# Multivariate Functional Time Series Forecasting: Application to Age-Specific Mortality Rates

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- 1)
- smooth the observed data in each population;
- 2)
- reduce the dimension of the functions in each population using functional principal component analysis (FPCA) separately;
- 3)
- fit the first set of principal component scores from all populations with VECM. Then, fit the second set of principal component scores with another VECM and so on. Produce forecasts using the fitted VECMs; and
- 4)
- produce forecasts of mortality curves.

## 2. Forecasting Models

#### 2.1. Univariate Autoregressive Integrated Moving Average Model

#### 2.2. Vector Autoregressive Model

#### 2.2.1. Model Structure

#### 2.2.2. Relationship between the Functional Autoregressive and Vector Autoregressive Models

#### 2.3. Vector Error Correction Model

#### 2.3.1. Fitting a Vector Error Correction Model to Principal Component Scores

#### 2.3.2. Estimation

#### 2.3.3. Expressing a Vector Error Correction Model in a Vector Autoregressive Form

#### 2.4. Product–Ratio Model

#### 2.5. Bootstrap Prediction Interval

- 1)
- Smooth the functions with ${y}_{t}^{(\omega )}({x}_{j})={f}_{t}^{(\omega )}({x}_{j})+{u}_{t}^{(\omega )}({x}_{j}),\phantom{\rule{1.em}{0ex}}\omega =1,2$, where ${u}_{t}^{(\omega )}$ is the smoothing error with mean zero and estimated variance ${\widehat{\sigma}}_{t}^{2}{({x}_{j})}^{(\omega )},\phantom{\rule{1.em}{0ex}}j=1,\cdots ,p$.
- 2)
- Perform FPCA on the smoothed functions ${f}_{t}^{(1)}$ and ${f}_{t}^{(2)}$ separately, and obtain K pairs of principal component scores ${\mathit{\xi}}_{t,k}={\left(\right)}^{{\xi}_{t,k}^{(1)}}\top $.
- 3)
- Fit K VECM models to the principal component scores. From the fitted scores ${\widehat{\mathit{\xi}}}_{t,k}$, for $t=1,\cdots ,n$ and $k=1,\cdots ,K$, obtain the fitted functions ${\widehat{\mathit{f}}}_{t},={\left(\right)}^{{\widehat{f}}_{t}^{(1)}}\top $.
- 4)
- Obtain residuals ${\mathit{e}}_{t}$ from ${\mathit{e}}_{t}={\mathit{f}}_{t}-{\widehat{\mathit{f}}}_{t}$.
- 5)
- Express the estimated VECM from step 3 in its VAR form: ${\mathit{\xi}}_{t,k}={\widehat{\mathit{A}}}_{1}{\mathit{\xi}}_{t-1,k}+{\widehat{\mathit{A}}}_{2}{\mathit{\xi}}_{t-2,k}+{\mathit{\u03f5}}_{t,k},$ $\phantom{\rule{1.em}{0ex}}t=1,\cdots ,n$ and $k=1,\cdots ,K$. Construct K sets of bootstrap principal component scores time series ${\mathit{\xi}}_{t,k}^{*}={\widehat{\mathit{A}}}_{1}{\mathit{\xi}}_{t-1,k}^{*}+{\widehat{\mathit{A}}}_{2}{\mathit{\xi}}_{t-2,k}^{*}+{\mathit{\u03f5}}_{t,k}^{*}$, where the error term ${\mathit{\u03f5}}_{t,k}^{*}$ is re-sampled with replacement from ${\mathit{\u03f5}}_{t,k}$.
- 6)
- Refit a VECM with ${\mathit{\xi}}_{t,k}^{*}$ and make h-step-ahead predictions ${\widehat{\mathit{\xi}}}_{n+h|n}^{*}$ and hence a predicted function ${\widehat{\mathit{f}}}_{n+h|n}^{*}$.
- 7)
- Construct a bootstrapped h-step-ahead prediction for the function by$$\begin{array}{c}\hfill {\widehat{\mathit{f}}}_{n+h|n}^{**}({x}_{j})={\widehat{\mathit{f}}}_{n+h|n}^{*}({x}_{j})+{\mathit{e}}_{t}^{*}+{\mathit{u}}_{t}^{*}({x}_{j}),\end{array}$$
- 8)
- Repeat steps 5 to 7 many times.
- 9)
- The $(1-\alpha )\times 100\%$ point-wise prediction intervals can be constructed by taking the $\frac{\alpha}{2}\times 100\%$ and $(1-\frac{\alpha}{2})\times 100\%$ quantiles of the bootstrapped samples.

