3. INTRODUCTION
The term life expectancy refers to the number of years a person can expect
to live. By definition life expectancy is based on an estimate of the average
age that members of a particular population group will be when they die. If a
child is born today in a country where the life expectancy is 75, they can
expect to live until they are 75. The statistic life expectancy actually refers to
the average number of years a new born is expected to live if mortality
patterns at the time of its birth remain constant in the future.
Despite improving healthcare, the gap in mortality between people with
Serious Mental Illness (SMI) and general population persists, especially for
younger age groups. The electronic database from a large and
comprehensive secondary mental healthcare provider in London was utilized
to assess the impact of SMI diagnoses on life expectancy at birth.
4. INTRODUCTION
We estimated life expectancy at birth for people with SMI and
each diagnosis, from national mortality returns between 2007–
09, using a life table method. A total of 31,719 eligible people,
aged 15 years or older, with SMI were analyzed. Among them,
1,370 died during 2007–09. Compared to national figures, all
disorders were associated with substantially lower life
expectancy 8.0 to 14.6 life years lost for men and 9.8 to 17.5 life
years lost for women. Highest reductions were found for men
with schizophrenia and women with schizoaffective disorders
5. LITERATURE REVIEW
• Life expectancy was calculated based on the interactions of season,
phase of the population cycle, population density, type of natal social
group, and weather conditions 21 and 30 d before and after birth. Life
expectancy was greatest for animals born in autumn, during the increase
phase of the population cycle, at population densities >100
• Muhammad et al. (1992) present a study. The aimed of study an
empirical study of modeling and forecasting time series data of sugarcane
production in Pakistan. The ARIMA model has been used for forecasting.
They are fit ARIMA (3, 2, and 2) model for forecasting. By using data years
of period from 1947 to 1989.
• Suresh and Priya (2011) studied the attempts forecasting the sugarcane
area, production and productivity of Tamilnadu. They are used ARIMA
model for forecasting. The data on sugarcane area, production and
productivity collected from 1950 to 2007 has been used for present
study. The ARIMA (1, 1, and 1) model was fitted for sugarcane area and
productivity.
6. Impact of Income on
Life Expectancy
• The World Health Report of 2008, by the World Health Organization,
shows a positive relationship between income and life
expectancy. The relationship appears to be one which shows that as
income per capita increases, life expectancy increases but at a
decreasing rate.
8. • Time Series Analysis:
The studies which relate the analysis of a variable with a specific period
of time (either long or short) come under the ambit of Time Series
Analysis. The analytical study of a Time Series is important so as to
forecast regarding the fluctuation of the data in future, on the basis of
the trend studied from the data. So, Time Series analysis may be
regarded as a decision-making factor of any concern, for their future
plan and estimate.
Components:
Now, let’s make an attempt to have a close look at the components of
Time Series. The major components are:
1. Secular trend
2. Seasonal variations
3. Cyclical fluctuations
4. Irregular variations
9. • Secular trend: The word trend means ‘tendency’. So, secular trend is
that component of the time series which gives the general tendency of
the data for a long period. It is smooth, regular and long-term
movement of a series.
• Seasonal variation: If we observe the sale structure of clothes in the
market, we will find that the sale curve is not uniform throughout the
year. It shows different trend in different seasons. It depends entirely
on the locality and the people who reside there.
• Cyclical fluctuations– Apart from seasonal variations, there is another
type of fluctuation which usually lasts for more than a year. This
fluctuation is the effect of business cycles. In every business there are
four important phases- I) prosperity, II) decline, III) depression, and IV)
improvement or regain.
• Irregular variations– These are, as the name suggests, totally
unpredictable. The effects due to flood, draughts, famines,
earthquakes, etc. are known as irregular variations.
10. AR: Auto regression. A model that uses the dependent relationship between an
observation and some number of lagged observations.
I: Integrated. The use of differencing of raw observations (e.g. subtracting an
observation from an observation at the previous time step) in order to make the time
series stationary.
MA: Moving Average. A model that uses the dependency between an observation and
a residual error from a moving average model applied to lag observations.
Each of these components are explicitly specified in the model as a parameter. A
standard notation is used of ARIMA (p, d, and q) where the parameters are
substituted with integer values to quickly indicate the specific ARIMA model being
used.
The parameters of the ARIMA model are defined as follows:
P: The number of lag observations included in the model, also called the lag order.
