人口推估(Population Projection)涉及國家的政策及規劃，精確的結果可協助國家適時制訂政策，提高國民福祉。臺灣現在使用的方法為人口變動要素合成法(The Cohort Component Method)，可算是情境推估(Scenario Forecast)的一種，其起源可追溯至1920年代(Whelpton, 1928)，參酌專家意見之後，使用高、中、低三種推計來描述其變動範圍。除了情境推估外，近年在人口變動要素合成方法上發展出的新方法大致可以分成三種：一為隨機推估(Stochastic Forecast Method)、一為模擬情境(Random Scenario Method)、一為推估誤差(ex post Method)，美國及聯合國已經不單單依賴專家提供的傳統高、中、低推計，轉而使用這些新的推估方法。 由於近年來生育率快速降低、平均餘命延長以及外籍新娘增多等因素，大為提高人口推估的難度，因此本文將機率的概念併入人口推估中，以預測區間（Prediction Interval）來捕捉人口各項特性的可能變動範圍。除了回顧幾種在人口變動要素合成法中發展出的隨機推估方法及合併專家意見的方針外，也使用區塊拔靴法(Block Bootstrap)電腦模擬，進行臺灣、美國、日本、法國四個國家的人口推估。另外，本文也採用以Stoto(1983)提出的預測誤差估計，評估區塊拔靴法和人力規劃處推估結果之異同，以提供使用專家意見與隨機方法的參考。最後則是比較臺灣以北中南東小區域推估和臺灣整體的推估結果，並合併專家意見進行臺灣地區人口推估。

政府制訂與規劃政策，需要未來人口數或人口結構的資訊，以期能針對國家需求對症下藥；預測未來人口總數及人口結構通稱為人口推估(Population Projection)，台灣的官方人口推估由行政院經濟建設委員會負責。人口推估通常使用人口變動要素合成法(The Cohort Component Method)，根據現有人口總數及結構，加上對未來生育(Fertility)、死亡(Death)、遷移(Migration)三要素的預期，推估出未來的人口數及其結構；除了參考歷史資料外，目前三要素大多參考專家意見(Expert opinion)決定變動範圍。 在三要素之中，生育和遷移由於變動趨勢較大，較不容易以歷史資料找出趨勢，過去大多倚賴專家意見進行生育和遷移的假設。但專家意見也有蒐集上的疑慮，例如意見沒有固定的形成格式，推估結果也難以機率角度詮釋發生可能性，亦即推估數值大多不隨時間波動。為了彌補專家意見的不足，近年不少國家的人口推估使用隨機方法(Stochastic Methods)，大致可分為三種：隨機推估法(Stochastic Forecast)、模擬情境法(Random Scenario Method)和推估誤差法(ex post Method)，這三種方法可組合使用，例如Lutz (1995)合併隨機推估和模擬情境，而郭孟坤與余清祥(2008)則是以隨機推估和推估誤差的組合來進行推估，使隨機方法更具有彈性，適用於不同情境的人口推估。 本文目的在於結合專家意見與隨機方法，希冀綜合兩者的優點，提出適用於台灣的人口推估方法。首先，本文提出將專家方法數量化的幾種作法，配合德菲法蒐集專家意見，先以電腦模擬找出較佳的數量化方法；接著再參考由歷史資料及隨機方法（區塊拔靴法）得出的推估結果，分別以主觀角度、貝氏分析(Bayesian Analysis)和貝氏可信度(Bayesian Credibility)的角度進行加權平均，結合專家意見及隨機方法的推估。

人口政策的制定需要人口推估作基礎。近年世界各國人口推估逐漸從專家意見推估走向機率推估，常見的機率推估分成三大類，隨機推估、模擬情境、推估誤差三種，本文所使用的人口推估方法為隨機推估法結合生育率之模擬情境方法，在人口變動要素組合法 (Cohort Component Method) 之下輔以電腦模擬的區塊拔靴法 (Block Bootstrap)，針對台灣地區與台灣北、中、南、東四地區進行人口推估。另外，本文試圖在隨機模型人口推估中加入遷移人口之考量，以期針對遷移人口在數量與其影響上都能有較深入的了解，比較區塊拔靴法與經建會推估之差異後發現遷移之考量確實會影響人口推估之結果。 / 針對與全區相符的小區域人口推估，本文亦提出可使得推估一致的方法，但其缺點為限制了生育、死亡人口要素之變動性。此推估在總數上與隨機推估方法差異不大，但在人口結構上則有明顯的差別，此差別可能是來自於死亡率在四區間差異造成。 / Population projection is important to policy making, and only with accurate population projection can the government achieve suitable policy planning and improve the welfare of the society. The most popular and well-known population projection method is the Cohort Component method, proposed since 1930’s. The trends of future fertility, mortality and migration are required, in order to apply the cohort component method. Currently in Taiwan, these trends are determined according to experts’ opinions (or scenario projection) and three future scenarios are assumed: high, median and low scenarios. One of the drawbacks in applying experts’ opinions is that the projection results of these three scenarios do not have the meaning in probability. / To modify the expert’ opinions and let the projection results carry the meaning in probability, many demographic researchers have developed stochastic projection methods. The proposed stochastic methods can be categorized into three groups: stochastic forecast, random scenario and ex post methods. In this study, we introduce these stochastic methods and evaluate the possibility of applying the methods in projecting the population in Taiwan. / In this study we use block bootstrap, a computer simulation and stochastic forecast method, to