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# Modeling and forecasting age-specific mortality: Lee-Carter method vs. Functional time series

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### Modeling and forecasting age-specific mortality: Lee-Carter method vs. Functional time series

1. 1. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecasting Modeling and forecasting age-speciﬁc mortality: Lee-Carter method vs. Functional time series Han Lin Shang Econometrics & Business Statistics http://monashforecasting.com/index.php?title=User:Han
2. 2. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingOutline 1 Lee-Carter model 2 Nonparametric smoothing 3 Functional principal component analysis 4 Functional time series forecasting
3. 3. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model 1 Lee and Carter (1992) proposed one-factor principal component method to model and forecast demographic data, such as age-speciﬁc mortality rates.
4. 4. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model 1 Lee and Carter (1992) proposed one-factor principal component method to model and forecast demographic data, such as age-speciﬁc mortality rates. 2 The Lee-Carter model can be written as ln mx,t = ax + bx × kt + ex,t , (1) where
5. 5. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model 1 Lee and Carter (1992) proposed one-factor principal component method to model and forecast demographic data, such as age-speciﬁc mortality rates. 2 The Lee-Carter model can be written as ln mx,t = ax + bx × kt + ex,t , (1) where ln mx,t is the observed log mortality rate at age x in year t,
6. 6. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model 1 Lee and Carter (1992) proposed one-factor principal component method to model and forecast demographic data, such as age-speciﬁc mortality rates. 2 The Lee-Carter model can be written as ln mx,t = ax + bx × kt + ex,t , (1) where ln mx,t is the observed log mortality rate at age x in year t, ax is the sample mean vector,
7. 7. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model 1 Lee and Carter (1992) proposed one-factor principal component method to model and forecast demographic data, such as age-speciﬁc mortality rates. 2 The Lee-Carter model can be written as ln mx,t = ax + bx × kt + ex,t , (1) where ln mx,t is the observed log mortality rate at age x in year t, ax is the sample mean vector, bx is the ﬁrst set of sample principal component,
8. 8. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model 1 Lee and Carter (1992) proposed one-factor principal component method to model and forecast demographic data, such as age-speciﬁc mortality rates. 2 The Lee-Carter model can be written as ln mx,t = ax + bx × kt + ex,t , (1) where ln mx,t is the observed log mortality rate at age x in year t, ax is the sample mean vector, bx is the ﬁrst set of sample principal component, kt is the ﬁrst set of sample principal component scores,
9. 9. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model 1 Lee and Carter (1992) proposed one-factor principal component method to model and forecast demographic data, such as age-speciﬁc mortality rates. 2 The Lee-Carter model can be written as ln mx,t = ax + bx × kt + ex,t , (1) where ln mx,t is the observed log mortality rate at age x in year t, ax is the sample mean vector, bx is the ﬁrst set of sample principal component, kt is the ﬁrst set of sample principal component scores, ex,t is the residual term.
10. 10. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model forecasts 1 There are a number of ways to adjust kt , which led to extensions and modiﬁcation of original Lee-Carter method.
11. 11. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model forecasts 1 There are a number of ways to adjust kt , which led to extensions and modiﬁcation of original Lee-Carter method. 2 Lee and Carter (1992) advocated to use a random walk with drift model to forecast principal component scores, expressed as kt = kt−1 + d + et , (2) where
12. 12. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model forecasts 1 There are a number of ways to adjust kt , which led to extensions and modiﬁcation of original Lee-Carter method. 2 Lee and Carter (1992) advocated to use a random walk with drift model to forecast principal component scores, expressed as kt = kt−1 + d + et , (2) where d is known as the drift parameter, measures the average annual change in the series,
13. 13. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model forecasts 1 There are a number of ways to adjust kt , which led to extensions and modiﬁcation of original Lee-Carter method. 2 Lee and Carter (1992) advocated to use a random walk with drift model to forecast principal component scores, expressed as kt = kt−1 + d + et , (2) where d is known as the drift parameter, measures the average annual change in the series, et is an uncorrelated error.
