eISSN: 2378315X BBIJ
Biometrics & Biostatistics International Journal
Research Article
Volume 6 Issue 1  2017
Modeling and Forecasting Norway Mortality Rates using the Lee Carter Model
Department of Statistics and Computer Science, University of Peradeniya, Sri Lanka
Received: April 19, 2017  Published: June 16, 2017
*Corresponding author:
Nawarathna LS, Department of Statistics and Computer Science, Faculty of Science, University of Peradeniya, Peradeniya 20400, Sri Lanka, Tel: +94 3894682; Email:
Citation:
Basnayake BMSC, Nawarathna LS (2017) Modeling and Forecasting Norway Mortality Rates using the Lee Carter Model. Biom Biostat Int J
6(1): 00158. DOI:
10.15406/bbij.2017.06.00158
Abstract
Mortality data is an important element in the fields of actuarial science, health, epidemiology and national planning. Mortality levels are generally regarded as indicators of a general welfare of a national population and its subgroups. It reflects the quality of life within quantity. Therefore, developing a model for forecasting mortality rate will help a nation to develop its quality of life. The Lee and Carter stochastic mortality model has been used for fitting and forecasting the mortality rate of Norway which is considered as the country with the highest living standards based on the human development index. The data set contains Norway mortality data from 19462014. The Singular Value Decomposition (SVD) approach is used for estimating the parameters of Lee Carter model. Auto Regressive Integrated Moving Average (ARIMA) time series model is used for forecasting the mortality values. In this study 97.5 % variance of Norway mortality data could be explained by the proposed Lee Carter model. The best fitting ARIMA model for Norway data is identified as ARIMA (3,2,1) with drift. The predicted Lee Carter model gives a good fit to Norway data over a wide range of ages but shows poor performance below age 4 years and after age 55 years. Therefore, improvements in the Lee Carter model is needed to obtain better predictions for the ages below 4 years and after 55 years. This proposed model can be used to construct the life tables, pension scheme planning and actuarial science applications.
Keywords: Auto regressive integrated moving average; Forecasting; Lee carter model; Mortality; Single value decomposition
Abbreviations
SVD: Singular Value Decomposition; ARIMA: Auto Regressive Integrated Moving Average; SSA: Singular Spectrum Analysis; LC: Lee and Carter; AIC: Akaiko Information Criterion; BIC: Bayesian Information Criterion; SSE: Sum of Squares of Errors
Introduction
In the fields of actuarial science, health, epidemiology and national planning, mortality data is an important element. Mortality levels reflect the quality of life within quantity. Population forecasting is essential for all long term planning for the provision of services of a nation. Therefore, developing a model for forecasting mortality rate can facilitate a nation to develop their quality of life [1].
The fundamental aspect of the mankind is to live healthy long life. Recent enormous advancements in technology have provided tremendous support to fulfill this aspect. However, wide disparities are visible in levels of mortality across countries and regions. The reduction of mortality, particularly infant and maternal mortality, is part of the internationally agreed development goals in the 21st century [2].
With the human lifespan increasing, several researchers in numerous fields have recently become inquisitive about finding out quantitative models of mortality rates [3]. Once we are able to model human aging, we are able to hunt for ways that to increase our life and counteract the negative aspects of aging [4].
Singular spectrum analysis (SSA) and HyndmanUllah models were used in the literature to obtain 10 forecasts for the period 20002009 in nine European countries including Belgium, Denmark, Finland, France, Italy, the Netherlands, Norway, Sweden and Switzerland [5]. Computational results show a superior accuracy of the SSA forecasting algorithms, when compared with the HyndmanUllah approach. In most previous studies Lee Carter model has been used for modeling mortality rate in other countries and has used Bayesian approach to forecast mortality rate. Lee carter method is used to model the variability [6]. With this approach, missing data is handled and the sampling error is automatically incorporated within the model and its mortality forecasts. Here the author has described the 20th century trends of mortality for developed countries with the US as an example. A Bayesian approach was used to model mortality data for males and females from England and Whales [7]. The author has developed Lee Carter mortality including ageperiod and agecohort interactions and random effects on mortality. Moreover, Poisson distribution in advanced statistical analysis of mortality is used in [8]. This technique helps to compare low numbers of deaths in a stratum, thereby deriving more meaningful conclusions from the information.