determine the trends of future fertility, mortality and migration in Taiwan, and combine it with the cohort component method for population projection in Taiwan. We compare the projection results with those from the Council for Economic Planning and Development (a scenario projection). We found that the block bootstrap is a possible alternative to the scenario projection in population projection, and the numbers of migration is small but have a non-ignorable influence on the future population. However, we also found that the block bootstrap alone might not be appropriate for population projection in small areas.

由於誤差與人口數成反比，資料多寡影響統計分析的穩定性及可靠性，因此常用於推估大區域人口的方法，往往無法直接套用至縣市及其以下層級，尤其當小區域內部地理、社會或經濟的異質性偏高時，人口推估將更為棘手。本文以兩個面向對臺灣小區域人口進行探討：其一、臺灣人口結構漸趨老化，勢必牽動政府政策與資源分配，且臺灣各縣市的人口老化速度不一，有必要針對各地特性發展適當的小區域人口推估方法；其二、因為壽命延長，全球皆面臨長壽風險(Longevity Risk)的挑戰，包括政府退休金制度規劃、壽險保費釐定等，由於臺灣各地死亡率變化不盡相同，發展小區域死亡率模型也是迫切課題。 小區域推估面臨的問題大致可歸納為四個方向：「資料品質」、「地區人數」、「資料年數」與「推估年數」，資料品質有賴資料庫與制度的建立，關於後三個問題，本文引進修勻(Smoothing, Graduation)等方法來提高小區域推估及小區域死亡模型的穩定性。人口推估方面結合修勻與區塊拔靴法(Block Bootstrap)，死亡率模型的建構則將修勻加入Lee-Carter與Age-Period-Cohort模型。由於小區域人口數較少，本文透過標準死亡比(Standard Mortality Ratio)及大區域與小區域間的連貫(Coherence)，將大區域的訊息加入小區域，降低因為地區人數較少引起的震盪。 小區域推估通常可用的資料時間較短，未來推估結果的震盪也較大，本文針對需要過去幾年資料，以及未來可推估年數等因素進行研究，希冀結果可提供臺灣各地方政府的推估參考。研究發現，參考大區域訊息有穩定推估的效果，修勻有助於降低推估誤差；另外，在小區域推估中，如有過去十五年資料可獲得較可靠的推估結果，而未來推估年數盡量不超過二十年，若地區人數過少則建議合併其他區域增加資料量後再行推估；先經過修勻而得出的死亡率模型，其效果和較為複雜的連貫模型修正相當。 / The population size plays a very important role in statistical estimation, and it is difficult to derive a reliable estimation for small areas. The estimation is even more difficult if the geographic and social attributes within the small areas vary widely. However, although the population aging and longevity risk are common phenomenon in the world, the problem is not the same for different countries. The aim of this study is to explore the population projection and mortality models for small areas, with the consideration of the small area’s distinguishing characteristic. The difficulties for small area population projection can be attributed into four directions: data quality, population size, number of base years, and projection horizon. The data quality is beyond the discussion of this study and the main focus shall be laid on the other three issues. The smoothing methods and coherent models will be applied to improve the stability and accuracy of small area estimation. In the study, the block bootstrap and the smoothing methods are combined to project the population to the small areas in Taiwan. Besides, the Lee-Cater and the age-period-cohort model are extended by the smoothing and coherent methods. We found that the smoothing methods can reduce the fluctuation of estimation and projection in general, and the improvement is especially noticeable for areas with smaller population sizes. To obtain a reliable population projection for small areas, we suggest using at least fifteen-year of historical data for projection and a projection horizon not more than twenty years. Also, for developing mortality models for small areas, we found that the smoothing methods have similar effects than those methods using more complicated models, such as the coherent models.