D: The number of times that the raw observations are differenced also called the
degree of differencing.
12. Parameter Estimate Stand. Error T P-value
AR(1) 1.76628 0.108279 16.3123 0.000000
AR(2) -0.798394 0.105919 -7.53775 0.000000
Table 3. ARIMA (2, 2, 0) Model Coefficient Summary
On the basis of Table 1, model coefficients the estimated life expectancy forecasted model is;
13. Models
(1) ARIMA (1, 2, 2)
(2) ARIMA (2, 2, 2)
(3) ARIMA (2, 2, 1)
(4) ARIMA (2, 2, 0)
(5) Brown's quadratic exp. smoothing with alpha = 0.9934
(6) Holt's linear exp. smoothing with alpha = 0.9999 and beta =
0.2872
(7) Brown's linear exp. smoothing with alpha = 0.9999
(8) Simple exponential smoothing with alpha = 0.9999
(9) Simple moving average of 2 terms
(10) S-curve trend = exp (4.09367 + -0.458884 /t)
(11) Exponential trend = exp (3.88653 + 0.00621067 t)
14. Model RMSE MAE MAPE ME MPE AIC HQC SBIC RUNM RUNS AUTO MEAN
(A) 0 0 0 0 0 -11.64 -11.6 -11.53 * OK * OK
(B) 0 0 0 0 0 -11.71 -11.67 -11.6 OK ** OK OK
(C) 0 0 0 0 0 -11.76 -11.73 -11.69 ** * OK OK
(D) 0 0 0 0 0 -11.8 -11.74 -11.65 OK ** OK OK
(E) 0.19 0.04 0.08 0 0 -3.31 -3.29 -3.27 * OK OK OK
(F) 0.04 0.03 0.06 -0.03 -0.05 -6.19 -6.16 -6.11 *** *** *** ***
(G) 0.18 0.05 0.1 0.02 0.05 -3.38 -3.36 -3.34 *** *** OK OK
(H) 0.45 0.4 0.72 0.4 0.72 -1.57 -1.55 -1.53 *** *** *** ***
(I) 0.67 0.6 1.07 0.6 1.07 -0.78 -0.76 -0.74 *** *** *** ***
(J) 4.16 3.55 6.23 0.16 -0.27 2.93 2.96 3 *** *** *** ***
(K) 1.4 1.17 2.06 0.01 -0.03 0.75 0.78 0.82 *** *** *** OK
(L) 0.39 0.31 0.55 0 -0.01 -1.78 -1.74 -1.67 *** *** *** *
(M) 1.15 0.93 1.66 0 -0.06 0.35 0.38 0.42 *** *** *** OK
(N) 5.85 4.87 8.67 0 -1.06 3.57 3.58 3.61 *** *** *** ***
(O) 0.19 0.15 0.27 0 0.03 -3.3 -3.28 -3.26 *** *** *** ***
(P) 0.45 0.41 0.73 0.41 0.73 -1.61 -1.61 -1.61 *** *** *** ***
Table 4. Model Selection and validity model testing criteria’s of life expectancy at birth Forecasting
based on 1960-2012
15. Figure 1. Residuals Normal Probability Plot of Life Expectancy at Birth Model for 1960-2012
16. Figure 2. Residuals Autocorrelation Plot of Life Expectancy at Birth of
Model ARIMA (2, 2, 0)
19. Conclusion
We use time series model to predict the life expectancy at birth time.
In this study, we developed time series models to forecasts “Life
expectancy at birth of Pakistan” on the basis of historical data i.e.
1960-2012. We have developed different time series models on life
expectancy at birth of Pakistan on this data. Best model is selected on
the basis of model selection criteria i.e. AIC and SBIC. Main interest
of developing time series model as other studies is that the model
fitted is also satisfied by residual assumptions i.e. normality,
independence and no autocorrelation. On the basis of these model
selection criteria, we have found that best model for forecasting life
expectancy at birth of Pakistan is ARIMA (2, 2, and 0). On the basis
of developed time series model, we have found that best time series
model for forecasting Life expectancy at birth of Pakistan is ARIMA
(2, 2, 0) because this model has lower AIC and SBIC as compared to
other fitted time series models. On the basis of this model, we have
found that life expectancy at birth of Pakistan would become 66.4489
percent in 2020 and would become 62.256 in 2042.
