...(a) First, determine the Queuing Models 1. Queuing Models Poisson Arrivals Standard (Infinite Queue) Exponential Service Times Single Server (M/M/1) Standard M/M/1 Model P0 = 1 – P (Probability of 0 customer in the system) P (n ≧k ) = ρk (Probability of Pn = P0ρn (Probability of exactly n customers in the system) Ls = λ / ( μ – λ ) (Mean no. of customers in the system) Lq = ρλ / ( μ – λ ) (Mean no. of customers in queue) Lb = λ / ( μ – λ ) (Mean no. of customers in queue for a busy system) Ws = 1 / ( μ – λ ) (Mean time customer spends in the system) Wq = ρ / ( μ – λ ) (Mean time customers spends in the queue) Wb = 1 / ( μ – λ ) (Mean time customers spends in queue for a busy system) Now, in our case, Λ (mean arrival rate) = 15/hr μ (mean service rate per busy server) Old machine’s mean service time= 3mins, 60/3 =20/hr New machine’s mean service time= 2mins, 60/2=30/hr Wq =ρ/ ( μ – λ ) (Mean time customers spends in the queue), Where ρ = λ / μ Old machine: (15/20)/ 20-15 = 0.1 hr = 9mins New machine: (15/30)/ 30-15= 0.03333 hr = 2mins According to the case, the average wage of the people who bring the documents to be copied is $8/hr, If the company rent the old machine and there are 15 arrivals/hr, There total mean time spend in the queue is ( 15 x 9 )mins = 135mins = 2.25hr, The total wage = $8 x 2.25 = $18 If the company rent the new machine and there are 15 arrivals/hr, The total mean time...
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...The Mathematics of Simple Macroeconomic Models There are several advantages in presenting economic models in the form of equations. Equations are very concise; they readily give quantitative answers; and they are the language of most practising economics. Unfortunately, for a large proportion of students beginning the study of macroeconomics, the first sight of equations in the course is a traumatic experience. Old fears about their mathematical ability can make them doubt their capacity to cope with macroeconomics. It may even result in their premature withdrawal from the subject. Fortunately, in the vast majority of cases such doubts are unfounded. Consequently, once fears are removed, these students cope quite happily with the equations in macroeconomics. This chapter is designed to allay these old fears about mathematics. It does this by means of a step-by-step examination and manipulation of the equations that a student first meets in macroeconomics. Once the student has mastered these equations, the new-found confidence should enable him or her to tackle most of the elementary mathematics that macroeconomics requires. THE FIRST MODEL The first model in equation form seen by macroeconomics students is usually: Y = C + 1 (16-1) C = Co + cY (16-2) I = Io (16-3) It is convenient to think of the symbols in these equations as words, and the equations themselves as sentences. Thus it is possible to translate an English sentence into an...
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...shocks| 2.|Use the IS/LM-AD/AS model to illustrate graphically how expansionary fiscal and monetary policy can help stabilize the output when economy is in a recession. | 3.|Use the IS-LM model to derive the AD curve and to show how expansionary fiscal and monetary policy can shift the AD curve. | 4.|A decrease in government spending reduces output more in the Keynesian-cross model than in the IS-LM model. Explain why this is true.| 5.|Use the IS/LM-AD/AS model to graphically analyze short-run & long-run effects of a negative IS Shock.| 6.|Assume that an economy is characterized by the following equations:C = 100 + (2/3)(Y – T)|T = 600|G = 500|I = 800 – (50/3)r|Ms/P = Md/P = 0.5Y – 50r|a.|Write the numerical IS curve for the economy, expressing Y as a numerical function of G, T, and r.|b.|Write the numerical LM curve for this economy, expressing r as a function of Y and M/P. |c.|Solve for the equilibrium values of Y and r, assuming P = 1.0 and M = 1,200. How do they change when P = 2.0? Check by computing C, I, and G.|d.|Write the numerical aggregate demand curve for this economy, expressing Y as a function of G, T, and M/P.|| 7.|Assume the following model of the economy, with the price level fixed at 1.0:C = 0.8(Y – T)|T = 1,000|I = 800 – 20r |G = 1,000|Y = C + I + G |Ms/P = Md/P = 0.4Y – 40r|Ms = 1,200||a.|Write a numerical formula for the IS curve, showing Y as a function of r alone. (Hint: Substitute out C, I, G, and T.)|b.|Write a numerical...