由於許多國家死亡率下降快速、壽命延長幅度超乎預期，加上生育率持續低於替代水準，人口老化現象愈發明顯，近年來個人生涯規劃及政府施政，都格外強調退休後經濟生活及老年相關社會資源分配的比重。以臺灣為例，行政院經濟建設委員會 (簡稱經建會) 從1990年代開始，每兩年公布一次未來的人口推估，但過去十年來經建會屢次修正歷年的推估假設，以因應生育率及死亡率變化快速，適時提醒臺灣日益加速的人口老化。正因為人口推估可能受到人口數、社會變遷、資料品質等因素，影響統計分析的可靠性，常用於國家層級的推估方法，往往無法直接套用至縣市及其以下的層級 (即小區域)，使得小區域人口推估較為棘手，需要更加謹慎面對。 本文延續王信忠等人 (2012) 的研究，以小區域人口推估為目標，著重在生育率推估研究，結合隨機模型與修勻方法，尋找適合臺灣縣市層級的小區域人口推估方法。本文考量的隨機模型計有區塊拔靴法 (Block Bootstrap) 和 Lee-Carter 模型 (Lee and Carter 1992)，以預測未來的生育率和死亡率，並套用年輪組成推計法 (或稱為人口要素合成法；Cohort Component Method) 及修勻 (Graduation) 方法，探討這些方法與人口規模之間的關係，評估用於小區域人口推估之可行性。 本文首先以電腦模擬，探討生育率的推估，討論是否可直接推估總生育率，類似增加樣本數的概念，取代各縣市的年齡別生育率，以取得較為穩定的推估。根據模擬結果，發現人口規模對出生數的推估沒有明顯的關係，只要使用總生育率、再結合區塊拔靴法，就足以提供穩定的推估結果。實證研究方面，以臺灣縣市層級的人口及其年齡結構 (例如：0-14歲、15-64歲、65歲以上) 為驗證對象，發現分析結果也與電腦模擬相似，發現以區塊拔靴法推估臺灣各縣市的總生育率、年齡組死亡率，其推估精確度不因人口規模而打折扣，顯示以區塊拔靴法推估總生育率、年齡組死亡率，可用於推估臺灣小地區的未來人口。 / Due to the rapid mortality reduction, prolonging human longevity is a common phenomenon and longevity risk receives more attention in 21st century. Many developed countries encounter many problems brought up by prolonging life, such as poor community infrastructure and insufficient financial pension funds for the elderly. Population Projection thus becomes essential in government planning in dealing with the population aging. However, rapid changes in mortality and fertility make the projection very tricky. It would be even more difficult to project areas with fewer populations (i.e., small areas) since it takes extra efforts to deal with the larger fluctuations in small population. The objective of the study is to construct a standard operating procedure (SOP) for small population projection. Unlike the previous study, e.g., Wang et al. (2012), we will take both the fertility and mortality into account (but set migration aside for simplicity). First, for the fertility projection, we evaluate if total fertility rates (TFR) are more appropriate than the age-specific fertility rates for small population. Also, we compare two fertility projection methods: Lee-Carter model and block bootstrap, and check which shows better results. Based on the computer simulation, we found that TFR performs better and the block bootstrap method is more sensitive to rapid fertility changes. As for mortality rate projection, we also recommend the standard operating procedure by Wang et al. (2012). However, the smoothing methods have limited impacts on mortality projection and can be ignored. In addition to simulation, we also apply the SOP for projecting the small population to Taiwan counties and it achieves satisfactory results. However, due to the availability of data, our method can only be used for short-term projection (at most 30 years) and these results might not apply to long-term projection. Also, similar to the previous work, the fertility rates have the larger impact on small population projection, although we think that the migration has large impact as well. In this study, only the stochastic projection is considered and we shall consider including expert opinions as the future study.

