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State-Space Modeling of the Relationship between
Air Quality and Mortality
a
b
Christian J. Murray & Charles R. Nelson
a
Department of Economics , University of Houston , Houston , Texas , USA
b
Department of Economics , University of Washington , Seattle , Washington , USA
Published online: 27 Dec 2011.
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between Air Quality and Mortality, Journal of the Air & Waste Management Association, 50:7, 1075-1080, DOI:
10.1080/10473289.2000.10464158
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TECHNICAL PAPER
ISSN 1047-3289 J. Air & Wa
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Copyright 2000 Air & Waste Management Association
State-Space Modeling of the Relationship between Air Quality
and Mortality
Christian J. Murray
Department of Economics, University of Houston, Houston, Texas
Charles R. Nelson
Department of Economics, University of Washington, Seattle, Washington
ABSTRACT
INTRODUCTION
A portion of a population is assumed to be at risk, with
the mortality hazard varying with atmospheric conditions
including total suspended particulates (TSP). This at-risk
population is not observed and the hazard function is
unknown; we wish to estimate these from mortality count
and atmospheric variables. Consideration of population
dynamics leads to a state-space representation, allowing
the Kalman Filter (KF) to be used for estimation. A har-
vesting effect is thus implied; high mortality is followed
by lower mortality until the population is replenished by
new arrivals.
An extensive literature documents evidence that episodes
of severe air pollution are associated with heightened mor-
tality in urban populations. However, the question of
whether mortality occurs within a relatively limited popu-
lation of frail individuals, or whether the risks are more
diffuse across the general population, has a long history
1-3
and is still being examined and debated by researchers.
In this paper, we explore a new approach to model-
ing the relationship between mortality and air quality and
apply it to daily data from Philadelphia, PA. We take as
given that a portion of the urban population is at risk,
subject to a probability of death, or hazard rate, that var-
ies with atmospheric conditions including total suspended
particulates (TSP). New entrants replenish the at-risk popu-
lation over time. Only mortality and atmospheric condi-
tions are observed; the at-risk population, new entrants,
and the hazard rate are unobserved. Because of the de-
pendence of mortality on the unobserved at-risk popula-
tion, the model is not directly estimable as a regression,
but fortunately it may be cast in state-space form. Using
the Kalman Filter (KF) we then infer the hazard rate over
time, its relationship to atmospheric variables, and the
implied path of the unobserved at-risk population. In the
state-space framework, one is able to address the follow-
ing questions: What is the size of the at-risk population?
What is the life expectancy of individuals in that popula-
tion? What is the effect of changes in air quality on that
life expectancy?
The model is applied to daily data for Philadelphia,
PA, 1973–1990. The estimated hazard function rises with
the level of TSP and at extremes of temperature and also
reflects a positive interaction between TSP and tempera-
ture. The estimated at-risk population averages about 480
and varies seasonally. We find that lags of TSP are statisti-
cally significant, but the presence of negative coefficients
suggests their role may be partially statistical rather than
biological. In the population dynamics framework, the
natural metric for health damage from air pollution is its
impact on life expectancy. The range of hazard rates over
the sample period is 0.07 to 0.085, corresponding to life
expectancies of 14.3 and 11.8 days, respectively.
IMPLICATIONS
Mortality in an at-risk population may be modeled in a
state-space framework. The relation of the hazard rate to
TSP and other variables and the unobserved population
are estimated using the KF and maximum likelihood. The
results for daily data from Philadelphia suggest that both
TSP and temperature are risk factors, the effect of both
rising at high temperature levels. The appropriate metric
for health damage is the effect of particulates on life ex-
pectancy, which in the context of daily variation is esti-
mated to be on the order of days.