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...Saving Function KM: The Multiplier Model - Principles of Macro - RIT - Dr. Jeffrey Burnette The Keynesian Model: The Multiplier Model Assumptions • Each market is analyzed on an aggregate scale (One interest rate, One type of labor, One type of output) Differences from the Classical Model • The Keynesian model does not assume that markets clear in the short-run. In fact, it believes that prices are sticky downward. That is, prices may not decrease to bring the market to equilibrium. This is why it focuses on the short-run. • In the short-run spending depends upon income. • Whereas the classical model focused on equilibrium in the labor market determining the level of output, the Keynesian model will focus on the domestic spending of consumers. The Keynesian Model w/out Government • Households have 2 choices for spending their income, Purchase goods and services (Consumption) or Save. • Firms have 2 types of expenditures, purchase goods and services (Consumption) or purchase equipment and structures from borrowed funds (Planned Investment). • In the short-run, planned investment is taken to be independent of income. • There is a funds market where savings is made available to firms that wish to borrow. Disposable Income = Income - Taxes Since there is no government, taxes are equal to zero. Therefore, real income is equal to real disposable income. Y = YD 1 KM: The Multiplier Model - Principles of Macro - RIT - Dr. Jeffrey Burnette Consumption Function - Algebraic...
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...Regression Analysis (Spring, 2000) By Wonjae Purposes: a. Explaining the relationship between Y and X variables with a model (Explain a variable Y in terms of Xs) b. Estimating and testing the intensity of their relationship c. Given a fixed x value, we can predict y value. (How does a change of in X affect Y, ceteris paribus?) (By constructing SRF, we can estimate PRF.) OLS (ordinary least squares) method: A method to choose the SRF in such a way that the sum of the residuals is as small as possible. Cf. Think of ‘trigonometrical function’ and ‘the use of differentiation’ Steps of regression analysis: 1. Determine independent and dependent variables: Stare one dimension function model! 2. Look that the assumptions for dependent variables are satisfied: Residuals analysis! a. Linearity (assumption 1) b. Normality (assumption 3)— draw histogram for residuals (dependent variable) or normal P-P plot (Spss statistics regression linear plots ‘Histogram’, ‘Normal P-P plot of regression standardized’) c. Equal variance (homoscedasticity: assumption 4)—draw scatter plot for residuals (Spss statistics regression linear plots: Y = *ZRESID, X =*ZPRED) Its form should be rectangular! If there were no symmetry form in the scatter plot, we should suspect the linearity. d. Independence (assumption 5,6: no autocorrelation between the disturbances, zero covariance between error term and X)—each individual should be independent 3. Look at the correlation between two variables by drawing...
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...Questions (No Solution will be Provided) 1. Suppose the data generating process (the true relationship) is y = Xβ + ε, where E[ε|X] = 0, E[εε |X] = σ 2 I n ; and X includes an intercept term. You do not observe the data set Z = [y X]. Instead you observe 150 15 50 Z Z = 15 25 0 50 0 100 2 Compute the least squares estimators β, s2 , R2 and RAdj (the adjusted R2 ). Is there anything to be gained by observing the full data set? 2. Suppose you have the simple regression model with no intercept: yi = xi β+ i for i = 1, 2. Suppose further that the true value of β is 1, the values of xi observed in the sample are x1 = 2 and x2 = 3, and the distribution of i is Pr( i = −2) = Pr( i = 2) = 1/2 with 1 independent of 2 . (a) Find the least squares estimator of β. (b) What is it mean and variance? Is it BLUE? (c) Consider the alternative estimator β ∗ = y /¯, where y is the sample mean ¯ x ¯ of yi and x is the sample mean of xi . What is the mean and variance of ¯ β ∗ ? Is it unbiased? (d) Which estimator is more efficient, the least squares estimator or β ∗ ? 3. Suppose x1 , x2 . . . xn is an independent but not identically distributed random sample from a population with E[xi ] = µ and Var[xi ] = σ 2 /i for i = 1, 2, . . . , n. Consider the following class of estimators for the population mean µ: n µ= ˆ ci xi where c1 , . . . , c n are constants i=1 Each sequence {c1 , c2 , . . . , cn } defines an estimator for µ. (a) Give a necessary...
