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# Demographic forecasting

Demographic forecasting using functional data analysis

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### Demographic forecasting

1. 1. Demographic forecastingusing functional dataanalysisRob J HyndmanJoint work with: Heather Booth, Han Lin Shang, Shahid Ullah, Farah Yasmeen.Demographic forecasting using functional data analysis 1
2. 2. Mortality rates France: male mortality (1816) 0 −2Log death rate −4 −6 −8 0 20 40 60 80 100 Age Demographic forecasting using functional data analysis 2
3. 3. Fertility rates Australia: fertility rates (1921) 250 200Fertility rate 150 100 50 0 15 20 25 30 35 40 45 50 Age Demographic forecasting using functional data analysis 3
4. 4. Outline 1 A functional linear model 2 Bagplots, boxplots and outliers 3 Functional forecasting 4 Forecasting groups 5 Population forecasting 6 ReferencesDemographic forecasting using functional data analysis 4
5. 5. Outline 1 A functional linear model 2 Bagplots, boxplots and outliers 3 Functional forecasting 4 Forecasting groups 5 Population forecasting 6 ReferencesDemographic forecasting using functional data analysis A functional linear model 5
6. 6. Some notationLet yt,x be the observed (smoothed) data in period tat age x, t = 1, . . . , n. yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 Estimate ft (x) using penalized regression splines. Estimate µ(x) as me(di)an ft (x) across years. Estimate βt,k and φk (x) using (robust) functional principal components. iid iid εt,x ∼ N(0, 1) and et (x) ∼ N(0, v(x)).Demographic forecasting using functional data analysis A functional linear model 6
7. 7. Some notationLet yt,x be the observed (smoothed) data in period tat age x, t = 1, . . . , n. yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 Estimate ft (x) using penalized regression splines. Estimate µ(x) as me(di)an ft (x) across years. Estimate βt,k and φk (x) using (robust) functional principal components. iid iid εt,x ∼ N(0, 1) and et (x) ∼ N(0, v(x)).Demographic forecasting using functional data analysis A functional linear model 6
8. 8. Some notationLet yt,x be the observed (smoothed) data in period tat age x, t = 1, . . . , n. yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 Estimate ft (x) using penalized regression splines. Estimate µ(x) as me(di)an ft (x) across years. Estimate βt,k and φk (x) using (robust) functional principal components. iid iid εt,x ∼ N(0, 1) and et (x) ∼ N(0, v(x)).Demographic forecasting using functional data analysis A functional linear model 6
9. 9. Some notationLet yt,x be the observed (smoothed) data in period tat age x, t = 1, . . . , n. yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 Estimate ft (x) using penalized regression splines. Estimate µ(x) as me(di)an ft (x) across years. Estimate βt,k and φk (x) using (robust) functional principal components. iid iid εt,x ∼ N(0, 1) and et (x) ∼ N(0, v(x)).Demographic forecasting using functional data analysis A functional linear model 6
10. 10. Some notationLet yt,x be the observed (smoothed) data in period tat age x, t = 1, . . . , n. yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 Estimate ft (x) using penalized regression splines. Estimate µ(x) as me(di)an ft (x) across years. Estimate βt,k and φk (x) using (robust) functional principal components. iid iid εt,x ∼ N(0, 1) and et (x) ∼ N(0, v(x)).Demographic forecasting using functional data analysis A functional linear model 6
11. 11. French mortality components 0.2 −1 0.20 0.1 −2 0.15 −3 φ1(x) φ2(x) 0.0µ(x) 0.10 −4 −0.1 −5 0.05 −6 −0.2 0.00 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Age Age Age 8 10 6 5 0 4 βt1 βt2 −5 2 −15 −10 0 −2 1850 1900 1950 2000 1850 1900 1950 2000 t t Demographic forecasting using functional data analysis A functional linear model 7
12. 12. French mortality components Residuals 100 80 60Age 40 20 0 1850 1900 1950 2000 Year Demographic forecasting using functional data analysis A functional linear model 7
13. 13. Outline 1 A functional linear model 2 Bagplots, boxplots and outliers 3 Functional forecasting 4 Forecasting groups 5 Population forecasting 6 ReferencesDemographic forecasting using functional data analysis Bagplots, boxplots and outliers 8
