Consider the D-optimal designs for the dth-degree polynomial regression model with a bounded and positive weight function on a compact interval. As the degree of the model goes to infinity, we show that the D-optimal design converges weakly to the arcsin distribution. If the weight function is equal to 1, we derive the formulae of the values of the D-criterion for five classes of designs including (i) uniform density design; (ii) arcsin density design; (iii) J_{1/2,1/2} density design; (iv) arcsin support design and (v) uniform support design. The comparison of D-efficiencies among these designs are investigated; besides, the asymptotic expansions and limits of their D-efficiencies are also given. It shows that the D-efficiency of the arcsin support design is the highest among the first four designs.

Experiments with both quantitative and qualitative factors always complicate the selections of experimental settings and the statistical analysis for data. Response surface methodology (RSM) provides the systematic procedures such as the steepest ascent method to develop and improve the response models through the optimal settings of quantitative factors. However the sequential method lacks of exploring the direction of the maximum increase in the response among the qualitative levels. In this dissertation the optimal designs for experiments with both qualitative and quantitative factors are investigated. Focused on the second-order response surface model for quantitative factors, which is widely used in RSM as a good approximation for the true response surface, the approximate and exact D-optimal designs are proposed for the model containing the qualitative effects. On spherical design regions, the D-optimal designs have particular structures for considering the qualitative effects to be fixed or random. In this study, the exact D-optimal designs for a second-order response surface model on a circular design region with qualitative factors are proposed. For this model, the interactions between the quantitative and qualitative factors are assumed to be negligible. Based on this design region, an exact D-optimal design with regular polygon structure is made up according to the remainder terms of the numbers of experimental trials at each qualitative levels divided by 6. The complete proofs of exact D-optimality for models including two quantitative factors and one 2-level qualitative factor are presented as well as those for a model with only quantitative factors. When the qualitative factor has more than 2 levels, a method is proposed for constructing exact designs based on the polygonal structure with high efficiency. Furthermore, a procedure for minimizing the number of support points for the quantitative factors of exact D-optimal designs is also proposed for practical consideration. There are no more than 13 support points for the quantitative factors at an individual qualitative level. When the effects between the quantitative and qualitative factors are taken into consideration, approximate D-optimal designs are investigated for models in which the qualitative effects interact with, respectively, the linear quantitative effects, or the linear effects and 2-factor interactions of the quantitative factors or quadratic effects of the quantitative factors. It is shown that, at each qualitative level, the corresponding D-optimal design consists of three portions as a central composite design but with different weights on the cube portion, star portion and center points. Central composite design (CCD) is widely applied in many fields to construct a second-order response surface model with quantitative factors to help to increase the precision of the estimated model. A chemical study is illustrated to show that the effects of the qualitative factor interacts with 2-factor interactions of the quantitative factors are important but absent in a second-order model including a qualitative factor treated as a coded variable. The verification of the D-optimality for exact designs has become more and more intricate when the qualitative levels or the number of quantitative factors increase, even when the patterns of the exact optimal designs have been speculated. The efficient rounding method proposed by Pukelsheim and Rieder (1992) is a model-free approach and it generates an exact design by apportioning the number of trials on the same support points of a given design. For constructing the exact designs with high efficiencies, a modified efficient rounding method is proposed and is based on the polygonal structure of the approximate D-optimal design on a circular design region. This modification is still based on the same rounding approach by apportioning the number of trials to the concentric circles where the support points of the given design are standing on. Then a regular polygon design will be assigned on the circles by the apportionments. For illustration, the exact designs for a third-order response surface model with qualitative factors are presented as well as those for the second-order model. The results show that nearly D-optimal designs are obtained by the modified procedure and the improvement in D-efficiency is very significant. When the factors with the levels selected randomly from a population, they are treated as with random effects. Especially for the qualitative effects caused by the experimental units that the experimenter is not interested in, one should consider the model with random block effects. In this model, the observations on the same unit are assumed to be correlated and they are uncorrelated between different units. Then the mean response surface is still considered as second-order for quantitative factors but the covariance matrix of the observations is different from the identity matrix. In the fourth part of this dissertation, the locally D-optimal designs on a circular design region are proposed for given the value of the correlations. These optimal designs with the structures based on the regular polygons are similar to the D-optimal designs for the uncorrelated model.