## 3. Forecast Evaluation

## 4. Simulation Studies

#### 4.1. With Co-Integration

#### 4.2. Without Co-Integration

#### 4.3. Results

## 5. Empirical Studies

#### 5.1. Swiss Age-Specific Mortality Rates

#### 5.2. Czech Republic Age-Specific Mortality Rates

## 6. Conclusions

## Supplementary Materials

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Functional Principal Component Analysis

## Appendix B. Functional Principal Component Regression

## References

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**Sample Availability:**Computational code in R are available upon request from the authors.

**Figure 1.**Simulated basis functions for the first and second populations. (

**a**) basis functions for population 1; (

**b**) basis functions for population 2.

**Figure 2.**The first row presents the mean squared prediction error (MSPE) and the mean interval scores for the two populations in a co-integration setting. The second row presents the MSPE and the mean interval scores for the two populations without the co-integration. (

**a**) $1\mathrm{st}$ population; (

**b**) $2\mathrm{nd}$ population; (

**c**) $1\mathrm{st}$ population; (

**d**) $2\mathrm{nd}$ population; (

**e**) $1\mathrm{st}$ population; (

**f**) $2\mathrm{nd}$ population; (

**g**) $1\mathrm{st}$ population; and (

**h**) $2\mathrm{nd}$ population.

**Figure 3.**Smoothed log mortality rates in Switzerland from 1950 to 2014. (

**a**) female population; (

**b**) male population.Short Caption

**Figure 4.**Czech Republic: forecast errors for female and male mortality rates (MSPE and interval scores are presented). (

**a**) MSPE for female data; (

**b**) mean interval score for female data; (

**c**) MSPE for male data; (

**d**) mean interval score for male data.

**Figure 5.**Czech Republic: p-values for the three tests comparing a functional VECM to the univariate, VAR, and product–ratio models, respectively (the horizontal line is the default level of significance $\alpha =0.05$). (

**a**) female population; (

**b**) male population.

**Table 1.**Mean squared prediction error (MSPE) for Swiss female and male rates (the smallest values are highlighted in bold).

h | Female | Male | ||||||
---|---|---|---|---|---|---|---|---|

UNI | VAR | PR | VECM | UNI | VAR | PR | VECM | |

1 | $0.081$ | $0.082$ | $0.076$ | $\mathbf{0}\mathbf{.}\mathbf{074}$ | $0.050$ | $\mathbf{0}\mathbf{.}\mathbf{048}$ | $0.049$ | $0.049$ |

2 | $0.085$ | $0.088$ | $0.079$ | $\mathbf{0}\mathbf{.}\mathbf{075}$ | $0.056$ | $\mathbf{0}\mathbf{.}\mathbf{052}$ | $0.053$ | $0.053$ |

3 | $0.090$ | $0.094$ | $0.084$ | $\mathbf{0}\mathbf{.}\mathbf{078}$ | $0.065$ | $\mathbf{0}\mathbf{.}\mathbf{059}$ | $0.060$ | $0.060$ |

4 | $0.096$ | $0.104$ | $0.091$ | $\mathbf{0}\mathbf{.}\mathbf{082}$ | $0.077$ | $\mathbf{0}\mathbf{.}\mathbf{067}$ | $0.070$ | $0.069$ |

5 | $0.103$ | $0.112$ | $0.098$ | $\mathbf{0}\mathbf{.}\mathbf{086}$ | $0.090$ | $\mathbf{0}\mathbf{.}\mathbf{078}$ | $0.080$ | $\mathbf{0}\mathbf{.}\mathbf{078}$ |

6 | $0.109$ | $0.119$ | $0.107$ | $\mathbf{0}\mathbf{.}\mathbf{090}$ | $0.107$ | $0.093$ | $0.093$ | $\mathbf{0}\mathbf{.}\mathbf{089}$ |

7 | $0.117$ | $0.130$ | $0.119$ | $\mathbf{0}\mathbf{.}\mathbf{096}$ | $0.129$ | $0.115$ | $0.109$ | $\mathbf{0}\mathbf{.}\mathbf{104}$ |

8 | $0.125$ | $0.140$ | $0.130$ | $\mathbf{0}\mathbf{.}\mathbf{102}$ | $0.149$ | $0.136$ | $0.124$ | $\mathbf{0}\mathbf{.}\mathbf{119}$ |

9 | $0.136$ | $0.151$ | $0.145$ | $\mathbf{0}\mathbf{.}\mathbf{111}$ | $0.171$ | $0.160$ | $0.139$ | $\mathbf{0}\mathbf{.}\mathbf{129}$ |

10 | $0.145$ | $0.163$ | $0.157$ | $\mathbf{0}\mathbf{.}\mathbf{116}$ | $0.198$ | $0.191$ | $0.160$ | $\mathbf{0}\mathbf{.}\mathbf{149}$ |

11 | $0.156$ | $0.171$ | $0.173$ | $\mathbf{0}\mathbf{.}\mathbf{125}$ | $0.224$ | $0.223$ | $0.178$ | $\mathbf{0}\mathbf{.}\mathbf{162}$ |