This study was mainly focused on modeling and forecasting mortality rates using Lee Carter Model for Norway which is considered as the country with the highest living standards based on the human development index. The predicted model can be used to construct life tables for Norway and also can be used in actuarial science applications and pension scheme planning.
This article is organized as follows. In Materials and methods section, the methodology and the statistical framework behind the analysis is discussed. We describe the Lee Carter model for modeling and forecasting mortality rates. Theories used are also discussed in this section. In the results and discussion section the methodology is illustrated by analyzing the Norway mortality data. Finally, the article is concluded with a discussion. The statistical software R [9] has been used for all the computations in this article.
Materials and Methods
Line graphs are used to visualize the mortality pattern of each age group and birth pattern of Norway. Through the graphs the outliers can be observed clearly. The Lee and Carter (LC) stochastic mortality model [1] has been used in this study for fitting and forecasting the mortality rate of Norway. The Singular Value Decomposition approach is used for estimating the ax and bx parameters of Lee Carter model [2].
The singular value decomposition is a factorization of a real or complex matrix. Generally, SVD of a m * n real or complex matrix A is a factorization of the form UDVT where U is a m*m matrix, V is a n*n matrix and D is a m*n rectangular diagonal matrix with nonnegative real numbers on the diagonal. The diagonal entries of D, are called the singular values of A. The columns of U and V are the left and right singular vectors of A [10].
Lee Carter Model for mortality data
The LeeCarter model is a numerical formula utilized in mortality prediction and life expectancy forecasting. The input to the model is a matrix of age specific mortality rates ordered monotonically by time, typically with ages in columns and years in rows. The output is another forecasted matrix of mortality rates [11,12]. The Lee Carter model for mortality data is as follows [11]
$ln\left({m}_{x,t}\right)\text{}=\text{}{a}_{x}+{b}_{x}{k}_{t}+{\epsilon}_{x,t}$
(1)
where, ${m}_{x,t}$
– central rate of mortality for age group x at time t, ${a}_{x}$
– coefficient which describes average age specific pattern of mortality, ${k}_{t}$
– time trend for the general mortality, ${b}_{x}$
– coefficient which measures sensitivity of $ln\left({m}_{x,t}\right)$
at age grouping x as the${k}_{t}$
varies,${\epsilon}_{x,t}$
– error associated with age grouping x and time t. Error terms assumed to follow a normal distribution with mean zero and to be independent of age x and time t [13].
The model uses the singular value decomposition (SVD) approach to find a univariate time series vector ${k}_{t}$
that describes mortality trend in a given time (t), captures 8090% of the mortality trend, a vector ${b}_{x}$
that describes the amount of mortality change at a given age(x) for a unit of yearly total mortality change. Life expectancy being fairly constant yearly is implied with the linearity of ${k}_{t}$
. First age specific mortality rates are transformed into ${a}_{x,t}$
which spans both age(x) and time(t) by taking their logarithms, and then centering them by subtracting their agespecific means which is calculated over time before being input to the SVD .
To forecast mortality, the above ${k}_{t}$
which may be adjusted into the future using ARIMA time series methods [11] the corresponding future ${a}_{x,t+n}$
is recovered by multiplying ${k}_{t+n}$
by ${b}_{x}$
and the appropriate diagonal element of S (when [U S V] = svd(mort)), and the actual mortality rates are recovered by taking exponentials of this vector. Because of the linearity of ${k}_{t}$
, it is generally modeled as a random walk with trend. Life expectancy and other life table measures can be calculated from this forecasted matrix after adding back the means and taking exponentials to yield regular mortality rates.
