[PDF] - ARCH 2013.1 Proceedings August 1- 4, 2012 Defang Wu, Xiaoming Liu, PDF Document

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Article from:

ARCH 2013.1 Proceedings

August 1- 4, 2012 Defang Wu, Xiaoming Liu, Yu Hao

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Assessing systematic bias in mortality prediction of the Lee-Carter model

Defang Wu, Xiaoming Liu, Hao Yu

Department of Statistics and Actuarial Science University of Western Ontario, London Ontario, Canada dwu87@uwo.ca

August 3, 2012

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Motivation

It has been found in literature that there exists bias when using the Lee-Carter(LC) model for mortality prediction.

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Motivation

It has been found in literature that there exists bias when using the Lee-Carter(LC) model for mortality prediction. An overall under-prediction of mortality decline and life expectancy gain is reported. For example, Lee and Miller(2001), Bell(1997), Booth(2002), etc.

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Motivation

It has been found in literature that there exists bias when using the Lee-Carter(LC) model for mortality prediction. An overall under-prediction of mortality decline and life expectancy gain is reported. For example, Lee and Miller(2001), Bell(1997), Booth(2002), etc. Possible reasons are put forward to explain such bias, such as: error correlation by age and horizon, changing age-shape of mortality, etc.

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Motivation

It has been found in literature that there exists bias when using the Lee-Carter(LC) model for mortality prediction. An overall under-prediction of mortality decline and life expectancy gain is reported. For example, Lee and Miller(2001), Bell(1997), Booth(2002), etc. Possible reasons are put forward to explain such bias, such as: error correlation by age and horizon, changing age-shape of mortality, etc. Corresponding modifications are developed to LC model.

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Motivation

In paper of Liu and Yu(2011), they found systematic bias of the LC model in forecast of life expectancy based on simulated data.

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Motivation

In paper of Liu and Yu(2011), they found systematic bias of the LC model in forecast of life expectancy based on simulated data. The advantage of using simulated data is to separate the potential model mis-specification issue from the model effectiveness test.

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Motivation

In paper of Liu and Yu(2011), they found systematic bias of the LC model in forecast of life expectancy based on simulated data. The advantage of using simulated data is to separate the potential model mis-specification issue from the model effectiveness test. Systematic bias in forecast of life expectancy were found even we eliminate potential model mis-specification.

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Objectives

The main purpose of our paper is to:

1 measure the magnitude of the bias using the bootstrap method 2 provide suggestions on how to correct the bias 3 illustrate the effectiveness of correction through examining the

forecast performance

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Brief review of LC model

The LC model log(mxt) = ax + bxkt + εxt kt = kt−1 + c + ξt, ξt ∼ N(0, σ2

ξ)

ax describes the age pattern of mortality averaged over time bx describes the deviations from the averaged pattern when kt varies kt describes the variation in the level of mortality over time εxt is the error term

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Brief review of LC model

Since the study bases on simulated data, we consider two situations when generating new sample paths of kt for LC data set:

1 simulate a random sample of ξt following normal distribution N(0, σ2) 2 simulate a random sample of ξt following normal distribution with

CDF: Fξ(x) = 0.95N(0, σ2/2.2) + 0.05N(0, (5σ)2/2.2) Case 1 follows original LC model and case 2 represents the situation where irregular large shocks may happen occasionally.

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Systematic bias of LC model

How do we find the systematic bias of LC model by bootstrap method?

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Systematic bias of LC model

How do we find the systematic bias of LC model by bootstrap method? For each given simulated data set, generate 10,000 sample paths of forecast kt by bootstrap method. Use ax, bx and forecast kt to obtain 10,000 sample of matrix of log(mxt). Use median as point forecast of log(mxt) and compare with the “real” mortality. Generate 10,000 simulated data set and take average of the difference between predicted value and its corresponding “real” value.

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Systematic bias of LC model

10 20 30 40 0.030 0.035 0.040 0.045 0.050 0.055 0.060

bias at age 30

forecast horizon Bias case 1 case 2 10 20 30 40 0.022 0.024 0.026 0.028 0.030 0.032 0.034

bias at age 40

forecast horizon Bias case 1 case 2 10 20 30 40 0.012 0.014 0.016 0.018 0.020 0.022

bias at age 60

forecast horizon Bias case 1 case 2 10 20 30 40 0.012 0.013 0.014 0.015 0.016 0.017 0.018

bias at age 80

forecast horizon Bias case 1 case 2

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Systematic bias of LC model

It is worth noticing that Overall values in the figure above for four different ages are positive. Positive bias indicates of under-prediction of decline of log(mxt) by LC model. This is consistent with the fact in Liu and Yu(2011) that bias of e0(t) is always negative and life expectancy gain is under-predicted. The systematic bias found in Liu and Yu(2011) is not the result of functional change of forecast variable from log(mxt) to e0(t) but effectiveness of LC model.

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Systematic bias of LC model

Further, we calculate the percentage of bias. Percentage of bias = bias/corresponding “real” value of log(mxt).

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Systematic bias of LC model

Further, we calculate the percentage of bias. Percentage of bias = bias/corresponding “real” value of log(mxt). Overall values of percentage of bias are negative, which seems “conflict” to the sign of bias.

