe 2/2019

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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.

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Note, however, that most studies are not conclusive in this respect and often produce implausible estimates of discount rates. However, see Gruber and Koszegi (2001) for a discussion of potential dynamic inconsistencies in preferences with respect to smoking. Note that our paper is concerned with embedding the Grossman model within the B–M framework and is not explicitly concerned with estimating discount rates.

Baltagi and other authors have also extensively explored the application of panel data methods on both tobacco and alcohol consumption, e.g. Baltagi (2007), Baltagi and Geishecker (2006) and Baltagi and Griffin (2001).

B–M models often include a wealth equation (e.g. Becker and Murphy 1988). This is omitted here as it is not essential to our narrative.

In this version of the model, where borrowing is not permitted, $C_t$ and $S_t$ are tied together by the budget constraint (3), which allows us to substitute $C_t$ out of the problem.

This assumption of linearity does not impact on the qualitative solution to the problem.

Some studies have used biological markers which are used to reflect the addictive stock (Adda and Cornaglia 2006, 2010).

More generally M can represent any good which is beneficial for health but yields no direct utility. We follow the standard approach in the literature and denote this as medical care.

The arguments of utility, for example, $U_C \left( t\right) $, denote the time period to which utility refers.

This is because the B–M problem is, as noted above, an inter-temporal optimisation problem, which is typically set up as an optimal control problem. The solution equations to an optimal control problem are necessary conditions for optimising the present value of the stream of future utilities, which will arise from future consumption decisions, taking account of how current consumption affects future addiction.

See “Appendix” for an illustration of which exogenous variables from the original system end up in each equation.

It might be argued that this assumption is unrealistic. However, relaxing the assumption requires either the identification of external instruments for $W_i$, or relying on instruments internal to the model. The latter might consist of transformations of $X_{it}, \ldots , X_{it-3}$ where a subset of these are assumed to be uncorrelated with the unobserved individual-specific effect in the spirit of Hausman- and Taylor-type estimators (Hausman and Taylor 1981). Alternatively, differences in the lags of the dependent variable can be used as instruments. It is not surprising, however, that such instruments are weak when used in this context. Estimates of the lagged dependent variables, $S_{it-1},\ldots ,S_{it-4}$ and $H_{it-1}, \ldots , H_{it-4}$, do not change dramatically for models estimated without the vector of time-invariant regressors.

In addition to the MA(3) structure, the error terms will also include cross-error elements. Note also that the equation-specific disturbance terms in (8) and (9) may contain the disturbance terms from the original system of four equations. See “Appendix” for an illustration of these issues.

The corresponding matrix of coefficients defining the data-generating process (DGP) for the MC exercise is:

$$\begin{aligned} \begin{bmatrix} Y_{1t} \\ Y_{2t} \end{bmatrix}= & {} \begin{bmatrix} 100\\ 300 \end{bmatrix} +\begin{bmatrix} 0.9&-0.4\\ .10&.95 \end{bmatrix} \begin{bmatrix} Y_{1t-1} \\ Y_{2t-1} \end{bmatrix} +\begin{bmatrix} 0.4&0\\ 0&0.6 \end{bmatrix} \begin{bmatrix} X_{1t} \\ X_{2t} \end{bmatrix} +\begin{bmatrix} \varepsilon _{1t} \\ \varepsilon _{2t} \end{bmatrix}\\ \begin{bmatrix} X_{1t} \\ X_{2t} \end{bmatrix}= & {} \begin{bmatrix} 0.6&0\\ 0&0.4 \end{bmatrix} \begin{bmatrix} X_{1t-1} \\ X_{2t-1} \end{bmatrix} +\begin{bmatrix} \epsilon _{1t} \\ \epsilon _{2t} \end{bmatrix} \end{aligned}$$

The matrix multiplying the vector of lagged Y’s contains the $\phi $’s, while the matrix multiplying the vector of X’s includes the $\theta $’s. For simplicity, we made this matrix of Y’s diagonal such that $\theta _{12}=\theta _{21}$= 0.

To run the experiment, we use the MC routine within the PcNaive module of the PcGive econometrics package. For the first equation, the initial values for the lagged Y’s were 10,000 for $Y_{t-1}$ and 9000 for $Y_{t-2}$. For the second equation, the initial values for the lagged Y’s were also 10,000 for $Y_{t-1}$ and 9000 for $Y_{t-2}$. The X’s were drawn from a normal distribution with mean 0 and variance 100 for both equations. Since we have a dynamically stable and unique equilibrium, regardless of what initial values we chose, the system would converge on that equilibrium. We simply chose these values to be sure we were starting the process a long way away from the equilibrium.