The implications of the at-risk population model in
state-space form differ fundamentally from those of lin-
4
ear models based on multiple regression. In the latter, a
risk factor such as TSP affects mortality directly, and a
persistent rise in the level of TSP would imply a persistent
rise in mortality count. In the state-space model the im-
pact of a risk factor such as TSP on mortality is indirect; it
is proportional to the size of the at-risk population on
Volume 50 July 2000
Journal of the Air & Waste Management Association 1075

Murray and Nelson
that day which is not observed directly. If the at-risk popu-
lation has been depleted by recent mortality, then the
impact of risk factors will be mitigated since it is a tempo-
rarily smaller population that is at risk. Indeed, it is this
zero. We seek to estimate the γ vector of coefficients and
the time series of the unobserved at-risk population P from
daily data on mortality and atmospheric variables.
Note that eq 2 implies that mortality is a nonlinear
function of Pt-1 and the atmospheric variables, the expected
number of deaths being the product of hazard and popu-
lation. This implies that mortality cannot be expressed as
a linear function of current and past atmospheric vari-
ables with constant coefficients. Indeed, the response of
mortality to a change in atmospheric conditions depends
on the time path of past conditions, reflected in the cur-
rent level of the at-risk population.
“
harvesting effect” which allows the unobserved at-risk
population to be estimated by the KF. If the higher haz-
ard rate persists, the mortality count will fall back toward
its previous level, since in the long run mortality is lim-
ited to the rate of new arrivals. However, the life expect-
ancy of those in the at-risk population will fall, and this
paper presents estimates of that effect.
MODEL SPECIFICATION AND ESTIMATION
To complete the model we need to say how N , the
t
Any population is diminished by deaths and replenished
by the arrival of new entrants; in this case, we studied the
population of individuals in Philadelphia who are at risk
unobserved number of new entrants to the at-risk popu-
lation, evolves through time. In any population, it is the
rate of arrival of new entrants that determines mortality
over a long time span. The time series of mortality of those
over 65 in Philadelphia displays remarkably little long-
term variation over the 14-yr period of our sample. The
annual averages fluctuate in a narrow range around the
long-run average of 35, ranging from 33 to 37, and end-
ing the period at the upper end of that range. Further,
Dickey-Fuller tests strongly reject non-stationarity or drift
over long time horizons. Given the long-run relationship
between arrivals and mortality, these conditions all sug-
gest that the rate of arrivals fluctuates around a long-run
mean during this period in the Philadelphia population.
With this background, we employ the specification
from changing air quality. Using P to denote the number
t
of individuals in the at-risk population at the end of day t,
D the number of deaths (mortality) during day t, and Nt
t
the number of new entrants to the at-risk population dur-
ing day t, the basic equation of population dynamics is
Pt = Pt-1 + N – Dt
(1)
t
Equation 1 may be viewed as simply an accounting rela-
tion that states that the at-risk population at time t is equal
to its previous value plus the number of new entrants less
the number of deaths. The only variable in this relation-
ship that is actually observed in the present context is the
mortality count.
N = N + ηt
(3)
t
Mortality is influenced by atmospheric variables
through a hazard function that operates on the at-risk
population. Listing atmospheric variables in a vector de-
where N is the average number of arrivals, and η is a ran-
dom deviation.
t
noted x , we assume the hazard function to be the linear
The fundamental problem of statistical inference in
this setting is that the at-risk population is an unobserved
variable. If Pt-1 were observed, then one could estimate
eq 2 by least squares, regressing mortality on variables
t
combination of these variables, denoted γ’x . The atmo-
t
spheric variables listed in x will include measures of air
t
pollution such as TSP and weather variables such as aver-
age temperature (AVGT). The elements of the vector γ are
coefficients that indicate how each atmospheric variable
affects the hazard function. These coefficients are un-
known and will have to be estimated from data. The haz-
ard rate is the value of the hazard function at period t,
and is the expected fraction of deaths in the at-risk popu-
lation on that date. Some deaths will result from other
factors that affect mortality but which we have not in-
cluded in x , so there will also be an error term, e , which
that would be cross-products of the x’s and P . However,
t-1
Pt-1 is not observed, so the model is non-linear and can-
not be expressed as a linear regression. Fortunately, this
model is a member of the class of state-space models that
can be made operational using the KF. Briefly, the state-
space representation consists of two equations, a measure-
ment equation and a state equation. The former shows
how the variable we observe and wish to explain depends
on unobserved variables called state variables. The latter
shows how those state variables evolve through time.
