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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Model averaging approach to forecasting thegeneral level of mortality
Marcin Bartkowiak1 Katarzyna Kaczmarek-Majer2
Aleksandra Rutkowska1 Olgierd Hryniewicz2
University of Economics and Business, Poznan, Niepodleglosci 10, Poland
Systems Research Institute, Polish Academy of Sciences, Warsaw, Newelska 6,Poland
4th European Actuarial Journal Conference10 September 2018
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Mortality ForecastingMotivation
The Problem of Mortality Forecasting
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Mortality ForecastingMotivation
The Problem of Mortality Forecasting
Problem with current models:
difficult-to-meet assumption of mortality models
data imprecise
Raw population data adjustments:
unknown age distribution,
splitting or aggregation,
smoothing.
Source: The Human Mortality Database(http://www.mortality.org/)
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Mortality ForecastingMotivation
Motivation
data imprecision
short time series for some countries
Improving the overall forecast accuracy of mortality rates by1%, leads to the decrease of insurers costs even by 3% [1]
[1]Fund Pension Protection: the Pensions Regulator (2006), The Purple Book: DB Pensions Universe Risk Profile,
(2008).
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Lee-Carter ModelNew approachAlgorithm steps
Let mx ,t denote the mortality rate in
an age group x = x1, . . . , xN
and time t = 1, 2, . . . ,T .
The Lee-Carter (LC) model [2] can be presented as follows:
ln(mx ,t) = αx + βxκt + εx ,t , (1)
where αx and βx are age-specific parameters and κt is atime-variant parameter.To obtain a unique solution the following constraints are imposed:αx = 1
T
∑Tt=1 ln(mx ,t),
∑Tt=1 κt = 0,
∑xNx=x1
βx = 1. [2] R.D. Lee, and L.
R. Carter.: Modeling and forecasting US mortality. Journal of the American statistical association 87.419 : 659-671
(1992)
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Lee-Carter ModelNew approachAlgorithm steps
Using LC model the corresponding future mx ,t is calculated:
it is assumed, that age dependent parameters are the same,
time dependent parameter κt is forecasted.
The state-of-the-art approaches assume modeling κt as a randomwalk with trend or as integrated autoregressive and moving average(ARIMA) process.The main disadvantage of ARIMA: the requirement of at least 50(but preferably more than 100 observations), which is often hard tomeet in case of mortality data.
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Lee-Carter ModelNew approachAlgorithm steps
Data-mining Approach to Finding Weights
[11]Hryniewicz, O., Kaczmarek-Majer, K.: Monitoring of short series of dependent observations using a XWAM
control chart. Frontiers in Statistical Quality Control 12. Knoth, S., Schmid, W. (eds.)(2018)
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Lee-Carter ModelNew approachAlgorithm steps
Predictive models
- We adapt stationary autoregressive processes of differentorders as predictive models M = {M1,M2, ...,MJ}.
- Autoregressive process (AR)
yi =
p∑i=1
φiyt−i + at (2)
where at ∼ N(0, σ2) are normally distributed independentrandom variables, σ2 ∈ (0, 1) and φi ∈ (-1, 1).
- For j ∈ {1, 2, ..., J} models (processes), its s realizations(training time series) are generated {yj ,i}si=1.
- The inspiration comes from the imaginary training samplesand the expected posterior priors introduced in [3].
[3] J. Perez and J. Berger, Expected-posterior prior distributions for model selection,Biometrica, vol. 89, pp.
491-511, 2002.
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Lee-Carter ModelNew approachAlgorithm steps
Step 1. Calculating similarities
- Dynamic Time Warping (DTW) [4] measures the distancebetween two time series
- The distances between the training time series and the timeseries for prediction y are calculated for j ∈ {1, ..., J} modelsand their i ∈ {1, ..., s} realizations (training time series).
distj ,i = DTW (yj ,i , y) (3)
- On small data sets elastic measures like DTW can besignificantly more accurate than Euclidean distance and otherlock-step measures because it takes into account thedilatation in time. [5]
[4] D. Berndt and J. Clifford, Aaai-94 workshop on knowledge discovery in databases, Using dynamic time warping
to find patterns in time series, pp. 359-370, 1994. [5] X. Wang, A. Mueen, H. Ding, G. Trajcevski, P.
Scheuermann, and E. Keogh,Experimental comparison of representation methods and distance measures for time
series data, Data Mining Knowledge Discovery, vol. 26(2), pp. 275-309, 2013.
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Lee-Carter ModelNew approachAlgorithm steps
Step 2. Aggregating similarities
The considered alternative approached to aggregate similarities:
1 Average. The average distance between the training timeseries of the same model j ∈ J and the considered short timeseries for prediction
distj =1
s
s∑i=1
distj ,i (4)
2 Number of most similar training time series with the useof quantiles. First, θ-quantiles from the sample of alldistances distj ,i for 1 6 i 6 s training time series of eachmodel 1 6 j 6 J are estimated. Secondly, the number numj ,θ
of training time series exceeding the θ-quantile is calculatedfor each model 1 6 j 6 J.