We have also explored extending the simulation exercise to a system of four first-order difference equations, results are available upon request.

Note that further lagged differences of the dependent variable are redundant when combined with instruments for the first-differenced equation (see Blundell and Bond 1998).

We have explored specifications where lags of smoking and health enter the same equation as well as the joint estimation of a system of two second-order difference equations, each including lags of health and smoking. The former were estimated by employing a similar GMM approach and overall did not appear to produce reliable estimates if compared to their OLS within fixed effects counterparts. (Estimates are available upon request.) The latter was estimated via GMM three-stage least squares (3SLS) and failed to achieve convergence.

Before 1999 (wave 9), Scottish individuals were only sampled if they resided South of the Caledonian Canal.

For further details on the BHPS sample structure, see Lynn (2006).

The lag structure imposed by serial correlation in the error term determines the exact number of lags required to construct valid instruments.

Additional and updated information on the SF-36 and its related literature are available on the SF-36 community web page (http://www.sf-36.org).

As specified in Brazier et al. (2002), to build the SF-6D (where 6D stands for six dimensions) they have excluded general health items and collapsed the two dimensions of role limitations due to physical and emotional problems into a single role limitations dimension.

These cover problems related to arms, legs or hands, sight, hearing, skin conditions, chest/breathing, heart/blood pressure, stomach or digestion, diabetes, anxiety/depression, alcohol or drug use, epilepsy, migraine.

Due to a change in wording and response categories in the SAH question at wave 9, we collapse the original five-category self-assessed variable (SAH) to a four-category measure. In waves 1–8 and 10–18, respondents are asked: Compared to people of your own age, would you say your health over the last 12 months on the whole has been: excellent, good, fair, poor or very poor?, whereas in wave 9, the question and possible answers are: In general, would you say your health is: excellent, very good, good, fair, poor?. Creating a SAH variable with four health categories (excellent, good or very good, fair, poor or very poor) allows common support over the two versions of the question.

We have also employed alternative specifications to compute predicted SF-6D values for all individuals in the sample such as linear fixed effects models. We have also estimated versions of these models (pooled OLS and linear fixed effects models) with lagged values of all regressors (health variables) to ease potential problems related to endogeneity. As predicted SF-6D scores and results from the main dynamic empirical models of health capital and smoking do not appear to differ across these specifications, we use results from the simple pooled OLS models to maximise the number of observations.

Other employment consists of looking after the family, maternity leave, government training, student or other jobs.

The set of regional dummy variables contains little variation across the waves, and accordingly these are categorised as the region in which a respondent was observed to reside the longest. The regions cover England (South East, South West, London, Midlands, Yorkshire, North West, North East), Scotland, Wales and Northern Ireland. The South East is taken as the baseline.

It is plausible that the inclusion of higher- order dynamics via four lags in our models may absorb any remaining serial correlation in the disturbances.

$S_{it-2}, \ldots , S_{it1}$ are potential instruments for $\Delta S_{it-1},\ldots , \Delta S_{it-4}$. Similarly, for $\Delta H_{it-1}, \ldots , \Delta H_{it-4}$.

These results are available on request.

The year dummies, which are not reported, indicate a decreasing trend in smoking across the waves for men but not for women and no discernible trends for health.

For example, it has been reported elsewhere that Black-Caribbean and Other South Asian women have a far greater prevalence of regular smoking (at levels slightly lower than white ethnic groups) compared to Bangladeshi, Indian, Pakistani , Chinese and other Black minority ethnic groups (Mellward and Karlson 2011). There is evidence of geographical variation, particularly for women where areas associated with decreased health (compared to the baseline of South East of England) are also associated with increased smoking.

Of the remaining variables, the lack of significance may well be a consequence of a lack of within-individual variation. Alternatively, it may indicate that these variables do not have a role in shifting the optimal trajectory for an individual. If these are variables which do not change often and which have relatively small impacts on the position of the individual’s optimal trajectory, it is not surprising that, conditional on the intrinsic dynamics, they do not appear important.

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Titel: Dynamic panel data estimation of an integrated Grossman and Becker–Murphy model of health and addiction
verfasst von: Andrew M. Jones
Audrey Laporte
Nigel Rice
Eugenio Zucchelli
Publikationsdatum: 18.01.2018
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 2/2019
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-017-1367-6

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