Thus, eq 2 is the measurement equation and eqs 1 and 3
are combined to form the state equation.
t
t
is the difference between actual and expected mortality.
We summarize this discussion by the expression
D = (γ’x ) • P + et
(2)
The random shocks in these two equations are as-
sumed to be uncorrelated in the state-space formulation.
Those shocks are errors in predicting mortality and ran-
dom arrivals to the at-risk population, respectively. The
t
t
t–1
A constant in the hazard function captures the average
level of mortality, so the mean value of the error term is
1076 Journal of the Air & Waste Management Association
Volume 50 July 2000

Murray and Nelson
assumption that they are uncorrelated is equivalent to
saying that there is a well-defined at-risk population in
the sense that the movement of individuals from the
population at large to the at-risk population does not de-
pend on mortality. We want the model to estimate the
size of the at-risk population, so we would not want to
exclude a portion of the population that is also at risk.
Given parameter values, we obtain an estimate of the se-
of TSP in the actual data. N was set at the mean mortal-
ity rate in the real data, and the standard deviations of
the error terms were set at 2 and 3, respectively. Series
length was 6000 to be comparable to the daily data we
studied. To increase the validity of the experiment, one
of us generated the data, providing the other with only
the mortality and TSP series to carry out the estimation.
The parameter values were estimated correctly to within
very small tolerances. The accuracy of estimates of the
unobserved population is apparent in Figure 1, where
the generated “actual” population is plotted along with
the KF estimates of population over an arbitrary sequence
of 400 days. The latter tracks both the local trend and
day-to-day fluctuations. The KF requires a prior distri-
5
,6
ries P by “running” the data through the KF. The result-
t
ing estimated level of P , denoted P , is the minimum
t
t|t
mean squared estimate of P based on data up to time t
t
and is referred to as the filtered estimate.
A byproduct of the KF is the construction of the
Gaussian likelihood function using Harvey’s prediction-
7
,8
error decomposition. Maximum likelihood estimation
is carried out using the OPTMUM module of the GAUSS
programming language. The prior distribution for the ini-
tial value of the unobserved at-risk population was given
a mean of 0 and variance of 1000, a very loose prior dis-
tribution that leaves determination of the at-risk popula-
tion to be determined by the data. No prior distribution
bution for the unobserved state variable P at t = 0. The
t
results were not sensitive to modifications of this prior
distribution. We found that the estimated P moved very
t
quickly from its initial guess value to the correct level.
These results give us added confidence in the validity of
our results.
is required for N since it is just a constant plus a random
Results for the Philadelphia Mortality Data
We apply the model to daily data for Philadelphia from
the period 1973–1990 provided to us by the Electric Power
Research Institute. The data set includes daily observations
on the number of non-accident deaths of individuals over
age 65, a measure of TSP, and other atmospheric variables
including temperature, barometric pressure, and ozone. Our
investigation focused on the relationship of mortality to
TSP. Before we were able to estimate the model, we had to
address the problem of missing observations in the TSP
series. Eliminating periods at the beginning and end of the
sample where there are large gaps in the TSP record leaves
a sequence of 5136 consecutive days within which there
are only 50 days missing TSP values. To fill in these gaps,
t
error. Maximum likelihood estimates of the parameters
are computed by maximizing the likelihood function over
the parameter space. This requires that we assume that
the error terms in eqs 2 and 3 are normal, serially ran-
dom, and uncorrelated with each other as well as with
variables in the model. The error terms cannot be exactly
normal in this context since mortality is integer-valued
and non-negative, so what we have is an approximation
to maximum likelihood.