14. 14. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingLee-Carter model forecasts 1 There are a number of ways to adjust kt , which led to extensions and modiﬁcation of original Lee-Carter method. 2 Lee and Carter (1992) advocated to use a random walk with drift model to forecast principal component scores, expressed as kt = kt−1 + d + et , (2) where d is known as the drift parameter, measures the average annual change in the series, et is an uncorrelated error. 3 From the forecast of principal component scores, the forecast age-speciﬁc log mortality rates are obtained using the estimated age eﬀects ax and estimated ﬁrst set of principal component bx .
15. 15. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingConstruction of functional data 1 Functional data are a collection of functions, represented in the form of curves, images or shapes. France: male log mortality rate (1899−2005) 0 −2 Log mortality rate −4 −6 −8 −10 0 20 40 60 80 100 Age
16. 16. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingConstruction of functional data 1 Functional data are a collection of functions, represented in the form of curves, images or shapes. 2 Let’s consider annual French male log mortality rates from 1816 to 2006 for ages between 0 and 100. France: male log mortality rate (1899−2005) 0 −2 Log mortality rate −4 −6 −8 −10 0 20 40 60 80 100 Age
17. 17. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingConstruction of functional data 1 Functional data are a collection of functions, represented in the form of curves, images or shapes. 2 Let’s consider annual French male log mortality rates from 1816 to 2006 for ages between 0 and 100. 3 By interpolating 101 data points in one year, functional curves can be constructed below. France: male log mortality rate (1899−2005) 0 −2 Log mortality rate −4 −6 −8 −10 0 20 40 60 80 100 Age
18. 18. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingSmoothed functional data 1 Age-speciﬁc mortality rates are ﬁrst smoothed using penalized regression spline with monotonic constraint.
19. 19. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingSmoothed functional data 1 Age-speciﬁc mortality rates are ﬁrst smoothed using penalized regression spline with monotonic constraint. 2 Assuming there is an underlying continuous and smooth function {ft (x); x ∈ [x1 , xp ]} that is observed with errors at discrete ages in year t, we can express it as mt (xi ) = ft (xi ) + σt (xi )εt,i , t = 1, 2, . . . , n, (3) where
20. 20. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingSmoothed functional data 1 Age-speciﬁc mortality rates are ﬁrst smoothed using penalized regression spline with monotonic constraint. 2 Assuming there is an underlying continuous and smooth function {ft (x); x ∈ [x1 , xp ]} that is observed with errors at discrete ages in year t, we can express it as mt (xi ) = ft (xi ) + σt (xi )εt,i , t = 1, 2, . . . , n, (3) where mt (xi ) is the log mortality rates,
21. 21. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingSmoothed functional data 1 Age-speciﬁc mortality rates are ﬁrst smoothed using penalized regression spline with monotonic constraint. 2 Assuming there is an underlying continuous and smooth function {ft (x); x ∈ [x1 , xp ]} that is observed with errors at discrete ages in year t, we can express it as mt (xi ) = ft (xi ) + σt (xi )εt,i , t = 1, 2, . . . , n, (3) where mt (xi ) is the log mortality rates, ft (xi ) is the smoothed log mortality rates,
22. 22. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingSmoothed functional data 1 Age-speciﬁc mortality rates are ﬁrst smoothed using penalized regression spline with monotonic constraint. 2 Assuming there is an underlying continuous and smooth function {ft (x); x ∈ [x1 , xp ]} that is observed with errors at discrete ages in year t, we can express it as mt (xi ) = ft (xi ) + σt (xi )εt,i , t = 1, 2, . . . , n, (3) where mt (xi ) is the log mortality rates, ft (xi ) is the smoothed log mortality rates, σt (xi ) allows the possible presence of heteroscedastic error,
23. 23. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingSmoothed functional data 1 Age-speciﬁc mortality rates are ﬁrst smoothed using penalized regression spline with monotonic constraint. 2 Assuming there is an underlying continuous and smooth function {ft (x); x ∈ [x1 , xp ]} that is observed with errors at discrete ages in year t, we can express it as mt (xi ) = ft (xi ) + σt (xi )εt,i , t = 1, 2, . . . , n, (3) where mt (xi ) is the log mortality rates, ft (xi ) is the smoothed log mortality rates, σt (xi ) allows the possible presence of heteroscedastic error, εt,i is iid standard normal random variable.
