Alternative methods for estimating systems of health equations

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Alternative methods for estimating systems of
(health) equations
Silvia, Andrew and Casey
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Motivating example
From Balia and Jones (2005) analysis of
jointly-determined dichotomous outcomes for
health, mortality and lifestyles.

• What are we motivating? A couple of things
• A generalisable method of estimating (at least
  reduced-form) systems of equations in which the
  data-generation process(es) can be identified at
  each equation,
• A method through which information can be
  retrieved from data via multivariate modelling.

Information on health, mortality and lifestyles
(eating breakfast, sleeping well, under-obese
weight, non-smoking, drinking alcohol prudently
and exercising) are taken from the original
(1984-5) Health and Lifestyle Survey with the
most recent follow-up (2005).
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Behavioural problem
Balia and Jones (2005) individuals are assumed
to maximise simultaneously the utility function

• where tth period utility is determined by
• the vector of l (in this case 6) lifestyles, Olt
• health Ht,
• exogenous variables XU, and
• unobserved µU

Time preferences are also affected by ßt with
mortality risk pt
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Econometric problem
This formulation of individual utility gives 3
elements to be estimated health, lifestyles and
mortality . In reduced-form the system to be
estimated is
Each of SAH (1 good), Mortality (1 dead) and
lifestyles breakfast (1 eating one), obese (1
not), non-smoker (1 is), prudent drinker (1
is), sleeping well (1 is) and exercise (1
undertaken) is dichotomous/dichotomised.
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Statistical problem
Estimated according to unobserved latent
variables yim, yih and yil, this presents us
with a fairly standard multivariate binary choice
model
Ordinary responses to this problem are the use of
the multivariate probit or logit.
Ours will be a multivariate copula.
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Copulas I revision
For every joint distribution H(x1,x2) whose
random variables are individually distributed
F1(x1), F2(x2) there exists a copula C, such that
H(x1,x2) C(F1(x1),F2(x2))
C is a function of F1(x1), F2(x2), not x1,x2.
Every copula is unique, and uniquely represents
the association between X1,X2.
There are many copulas C(F1(x1), F2(x2)), but
only one true joint distribution H(x1,x2), as for
F1(x1) and F2(x2) in a univariate framework.
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Copulas II example
Consider the Farlie-Gumbel-Morgenstern family.
The bivariate FGM copula/distribution is given by
The essential property of the copula is that
F1(x1) can be of any type. A secondary property
is the separation of each F1(x1) from the
dependence parameter ?. This is so for the
density also, given by
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Multivariate Copulas
In (very) basic statistical terms there are 4-5
multivariate copula(type)s available to us the
multivariate FGM, multivariate Archimedean,
Gaussian/t, Mixture-of-powers and
Mixture-of-Max-ID.

• In practical terms, however,
• The FGM and the Archimedean n-copulas are too
  limiting in their capture of dependence
• The Mixture-of-Powers copula is a combination of
  limited and unworkable, in dimensions this high.

By reduction our analysis will be confined to the
Gaussian, t and a mixture of Max-ID copulas.
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Gaussian and t Copulas
The Gaussian copula is based on the so-called
method of inversion. It is the copula C such
that
Thus
We still retain autonomy over each marginal
distribution type, and dependence is captured
generally by Pearsons correlation between
functions (rather than random variables).
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Gaussian and t Copulas
Similarly the t-copula is H(.) such that
Essentially this is the same. Thanks to the
nature of the t-distribution however, the
multivariate t-copula (or distribution) is a mite
narrower, with more tail dependence, or rather
better capture of tail dependence, if it is
there.
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Mixture of Max-ID Copulas
The Mixture of Max-ID copula has two essential
elements the mixture part, and the Max-ID part.
Max-Infinite Divisibility equivalent to Total
Positivity of Order 2 where, for all x1 lt y1 and
x2 lt y2 with x, y ? R, any non-negative function
g will give
g(x1, x2)g(y1,y2) gt g(x1, y2)g(x2,y1)
I.e. we demand strict monotonicity.
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Mixture of Max-ID Copulas
The mixture is a method by which Laplace
transformations are used to construct
multivariate distributions from univariate (the
mixture of powers) or lower-dimensionally
multivariate (the mixture of Max-ID) ones.
For some CDF M of a positive random variable a,
and Laplace transform ?, the multivariate
distribution function F can be written in terms
of some max-ID distribution H, thus
Although we dont need to know this, necessarily.
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Mixed-max-ID Frank copula
The mixed-distribution is then
The mixture-of-max-ID copula used in this
analysis is a mixture of the Frank copula
Using Joes Laplace transform A
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Mixed-max-ID Frank copula
Finally the copula is
Although we are more likely to use
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(non-)normality I
The desire to go to all this trouble is
(potentially) several-fold (two, at least). One
is control over univariate random variation, the
other is the multivariate behaviour.
These arguments tend to be relative to
(multivariate) normality. Although the normality
assumption is fairly robust to departures from
normality, it is less so with higher dimensions.
In systems of (structural) equations, such Balia
and Jones multivariate non-normality can also
lead to erroneous rejection of some models within
the structure.
As well as protecting against this, sometimes
information at hand suggest non-normality (in
dichotomous-variables estimation, we may have
reason to prefer logit over a probit, for some
margins
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(non-)normality II