Standard errors for parameter estimates are based on
asymptotic distribution theory that may not be exact in
finite samples, but sample size here is very large. The pa-
rameters to be estimated are the γ vector, the average ar-
rival rate N, and the standard deviations of the two error
terms e and η denoted σ and σ , respectively. The t-statis-
e
η
tic shown in parentheses beside each point estimate in
the tables is based on numerical second derivatives to
obtain standard errors. The value of the log of the likeli-
hood function is also reported, since it gives us a basis for
comparing models using likelihood ratio tests or infor-
mation criteria. For nested models, the likelihood ratio
test is that 2 times the difference in log likelihood will be
approximately chi-square with degrees of freedom equal
to the number of additional parameters in the more com-
plex model.
We have verified that the algorithm does in fact un-
cover the underlying parameters and the at-risk popula-
tion in a system with the structure of eqs 1–3 by
confronting it with artificial mortality data that we gen-
erated. Our hazard function was (0.1 + 0.003•x ) where
Figure 1. Comparison of actual at-risk population and KF estimate
t
the sequence {x } was constructed to mimic the behavior
for simulated data.
t
Volume 50 July 2000
Journal of the Air & Waste Management Association 1077

Murray and Nelson
we regressed TSP on other atmospheric variables, using the
integer part of the fitted series.
Thus, we regard Model 6 as a reasonable baseline specifi-
cation. Note that to interpret the effect of TSP on the haz-
ard rate in Model 6, one needs to keep in mind that the
coefficient of TSP is a linear function of AVGT, so the ef-
fect is positive in spite of the negative coefficient of TSP
alone. We also note that the residuals in the state equa-
tion lack serial correlation as assumed. For Model 6, those
residuals display serial correlation of -0.012 at lag 1 day,
-0.019 at lag 7 days, and 0.026 at lag 365 days, and none
is statistically significant.
How do TSP and temperature affect the hazard rate?
The hazard function of Model 6 implies that both extremes
of temperature are harmful and that TSP is also harmful,
with that effect rising with temperature. At a typical level
of TSP, the effect of an increase in temperature from 40 °F
to 90 °F is an increase in the hazard rate from 0.07 to
0.08. At a high temperature of 90 °F, the effect of an in-
crease in TSP from zero to the highest levels observed in
the sample is to raise the hazard rate from 0.075 to 0.091.
Any statements about the effect of TSP on the hazard rate
need to be tempered by the uncertainty connected with
these estimates.
Figure 2 plots the KF estimate of the at-risk popula-
tion and the estimated hazard rate in Model 6. The esti-
mated at-risk population averages 484 and varies
seasonally. Since average mortality is 35 per day, this im-
plies that about 7% of the at-risk population dies on aver-
age per day. The corresponding time series of the hazard
rate in Figure 2 is stable around its mean of 0.073, re-
maining well within the required (0,1) interval for a prob-
ability. The hazard rate fluctuates seasonally as periods of
high TSP and temperature extremes reap a grim harvest,
followed by less lethal conditions. As noted above, the
mortality series is essentially trendless, so it is of some
interest that the estimated at-risk population series does
move higher with time, from an annual average of around
470 in early years to around 500 at the end. There is a
Table 1 displays maximum likelihood estimates of the
observation and state equations for six baseline specifica-
tions of the x vector that include various combinations of
TSP and AVGT, the square of temperature, and the multi-
plicative interaction of temperature with TSP. We chose
AVGT as a baseline variable because of its strong linear
association with daily mortality. A constant term is in-
cluded in every case. Model 1 uses only TSP, which is
highly significant. Model 2 adds AVGT, which is also
highly significant. Model 3 adds the square of AVGT to
allow for a hazard rate that increases at both extremes of
temperature, which is also significant. Model 4 allows the
effect of TSP and AVGT to each depend on the value of
the other by adding the interaction variable AVGT*TSP.
