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.

迴歸模式的變數選取是很重要的課題，Tibshirani於1996年提出最小絕對壓縮挑選機制（Least Absolute Shrinkage and Selection Operator；簡稱Lasso），主要特色是能在估計的過程中自動完成變數選取。但因為Lasso本身並沒有牽扯到統計推論的層面，因此2014年時Lockhart et al.所提出的Lasso顯著性檢定是重要的突破。由於Lasso顯著性檢定的建構過程與傳統向前逐步迴歸相近，本研究接續Lockhart et al.(2014)對兩種變數選取方法的比較，提出以Bootstrap來改良傳統向前逐步迴歸；最後並比較Lasso、Lasso顯著性檢定、傳統向前逐步迴歸、以AIC決定變數組合的向前逐步迴歸，以及以Bootstrap改良的向前逐步迴歸等五種方法變數選取之效果。最後發現Lasso顯著性檢定雖然不容易犯型一錯誤，選取變數時卻過於保守；而以Bootstrap改良的向前逐步迴歸跟Lasso顯著性檢定一樣不容易犯型一錯誤，而選取變數上又比起Lasso顯著性檢定更大膽，因此可算是理想的方法改良結果。 / Variable selection of a regression model is an essential topic. In 1996, Tibshirani proposed a method called Lasso (Least Absolute Shrinkage and Selection Operator), which completes the matter of selecting variable set while estimating the parameters. However, the original version of Lasso does not provide a way for making inference. Therefore, the significance test for lasso proposed by Lockhart et al. in 2014 is an important breakthrough. Based on the similarity of construction of statistics between Lasso significance test and forward selection method, continuing the comparisons between the two methods from Lockhart et al. (2014), we propose an improved version of forward selection method by bootstrap. And at the second half of our research, we compare the variable selection results of Lasso, Lasso significance test, forward selection, forward selection by AIC, and forward selection by bootstrap. We find that although the Type I error probability for Lasso Significance Test is small, the testing method is too conservative for including new variables. On the other hand, the Type I error probability for forward selection by bootstrap is also small, yet it is more aggressive in including new variables. Therefore, based on our simulation results, the bootstrap improving forward selection is rather an ideal variable selecting method.

In this paper, we propose two types of sieve bootstrap, univariate and multivariate approach, for the generalized method of moments estimators of time series data. Compared with the nonparametric block bootstrap, the sieve bootstrap is in essence parametric, which helps fitting data better when researchers have prior information about the time series properties of the variables of interested. Our Monte Carlo experiments show that the performances of these two types of sieve bootstrap are comparable to the performance of the block bootstrap. Furthermore, unlike the block bootstrap, which is sensitive to the choice of block length, these two types of sieve bootstrap are less sensitive to the choice of lag length.

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.

本文採用可加性模型分析法建立台灣地區小型商用車市場之需求模型，並 引進Box-Jenkins時間序列模型處理具自我相關之誤差項，以利進行拔靴 推論設計時，能拔靴白干擾(bootstrapping white noise)，即重抽樣白 干擾的經驗分配。在此次研究過程中，除配適Box-Jenkins時間序列模型 外，所有分析步驟都是完全自動的，不須作假設和檢驗的工作，所以可降 低傳統上因統計人員主觀判斷錯誤所造成的估計偏誤。可加性模型改進傳 統迴歸模型須先假設模型形式的限制，可從商用車實證分析中，直接由資 料配適平滑函數，顯見其合理性。拔靴法免除傳統推論程序須強使隨機干 擾項分配為常態分配或漸近常態分配之束縛，改由殘差經驗分配模擬隨機 干擾項分配行為，在推論商用車市場上，也獲得不錯的結果。

台灣地區的人口隨著生育率及死亡率的雙重下降而呈現快速老化，其中生育率的降低影響尤為顯著。民國50年時，台灣平均每位婦女生育5.58個小孩，到了民國70年卻只生育1.67個小孩，去年（民國90年）生育率更創歷年新低，只有1.4。死亡率的下降可由平均壽命的延長看出，民國75年時男性為70.97歲，女性為75.88歲；到了民國90年，男性延長到72.75歲，女性延長到78.49歲。由於生育率的變化幅度高於死亡率，對人口結構的影響較大，因此本文分成兩個部份，主要在研究台灣地區15至49歲婦女生育率的變化趨勢，再將研究結果用於台灣地區未來人口總數及其結構的預測。 　　本研究第一部分是生育率的研究，引進Gamma函數、Gompertz函數、Lee-Carter法三種模型及單一年齡組個別估計法，以民國40年至84年（西元1951年至1995年）的資料為基礎，民國85年至89年（西元1996年至2000年）資料為檢測樣本，比較模型的優劣，尋求較適合台灣地區生育率的模型，再以最合適的模型預測民國91年至140年（西元2002年至2051年）的生育率。第二部分是人口推估，採用人口變動要素合成方法（Cohort Component Projection Method）推估台灣地區未來50年的人口總數及其結構，其中生育率採用上述最適合台灣地區的模型、死亡率則引進國外知名的Lee-Carter法及SOA法（Society of Actuaries），探討人口結構，並與人力規劃處的結果比較之。 / Both the fertility rate and mortality rate have been experiencing dramatic decreases in recent years. As a result, the population aging has become one of the major concerns in Taiwan area, and the proportion of the elderly (age 65 and over) increases promptly from 2.6% in 1965 to 8.8% in 2001. The decrease of fertility rate is especially significant. For example, the total fertility rate was 5.58 in 1961, and then decreases dramatically to 1.67 in 1981 (1.4 in 2001), a reduction of almost 70% within 20 years. 　　The goal of this paper is to study the population aging in Taiwan area, in particular, the fertility pattern. The first part of this paper is to explore the fertility models and decide which model is the most suitable based on age-fertility fertility rates in Taiwan. The models considered are Gamma function, Gompertz function, Lee-Carter method and individual group estimation. We use the data from 1951 to 1995 as pilot data and 1996 to 2000 as test data to judge which model fit well. The second part of this study is to project the Taiwan population for the next 50 years, i.e. 2002-2051. The projection method used is Cohort Component Projection method, assuming the population in Taiwan area is closed. We also compare our projection result to that by Council for Economic Planning and Development, the Executive Yuan of the Republic of China.