• In a multivariate model, the nature of the
  multivariate association is of (more or less)
  importance
• even entirely normally-distributed margins may
  not have a multivariate normal distribution,
  necessarily
• many (non-normal) distributions do not extend to
  a multivariate form (e.g. the Beta)
• in our analysis, we may prefer the logit but
  face its multivariate correlation restrictions
• or want the complementary log-logs asymmetry
  but have no multivariate distribution.

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Marginal distributions I
In fact we find no consistent results favouring
univariate probits in our multivariate model
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Marginal distributions III
Nor do we consider a multivariate (latent)
logistic framework preferable
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Marginal distributions II
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Marginal distributions IV
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Marginal distributions V
Ergo we would like to accommodate H(ym, yh, yl1,
yl2, yl3, yl4, yl5, yl6) such that
F1(ym) Logit F2(yh) Complementary
log-log F3(yl1) Complementary log-log F4(yl2)
Logit F5(yl3) Logit F6(yl4) Complementary
log-log F7(yl5) Logit F8(yl6) Complementary
log-log
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Univariate estimates I
What will this mean for results and
interpretations?
Pr (drinking prudently) marginal effects
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Univariate estimates II
What will this mean for results and
interpretations?
Pr (drinking prudently) p-values
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Estimating copulas
Ordinarily, the method of maximum likelihood is
to solve (?L/?ß1,..., ?L/?ßn, ?L/??) 0.
We use the method of Inference Functions for
Margins (IFM) to solve (?L1/?ß1 0,..., ?Ln/?ßn
0, ?L/?? 0).

• This is then a 2-step procedure in which we
• First maximise each L1(ß1),..., Ln(ßn),
  retrieving ML estimates.
• Then maximise L(ß1,..., ßn, ?) to get the ML
  estimate of ?.

This facilitates the use of the Gaussian/t
copulas, as well as simplifying analysis of
ordinary copulas.
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Estimating the Gaussian
In accordance with the method of IFM, the
predicted probabilities (of choice) from each
margin is used
e.g.
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Estimating the t
The t-copula is similarly determined
e.g.
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Estimating the Max-ID I
The Max-ID copula requires a 3-step procedure,
which is achieved with some statistical
sleight-of-hand
Also a survival function for mortality.
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Estimating the Max-ID II

• For this
• First estimate the margins, as before,
• Then estimate the bivariate margins separately,
  replacing ? with some constant (gt 1),
• Finally, replace ? as a random variable, with
  numerical values for ?, and get the ML estimate
  of global dependence.

What are the implications for bootstrapping the
standard errors of ß? Unknown.
Bootstrapping may not be employed analytic
derivatives will be available (maybe).
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Comparing models I
Now the use of information criteria are more
informative.
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Comparing models II
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Univariate estimates II
What will this mean for results and
interpretations?
Pr (drinking prudently) p-values
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Further analysis
Going to try out-of-sample predictions.

• Copulas allow us to get away with a lot can we
  get away with more? Such as
• Hazard models for death,
• Ordered models for health,
• Count models for lifestyles

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Marginal distributions IV
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Marginal distributions II