That interaction is significant, while TSP and AVGT alone
are much less significant than in Model 3. Model 5 is in-
cluded for comparison purposes and uses only the AVGT
variables. Model 5 has roughly the same log likelihood as
Model 1, which used only TSP. Finally, Model 6 uses all of
the variables considered in this table: TSP and AVGT, the
square of AVGT, and the interaction AVGT*TSP.
How important is TSP as a predictor of mortality?
When comparing the log likelihood for Model 6 with that
of Model 5, which does not include either TSP-related
variable, the difference in value is about 14, which is
highly significant (p-value is off the chi-square table). How
important are the temperature-related variables? When
comparing Model 6 with Model 1, the difference in log
likelihoods is again about 14, and even with three addi-
tional parameters that difference is overwhelmingly sig-
nificant. We also note that, although the interaction
variable AVGT*TSP has a t-statistic of only 1.42 in Model
6, when we compare the log likelihood with that of Model
3
, the difference is about 2. That corresponds to a p-value
of about 0.05, supporting a role for the interaction term.
Table 1. Parameter estimates for baseline state-space models, Philadelphia data.
Coefficient
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
1
0.0632924 (8.61)
0.0000534 (5.60)
n.a.
0.0650052 (9.34)
0.0000480 (5.50)
0.0001002 (2.96)
n.a.
0.0696869 (8.95)
0.0000412 (4.09)
-0.000167 (-2.95)
0.0000030 (4.34)
n.a.
0.0686944 (7.72)
-0.000023 (-0.69)
0.0000079 (0.62)
n.a.
0.0765619 (8.94)
n.a.
0.0711416 (1.89)
-0.000007 (-0.21)
-0.000168 (-0.21)
0.0000023 (0.28)
0.0000009 (1.42)
5.754223 (49.13)
19.15221 (5.96)
35.08867 (131.02)
-16910.97
TSP
AVGT
-0.000213 (-1.88)
0.0000041 (3.35)
n.a.
2
AVGT
n.a.
AVGT*TSP
n.a.
n.a.
0.0000013 (2.28)
5.755896 (73.14)
19.18479 (13.39)
35.08771 (130.85)
-16914.00
σ
5.80991 (76.69)
19.7923 (13.63)
35.0947 (126.84)
-16924.9
5.756863 (74.90)
19.04966 (14.81)
35.08667 (131.80)
-16919.03
5.754371 (74.15)
19.05996 (13.95)
35.08826 (131.73)
-16913.12
5.731345 (73.79)
18.58655 (14.78)
35.08178 (135.08)
-16924.72
e
ση
N
lnL
Notes: t-statistics for the null hypothesis that an individual coefficient is zero are in parentheses beside it. The statistic lnL is the log of the likelihood function.
1078 Journal of the Air & Waste Management Association
Volume 50 July 2000

Murray and Nelson
Does TSP have a cumulative effect on mortality over
several days? We investigated this possibility by adding
the level of TSP in preceding days to the list of variables
in the hazard function in Models 1–3. The results for add-
ing 4 or 8 lags of TSP are displayed in Table 2, which dif-
fers from Table 1 by adding another line listing the
likelihood ratio statistic for testing the null hypothesis
that all the added lagged values of TSP have a zero coeffi-
cient. What the test results indicate is that lags of TSP do
have explanatory power that is statistically significant,
but the coefficient estimates suggest that the effect is not
large. The coefficient of current-day TSP is reduced when
lags are added; the first lag is positive and significant, but
the second lag is negative and remaining lags have small
estimated coefficients. The sum of all TSP coefficients for
Model 3 is raised, however, by almost half when 8 lags
are added. How do we understand a negative effect of TSP
after two days? One explanation is that the apparent ef-
fect of lags of TSP is partially statistical rather than bio-
logical. If we measure TSP imperfectly, due to aggregation
over the 24-hr day or extrapolation of atmospheric mea-
surements across a geographical region, then TSP on ad-
jacent days may contain additional information relevant
to the explanation of mortality. The lag pattern we ob-
tain then may represent, at least in part, a kind of moving
average measure of TSP rather than a biological process.