14. 14. French male mortality rates France: male death rates (1900−2009) 0 −2Log death rate −4 −6 −8 0 20 40 60 80 100 Age Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 9
15. 15. French male mortality rates France: male death rates (1900−2009) 0 War years −2Log death rate −4 −6 −8 0 20 40 60 80 100 Age Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 9
16. 16. French male mortality rates France: male death rates (1900−2009) 0 War years −2Log death rate −4 −6 −8 Aims 1 “Boxplots” for functional data 0 20 240 Tools for detecting outliers in 60 80 100 functional data Age Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 9
17. 17. Robust principal componentsLet {ft (x)}, t = 1, . . . , n, be a set of curves. 1 Apply a robust principal component algorithm n−1 ft (x) = µ(x) + βt,k φk (x) k =1 µ(x) is median curve {φk (x)} are principal components {βt,k } are PC scores 2 Plot βi ,2 vs βi ,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 10
18. 18. Robust principal componentsLet {ft (x)}, t = 1, . . . , n, be a set of curves. 1 Apply a robust principal component algorithm n−1 ft (x) = µ(x) + βt,k φk (x) k =1 µ(x) is median curve {φk (x)} are principal components {βt,k } are PC scores 2 Plot βi ,2 vs βi ,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 10
19. 19. Robust principal componentsLet {ft (x)}, t = 1, . . . , n, be a set of curves. 1 Apply a robust principal component algorithm n−1 ft (x) = µ(x) + βt,k φk (x) k =1 µ(x) is median curve {φk (x)} are principal components {βt,k } are PC scores 2 Plot βi ,2 vs βi ,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 10
20. 20. Robust principal componentsLet {ft (x)}, t = 1, . . . , n, be a set of curves. 1 Apply a robust principal component algorithm n−1 ft (x) = µ(x) + βt,k φk (x) k =1 µ(x) is median curve {φk (x)} are principal components {βt,k } are PC scores 2 Plot βi ,2 vs βi ,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 10
21. 21. Robust principal componentsLet {ft (x)}, t = 1, . . . , n, be a set of curves. 1 Apply a robust principal component algorithm n−1 ft (x) = µ(x) + βt,k φk (x) k =1 µ(x) is median curve {φk (x)} are principal components {βt,k } are PC scores 2 Plot βi ,2 vs βi ,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 10
22. 22. Robust principal componentsLet {ft (x)}, t = 1, . . . , n, be a set of curves. 1 Apply a robust principal component algorithm n−1 ft (x) = µ(x) + βt,k φk (x) k =1 µ(x) is median curve {φk (x)} are principal components {βt,k } are PC scores 2 Plot βi ,2 vs βi ,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 10
23. 23. Robust principal componentsLet {ft (x)}, t = 1, . . . , n, be a set of curves. 1 Apply a robust principal component algorithm n−1 ft (x) = µ(x) + βt,k φk (x) k =1 µ(x) is median curve {φk (x)} are principal components {βt,k } are PC scores 2 Plot βi ,2 vs βi ,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 10
24. 24. Robust principal componentsLet {ft (x)}, t = 1, . . . , n, be a set of curves. 1 Apply a robust principal component algorithm n−1 ft (x) = µ(x) + βt,k φk (x) k =1 µ(x) is median curve {φk (x)} are principal components {βt,k } are PC scores 2 Plot βi ,2 vs βi ,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 10
25. 25. Robust principal componentsLet {ft (x)}, t = 1, . . . , n, be a set of curves. 1 Apply a robust principal component algorithm n−1 ft (x) = µ(x) + βt,k φk (x) k =1 µ(x) is median curve {φk (x)} are principal components {βt,k } are PC scores 2 Plot βi ,2 vs βi ,1 ¯ Each point in scatterplot represents one curve. ¯ Outliers show up in bivariate score space.Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 10
26. 26. Robust principal components Scatterplot of first two PC scores q q 6 q q q q 4 q qPC score 2 q 2 q q q q q qq qq q q q qq q qq q q q q q q q q qq q q q q q q qq q q qq qq qq qqq q q q q q q qq q qq q qq q q q qq q q qq 0 q q qq q q q q q q q q q qq q q q q q q q q q q q −2 −10 −5 0 5 10 15 PC score 1 Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 11
27. 27. Robust principal components Scatterplot of first two PC scores 1915 1914q q 6 1916q 1918q 1944q q 1917 4 1940q 1943qPC score 2 1919q 2 q q q q q qq qq q 1945q q qq q qq q q q q q q q qq q 1942q q q q q q qq 1941qqqqqqq q qq q qq qqq qq q q qq q q q qqq qqq q qq q 0 q q qq q q q q q q q q q qq q q q q q q q q q q q −2 −10 −5 0 5 10 15 PC score 1 Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 11