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Systematic bias of LC model

10 20 30 40 −0.0070 −0.0060 −0.0050 −0.0040

percentage of bias at age 30

forecast horizon Percentage of bias case 1 case 2 10 20 30 40 −0.0050 −0.0045 −0.0040 −0.0035 −0.0030

percentage of bias at age 40

forecast horizon Percentage of bias case 1 case 2 10 20 30 40 −0.0045 −0.0040 −0.0035 −0.0030 −0.0025

percentage of bias at age 60

forecast horizon Percentage of bias case 1 case 2 10 20 30 40 −0.0055 −0.0050 −0.0045 −0.0040

percentage of bias at age 80

forecast horizon Percentage of bias case 1 case 2

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Bias correction

Though percentage of bias in forecast of log(mxt) are relatively small, the bias in forecast of e0(t) could be significant because exponential functions are applied to log(mxt).

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Bias correction

Though percentage of bias in forecast of log(mxt) are relatively small, the bias in forecast of e0(t) could be significant because exponential functions are applied to log(mxt). By applying bias correction to forecast variable, we want forecast performance to be more accurate in forecast of e0(t). The main idea of bias correction is:

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Bias correction

Though percentage of bias in forecast of log(mxt) are relatively small, the bias in forecast of e0(t) could be significant because exponential functions are applied to log(mxt). By applying bias correction to forecast variable, we want forecast performance to be more accurate in forecast of e0(t). The main idea of bias correction is: Bias Correction new predicted value = predicted value - estimated bias, where estimated bias = E[ predicted value - value from model predict line ]

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Bias correction

Remarks: We fit LC model to mortality data set to obtain parameters of ax, bx and kt, where kt is modeled by c and σ2

ξ.

Predicted value: for forecast purpose, we generate sample path of kt at time t0+i given the data available up to time t0 by: kt0+i = kt0 + i · c + i

j=1 ξj.

Value from model predict line: generate sample path of kt at time t0+i given the data available up to time t0 by: kt0+i = kt0 + i · c.

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Bias correction

Apply bias correction to two forecast variables:

1

applying bias correct to final forecast value of e0(t)

2

applying bias correct to log(mxt) and then calculating e0(t) with corrected log(mxt).

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Bias correction

Apply bias correction to two forecast variables:

1

applying bias correct to final forecast value of e0(t)

2

applying bias correct to log(mxt) and then calculating e0(t) with corrected log(mxt).

We illustrate the effectiveness of correction under these two methods through examining the forecast performance. Forecast performance is evaluated in RMSE, MAPE, bias, Kolmogorov-Smirnov(KS) statistics, coverage and average confidence interval width.

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Evaluation

10 20 30 40 0.0 0.5 1.0 1.5 forecast horizon RMSE case 1 case 2 10 20 30 40 0.0 0.5 1.0 1.5 1:len RMSE case 1 case 2 10 20 30 40 0.004 0.006 0.008 0.010 0.012 0.014 forecast horizon MAPE case 1 case 2 10 20 30 40 0.004 0.006 0.008 0.010 0.012 0.014 forecast horizon MAPE case 1 case 2 10 20 30 40 −0.005 0.000 0.005 0.010 forecast horizon Bias case 1 case 2 10 20 30 40 −0.10 −0.08 −0.06 −0.04 −0.02 0.0 forecast horizon Bias case 1 case 2

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Evaluation

10 20 30 40 0.0 0.5 1.0 1.5 2.0 2.5 3.0 forecast horizon KS Statistics case 1 case 2 10 20 30 40 0.0 0.5 1.0 1.5 2.0 2.5 3.0 forecast horizon KS Statistics case 1 case 2 10 20 30 40 80 85 90 95 100 forecast horizon Prediction Coverage case 1 case 2 10 20 30 40 80 85 90 95 100 forecast horizon Prediction Coverage case 1 case 2 10 20 30 40 1 2 3 4 5 6 forecast horizon Averyage Confidence Width case 1 case 2 10 20 30 40 1 2 3 4 5 6 forecast horizon Averyage Confidence Width case 1 case 2

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Evaluation

It is remarkable to notice that:

RMSE, MAPE with bias correct applied to e0(t) is slightly smaller than that with bias correct applied to log(mxt) Applying correction to e0(t) makes bias randomly distribute around zero for both case 1 and case 2. KS statistics of case 1 with bias correct applied to e0(t) are larger than critical value while that with bias correct applied to log(mxt) is smaller.

Applying bias correct to e0(t) is more effective in terms of first three evaluation measurements for forecast performance.

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Evaluation

Real mortality data are chosen from Canada(1922-1950) and forecast to year 1995.

20 40 60 60 65 70 75

Canada from 1950

horizon e(0) with bias correction without bias correction true value of e0

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Conclusion and Limitation

It’s reported that bias in forecast log(mxt) for LC model are positive in general. Positive bias indicates under-prediction of mortality decline but the deviations are less 1% in general. In order to obtain more accurate forecast performance of e0(t), two kinds of bias correction methods are suggested. Applying correction to e0(t) is more effective to obtain better forecast performance based on simulated data. Due to dramatically increasing e0(t) in reality, forecast by LC model is still under-predicted even we try to correct the systematic bias of the model.

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Reference

Lee, R. D. and Carter, L. R. 1992. Modeling and Forecasting U.S. Mortality. Journal of the American Stotistical Association 87: 659-675. Lee, R. D. and Miller, T. 2001. Evaluating the Performance of the Lee-Carter Method for Forecasting Mortality. Demography 38: 537-549. Liu, X. and Yu, H. Assessing and extending the lee-carter model for long-term mortality prediction. Orlando, Fla., January 2011. Living to 100 Symposium.

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Thank you for your listening!

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