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...Classical Probabilistic Models and Conditional Random Fields Roman Klinger Katrin Tomanek Algorithm Engineering Report TR07-2-013 December 2007 ISSN 1864-4503 Faculty of Computer Science Algorithm Engineering (Ls11) 44221 Dortmund / Germany http://ls11-www.cs.uni-dortmund.de/ Classical Probabilistic Models and Conditional Random Fields Roman Klinger∗ Katrin Tomanek∗ Fraunhofer Institute for Algorithms and Scientific Computing (SCAI) Schloss Birlinghoven 53754 Sankt Augustin, Germany Jena University Language & Information Engineering (JULIE) Lab F¨rstengraben 30 u 07743 Jena, Germany Dortmund University of Technology Department of Computer Science Chair of Algorithm Engineering (Ls XI) 44221 Dortmund, Germany katrin.tomanek@uni-jena.de roman.klinger@scai.fhg.de roman.klinger@udo.edu Contents 1 Introduction 2 2 Probabilistic Models 2.1 Na¨ Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ıve 2.2 Hidden Markov Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Maximum Entropy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 5 6 3 Graphical Representation 10 3.1 Directed Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Undirected Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Conditional Random Fields 4.1 Basic Principles . . . . . . . . 4.2 Linear-chain CRFs . . . . . . 4.2.1 Training . . . ...
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...Fliptop Model(X) | 3 | 5 | 5 | 1,000 | Tiptop Model(Y) | 4 | 4 | 2 | 1,000 | Total | 36 | 40 | 30 | | Linear Model: Maximize= 1,000 X+ 1,000 Y S.T. 3X + 4Y ≤ 36 5X + 4Y ≤ 40 5X + 2Y ≤ 30 X, Y ≥ 0 Solution: 3X +4Y =36 5X + 4Y= 40 5X + 2Y= 30 Constraint 1 | Constraint 2 | Constraint 3 | 3X +4Y=36 | 5X + 4Y= 40 | 5X +2Y= 30 | IF X= 0 | IF X=0 | IF X=0 | Y=9 | Y=10 | Y=15 | IF Y=0 | IF Y=0 | IF Y=0 | X=12 | X=8 | X=6 | Corners: A→6, 0 B→3.2, 6.7 C→0, 10 Optimal solution: Z= 1,000X + 1,000Y A: 1,000(6) + 1,000 (0) = 6,000$ B: 1,000(3.2) + 1,000 (6.7) =9,900$ C: 1,000(0) + 1,000 (10) = 10,000$ Optimal Combination: 0 Fliptop Model 10 Tiptop Model Question#2: | | Resources | | Profit | Decision Variable | Synthetic | Hours | Foam | | Grade X | 50 | 25 | 20 | 200 | Grade Y | 40 | 28 | 15 | 160 | Total | 3,000 | 1,800 | 1,500 | | Linear Model: Maximize = 200X + 160Y S.T. 50X + 40Y ≤ 3,000 25X + 28Y ≤ 1,800 20X + 15Y ≤ 1,500 X, Y ≥0 Solution: 50x + 40y=3,000 25X + 28Y = 1,800 20X + 15Y = 1,500 Constraint 1 | Constraint 2 | Constraint 3 | 50X +40Y=3,000 | 25X + 28Y= 1,800 | 20X +15Y= 1,500 | IF X= 0 | IF X=0 | IF X=0 | Y=75 | Y=64.2 | Y=100 | IF Y=0 | IF Y=0 | IF Y=0 | X=60 | X=72 | X=75 | Corners: A→60, 0 B→32, 37 C→0, 65 Optimal solution: Z= 200X + 160Y A: 200(60) + 160 (0) = 12,000$ B: 200(32) + 160(37) =12,320$ C: 200(0)...