This Lee Carter model is considered as the golden model for modeling mortality due to the simplicity in parameter estimation and it gives a good fit over a wide range of ages. Lee and Carter used U.S. mortality rates for conventional 5year age groups [1]. The same procedure is used in this study. As the initial step mortality data of 158 years is plotted for 19 age groups.
The estimated parameter vector ${\stackrel{\u2322}{a}}_{x}$
is determined as the average over time of the logarithm of the central death rates as
${\stackrel{\u2322}{a}}_{x}=1/\left(158{\displaystyle \sum}_{t}ln\left({m}_{x,t}\right)\right)$
(2)
The results were stored in a matrix of 158 years by 19 age groups. Then subtract the average age pattern ${\stackrel{\u2322}{a}}_{x}$
x from all years. To obtain estimated parameters ${b}_{x}$
and ${k}_{t}$
, singular value decomposition is applied on matrix Z, where$Z=ln\left({m}_{x,t}\right)$
–${\stackrel{\u2322}{a}}_{x}$
. By applying SVD to matrix Z, $SVD\text{}\left(Z\right)\text{}={\lambda}_{1}{P}_{x,1}{Q}_{t,1}+{\lambda}_{2}{P}_{x,2}{Q}_{t,2}+\text{}\mathrm{.....}\text{}={\lambda}_{1}{P}_{x,k}{Q}_{t,k}$
where$k\text{}=\text{}rank\left(z\right)$
, ${\lambda}_{i}\left(i=\text{}1,2,\text{}\dots \dots ,k\right)$
is the singular values with increasing order ${P}_{x,i}$
and${Q}_{t,i}$
left and right singular vectors respectively. Also,
${\stackrel{\u2322}{b}}_{x}={P}_{x,\text{}1}$
and (3)
${\stackrel{\u2322}{k}}_{t}=\text{}{\lambda}_{1}{Q}_{t,1}$
(4)
This ${\stackrel{\u2322}{b}}_{x}$
vector models how the different age groups react to mortality change. Moreover, this ${\stackrel{\u2322}{k}}_{t}$
vector captures overall mortality change over time. The proportion of variance described by the 1st component of SVD is calculated as
${\lambda}_{1}{}^{2}/\Sigma {\lambda}_{i}{}^{2}$
(5)
This is used as a diagnostic test for Lee Carter model [14]. Model validation is presumably the foremost vital step within the model building sequence. Information criteria are model selection tools which are used to compare any models fit to the same data. Basically, information criteria are likelihoodbased measures of model fit that embody a penalty for complexity, specifically, the number of parameters. Different information criteria are distinguished by the form of the penalty, and can prefer different models. Akaiko Information Criterion (AIC) and Bayesian Information Criterion (BIC) were used to identify the best fitting parameters. For accurate prediction, predicted values are compared with the actual values. The Human development index is a composite statistic developed by the United Nations to measure and rank the level of social and economic development based on life expectancy, education and income per capita in countries.
The mortality data was extracted from the life tables. Life tables are tables of statistics relating to life expectancy and mortality for a given category of people [15]. First the data set was divided in to training and testing set. The training data set contained the data from 1846 to 2004. Then the data from 2005  2014 were used as test set. The Singular Value Decomposition (SVD) approach is used for estimating the parameters of LC model. Auto Regressive Integrated Moving Average (ARIMA) time series model is used for forecasting the mortality values.
To produce mortality forecasts, Lee and Carter assume that ${b}_{x}$
remains constant over time and use forecasts of from a univariate time series model [16]. After testing several ARIMA specifications most appropriate model was identified using AIC and BIC values. Then the Actual and Predicted Value plots were used to identify how well the model behaved [17].
Results and Discussion
In this section birth and death pattern of Norway was plotted to identify how the death rate varied in each age group for the past 168 years. To identify the trend and outliers of mortality rate the preliminary analysis was carries out.
First log (number of births) were plotted against Year. The resulting plot is shown in Figure 1. Dramatic decline around 1940s and 1980s was visible.
Figure 1: Birth Pattern of Norway.