24. 24. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingSmoothed functional data 1 Smoothness (also known ﬁltering) allows us to analyse derivative information of curves. France: male log mortality rate (1899−2005) 0 −2 Log mortality rate −4 −6 −8 −10 0 20 40 60 80 100 Age
25. 25. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingSmoothed functional data 1 Smoothness (also known ﬁltering) allows us to analyse derivative information of curves. 2 We transform n × p data matrix to n vector of functions. France: male log mortality rate (1899−2005) 0 −2 Log mortality rate −4 −6 −8 −10 0 20 40 60 80 100 Age
26. 26. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingFunctional principal component analysis (FPCA) 1 FPCA can be viewed from both covariance kernel function and linear operator perspectives.
27. 27. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingFunctional principal component analysis (FPCA) 1 FPCA can be viewed from both covariance kernel function and linear operator perspectives. 2 It is a dimension-reduction technique, with nice properties:
28. 28. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingFunctional principal component analysis (FPCA) 1 FPCA can be viewed from both covariance kernel function and linear operator perspectives. 2 It is a dimension-reduction technique, with nice properties: FPCA minimizes the mean integrated squared error, K 2 E f c (x) − βk φk (x) dx, K < ∞, (4) I k=1 where f c (x) = f (x) − µ(x) represents the decentralized functional curves, and x ∈ [x1 , xp ].
29. 29. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingFunctional principal component analysis (FPCA) 1 FPCA can be viewed from both covariance kernel function and linear operator perspectives. 2 It is a dimension-reduction technique, with nice properties: FPCA minimizes the mean integrated squared error, K 2 E f c (x) − βk φk (x) dx, K < ∞, (4) I k=1 where f c (x) = f (x) − µ(x) represents the decentralized functional curves, and x ∈ [x1 , xp ]. FPCA provides a way of extracting a large amount of variance, ∞ ∞ ∞ Var[f c (x)] = Var(βk )φ2 (x) = k λk φ2 (x) = k λk , (5) k=1 k=1 k=1 where λ1 ≥ λ2 , . . . , ≥ 0 is a decreasing sequence of eigenvalues and φk (x) is orthonormal.
30. 30. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingFunctional principal component analysis (FPCA) 1 FPCA can be viewed from both covariance kernel function and linear operator perspectives. 2 It is a dimension-reduction technique, with nice properties: FPCA minimizes the mean integrated squared error, K 2 E f c (x) − βk φk (x) dx, K < ∞, (4) I k=1 where f c (x) = f (x) − µ(x) represents the decentralized functional curves, and x ∈ [x1 , xp ]. FPCA provides a way of extracting a large amount of variance, ∞ ∞ ∞ Var[f c (x)] = Var(βk )φ2 (x) = k λk φ2 (x) = k λk , (5) k=1 k=1 k=1 where λ1 ≥ λ2 , . . . , ≥ 0 is a decreasing sequence of eigenvalues and φk (x) is orthonormal. The principal component scores are uncorrelated, that is cov(βi , βj ) = E(βi βj ) = 0, for i = j.