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...measurable); continuous 2. Now: variables that take small number of values; discrete a) b) c) d) New type of variable Gender Market size Region of country Marital status (married vs. not), etc Slide #2 Introduction (cont.) B. C. Introduction (cont.) Institute Used as IV in this section Used as DV later in course of Management Accountants (IMA) publishes an annual Salary Guide In Strategic Finance magazine sfmag@imanet.org Annual survey of members “…based on a regression equation derived from survey results.” Slide #3 Slide #4 IMA Salary Guide (cont.) SALARY = 35,491 + 18393TOP + 8392SENIOR – 10615ENTRY +914YEARS +10975ADVDEGREE – 8684NODEGREE + 9195PROFCERT + 8417MALE IMA Salary Guide (cont.) Average IMA member (1999) TOP=1 if top level mgmt, 0 if not SENIOR=1 if senior level mgmt , 0 if not ENTRY=1 if entry level , 0 if not ADVDEGREE=1 if advanced degree , 0 if not NODEGREE=1 if no degree , 0 if not PROFCERT=1 if hold professional certification , 0 if not MALE=1 if male , 0 if not YEARS=years of experience Slide #5 Male 14.5 years experience Professional certification Salary = $66,356 Figure obtained from substituting values into regression equation Slide #6 1 Are Wins Worth More in a Large Market? Introduction (cont.) D. Example #1 1. Y = + X2 + 2. Y: social program expenditures per state 3. X2: state’s total revenue 4. Suppose legislatures...
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...3rd edition Solutions to Selected Exercises Alan Agresti Version August 3, 2012, c Alan Agresti 2012 This file contains solutions and hints to solutions for some of the exercises in Categorical Data Analysis, third edition, by Alan Agresti (John Wiley, & Sons, 2012). The solutions given are partly those that are also available at the website www.stat.ufl.edu/~ aa/ cda2/cda.html for many of the odd-numbered exercises in the second edition of the book (some of which are now even-numbered). I intend to expand the document with additional solutions, when I have time. Please report errors in these solutions to the author (Department of Statistics, University of Florida, Gainesville, Florida 32611-8545, e-mail AA@STAT.UFL.EDU), so they can be corrected in future revisions of this site. The author regrets that he cannot provide students with more detailed solutions or with solutions of other exercises not in this file. Chapter 1 1. a. nominal, b. ordinal, c. interval, d. nominal, e. ordinal, f. nominal, 3. π varies from batch to batch, so the counts come from a mixture of binomials rather than a single bin(n, π). Var(Y ) = E[Var(Y | π)] + Var[E(Y | π)] > E[Var(Y | π)] = E[nπ(1 − π)]. 7. a. ℓ(π) = π 20 , so it is not close to quadratic. b. π = 1.0. Wald statistic z = (1.0−.5)/ 1.0(0)/20 = ∞. Wald CI is 1.0 ±1.96 1.0(0)/20 = ˆ 1.0 ± 0.0, or (1.0, 1.0). These are not sensible. c. z = (1.0 − .5)/ .5(.5)/20 = 4.47, P < 0.0001. Score CI is (0.839, 1.000). d. Test statistic...
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...Australia ** Tinbergen Institute, Free University Amsterdam, Netherlands August 2002 (preliminary draft) Abstract This paper looks inside the “black box” of the family and examines the determinants of inter vivos transfers in the form of allowances given to children. We consider in a simple model two main competing explanations for the transfer of money from parents to children in the form of regular allowances, namely altruism and exchange. We also extend the altruism framework to include unobserved child heterogeneity in monetary autonomy or the 'value of independence'. We use a unique dataset drawn from the British Family Expenditure Survey, which enables us to explicitly test both the inter-generational predictions of the various models, and through a study of siblings, we are also able to consider the intra-household aspects of such payments. Using both random (inter-household) and fixed-effect (intra-household) estimators, we find robust evidence of an nshape relationship between a child's external income and the receipt of allowances from parents. Importantly, this estimated profile does not fit the predications of simple models of altruism or exchange, but does fit an altruism model with unobserved heterogeneity. Further support for the importance of the value of independence is that girls and those with higher birth orders obtain much higher allowances, whereby we argue both girls and those born later mature earlier and are therefore likely to be causally related to a...
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...American Psychological Association, Inc. 0021-843X/03/$12.00 DOI: 10.1037/0021-843X.112.4.558 Testing Mediational Models With Longitudinal Data: Questions and Tips in the Use of Structural Equation Modeling David A. Cole Vanderbilt University Scott E. Maxwell University of Notre Dame R. M. Baron and D. A. Kenny (1986) provided clarion conceptual and methodological guidelines for testing mediational models with cross-sectional data. Graduating from cross-sectional to longitudinal designs enables researchers to make more rigorous inferences about the causal relations implied by such models. In this transition, misconceptions and erroneous assumptions are the norm. First, we describe some of the questions that arise (and misconceptions that sometimes emerge) in longitudinal tests of mediational models. We also provide a collection of tips for structural equation modeling (SEM) of mediational processes. Finally, we suggest a series of 5 steps when using SEM to test mediational processes in longitudinal designs: testing the measurement model, testing for added components, testing for omitted paths, testing the stationarity assumption, and estimating the mediational effects. Tests of mediational models have been an integral component of research in the behavioral sciences for decades. Perhaps the prototypical example of mediation was Woodsworth’s (1928) S-O-R model, which suggested that active organismic processes are responsible for the connection between stimulus and response...