The plots of mortality rate versus age indicated a decreasing trend of mortality in each age group. (Plots for each age group are included in the Figure 714).
And it was clearly visible that male mortality rate is always higher than the female rate. After the 1950s there was a significant increase in male mortality. This was directly linked with the diseases due to tobacco use like cancers and chronic lower respiratory diseases. Child mortality dropped rapidly after 1950s. The dominant factor was that people were aware of the personal hygiene. Specially the infant mortality was directly connected with the mothers’ education status. With the growth of mothers’ education status infant and maternal mortality declined rapidly [14].
Figure 2: Death rate of Age 2.
Figure 3: General Mortality Pattern of Norway.
In this research Lee and Carter model stated in the equation1 was used to model the Norway data. The parameters were estimated according to the equation 2, 3 and 4. The values obtained for the parameters ${a}_{x}$
and ${b}_{x}$
of Lee Carter model for Norway data are summarized in the Table 1.
Age Group 
${a}_{x}$

${b}_{x}$

0004 
3.3329 
0.07864 
0509 
5.3973 
0.1156 
1014 
6.4578 
0.10207 
1519 
6.693 
0.0935 
2024 
6.0772 
0.07436 
2529 
5.7985 
0.07753 
3034 
5.7849 
0.07694 
3539 
5.7175 
0.0716 
4044 
5.5552 
0.06395 
4449 
5.3472 
0.053 
5054 
5.079 
0.04177 
5559 
4.7593 
0.03331 
6064 
4.407 
0.02671 
6569 
3.9991 
0.02254 
7074 
3.5596 
0.01992 
7579 
3.0991 
0.01668 
8084 
2.6013 
0.0154 
8589 
2.1478 
0.01087 
90+ 
1.5219 
0.00564 
Table 1: Values of the ${a}_{x}$
and ${b}_{x}$
parameters for 19 age groups.
Figure 4: Forecasted values from ARIMA (3,2,1) with drift.
Figure 5: Trend of Mortality Index kt over the period 19462004.
Figure 6: Fitted and Actual value plot.
Figure 7: Morality pattern of Norway for age 2, 7, 12, 17.
Then the estimated parameters were plotted against the age group. Differences in relative rates of change by age are captured by${b}_{x}$
. Differences in mortality by age are captured by${a}_{x}$
.
Figure 8: Morality pattern of Norway for ages 22, 27, 32, 37.
Figure 9: Morality pattern of Norway for ages 42, 47, 52, 57.
Figure 10: Morality pattern of Norway for ages 62, 67, 72, 77.
Figure 11: Morality pattern of Norway for ages 82, 87, 92.
Figure 12: Actual Vs Fitted Value plots for 2005 to 2008.
Figure 13: Actual and Fitted Value plots for 2009 to 2012.
Figure 14: Actual and Fitted Value plots for 2013 to 2014.
From the Figure 3, higher child mortality below age 5 years is observed and after the age of 25 mortality increases nearly exponentially. Lowest mortality rate is observed at the age group of 1519.
The proportion of variance described by the 1st component of SVD is calculated using the equation 5. By applying SVD 97.5462 % temporal variance of Norway mortality data could be explained by the 1st component of SVD. After testing AIC and BIC values for several ARIMA specifications the best fitting parameters of ARIMA model for Norway data was identified. The AIC and BIC values for fitted ARIMA models are shown in Table 2.