28. 28. Functional bagplot Bivariate bagplot due to Rousseeuw et al. (1999). Rank points by halfspace location depth. Display median, 50% convex hull and outer convex hull (with 99% coverage if bivariate normal). 1915 1914q q q 6 q 1916qq 1918q q 1944q q 1917 q q 4 1940q q 1943q q PC score 2 1919q 2 q q q q q qq q q q q qqq q q q q q q q q 1945q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q qq q qq 0 q q q q q q q q qq q q q q q q qqq q q q q q q q q q q q −2 −10 −5 0 5 10 15 PC score 1Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 12
29. 29. Functional bagplot Bivariate bagplot due to Rousseeuw et al. (1999). Rank points by halfspace location depth. Display median, 50% convex hull and outer convex hull (with 99% coverage if bivariate normal). Boundaries contain all curves inside bags. 95% CI for median curve also shown. 0 1915 1914q q q 6 q 1916qq 1918q q −2 1944q q 1917 q q 4 1940q q 1943q q Log death ratePC score 2 −4 1919q 2 q q q q qq q q q q qqq q q q q q q q 1945q q −6 q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q qq q qq 0 q q q q q q q q qq q q q q 1914 1918 −8 q q q qqq q q q q q q 1915 1940 q q q q 1916 1943 1917 1944 −2 −10 −5 0 5 10 15 0 20 40 60 80 100 PC score 1 Age Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 12
30. 30. Functional bagplot 0 −2Log death rate −4 −6 1914 1918 −8 1915 1940 1916 1943 1917 1944 0 20 40 60 80 100 Age Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 13
31. 31. Functional HDR boxplot Bivariate HDR boxplot due to Hyndman (1996). Rank points by value of kernel density estimate. Display mode, 50% and (usually) 99% highest density regions (HDRs) and mode. 99% outer region q 6 q q q q q 4 q 1943q q PC score 2 1919q 2 q q q q q q q q qqq q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q qq o q q q q q qq q qqq q 0 q q q q q q q q qq q q q q q q qqq q q q q q q q q q q q −2 −10 −5 0 5 10 15 PC score 1Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 14
32. 32. Functional HDR boxplot Bivariate HDR boxplot due to Hyndman (1996). Rank points by value of kernel density estimate. Display mode, 50% and (usually) 99% highest density regions (HDRs) and mode. 90% outer region 1915 1914q q q 6 q 1916qq 1918q q 1944q q 1917 q q 4 1940q q 1943q q PC score 2 1919q 2 q q q q qq q q q q qqq q q q q q q q 1945q q q q q q q q q q 1942q q q qq q q q q q q q q q q q q q qq q q q q q q q q q qq o q q q q q qq q qqq q 0 q q q q q q q q qq q q q q q q qqq q q q q q q q q q q q −2 −10 −5 0 5 10 15 PC score 1Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 14
33. 33. Functional HDR boxplot Bivariate HDR boxplot due to Hyndman (1996). Rank points by value of kernel density estimate. Display mode, 50% and (usually) 99% highest density regions (HDRs) and mode. Boundaries contain all curves inside HDRs. 0 90% outer region 1915 1914q q q 6 q 1916qq 1918q q −2 1944q q 1917 q q 4 1940q q Log death rate 1943q qPC score 2 −4 1919q 2 q q q q qq q q q q qqq q q q q q q q 1945q q −6 q q q q q q q q 1942q q q qq q q q q q q q q q q q 1914 1940 q q qq q q q q q qq o q qq qq q q q q qq q qqq q 1915 1943 0 q q q q q q q q qq q q q q 1916 1944 −8 q q q qqq q q q q q q 1917 1945 q q q q 1918 1948 1919 −2 −10 −5 0 5 10 15 0 20 40 60 80 100 PC score 1 Age Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 14
34. 34. Functional HDR boxplot 0 −2Log death rate −4 −6 1914 1940 1915 1943 1916 1944 −8 1917 1945 1918 1948 1919 0 20 40 60 80 100 Age Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 15
35. 35. Outline 1 A functional linear model 2 Bagplots, boxplots and outliers 3 Functional forecasting 4 Forecasting groups 5 Population forecasting 6 ReferencesDemographic forecasting using functional data analysis Functional forecasting 16
36. 36. Functional time series model yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 The eigenfunctions φk (x) show the main regions of variation. The scores {βt,k } are uncorrelated by construction. So we can forecast each βt,k using a univariate time series model. Outliers are treated as missing values. Univariate ARIMA models are used for forecasting.Demographic forecasting using functional data analysis Functional forecasting 17