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Lee-Carter ModelNew approachAlgorithm steps
Step 3. Weighted model averaging
Individual forecasts of the selected k most similar models arecalculated fi (t0) according to their prior definitions in M.
The final forecast is calculated as the weighted average of theindividual forecasts fi (t0) and the corresponding weights{w1, ...,wk}:
f (t0) =k∑
i=1
wi fi (t0). (5)
The concept of model averaging was promoted by Geweke,see i.e. Bayesian model averaging in [15].
p(ω|y ,M) =J∑
j=1
p(ω|y ,Mj)p(Mj |M) (6)
[15] Geweke, J.: Contemporary Bayesian econometrics and statistics, J.Wiley, Hoboken NJ, 2005.
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
DataResults
Data
Country period n
Bulgaria 1947-2010 63Croatia 2002-2015 14Czechia 1950-2014 64Estonia 1959-2014 55Hungary 1950-2014 64Latvia 1959-2014 55Lithuania 1959-2014 55Poland 1958-2014 56Slovakia 1950-2014 64Slovenia 1983-2014 31
The study is conducted separately for the male and femalepopulation for ages 0 to 100.
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
DataResults
Illustrative example for Poland
Figure: Observed and forecast of log mortality rates for female of Polandby F-DM* model averaging method in 2005 (left) and 2014 (right)
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
DataResults
Figure: Observed and forecast of mortality rates for female aged 40 inPoland
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
DataResults
Figure: Observed and forecast of mortality rates for female aged 80 inPoland
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
DataResults
Figure: Observed and forecast of mortality rates for male aged 40 inPoland
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
DataResults
Figure: Observed and forecast of mortality rates for male aged 80 inPoland
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
DataResults
Forecasting accuracy
mae mse mdae mdape smape smdapePOL female 4.42% 3.67% 0.00% 7.44% -3.46% 5.76%
male 3.49% 3.33% 0.00% 2.49% -0.30% 2.87%CZE female 8.21% 5.34% 0.00% 10.70% -1.48% 11.01%
male -3.56% -1.59% -10.00% -3.76% 1.30% -5.34%SVK female -2.27% 8.79% 0.00% -15.65% 7.79% -14.60%
male 3.05% 7.65% -12.50% 1.31% 4.56% 0.92%HUN female 1.79% 5.41% 0.00% 0.49% 0.93% -1.15%
male -5.42% -14.84% 16.22% -0.78% -2.58% -0.19%BGR female 5.21% 11.28% 25.00% 4.36% 7.82% 6.91%
male -2.67% -4.24% 0.00% -2.44% -1.23% -1.53%LTU female -0.2%2 -0.38% 0.00% -0.48% -0.20% -0.65%
male 0.61% -4.96% 15.79% 7.42% 2.57% 3.99%LVA female 0.09% -0.73% 0.00% 0.16% 0.36% 0.72%
male 2.97% -0.83% 8.33% 7.86% 3.66% 7.02%EST female 0.21% 1.15% -8.33% 2.26% 0.18% 0.93%
male -0.16% -0.21% -2.44% 0.94% 0.66% 1.42%SVN female 0.68% 1.09% 0.00% -1.94% -0.06% -0.82%
male 2.95% 9.27 -16.67% -5.20% -0.96% -5.02%HRV female -1.79% -7.18% 0.00% -11.26% -4.92% -7.21%
male 3.45% 8.47% 0.00% 3.68% 1.70% -0.19%
avg 1.05% 1.52% 0.77% 0.38% 0.82% 0.24%median 0.65% 1.12% 0.00% 0.71% 0.27% 0.26%
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
DataResults
Figure: Relative differences of errors for M0 vs F-DM* approach.
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
DataResults
Relative differences of errors
mae mse mdae mdape smape smdape
full sample (1-10 years ahead)
avg 1.18% 1.79% 0.72% 0.90% 0.75% 0.97%median 0.87% 1.39% 0.00% 1.45% 0.31% 0.80%
1st period (1-5 years ahead)
avg 0.43% 0.73% 1.11% 0.19% 0.48% -0.04%median 0.56% 0.66% 0.00% 0.40% 0.18% 0.11%
2nd period (6-10 years ahead)
avg 1.31% 1.41% 1.28% 0.89% 0.76% 1.19%median 1.22% 1.46% 0.00% 0.15% 0.28% 0.06%
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Concluding RemarksFuture works
Summary
The improvement in forecast accuracy (relative error forh=10) ranges from 0.72% according to MDAE to 1.79%according to MSE.
We obtain in 55.83% of cases the smaller errors thanauto.arima and in 30.83% the larger ones
The Diebold-Mariano test rejected the null hypothesis in favorof the alternative hypothesis that proposed method is moreaccurate in 11 cases, i.e.
for men in: Croatia, Poland, Slovenia, Slovakiafor women in: Estonia, Poland, Bulgaria, Czechia, Slovenia,Slovakia and Hungary.
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Background and MotivationThe model averaging approach
Empirical ResultsConclusion and Future Work
Concluding RemarksFuture works
Future works
extension of the database with alternative models,
extension of the LC model to reflect cohort effects and furtheranalysis of optimal weights establishment.
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