12 | $0.167$ | $0.186$ | $0.195$ | $\mathbf{0}\mathbf{.}\mathbf{133}$ | $0.261$ | $0.269$ | $0.206$ | $\mathbf{0}\mathbf{.}\mathbf{184}$ |

13 | $0.174$ | $0.192$ | $0.210$ | $\mathbf{0}\mathbf{.}\mathbf{137}$ | $0.299$ | $0.317$ | $0.232$ | $\mathbf{0}\mathbf{.}\mathbf{201}$ |

14 | $0.188$ | $0.203$ | $0.238$ | $\mathbf{0}\mathbf{.}\mathbf{145}$ | $0.344$ | $0.361$ | $0.260$ | $\mathbf{0}\mathbf{.}\mathbf{213}$ |

15 | $0.183$ | $0.209$ | $0.254$ | $\mathbf{0}\mathbf{.}\mathbf{141}$ | $0.396$ | $0.414$ | $0.293$ | $\mathbf{0}\mathbf{.}\mathbf{228}$ |

16 | $0.197$ | $0.219$ | $0.281$ | $\mathbf{0}\mathbf{.}\mathbf{152}$ | $0.460$ | $0.444$ | $0.332$ | $\mathbf{0}\mathbf{.}\mathbf{239}$ |

17 | $0.209$ | $0.223$ | $0.327$ | $\mathbf{0}\mathbf{.}\mathbf{164}$ | $0.538$ | $0.556$ | $0.373$ | $\mathbf{0}\mathbf{.}\mathbf{251}$ |

18 | $0.209$ | $0.233$ | $0.354$ | $\mathbf{0}\mathbf{.}\mathbf{165}$ | $0.649$ | $0.652$ | $0.416$ | $\mathbf{0}\mathbf{.}\mathbf{263}$ |

19 | $0.197$ | $0.232$ | $0.457$ | $\mathbf{0}\mathbf{.}\mathbf{162}$ | $0.792$ | $0.733$ | $0.502$ | $\mathbf{0}\mathbf{.}\mathbf{253}$ |

20 | $\mathbf{0}\mathbf{.}\mathbf{144}$ | $0.249$ | $0.493$ | $0.175$ | $0.904$ | $0.753$ | $0.525$ | $\mathbf{0}\mathbf{.}\mathbf{270}$ |

Mean | $0.145$ | $0.165$ | $0.203$ | $\mathbf{0}\mathbf{.}\mathbf{120}$ | $0.298$ | $0.286$ | $0.213$ | $\mathbf{0}\mathbf{.}\mathbf{158}$ |

Median | $0.145$ | $0.265$ | $0.173$ | $\mathbf{0}\mathbf{.}\mathbf{120}$ | $0.224$ | $0.223$ | $0.178$ | $\mathbf{0}\mathbf{.}\mathbf{158}$ |

**Table 2.**Mean interval score (80%) for Swiss female and male rates (the smallest values are highlighted in bold).

h | Female | Male | ||||||
---|---|---|---|---|---|---|---|---|

UNI | VAR | PR | VECM | UNI | VAR | PR | VECM | |

1 | $1.089$ | $1.042$ | $0.865$ | $\mathbf{0}\mathbf{.}\mathbf{852}$ | $0.871$ | $0.767$ | $\mathbf{0}\mathbf{.}\mathbf{657}$ | $0.715$ |

2 | $1.114$ | $1.042$ | $0.878$ | $\mathbf{0}\mathbf{.}\mathbf{864}$ | $0.964$ | $0.786$ | $\mathbf{0}\mathbf{.}\mathbf{699}$ | $0.748$ |

3 | $1.153$ | $1.059$ | $0.909$ | $\mathbf{0}\mathbf{.}\mathbf{880}$ | $1.088$ | $0.852$ | $\mathbf{0}\mathbf{.}\mathbf{759}$ | $0.791$ |

4 | $1.204$ | $1.102$ | $0.954$ | $\mathbf{0}\mathbf{.}\mathbf{902}$ | $1.243$ | $0.911$ | $\mathbf{0}\mathbf{.}\mathbf{838}$ | $0.839$ |

5 | $1.254$ | $1.136$ | $0.997$ | $\mathbf{0}\mathbf{.}\mathbf{926}$ | $1.407$ | $1.011$ | $0.909$ | $\mathbf{0}\mathbf{.}\mathbf{887}$ |

6 | $1.306$ | $1.169$ | $1.046$ | $\mathbf{0}\mathbf{.}\mathbf{964}$ | $1.594$ | $1.134$ | $1.005$ | $\mathbf{0}\mathbf{.}\mathbf{954}$ |