31. 31. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingKarhunen-Lo`ve (KL) expansion e By KL expansion, a stochastic process f (x), x ∈ [x1 , xp ] can be expressed as ∞ f (x) = µ(x) + βk φk (x), (6) k=1 K = µ(x) + βk φk (x) + e(x), (7) k=1 where 1 µ(x) is the population mean,
32. 32. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingKarhunen-Lo`ve (KL) expansion e By KL expansion, a stochastic process f (x), x ∈ [x1 , xp ] can be expressed as ∞ f (x) = µ(x) + βk φk (x), (6) k=1 K = µ(x) + βk φk (x) + e(x), (7) k=1 where 1 µ(x) is the population mean, 2 βk is the k th principal component scores,
33. 33. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingKarhunen-Lo`ve (KL) expansion e By KL expansion, a stochastic process f (x), x ∈ [x1 , xp ] can be expressed as ∞ f (x) = µ(x) + βk φk (x), (6) k=1 K = µ(x) + βk φk (x) + e(x), (7) k=1 where 1 µ(x) is the population mean, 2 βk is the k th principal component scores, 3 φk (x) is the k th functional principal components,
34. 34. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingKarhunen-Lo`ve (KL) expansion e By KL expansion, a stochastic process f (x), x ∈ [x1 , xp ] can be expressed as ∞ f (x) = µ(x) + βk φk (x), (6) k=1 K = µ(x) + βk φk (x) + e(x), (7) k=1 where 1 µ(x) is the population mean, 2 βk is the k th principal component scores, 3 φk (x) is the k th functional principal components, 4 e(x) is the error function, and
35. 35. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingKarhunen-Lo`ve (KL) expansion e By KL expansion, a stochastic process f (x), x ∈ [x1 , xp ] can be expressed as ∞ f (x) = µ(x) + βk φk (x), (6) k=1 K = µ(x) + βk φk (x) + e(x), (7) k=1 where 1 µ(x) is the population mean, 2 βk is the k th principal component scores, 3 φk (x) is the k th functional principal components, 4 e(x) is the error function, and 5 K is the number of retained principal components.
36. 36. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingEmpirical FPCA 1 Because the stochastic process f is unknown in practice, the population mean and eigenfunctions can only be approximated through realizations of {f1 (x), f2 (x), . . . , fn (x)}.
37. 37. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingEmpirical FPCA 1 Because the stochastic process f is unknown in practice, the population mean and eigenfunctions can only be approximated through realizations of {f1 (x), f2 (x), . . . , fn (x)}. 2 A function ft (x) can be approximated by K ¯ ft (x) = f (x) + βt,k φk (x) + e(x), (8) k=1 where
38. 38. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingEmpirical FPCA 1 Because the stochastic process f is unknown in practice, the population mean and eigenfunctions can only be approximated through realizations of {f1 (x), f2 (x), . . . , fn (x)}. 2 A function ft (x) can be approximated by K ¯ ft (x) = f (x) + βt,k φk (x) + e(x), (8) k=1 where ¯ 1 n f (x) = n t=1 ft (x) is the sample mean function,
39. 39. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingEmpirical FPCA 1 Because the stochastic process f is unknown in practice, the population mean and eigenfunctions can only be approximated through realizations of {f1 (x), f2 (x), . . . , fn (x)}. 2 A function ft (x) can be approximated by K ¯ ft (x) = f (x) + βt,k φk (x) + e(x), (8) k=1 where ¯ 1 n f (x) = n t=1 ft (x) is the sample mean function, βk is the k th empirical principal component scores,
40. 40. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingEmpirical FPCA 1 Because the stochastic process f is unknown in practice, the population mean and eigenfunctions can only be approximated through realizations of {f1 (x), f2 (x), . . . , fn (x)}. 2 A function ft (x) can be approximated by K ¯ ft (x) = f (x) + βt,k φk (x) + e(x), (8) k=1 where ¯ 1 n f (x) = n t=1 ft (x) is the sample mean function, βk is the k th empirical principal component scores, φk (x) is the k th empirical functional principal components,
41. 41. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingEmpirical FPCA 1 Because the stochastic process f is unknown in practice, the population mean and eigenfunctions can only be approximated through realizations of {f1 (x), f2 (x), . . . , fn (x)}. 2 A function ft (x) can be approximated by K ¯ ft (x) = f (x) + βt,k φk (x) + e(x), (8) k=1 where ¯ 1 n f (x) = n t=1 ft (x) is the sample mean function, βk is the k th empirical principal component scores, φk (x) is the k th empirical functional principal components, e(x) is the residual function.