Both factors may well be at work in producing the pat-
tern seen here.
Figure 2. Estimated hazard rate and KF estimate of at-risk population
in Model 6.
corresponding and offsetting decline in the hazard rate,
moving downward from an annual average of about 0.072
in early years to 0.0716 in later years. This decline in the
hazard rate is driven by TSP declining from annual aver-
ages of about 80 to about 70, the result of efforts to clean
up the air around Philadelphia. A lower hazard rate length-
ens life expectancy and allows individuals to stay longer
in the at-risk population, thereby making that popula-
tion larger. Finally, we note that output from all six mod-
els is qualitatively similar in producing population and
hazard rate estimates that are stable and positive.
Table 2. Parameter estimates for state-space models with lags of TSP, Philadelphia data.
Coefficient
Model 1a
Model 1b
Model 2a
Model 2b
Model 3a
Model 3b
1
0.061616 (8.62)
0.000043 (4.64)
0.000034 (4.11)
-0.000011 (n.a.)
0.000009 (2.26)
0.000007 (3.20)
n.a.
0.057272 (8.30)
0.000042 (1.62)
0.000033 (0.94)
-0.000011 (-0.22)
0.000009 (0.51)
0.000004 (0.41)
0.000005 (0.39)
0.000010 (1.26)
-0.000006 (-0.68)
0.000000 (0.01)
n.a.
0.063138 (9.86)
0.000040 (4.76)
0.000030 (3.73)
-0.000013 (n.a.)
0.000009 (6.11)
0.000005 (n.a.)
n.a.
0.058708 (8.76)
0.000040 (4.43)
0.000030 (3.18)
-0.000013 (-4.51)
0.000008 (n.a.)
0.000002 (0.38)
0.000004 (0.68)
0.000010 (1.58)
-0.000006 (-1.31)
-0.000000 (0.05)
0.000058 (1.22)
n.a.
0.067901 (10.85)
0.000033 (4.31)
0.000028 (3.33)
-0.000015 (-2.38)
0.000007 (n.a.)
0.000004 (3.03)
n.a.
0.0632121 (16.03)
0.000033 (4.58)
0.000027 (n.a.)
-0.000014 (n.a.)
0.000007 (n.a.)
0.000002 (n.a.)
0.000004 (n.a.)
0.000011 (n.a.)
-0.000006 (n.a.)
0.000000 (n.a.)
-0.000177 (2.44)
0.000003 (3.41)
5.79049 (81.75)
19.5993 (15.91)
35.0935 (128.01)
16877.2
TSP
TSP(-1)
TSP(-2)
TSP(-3)
TSP(-4)
TSP(-5)
TSP(-6)
TSP(-7)
TSP(-8)
AVGT
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
0.000070 (2.58)
n.a.
-0.000018 (0.00)
0.000003 (3.06)
5.76622 (78.30)
19.1418 (16.43)
35.0877 (131.23)
-16898.6
2
AVGT
n.a.
n.a.
σ
5.80483 (76.22)
19.5818 (13.79)
35.0924 (128.18)
-16906.7
5.82535 (79.24)
20.0188 (13.54)
35.0979 (124.58)
-16884.9
5.73976 (77.88)
19.1180 (15.99)
35.0866 (131.30)
-16904.1
5.79522 (74.44)
19.5942 (14.15)
35.0924 (128.04)
-16882.8
e
ση
N
lnL
LR
*
**
***
***
***
***
***
36.4
80.0
38.4
72.46
29.04
67.54
Notes: LR is the likelihood ratio statistic for testing the null hypothesis that the coefficients on all lags of TSP are zero, and in each case LR is significant at a p-value below 0.01.
Individual t-statistics are not available in cases where the standard error could not be computed. See also the notes to Table 1.