思及當前捉襟見肘的財政窘境，無可避免地，債台高築的臺灣實陷入飲鴆止渴般以債養債之無限迴圈中，導致政府政策效能不彰、社會福利運作生弊亦無可厚非；於『公共債務法』之財政規範下，臺灣業已瀕臨法定舉債門檻，故不論是對短期政府支出之排擠、扭曲性稅率之稽徵抑或對長期經濟成長的斲傷，皆是身為中華民國國民真正惶悚不安之所在。 職是之故，本研究係採用一納入政府財政部門及貨幣當局之擴充『實質景氣循環模型』，藉以Sidrauski（1967）所提出的貨幣效用函數為出發點，將實質餘額引進理論模型，並透過計量操作捕捉實證期間起於西元1971年第一季迄至2007年第四季之政府政策函數，過程中，我們不難發現政府購買性支出及稅率皆存在相當的持續性，且對政府未償公債餘額之高低作出某種程度的反應。亦即，若政府實施公債融通政策，俾使期初公債餘額較高之際，則本期甚或往後各期的政府支出將遭受抑制和排擠，尤有甚者，政府勢必擬以提高未來稅率以茲挹注該債務之還本付息所造成的財政缺口；是以，本研究著眼於引進公債餘額對政府支出及稅率存在反饋作用下，財政政策與貨幣政策之總體效果及各總體變數之動態調整過程的風貌。即便公債發行或賒借為政府提供一財務週轉工具以裨益財政政策保有更靈活之彈性，然據模型所產生的結果顯示，就長期而論，政府必須維持一穩定之未償公債餘額，即公債水準具備『均數復歸』性質，而該財政目標係透過削減未來政府支出、調整扭曲性稅率及鑄幣稅融通政策方得以達成預算平衡，準此，該設定將造成公債融通之減稅政策對經濟體系具有實質效果，『公債融通』管道亦『非中立性政策』，從而傳統『李嘉圖等值定理』於本模型中無法成立。 就政策面層次而言，本研究試圖放寬『反饋法則』與政策係數之設定，以檢視透過不同程度之政府支出、稅率甚至貨幣供給途徑的改變來平衡因增加公債發行所造成的財政赤字，對經濟體系之長短期效果有何迥異處；是文亦藉由衝擊反應函數分別探討於政府支出增加、減稅措施及貨幣擴張之下，政策的傳遞機制與各總體變數之動態性質，顯然地，就高債務比率前提下，當政府戮力於刺激景氣而欲積極實施立竿見影的總體經濟政策之際，卻常因狃於急效而欲速不達，非但政策效果有限，亦可能使體系落入更為不景氣的田地，從而，財政惡化不啻為經濟危機的導火線也就不言而喻。再者，貨幣政策對體系之實質變數具有一定程度的作用，是故，本模型於短期內無法一窺『貨幣中立性』之堂奧，唯長期始得以復見。總括言之，政府亟須奉『健全財政』為圭臬，擬定政策時更得戒慎恐懼，並適切權衡利弊得失，以茲裨益有更具信心的經濟表現。 此外，本研究亦透過『效準』實驗以評估模型『配適度』之良窳，即便於反覆疊代法下，該模擬表現係瑕瑜互見而不盡完美，卻也大抵符合景氣循環之『典型化特徵』；然就實質景氣循環模型所為人詬病之勞動市場一隅而論，引進公債之反饋法則下的財政政策操作，無疑地改善了傳統工時與工資率動輒高度正相關之本質，從而獲致相對較低之理論相關係數，亦朝實證資料所呈現工時與工資率存在幾近零相關甚或低度負相關之表徵更邁進一大步。 / With current financial difficulties beyond government capability, it is inevitable that the already deep-in-debt Taiwan opted for momentary relief by paying debt through debt financing and ended up in an infinite loop, causing spiral-down performances in government policies and faulty operations of social welfare instruments. Taiwan has been on the verge of reaching the statutory upper limit of debt financing according to “The Public Debt Act” regulations and all nationals are becoming anxious about such impacts as crowding out of short-run government spending, levying of distorting taxes, and damages on long-run economic growth. To better understand the debt’s impacts, this research uses the “Real Business Cycle Model” extended by taking government treasury agency and monetary institution into account. Starting with Money In Utility Function (MIUF) as proposed by Sidrauski (1967) to introduce real money balance into the theoretical model and, in the process of econometric manipulation, to detect empirical governmental policy functions in the period between the first quarter, 1971 and the fourth quarter, 2007, it is not hard to discover that there are considerable persistence in both government purchases and tax rates, with manifestation of certain degree of responses to the total amount of outstanding bonds the government has yet to pay. In other words, a governmental bond financing policy designed to render high initial bonds outstanding tends to cause suppression and crowding out of government spending in current and even later periods. Furthermore, the government is bound to plan on raising taxes in the future in order to cut financial deficit gap caused by paying back the principles and interests of the debt. Therefore, this study focuses on presenting the macroeconomic effects of fiscal policies and monetary policies, as well as the dynamic adjustment processes of macroeconomic variables based on the