國家政策之制定須配合未來人口總數及其結構等特性，藉以達到提高國民福 祉的願景，因此各國均定期公佈人口推估(Population Projection)的結果，目前臺 灣官方人口推估為每兩年公布一次。人口推估主要考量三個要素：出生、死亡、 遷移，以國家層級而言，通常遷徙對未來人口的影響遠小於出生與死亡，所以過 去行政院經濟建設委員會的官方全國人口推估一般專注於出生與死亡。然而，各 國研究發現遷徙是小區域人口推估為最重要的因素，人口數愈少、影響程度有愈 大的傾向，但文獻中較缺乏臺灣內部遷移的研究。如能掌握臺灣小區域人口遷徙 的變遷，將能使政策因地制宜，有助地方政府提高推行政策的有效性，也是本研 究之目標。 由於缺乏完整的縣市、鄉鎮市區層級的詳細遷移資料，本研究以人口平衡公 式反推淨遷移人數，找出各地區的遷移特性後，代入人口變動要素合成法(Cohort Component Method)，搭配屬於機率推估的區塊拔靴法(Block Bootstrap)，推估小 區域的未來人口。關於出生及死亡的推估，過去研究發現使用區塊拔靴法用於小 區域的生育率（曹育欣，2012）及死亡率（金碩，2011），皆有不錯的推估結果。 本研究以臺北市為範例，討論區塊拔靴法在小區域遷徙人口數、年齡別遷徙人口 的推估效果，及是否適合運用在其他不同縣市。 / The population projection is used to provide information for the policy planning of governments. In Taiwan, the Council for Economic Planning and Development is in charge of the official population projection and it release projection results every two years. Basically, three factors are considered in population projection: birth, death, and migration. Since the migration has little impacts in country-level projection, many countries (including Taiwan) assume the future migration is zero or close to zero, and the focus of projection is usually on the birth and death. However, for the projection of small area (such as county- or township-level), past studies found that the effect of migration cannot be ignored. But, partly due to the limitation of migration data, there are not many studies explore the migration patterns of counties or townships in Taiwan. In this study, we use the population records (births and deaths) and the population equation to derive the county-level records of internal migration in Taiwan. We use these data to explore the migration patterns of all counties in Taiwan, and then applying block bootstrap method to modify the county-level population projection. Note that, the block bootstrap is shown to be reliable in forecasting fertility (Tsao, 2012) and mortality (Jin, 2011) for small areas. In this study, we also use the Taipei City to demonstrate the population projection which includes the internal migration, and the result is promising.

臺灣各縣市人口結構差異明顯，各縣市的人口出生、老化程度都不盡相同，而且在醫療分配及社會資源的使用也有很大的差異，因此各縣市應因應各地特性發展不同的小區域人口推估方法。由於樣本數與變異數成反比，人數較少者的死亡率（像是高齡人口）通常震盪較大，藉由適當的修勻(Graduation)調整，通常可降低年齡層間的死亡率震盪。然而，當縣市層級的人數太少時，只依賴修勻往往不足，多半會再參考人口較多的大母體之死亡率。例如：傳統的的貝氏修勻，使用Lee-Carter之類的參數死亡模型(Lee and Carter, 1992)，或是透過小區域及大母體的死亡率比值（王信忠, 2012）。然而過去研究較少全面性的比較這些方法，尤其是用於人數較少（如：十萬人）的地區。 本文以探討小區域生命表及死亡率推估為目標，著眼於人數不多於五萬人，尋求較為適合臺灣及類似國家的死亡率編算方法。由於修勻或貝氏等方法可視為增加樣本數，本文將擴大樣本分為四種方式：「同地同時」、「同地異時」、「異地同時」、「異地異時」，亦即將死亡資料的整併分成是否限定於小區域，以及是否可擴及其他年度。本文藉由電腦模擬測試，提供在各種限制之下，最合適小區域生命表建構的準則。其中，本文假設大、小區域的死亡率間存有三種情境的關係：定值、遞增、V字型，藉由調整大小區域死亡率比值間的幅度，探討大母體及小區域間的差異對實務使用的影響。研究發現，Partial SMR方法是一個值得參考的方法，當大小區域死亡率類型接近時的效果不錯，甚至可用於人數小於一萬人，但若死亡率類型差異過大，修勻方法會有限制，使用時需格外謹慎。 / The population structure, life expectancy (and age-specific mortality rates), and the speed of population aging vary a lot in different county of Taiwan. Each county has its own policy planning according to the needs. However, the county level population is usually not enough to provide stable estimates, such as of the life expectancies and mortality rates at the county level. Thus, certain graduation methods are applied to stabilize these estimates. However, only a few studies focus on comparing different types of graduation methods, including traditional graduation methods, Bayesian methods, and parametric mortality models. In this study, we separate the graduation methods into four types, according to if using only the small area data and if one year or multiple years of data are used, and explore which methods are appropriate to the areas with population fewer than 100,000. We use computer simulation to evaluate the graduation methods. We found that the Standard Mortality Ratio is promising when the mortality profiles of small and large populations are similar, and it is a feasible solution even for the areas with population fewer than 10,000. However, if the mortality profiles differ significantly, all graduation methods need to be applied with care.

