The main objective of this research is to study the relationship between intensity of
exercise and mental health. Utilizing a panel data from SHLSE, the fixed-effect model
is estimated. We will begin the discussion of this section by introducing the formal
specification of each econometric model, and then we pay special attention on
fixed-effect Tobit and fixed-effect probit models in section 4.2 and 4.3.
4.1 Econometric specifications
To reach our goal, several econometric issues need to be tackled. First, it is
necessary to control for unobserved heterogeneity problem to address the effects of
exercise on mental health. Because there are likely to be some unobservable
components which jointly determine both exercise behavior and mental health, without
controlling for such effects will lead to be biased estimations. For instance,
unobservable personal characteristics, such as personality and motivation, may be
related to mental health. This would be true if those who are more introverted or less
motivated had less chance to work out and a higher depression tendency.4 Due to the
4 If such variables are unobserved, they will be captured by random error term, which results in a
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lack of valid and convincing instruments of these variables, we use the
individual-specific fixed effects model to control for the effects of unobserved
heterogeneity on mental health of the elderly. Second, it is a need to correct the
censoring problem which violates the basic assumption of the least-squared estimation.
If a specific individual was not depressed at all, the estimations of the CES-D equations
will be zero. Therefore, treating those groups of people as the same as other groups of
respondents are not appropriate. Third, since our third dependent variable is classified
into two classes, it is reasonable to use a binary regression model rather than ordinary
least squares estimation.
In a panel data model, a popular estimation method is to use fixed-effect model.
Unlike the general linear panel data model, the random error term uit is decomposed to
individual-specific effects and an idiosyncratic error in fixed-effect model.5 Using a
fixed-effect estimation, we are able to control for all unobserved time-invarying
individual-specific factors that are related to mental health.6
Assuming that the individual's continuous CES-D score of mental health (yit) is
determined by a linear combination of exercise behavior and other time-varying
independent variables, the effect of exercise intensity on mental health can be estimated
correlation between exercise variables and random error term.
5 The general linear panel data model is specified as: yit=α+xitβ+uit, where yit and xit are dependent variable and independent variables respectively; uit is a random error term for individual i at time t.
6 In contrast to fixed-effect model, the random-effect model assumes that vi are random variables.
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using the following fixed-effect OLS model:
_ β _ β β ,
it it M it L it i it
y M exercise L exercise x v
(1)where the subscript i and t refer to individual and time period respectively; M_exerciseit
and L_exerciseit represent the indicators of moderate and light exercise intensities
respectively. The reference group is those who don't work out regularly at least once per
week. Thus, the parameters βM and βL represent the effects on mental health score of
each corresponding group compared to the reference group; xit is a vector of other
independent variables with corresponding vector of the parameters β; vi is
individual-specific time-constant effect and εit is assumed to be a mean-zero,
constant-variance random error term that is uncorrelated with the independent variables.
4.2 Fixed-effect Tobit model
About 28% people reported zero CES-D scores in each year. Due to the censoring
nature of the second defined dependent variable, a fixed effect Tobit model is specified
with the following equation. In our sample,
* _ β _ β β ,
it it M it L it i it
y M exercise L exercise x v
(2) where y*it is a unobservable latent variable of the censored CES-D score of eachindividual's mental health; M_exerciseit, L_exerciseit, xit, vi and εit are defined as above.
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density function (CDF) and probability density function of standard normal distribution
respectively; note that we use z itα here to represent whole independent variables for
convenience.
As indicated in Greene (2003), the quantitative effects of the exogenous variables
are better presented by the marginal effects: which measure the changes in exogenous
variables on the conditional expected means of the mental health scores. Note that the
calculation of marginal effects is not trivial in nonlinear models, because taking direct
partial derivatives are not applicable for exogenous dummy variables (Greene 2003).
The appropriate formulates of the marginal effects for continuous and binary
independent variables are presented below. Here we distribute the independent variables
into a specific dummy variable (dit) and all the other variables (cit) for convenience.
Given the conditional mean function:
30 standard errors are computed by delta method (Greene 2003).
4.3 Fixed-effect probit model
Our third dependent variable is a binary indicator of depression which is defined
using the appropriate cut-off point (i.e. CES-D score ≥ 10 for depression; and 0
otherwise). If the unobserved latent variable is indicated by the continuous variable Iit
*,
we incorporate the same independent variables discussed above and use the following
fixed-effect probit model:
* _ β _ β β .
it it M it L it i it
I M exercise L exercise x v
(8)31
The mean function for fixed-effect probit model is
[ it it, it] ( itβc itβd i).
E I c d
c
d
v
(11)The corresponding marginal effects for fixed-effect probit model are
[ , , ]
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