Volume 50 July 2000
Journal of the Air & Waste Management Association 1079

Murray and Nelson
We have also experimented with adding other atmo-
spheric variables to baseline Model 6. Barometric pres-
sure on the same day enters the model with a negative
and significant coefficient, while the coefficients on TSP
and AVGT change only slightly. The resulting estimated
population and hazard series are essentially indistinguish-
able from those of Model 6.
the latter to 11.8 days, a difference of about 2.5 days on
average for the roughly 500 at-risk individuals in Phila-
delphia.
In summary, we feel that the state-space approach
represents a very promising avenue of research in model-
ing the effects of air pollution. Its primary advantage
comes from explicit recognition of the role of population
dynamics in mortality and from making use of the KF to
estimate the population model directly. The results re-
ported here are preliminary but suggest that the approach
is practical and provides explanations of otherwise puz-
zling results from conventional analysis.
DISCUSSION AND CONCLUSIONS
In thinking about the dynamics of the state-space model,
we have found it useful to compare it to the more famil-
iar linear regression or distributed lag model linking mor-
tality to risk factors. The latter have the form
ACKNOWLEDGMENTS
(
4)
This research was supported by the Electric Power Research
Institute. We are grateful to Fred Lipfert and Ron Wyzga,
who provided invaluable information, suggestions, and
insights throughout this project. We also received help-
ful comments from Richard Burnett, Robert Engle, Levis
Kochin, Jeremy Piger, Richard Startz, Mark Wohar, and
anonymous referees. The views expressed, however, are
those of the authors who are solely responsible for the
content of this article.
The impact of xt-k on mortality is given by fixed coef-
ficients in eq 4, while in the state-space model the impact
depends on the unobserved state variable P. Because this
functional relationship is multiplicative, the state-space
model is highly non-linear. In particular, the impact will
be smaller if recent mortality has been high, since the at-
risk population will have been reduced. This harvesting
effect is small in the models we estimated, but it is very
persistent since the rate of replenishment of the popula-
tion by new arrivals is slow. Thus, the relation of past
atmospheric conditions to mortality is highly non-linear
in the state-space model.
If we nevertheless run regressions of the form of eq 4
on the Philadelphia data set with TSP as the risk factor,
there is a consistent pattern in the coefficients obtained.
For example, with lags up to 7 days the lag zero coeffi-
cient is 0.013, but the sum over lags 0 through 7 days is
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-
0.041. The character of the results is not sensitive to in-
cluding more lags and other variables; the leading coeffi-
cient on TSP is positive but the sum βk is small or
Σ
negative. Thus, a positive association between mortality
and TSP on the same day is more than offset by a sequence
of negative coefficients, as predicted by the harvesting
effect. The state-space model also predicts that the sum
of coefficients will be small, since over the long run the
level of mortality is only determined by the rate of arriv-
als, not by risk factors. Thus the state-space model pro-
posed here helps explain why multiple regression produces
results that are otherwise hard to explain, namely an ap-
parent negative effect of TSP on mortality after a lag.
While an increase in risk factors cannot increase
mortality in the long run, life expectancy is the inverse of
the hazard rate, so hazard-causing agents will shorten it.
The range of hazard rates observed over the sample pe-
riod is roughly 0.07 to 0.085. Since life expectancy is the
reciprocal of the hazard rate, the former corresponds to a
life expectancy in the at-risk population of 14.3 days and
About the Authors
Christian J. Murray is assistant professor of Economics at
the University of Houston (Department of Economics, Hous-
ton, TX 77204; cjmurray@uh.edu). Charles R. Nelson is the
Ford and Louisa Van Voorhis Professor of Economics at the
University of Washington (Department of Economics, Box
353330, Seattle, WA 98195; cnelson@u.washington.edu).
Both authors work in time-series econometrics and have
collaborated on papers that investigate the nature of trend
and cycle in GDP. They were consultants to the Electric
Power Research Institute on this project.
1080 Journal of the Air & Waste Management Association
Volume 50 July 2000