37. 37. Functional time series model yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 The eigenfunctions φk (x) show the main regions of variation. The scores {βt,k } are uncorrelated by construction. So we can forecast each βt,k using a univariate time series model. Outliers are treated as missing values. Univariate ARIMA models are used for forecasting.Demographic forecasting using functional data analysis Functional forecasting 17
38. 38. Functional time series model yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 The eigenfunctions φk (x) show the main regions of variation. The scores {βt,k } are uncorrelated by construction. So we can forecast each βt,k using a univariate time series model. Outliers are treated as missing values. Univariate ARIMA models are used for forecasting.Demographic forecasting using functional data analysis Functional forecasting 17
39. 39. Functional time series model yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 The eigenfunctions φk (x) show the main regions of variation. The scores {βt,k } are uncorrelated by construction. So we can forecast each βt,k using a univariate time series model. Outliers are treated as missing values. Univariate ARIMA models are used for forecasting.Demographic forecasting using functional data analysis Functional forecasting 17
40. 40. Functional time series model yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 The eigenfunctions φk (x) show the main regions of variation. The scores {βt,k } are uncorrelated by construction. So we can forecast each βt,k using a univariate time series model. Outliers are treated as missing values. Univariate ARIMA models are used for forecasting.Demographic forecasting using functional data analysis Functional forecasting 17
41. 41. Forecasts yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1Demographic forecasting using functional data analysis Functional forecasting 18
42. 42. Forecasts yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 K E[yn+h ,x | y] = µ(x) + ˆ ˆ ˆ βn+h ,k φk (x) k =1 K ˆ2Var[yn+h ,x | y] = σµ (x) + ˆ2 vn+h ,k φk (x) + σt2 (x) + v(x) k =1where vn+h ,k = Var(βn+h ,k | β1,k , . . . , βn,k )and y = [y1,x , . . . , yn,x ].Demographic forecasting using functional data analysis Functional forecasting 18
43. 43. Forecasting the PC scores Main effects Interaction 0 0.2 0.20 −2 Basis function 1 Basis function 2 0.1 0.15Mean 0.0 −4 0.10 −0.1 0.05 −6 −0.2 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 Age Age Age 10 15 q 2 q qq q q q q 0 5 Coefficient 1 Coefficient 2 0 −2 q q −10 q −4 q q −6 q −20 q q 1900 1940 1980 2020 1900 1940 1980 2020 Year Year Demographic forecasting using functional data analysis Functional forecasting 19
44. 44. Forecasts of ft (x) France: male death rates (1900−2009) 0 −2Log death rate −4 −6 −8 −10 0 20 40 60 80 100 Age Demographic forecasting using functional data analysis Functional forecasting 20
45. 45. Forecasts of ft (x) France: male death rates (1900−2009) 0 −2Log death rate −4 −6 −8 −10 0 20 40 60 80 100 Age Demographic forecasting using functional data analysis Functional forecasting 20
46. 46. Forecasts of ft (x) France: male death forecasts (2010−2029) 0 −2Log death rate −4 −6 −8 −10 0 20 40 60 80 100 Age Demographic forecasting using functional data analysis Functional forecasting 20
47. 47. Forecasts of ft (x) France: male death forecasts (2010 & 2029) 0 −2Log death rate −4 −6 −8 −10 80% prediction intervals 0 20 40 60 80 100 Age Demographic forecasting using functional data analysis Functional forecasting 20
48. 48. Fertility application Australia fertility rates (1921−2009) 250 200Fertility rate 150 100 50 0 15 20 25 30 35 40 45 50 Age Demographic forecasting using functional data analysis Functional forecasting 21
49. 49. Fertility model 0.2 15 0.25 0.1 10 0.0 Φ1(x) Φ2(x) 0.15Μ −0.1 5 0.05 0 −0.3 15 20 25 30 35 40 45 50 15 20 25 30 35 40 45 50 15 20 25 30 35 40 45 50 Age Age Age 10 8 6 5 4 0 Βt1 Βt2 2 −5 0 −4 −2 −10 1970 1980 1990 2000 2010 1970 1980 1990 2000 2010 t t Demographic forecasting using functional data analysis Functional forecasting 22
50. 50. Fertility model Residuals 45 40 35Age 30 25 20 15 1970 1980 1990 2000 Year Demographic forecasting using functional data analysis Functional forecasting 23