ARIMA 
(0,0,0) 
(0,0,1) 
(0,1,0) 
(0,1,1) 
(1,0,0) 
(1,1,0) 
(1,1,1) 

AIC 
1282.56 
1084 
463.2 
465.2 
475.8 
465.2 
461 

BIC 
1288.7 
463.2 
466.3 
471.3 
485 
471.3 
470.2 

ARIMA 
(0,2,0) 
(0,2,2) 
(0,0,2) 
(2,2,0) 
(2,0,0) 
(2,0,2) 
(1,2,1) 

AIC 
565.25 
460.8 
928.5 
505.3 
477.8 
475.2 
460.7 

BIC 
568.3 
470 
940.8 
514.4 
490.1 
493.6 
469.9 

ARIMA 
(1,2,2) 
(1,1,2) 
(2,2,1) 
(2,2,2) 
(2,1,1) 
(2,1,2) 
(3,2,1) 
(1,2,3) 
AIC 
460.1 
462.7 
461.4 
460.7 
463.4 
464.4 
457.5 
460.1 
BIC 
472.32 
474.9 
473.6 
476 
475.6 
479.7 
472.8 
475.3 
Table 2: AIC and BIC values for fitted ARIMA models.
The best fitting parameters of ARIMA model were identified as ARIMA (3,2,1) with drift which gave the lowest AIC and BIC values 457.47, 472.75 respectively.
Values of k for next 10 years are forecasted from ARIMA (3,2,1) with drift with the 80% and 95% confidence interval. This was done using the “FORECAST” package of R. The forecasted values are shown in the Table 3.


Confidence Interval 


80% 
95% 
Year 
Forecast 
Lower Bound 
Upper Bound 
Lower Bound 
Upper Bound 
2005 
25.595 
26.8876 
24.3025 
27.5718 
23.6183 
2006 
25.869 
27.6966 
24.0422 
28.6638 
23.0749 
2007 
26.047 
28.3651 
23.728 
29.5925 
22.5007 
2008 
26.305 
28.9161 
23.6947 
30.2981 
22.3126 
2009 
26.629 
29.5247 
23.7327 
31.0577 
22.1996 
2010 
26.976 
30.1275 
23.8243 
31.7958 
22.1559 
2011 
27.311 
30.7315 
23.8905 
32.5422 
22.0798 
2012 
27.635 
31.3132 
23.9575 
33.2602 
22.0105 
2013 
27.954 
31.8881 
24.0205 
33.9706 
21.938 
2014 
28.275 
32.4556 
24.0944 
34.6687 
21.8813 
Table 3: Forecasted Values for ${k}_{t}$
and Confidence Interval.
The trajectory of k is shown in the Figure 5. ${k}_{t}$
reflects yeartoyear changes in the general level of mortality.
From the Figure 5 decreasing trend is observed for mortality index (${k}_{t}$
). Two distinguishable peaks are observed during 19141918 and 1939 1945 due to the world war I and World War II respectively. The forecasted k values are used together with calculated a and b values to predict the mortality rate for next 10 years. The fitted and actual value plots are shown in the Figure 6.
(Actual and fitted value plots and values for all predicted 10 years are shown in the Figure 57. Then for each age group Sum of Squared error values were calculated as below.
From the fitted and actual value plots shown in Figure 6 and from the age group wise Sum of Squares of Errors (SSE) values in Table 4 it was clear that higher deviation is observed for 0004 years and after 50 years (Tables 57).
Age Group 
SSE 
0004 
2.47E05 
0509 
3.00E08 
1014 
3.74E09 
1519 
5.72E09 
2024 
4.69E08 
2529 
2.78E07 
3034 
3.54E07 
3539 
1.95E07 
4044 
1.08E07 
4449 
5.93E08 
5054 
5.82E07 
5559 
2.69E06 
6064 
1.52E05 
6569 
5.23E05 
7074 
0.000182498 
7579 
0.000776518 
8084 
0.001869918 
8589 
0.005167924 
90+ 
0.006279726 
Table 4: Age group wise SSE (Sum of Squared Error).