impact of feedback effect of bonds outstanding on government spending and tax rates. Even thought public bonds issuance or debt financing serves as a governmental fiscal instrument for financial turnover to ensure flexibility of fiscal policies, our model shows that the government should, from a long-run perspective, maintain a stable amount of bonds outstanding. Put in a different way, the level of bonds outstanding shows “mean-reverting” characteristics which rely on future government spending cut, distorting tax adjustment and seigniorage financing policy to achieve balance of budget. As a result, such setup would cause the bond-financing backed tax deduction policies to create practical effects on economies and, as the bond financing instruments are “Non-Neutrality” policies, would render the “Ricardian Equivalence Theorem” invalid in our model. In the policy aspect, this study tries to relax both “feedback rules” and setup of policy parameters for investigating the differences between long-run and short-run effects on the economy by different degrees of changes in government spending, tax rates and even money supply channels which are used to balance the fiscal deficit caused by increased bond issuance. This article also studies, through the impulse response function, the policy propagation mechanism and the dynamics of key macroeconomic variables under the situation of government spending increase, tax deduction and monetary expansion. It is obvious that the government, in the case of high debt ratios and when making all endeavors to spur economy by implementing macroeconomic policies aimed for instant results, is accustomed to seeking quick fixes only to achieve very limited effects, sometimes even to drive the economy into further recession. It is therefore evident that fiscal degradation could lead to economic disaster. Moreover, as the monetary policies have certain degrees of influence on real variables of the economy, this model will not be able to clearly analyze the “neutrality of money” in such a short period of time. The effect will only reveal in the long run. In summary, the government should keep “sound finance” as the highest guiding principle and be extremely cautious in formulating policies in order to weigh all pros and cons discreetly, thus help to achieve a benefiting economic performance that generates more confidence. Furthermore, this study assesses “goodness of fit” of the model through a “calibration” experiment. Although the simulation results show, under recursive method, intermingled good and poor occasions that are beyond satisfaction, they generally agree with the “typical characteristics” of business cycles. However, in the aspect of long-criticized labor market of the real business cycle model, the fiscal policy operation under feedback rules with introduction of public debts for sure has greatly improved on the conventional intrinsic property of high correlation between labor hours and real wage rates, by delivering a relatively low theoretical correlation coefficient, which is a big step towards the empirical results of almost zero or even weakly negative correlation between labor hours and real wage rates.