51. 51. Fertility model Main effects Interaction 0.2 15 0.25 0.1 Basis function 1 Basis function 2 10 0.0Mean 0.15 −0.1 5 0.05 0 −0.3 15 20 25 30 35 40 45 50 15 20 25 30 35 40 45 50 15 20 25 30 35 40 45 50 Age Age Age 10 15 20 25 8 6 Coefficient 1 Coefficient 2 4 2 5 0 0 −4 −2 −10 1970 1990 2010 2030 1970 1990 2010 2030 Year Year Demographic forecasting using functional data analysis Functional forecasting 24
52. 52. Forecasts of ft (x) Australia fertility rates (1921−2009) 250 200Fertility rate 150 100 50 0 15 20 25 30 35 40 45 50 Age Demographic forecasting using functional data analysis Functional forecasting 25
53. 53. Forecasts of ft (x) Australia fertility rates (1921−2009) 250 200Fertility rate 150 100 50 0 15 20 25 30 35 40 45 50 Age Demographic forecasting using functional data analysis Functional forecasting 25
54. 54. Forecasts of ft (x) Australia fertility rates: 2010−2029 250 200Fertility rate 150 100 50 0 15 20 25 30 35 40 45 50 Age Demographic forecasting using functional data analysis Functional forecasting 25
55. 55. Forecasts of ft (x) Australia fertility rates: 2010 and 2029 80% prediction intervals 250 200Fertility rate 150 100 50 0 15 20 25 30 35 40 45 50 Age Demographic forecasting using functional data analysis Functional forecasting 25
56. 56. Outline 1 A functional linear model 2 Bagplots, boxplots and outliers 3 Functional forecasting 4 Forecasting groups 5 Population forecasting 6 ReferencesDemographic forecasting using functional data analysis Forecasting groups 26
57. 57. The problemLet ft,j (x) be the smoothed mortality rate for age x ingroup j in year t. Groups may be males and females. Groups may be states within a country. Expected that groups will behave similarly. Coherent forecasts do not diverge over time. Existing functional models do not impose coherence.Demographic forecasting using functional data analysis Forecasting groups 27
58. 58. The problemLet ft,j (x) be the smoothed mortality rate for age x ingroup j in year t. Groups may be males and females. Groups may be states within a country. Expected that groups will behave similarly. Coherent forecasts do not diverge over time. Existing functional models do not impose coherence.Demographic forecasting using functional data analysis Forecasting groups 27
59. 59. The problemLet ft,j (x) be the smoothed mortality rate for age x ingroup j in year t. Groups may be males and females. Groups may be states within a country. Expected that groups will behave similarly. Coherent forecasts do not diverge over time. Existing functional models do not impose coherence.Demographic forecasting using functional data analysis Forecasting groups 27
60. 60. The problemLet ft,j (x) be the smoothed mortality rate for age x ingroup j in year t. Groups may be males and females. Groups may be states within a country. Expected that groups will behave similarly. Coherent forecasts do not diverge over time. Existing functional models do not impose coherence.Demographic forecasting using functional data analysis Forecasting groups 27
61. 61. The problemLet ft,j (x) be the smoothed mortality rate for age x ingroup j in year t. Groups may be males and females. Groups may be states within a country. Expected that groups will behave similarly. Coherent forecasts do not diverge over time. Existing functional models do not impose coherence.Demographic forecasting using functional data analysis Forecasting groups 27
62. 62. The problemLet ft,j (x) be the smoothed mortality rate for age x ingroup j in year t. Groups may be males and females. Groups may be states within a country. Expected that groups will behave similarly. Coherent forecasts do not diverge over time. Existing functional models do not impose coherence.Demographic forecasting using functional data analysis Forecasting groups 27
63. 63. Forecasting the coeﬃcients yt,x = ft (x) + σt (x)εt,x K ft (x) = µ(x) + βt,k φk (x) + et (x) k =1 We use ARIMA models for each coeﬃcient {β1,j ,k , . . . , βn,j ,k }. The ARIMA models are non-stationary for the ﬁrst few coeﬃcients (k = 1, 2) Non-stationary ARIMA forecasts will diverge. Hence the mortality forecasts are not coherent.Demographic forecasting using functional data analysis Forecasting groups 28

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