Year 
2005 
2006 
2007 
k 
25.59506215 
25.86935037 
26.04657306 
Age group 
Fitted 
Actual 
Fitted 
Actual 
Fitted 
Actual 
04 
0.004768 
0.00307 
0.004667 
0.0032 
0.004602 
0.00307 
59 
0.000235 
0.00021 
0.000228 
0.00018 
0.000223 
0.00016 
1014 
0.000115 
0.00011 
0.000112 
0.00012 
0.00011 
0.00007 
1519 
0.000113 
0.00012 
0.00011 
0.00006 
0.000109 
0.00012 
2024 
0.000342 
0.00036 
0.000335 
0.00037 
0.000331 
0.0003 
2529 
0.000417 
0.0007 
0.000408 
0.00058 
0.000403 
0.00056 
3034 
0.000429 
0.00061 
0.00042 
0.00062 
0.000414 
0.00065 
3539 
0.000526 
0.00066 
0.000516 
0.00069 
0.000509 
0.00057 
4044 
0.000753 
0.00096 
0.00074 
0.00084 
0.000731 
0.00077 
4449 
0.001226 
0.00128 
0.001209 
0.00111 
0.001197 
0.00112 
5054 
0.002137 
0.00205 
0.002113 
0.00206 
0.002097 
0.00199 
5559 
0.003654 
0.00327 
0.003621 
0.00319 
0.0036 
0.00311 
6064 
0.006154 
0.00518 
0.006109 
0.00502 
0.00608 
0.00487 
6569 
0.010296 
0.0083 
0.010233 
0.0085 
0.010192 
0.00818 
7074 
0.017088 
0.01318 
0.016995 
0.01259 
0.016935 
0.01311 
7579 
0.029421 
0.02214 
0.029287 
0.02106 
0.0292 
0.0211 
8084 
0.050015 
0.03884 
0.049805 
0.03711 
0.049668 
0.03776 
8589 
0.088397 
0.06745 
0.088134 
0.06781 
0.087964 
0.06839 
90+ 
0.188957 
0.167835 
0.188665 
0.167123 
0.188476 
0.167564 
Table 5: Actual and Fitted Values for 2005 to 2007.
Year 
2008 
2009 
2010 
k 
26.30536036 
26.62866133 
26.97586597 
Age group 
Fitted 
Actual 
Fitted 
Actual 
Fitted 
Actual 
04 
0.004509 
0.00273 
0.004396 
0.00313 
0.004278 
0.00276 
59 
0.000216 
0.00017 
0.000209 
0.00015 
0.0002 
0.00012 
1014 
0.000107 
0.0001 
0.000104 
0.00011 
9.99E05 
0.00009 
1519 
0.000106 
0.0001 
0.000103 
0.0001 
9.95E05 
0.00009 
2024 
0.000325 
0.00036 
0.000317 
0.00039 
0.000309 
0.00031 
2529 
0.000395 
0.00061 
0.000385 
0.00051 
0.000375 
0.00058 
3034 
0.000406 
0.00062 
0.000396 
0.00063 
0.000386 
0.00064 
3539 
0.0005 
0.0006 
0.000489 
0.00064 
0.000477 
0.00059 
4044 
0.000719 
0.00081 
0.000704 
0.00082 
0.000689 
0.00068 
4449 
0.001181 
0.00118 
0.001161 
0.00115 
0.00114 
0.0011 
5054 
0.002075 
0.00181 
0.002047 
0.00178 
0.002017 
0.00177 
5559 
0.003569 
0.00311 
0.003531 
0.00316 
0.00349 
0.00301 
6064 
0.006038 
0.0047 
0.005986 
0.00477 
0.005931 
0.0048 
6569 
0.010133 
0.00812 
0.010059 
0.00779 
0.009981 
0.00776 
7074 
0.016848 
0.01284 
0.01674 
0.01256 
0.016624 
0.01262 
7579 
0.029074 
0.02081 
0.028918 
0.02 
0.028751 
0.01994 
8084 
0.049471 
0.03713 
0.049225 
0.03519 
0.048963 
0.0351 
8589 
0.087718 
0.06697 
0.08741 
0.06535 
0.08708 
0.06402 
90+ 
0.188202 
0.164542 
0.187859 
0.163347 
0.187491 
0.161117 
Table 6: Actual and Fitted Values for 2008 2010.