壽命延長及生育率下降使得人口老化日益明顯，成為全球多數國家在21世紀必須面對的議題，由於各區域人口老化的速度不同，必須根據各地特性而調整因應對策。其中研究死亡率變化為面對人口老化的必備課題，尤其是高齡族群的死亡率，這也是近年高齡死亡模型廣受重視的主因之一。因為樣本數與變異數成反比，人口較少的區域或是高齡人口，死亡率的觀察值通常會有較大震盪，為了降低震盪多半會經過修勻，以取得較為穩定的死亡率推估值（王信忠等人，2012）。此外，Li and Lee (2005)的Coherent Lee-Carter模型也是另一種可行方法，透過參考大區域的資訊降低小區域的估計誤差。 本文探討結合上述修勻、死亡率模型的可能，希冀能綜合兩者的優點，提高小區域死亡率推估的精確性。因為Coherent Lee-Carter模型的想法類似增加小區域的人數（加入大區域的人數），本文探討人口數與Lee-Carter模型參數估計值的關係，再以修勻調整大小區域的差異，透過電腦模擬及資料分析，驗證本文提出方法是否有效。其中，仿造王信忠等人的作法，假設小區域與大區域死亡率間的七種可能情境，以平均絕對百分誤差(Mean Absolute Percentage Error)為衡量標準，找出調整修勻、相關模型的方法。另外，本文也以臺灣縣市為研究區域，驗證本文方法的估計結果。研究發現適當地使用修勻方法，可降低小區域的死亡率估計值，其效果優於Coherent Lee-Carter模型。

一個國家對全國人口有充分瞭解，方能依據國情制定適合的政策，地方發展更是如此，更須洞悉各地的人口結構，以善用有限的資源。台灣近年人口老化日益明顯，各縣市的老化速度及人口問題也不盡相同，若可獲得各地區未來的人口相關數值 (亦即人口推估)，當能減輕未來人口老化對台灣造成的衝擊。本文以縣市層級的人口推估，也就是小區域人口推估為研究目標，探討需注意的事項，尋找適合台灣地區的小區域推估方法。 本文整理小區域人口推估方法，並使用人口要素變動合成法 (Cohort Component Method)，以雲嘉兩縣、臺北市、澎湖縣為範例，測試縣市層級的人口推估。人口推估與生育、死亡、遷移三者的假設有密切關係，我們以死亡率為目標，比較不同模型的優劣，考慮的模型包括 Lee-Carter 模型、區塊拔靴法 (Block Bootstrap)、篩網拔靴法 (Sieve Bootstrap) 以及泛函資料分析 (Functional Data Analysis) 中的主成份分析 (Principle Component Analysis)，以估計誤差為衡量方法優劣的標準。分析發現篩網拔靴法、區塊拔靴法、Lee-Carter 模型三者的結果較佳，因此在小區域推估中使用較簡便的區塊拔靴法。研究發現對小區域的人口推估而言，遷移假設扮演非常重要的角色，此與全國規模的人口推估結果截然不同。研究過程亦發現人口三要素對人口推估有明顯的影響，若假設三要素間互相獨立 (也就是傳統推估時的假設)，推估結果的預測區間遠小於三要素不獨立。 / The government can make policy according to the population change in this country, while the local government can develop their district by using their limited resources well after realizing the populaton structure. The population ageing is becoming more serious and being more different among every counties in Taiwan day by day. If we can get the relative numbers of population in the future (population projection), we can decrease the attack of population ageing for Taiwan. The aim of this paper is to find an appropriate method and some notations of small area population projection in Taiwan. The paper includes the summary of methods of small area population projection and the results by using cohort component method on three areas in Taiwan, YunLin & ChiaYi, Taipein City and PengHu. Population projection is highly related with birth, death and migration, hence we test the mortality rate by using several methods, Lee-Carter, block bootstrap, sieve bootstrap and principal component analysis of functional data analysis are included. We found that the result of sieve bootstrap, block bootstrap and Lee-Carter are much better than the others, therefore, we take block bootstrap which is much simpler than the other two to analysis the effect of birth, death and migration in population projection. The sutdy found that, in small area population projecton, migration plays an important role, which is totally different from the whole country population projection.