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...REVISED M04_REND6289_10_IM_C04.QXD 5/7/08 2:49 PM Page 46 C H A P T E R Regression Models 4 15 9 40 20 25 25 15 35 6 4 16 6 13 9 10 16 TEACHING SUGGESTIONS Teaching Suggestion 4.1: Which Is the Independent Variable? We find that students are often confused about which variable is independent and which is dependent in a regression model. For example, in Triple A’s problem, clarify which variable is X and which is Y. Emphasize that the dependent variable (Y ) is what we are trying to predict based on the value of the independent (X) variable. Use examples such as the time required to drive to a store and the distance traveled, the totals number of units sold and the selling price of a product, and the cost of a computer and the processor speed. Teaching Suggestion 4.2: Statistical Correlation Does Not Always Mean Causality. Students should understand that a high R2 doesn’t always mean one variable will be a good predictor of the other. Explain that skirt lengths and stock market prices may be correlated, but raising one doesn’t necessarily mean the other will go up or down. An interesting study indicated that, over a 10-year period, the salaries of college professors were highly correlated to the dollar sales volume of alcoholic beverages (both were actually correlated with inflation). Teaching Suggestion 4.3: Give students a set of data and have them plot the data and manually draw a line through the data. A discussion of which line is “best” can help them appreciate...
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...additional exercises. (2) I will give solutions of the assignment and the “extra-problem” set after the due date. Page 1 Problem 1~25: Multiple choice problems 1. C 2. A 3. D This problem helps you pay attention to the difference between variables and functions. First, whether prices (wages are prices for labor) are assumed to be flexible or 4. C sticky depends on the time horizon. In other words, the assumption of flexible prices is proper in a long-run model while the assumption of sticky prices makes sense in a short-run model. Second, a large part of macro-theory is based on the optimizing behavior of individual agents, such as households and firms, but the main concern of macroeconomics is the movement of the whole economy aggregated from those individual decisions. 5. B Economists, as well as most other social scientists, cannot do controlled experiments. 6. A Consider the National Income Accounts Identity: Y = C + I + G + NX 7. B 8. C Inventory is a kind of investment, so the sale of inventory can be understood as a decrease of investment, but there is an increase in consumption in the same amount due to this transaction. 9. D 10. B 11. A 12. B 13. D 14. C 15. D 16. A This is an important conclusion from the benchmark model in Ch.3. 17. D 18. B 19. A 20. C See page 35 of the slides for Ch.3. 21. B 22. A 23. A The nominal interest rate equals the real interest rate plus the inflation rate. 24. B 25. B One way to solve for...
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...Asymmetric Shocks, Long-term Bonds and Sovereign Default1 Junjun Zhu, Shiyu Xie School of Economics, Fudan University January 2011 Abstract: We present a sovereign default model with asymmetric shocks and long-term bonds, and solve the model using discrete state dynamic programming. As result, our model matches the Argentinean economy over period 1993Q1-2001Q4 quite well. We show that our model can match high default frequency, high debt/output ratio and other cyclical features, such as countercyclical interest rate and trade balance in emerging countries. Moreover, with asymmetric shocks we are able to match high sovereign spread level and low spread volatility simultaneously in one model, which is till now not well solved. As another contribution of our paper, we propose a simulation-based approach to approximate transition function of output shocks between finite states, which is an indispensable step in discrete state dynamic programming. Comparing to Tauchen’s method, our approach is very flexible in transforming various econometric models to finite state transition function, so that our approach can be widely used in simulating different kinds of discrete state shocks. JEL Classification: E44, F32, F34 Keywords: Sovereign Default, Asymmetric Shocks, Transition Function, Long-term Bonds 1 Corresponding authors: Junjun Zhu, Ph.D candidate at School of Economics, Fudan University, No. 200 Guoquan Road, Shanghai 200433, China. Shiyu Xie, professor at School of Economics...
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