Year 
2011 
2012 
2013 
2014 
k 
27.31098443 
27.63534578 
27.95428909 
28.27498506 
Age group 
Fitted 
Actual 
Fitted 
Actual 
Fitted 
Actual 
Fitted 
Actual 
04 
0.004166 
0.00232 
0.004061 
0.00248 
0.003961 
0.00245 
0.003862 
0.00243 
5_9 
0.000193 
0.00014 
0.000186 
0.00014 
0.000179 
0.00012 
0.000172 
0.00012 
10_14 
9.66E05 
0.00007 
9.34E05 
0.00008 
9.04E05 
0.00006 
8.75E05 
0.00008 
15_19 
9.64E05 
0.00011 
9.36E05 
0.00012 
9.08E05 
0.00005 
8.81E05 
0.00007 
20_24 
0.000301 
0.00047 
0.000294 
0.00023 
0.000287 
0.00027 
0.00028 
0.00021 
25_29 
0.000365 
0.00053 
0.000356 
0.00042 
0.000347 
0.00042 
0.000339 
0.00039 
30_34 
0.000376 
0.00054 
0.000367 
0.00046 
0.000358 
0.00047 
0.000349 
0.00046 
35_39 
0.000465 
0.0007 
0.000455 
0.00057 
0.000444 
0.00056 
0.000434 
0.00056 
40_44 
0.000674 
0.00075 
0.000661 
0.00074 
0.000647 
0.00075 
0.000634 
0.00073 
44_49 
0.00112 
0.00103 
0.001101 
0.001 
0.001082 
0.00108 
0.001064 
0.00092 
50_54 
0.001989 
0.00186 
0.001963 
0.00164 
0.001937 
0.0016 
0.001911 
0.00156 
55_59 
0.003451 
0.0028 
0.003414 
0.00282 
0.003378 
0.00275 
0.003342 
0.00274 
6064 
0.005878 
0.00479 
0.005827 
0.0044 
0.005778 
0.00464 
0.005729 
0.00414 
65_69 
0.009906 
0.00737 
0.009834 
0.00749 
0.009763 
0.00719 
0.009693 
0.00677 
70_74 
0.016514 
0.01202 
0.016407 
0.01199 
0.016303 
0.01193 
0.0162 
0.01122 
75_79 
0.028591 
0.01917 
0.028436 
0.01936 
0.028285 
0.01875 
0.028135 
0.018 
80_84 
0.048711 
0.03452 
0.048468 
0.03378 
0.048231 
0.03329 
0.047993 
0.03186 
85_89 
0.086764 
0.06267 
0.086459 
0.06349 
0.086159 
0.06083 
0.08586 
0.05876 
90+ 
0.187137 
0.160931 
0.186795 
0.161875 
0.18646 
0.15791 
0.186123 
0.15525 
Table 7: Actual and Fitted Values for 2011 to 2014.
Conclusion
The Lee and Carter (LC) stochastic mortality model has been used in this study for fitting and forecasting the mortality rate of Norway which is considered as the country with the highest living standards based on the human development index. LC model is used since it is regarded as the golden model for mortality data due to the simplicity in parameter estimation and it gives a good fit over a wide range of ages. The data set contains Norway mortality data from 18462014. The Singular Value Decomposition (SVD) approach is used for estimating the parameters of LC model. Auto Regressive Integrated Moving Average (ARIMA) time series model is used for forecasting the mortality values.
In this study 97.5 % temporal variance of Norway mortality data could be explained by the 1st SVD component. The best fitting ARIMA model for Norway data is identified as ARIMA (3,2,1) with drift which gave the lowest Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) values. The general pattern of mortality showed higher child mortality for ages below 4 years and an accidental hump around ages 20 and nearly exponential increase after the age of 25. The sensitivity of mortality has shown mortality decline at high rate for ages 2025 years. Mortality index has shown decreasing trend and two spikes due to World War I and World War II. The predicted Lee Carter model gives a good fit to Norway data over a wide range of ages but shows poor performance below age of 4 years and after age of 55 years. Therefore, an improvement in the LC model is needed to obtain better predictions for these two age categories.
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