7 | $1.358$ | $1.234$ | $1.113$ | $\mathbf{0}\mathbf{.}\mathbf{996}$ | $1.789$ | $1.289$ | $1.113$ | $\mathbf{1}\mathbf{.}\mathbf{059}$ |

8 | $1.413$ | $1.276$ | $1.166$ | $\mathbf{1}\mathbf{.}\mathbf{026}$ | $1.969$ | $1.430$ | $\mathbf{1}\mathbf{.}\mathbf{190}$ | $1.133$ |

9 | $1.483$ | $1.349$ | $1.241$ | $\mathbf{1}\mathbf{.}\mathbf{088}$ | $2.134$ | $1.587$ | $1.282$ | $\mathbf{1}\mathbf{.}\mathbf{204}$ |

10 | $1.532$ | $1.426$ | $1.287$ | $\mathbf{1}\mathbf{.}\mathbf{113}$ | $2.326$ | $1.798$ | $1.388$ | $\mathbf{1}\mathbf{.}\mathbf{338}$ |

11 | $1.608$ | $1.479$ | $1.358$ | $\mathbf{1}\mathbf{.}\mathbf{170}$ | $2.476$ | $2.012$ | $1.475$ | $\mathbf{1}\mathbf{.}\mathbf{458}$ |

12 | $1.661$ | $1.591$ | $1.437$ | $\mathbf{1}\mathbf{.}\mathbf{209}$ | $2.655$ | $2.303$ | $\mathbf{1}\mathbf{.}\mathbf{609}$ | $1.628$ |

13 | $1.716$ | $1.647$ | $1.463$ | $\mathbf{1}\mathbf{.}\mathbf{237}$ | $2.819$ | $2.618$ | $\mathbf{1}\mathbf{.}\mathbf{706}$ | $1.767$ |

14 | $1.766$ | $1.723$ | $1.540$ | $\mathbf{1}\mathbf{.}\mathbf{281}$ | $3.001$ | $2.892$ | $\mathbf{1}\mathbf{.}\mathbf{793}$ | $1.891$ |

15 | $1.705$ | $1.775$ | $1.571$ | $\mathbf{1}\mathbf{.}\mathbf{262}$ | $3.145$ | $3.082$ | $1.892$ | $1.963$ |

16 | $1.774$ | $1.790$ | $1.638$ | $\mathbf{1}\mathbf{.}\mathbf{304}$ | $3.309$ | $3.180$ | $\mathbf{1}\mathbf{.}\mathbf{957}$ | $1.986$ |

17 | $1.852$ | $1.860$ | $1.760$ | $\mathbf{1}\mathbf{.}\mathbf{352}$ | $3.521$ | $3.692$ | $2.041$ | $\mathbf{2}\mathbf{.}\mathbf{011}$ |

18 | $1.819$ | $1.884$ | $1.767$ | $\mathbf{1}\mathbf{.}\mathbf{368}$ | $3.632$ | $4.148$ | $\mathbf{2}\mathbf{.}\mathbf{036}$ | $2.051$ |

19 | $1.795$ | $1.986$ | $1.941$ | $\mathbf{1}\mathbf{.}\mathbf{360}$ | $3.683$ | $4.254$ | $2.175$ | $\mathbf{1}\mathbf{.}\mathbf{974}$ |

20 | $1.679$ | $2.347$ | $2.176$ | $\mathbf{1}\mathbf{.}\mathbf{398}$ | $3.873$ | $3.595$ | $2.375$ | $\mathbf{1}\mathbf{.}\mathbf{978}$ |

Mean | $1.514$ | $1.496$ | $1.355$ | $\mathbf{1}\mathbf{.}\mathbf{128}$ | $2.375$ | $2.167$ | $1.445$ | $\mathbf{1}\mathbf{.}\mathbf{419}$ |

Median | $1.532$ | $1.479$ | $1.355$ | $\mathbf{1}\mathbf{.}\mathbf{128}$ | $2.375$ | $2.012$ | $1.445$ | $\mathbf{1}\mathbf{.}\mathbf{419}$ |

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## Share and Cite

**MDPI and ACS Style**

Gao, Y.; Shang, H.L.
Multivariate Functional Time Series Forecasting: Application to Age-Specific Mortality Rates. *Risks* **2017**, *5*, 21.
https://doi.org/10.3390/risks5020021

**AMA Style**

Gao Y, Shang HL.
Multivariate Functional Time Series Forecasting: Application to Age-Specific Mortality Rates. *Risks*. 2017; 5(2):21.
https://doi.org/10.3390/risks5020021

**Chicago/Turabian Style**

Gao, Yuan, and Han Lin Shang.
2017. "Multivariate Functional Time Series Forecasting: Application to Age-Specific Mortality Rates" *Risks* 5, no. 2: 21.
https://doi.org/10.3390/risks5020021