42. 42. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingDecomposition 0.2 0.2 −1 0.2 0.20 0.1 0.1 −2 Basis function 1 Basis function 2 Basis function 3 Basis function 4 Mean function 0.15 0.0 0.1 −3 0.0 −0.1 0.10 −4 0.0 −0.1 −0.2 −5 0.05 −0.3 −0.1 −6 −0.2 0.00 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Age Age Age Age Age 8 10 0.5 6 1 5 Coefficient 1 Coefficient 2 Coefficient 3 Coefficient 4 0 4 0.0 0 −5 2 −0.5 −1 −10 0 −15 −1.0 −2 −2 1850 1900 1950 2000 1850 1900 1950 2000 1850 1900 1950 2000 1850 1900 1950 2000 Year Year Year Year 1 The principal components reveal underlying characteristics across age direction.
43. 43. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingDecomposition 0.2 0.2 −1 0.2 0.20 0.1 0.1 −2 Basis function 1 Basis function 2 Basis function 3 Basis function 4 Mean function 0.15 0.0 0.1 −3 0.0 −0.1 0.10 −4 0.0 −0.1 −0.2 −5 0.05 −0.3 −0.1 −6 −0.2 0.00 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Age Age Age Age Age 8 10 0.5 6 1 5 Coefficient 1 Coefficient 2 Coefficient 3 Coefficient 4 0 4 0.0 0 −5 2 −0.5 −1 −10 0 −15 −1.0 −2 −2 1850 1900 1950 2000 1850 1900 1950 2000 1850 1900 1950 2000 1850 1900 1950 2000 Year Year Year Year 1 The principal components reveal underlying characteristics across age direction. 2 The principal component scores reveal possible outlying years across time direction.
44. 44. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingPoint forecast Because orthogonality of the estimated functional principal components and uncorrelated principal component scores, point forecasts are obtained by K ¯ fn+h|n (x) = E[fn+h (x)|I, Φ] = f (x) + βn+h|n,k φk (x), (9) k=1 where 1 fn+h|n (x) is the h-step-ahead point forecast,
45. 45. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingPoint forecast Because orthogonality of the estimated functional principal components and uncorrelated principal component scores, point forecasts are obtained by K ¯ fn+h|n (x) = E[fn+h (x)|I, Φ] = f (x) + βn+h|n,k φk (x), (9) k=1 where 1 fn+h|n (x) is the h-step-ahead point forecast, 2 I represents the past data,
46. 46. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingPoint forecast Because orthogonality of the estimated functional principal components and uncorrelated principal component scores, point forecasts are obtained by K ¯ fn+h|n (x) = E[fn+h (x)|I, Φ] = f (x) + βn+h|n,k φk (x), (9) k=1 where 1 fn+h|n (x) is the h-step-ahead point forecast, 2 I represents the past data, 3 Φ = (φ1 (x), . . . , φK (x)) is a set of ﬁxed estimated functional principal components,
47. 47. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingPoint forecast Because orthogonality of the estimated functional principal components and uncorrelated principal component scores, point forecasts are obtained by K ¯ fn+h|n (x) = E[fn+h (x)|I, Φ] = f (x) + βn+h|n,k φk (x), (9) k=1 where 1 fn+h|n (x) is the h-step-ahead point forecast, 2 I represents the past data, 3 Φ = (φ1 (x), . . . , φK (x)) is a set of ﬁxed estimated functional principal components, 4 βn+h|n,k is the forecast of principal component scores by a univariate time series method, such as exponential smoothing.
48. 48. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingPoint forecast Point forecasts (2007−2026) 0 −2 Log mortality rate −4 −6 −8 Past data −10 Forecasts 0 20 40 60 80 100 Age Figure: 20-step-ahead point forecasts. Past data are shown in gray, whereas the recent data are shown in color.
49. 49. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingConclusion 1 We revisit the Lee-Carter model and functional time series model for modeling age-speciﬁc mortality rates,
50. 50. Lee-Carter model Nonparametric smoothing Functional principal component analysis Functional time series forecastingConclusion 1 We revisit the Lee-Carter model and functional time series model for modeling age-speciﬁc mortality rates, 2 We show how to compute point forecasts for both models.