Model averaging approach to forecasting the general level

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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Mortality ForecastingMotivation

The Problem of Mortality Forecasting

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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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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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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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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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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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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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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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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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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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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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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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DataResults

Figure: Observed and forecast of mortality rates for female aged 40 inPoland

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DataResults

Figure: Observed and forecast of mortality rates for female aged 80 inPoland

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DataResults

Figure: Observed and forecast of mortality rates for male aged 40 inPoland

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DataResults

Figure: Observed and forecast of mortality rates for male aged 80 inPoland

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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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DataResults

Figure: Relative differences of errors for M0 vs F-DM* approach.

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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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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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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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Documents

Model averaging approach to forecasting the general level