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ECO6133 Topics in the Theory of Public Economy
创建日期：2024-07-20 14:59:11
所属分类：经济Economics
标签： ECO6133
ECO6133 Topics in the Theory of Public Economy

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ECO6133 Topics in the Theory of Public Economy: Empirical Welfare Analysis

 For this take home exam, you will need to use the data set CIS.RData. This is a cleaned version of the Canadian Income Survey of 2017.

Take Home Midterm Exam 

For this take home exam, you will need to use the data set CIS.RData. This is a cleaned version of the Canadian Income Survey of 2017. The Canadian Income Survey (CIS) is a cross-sectional survey developed to provide a portrait of the income and income sources of Canadians, with their individual and household characteristics.  The target population of   CIS is all individuals in Canada, excluding residents of the Yukon, the Northwest Territories and Nunavut, residents of institutions, persons living on reserves and other Aboriginal settlements in the provinces, and members of the Canadian Forces living in military camps. Overall, these exclusions amount to less than 3 percent of the population.  The description   of the variables can be found in the file CIS2017(description of variables).pdf.

Note: The answers to problems of this section should be formatted properly and presented in a pdf file. In addition, you should submit two R script files containing your codes. One should named Midterm.R and the other MidtermFunctions.R

Make sure the following packages are installed on your version of R: (1) Rcpp and (2) ineq. You can do this by taping this in your R Console:

If one of these commands return a FALSE value then install the package using

Make sure you save the C++ function Fhat.cpp that I have created for you in your working directory. This function gives an estimate of the empirical distribution function at each point of the data set of interest. Since it involves n × n operations, it will save you some cpu time to run it in C++. 

It is coded for you, you only need to call the function in R as any other R function.  The function has only one argument which is a matrix.  The first row of        the matrix should contain all income observations yi  (i ∈ {1, 2, . . . , n}) and the second row  of the matrix should contain the vector of associated statistical weight wi (i ∈ {1, 2, . . . , n}). The  function  will  return  for  each  observation  i,  pi  =  F (yi).   This  function  will  be useful when you will build the Lorenz curves.

Your Midterm.R script should start with

For each question, I will tell you if your script should be in MidtermFunctions.R or Midterm.R

Problem 1: Inequality rankings (45 marks) 

In the ineq R library, the Atkinson function generates the point estimate of the Atkinson index without its estimated variance and without accounting for survey weights. The  objective of this problem is to create a function Atkinson1 that will generate the point estimate and the estimated variance of the Atkinson index for an aversion to inequality equal to 1, accounting for survey weights. Let F (y) be the distribution on income of interest. For this level of aversion to inequality, the expression for the point estimate of the Atkinson index is

We focus on Eastern Canada in our exam. In your Midterm.R script,  create three (3)  regions: Atlantic Canada (Newfoundland and Labrador, Prince Edward Island, Nova Scotia and New Brunswick), Qu´ebec, and Ontario.  For each region create a 2 × Nr  matrix where Nr is the number of observations for the region. In the first row of each matrix, include theobservations of income and in the second row, the associated survey weights.

For Ontario, create a vector of 1s of dimension NON .

a)Inyour R script, create a function called Atkinson1 that is a func- tion of two vectors. The first argument of the function is a vector containing the observations on income and the second argument of the function is another vector containing survey weights (essentially row 1 and row 2 of the matrix of the province). The function should return a vector of two elements. The first element of this vector is the point estimate of the Atkinson index and the second element is its variance. (10 marks)公共经济理论代考

b)In your Midterm.R script, compute the point estimate of the Atkinson index  for Ontario using the Atkinson function included in the ineq library (don’t forget to set your parameter to 1 because the default value  is 0.5).  What is the point estimate?     (5marks)

c)In your Midterm.R script, compute the point estimate of the Atkinson index forOntario using your Atkinson1 function assuming that all observation have a weight equal to 1 (use the vector of one that you have created for Ontario as your survey weights). What is the value of this point estimate? How does this compare with the answer in (b). If they are different, you should rework (a). (5 marks)

d)In your Midterm.R script, compute the point estimate and the variance of the Atkin- sonindex for Ontario, Qu´ebec, and the Atlantic region using your Atkinson1  Also construct their 95% asymptotic confidence intervals. What are the values you get? What ranking of these three regions do you get using the point estimates? If you look  at  their  confidence  intervals  does  the  inequality  between  Ontario  and  Qu´ebec, Ontario  and  the  Atlantic  and  Qu´ebec  and  the  Atlantic  seem  statistically  different? (5 marks)

e)Let us now run some statistical tests. We want totest:

H0 : W1 = W2

H1 : W1 > W2

What I will ask you to do here is not 100% accurate. You will run a bootstrap using equal probability sampling but then use the survey weights for computing the statis- tics at each step of the bootstrap. You will proceed similarly each time I ask you to code a bootstrap in this exam. In your MidtermFunctions.R script, create a function called boot.Atkinson that is a function of two matrices and a scalar. The first matrix is the 2 × N1 matrix of the region 1 (data1) and the second matrix is the 2 × N2

matrix of the region 2 (data2). The scalar is the number of bootstrap repetitions.公共经济理论代考(10 marks)

A useful hint: If you want to sample with replacement use the following commands to create the bootstrap samples 1 (draw1) and 2 (draw2):

f)In your Midterm.R script,  compute  the  p-values  with  999  bootstrap  replications,  p1, p2, and p3for the following tests: (5 marks)

g)Whatcan you conclude on inequality comparisons between Ontario, Qu´ebec and the Atlantic region. (5 marks)

Problem 2: Lorenz dominance (55 marks) 

a)In your MidtermFunctions.R script, create a function named LorenzCoordinates that generates the Lorenz curve coordinates. This function has three arguments: incomes, survey weights, and grid size (hint: since you want to include the coordinate L(0) = 0, it is useful to work with  g + 1 as the real grid size and grid size,  g  as the number     of points other than (0, L(0))). This function should return a vector of size  g + 1 of  the values (L(0), L(p1), L(p2), . . . , L(pg−1), L(1)). The expression of the estimatorof

these coordinates is

This is where the Fhat.cpp function will help you to estimate F^(yi) =   1Nj=1wj1(yj ≤yi) for each observation.  This involves N × N simple operations (this is morethan

80 million operations for Ontario alone) at each step of the bootstrap in the following subquestion.

A compiled language such as C++ outperforms an interpreted language such R in doing this. (10 marks)

b)Inyour R script, create a function named boot.LorenzCoordinates that will bootstrap the function created in (a). This function has four arguments: incomes, survey weights, grid size and, number of bootstrap replications. This func- tion should return a matrix of 3 rows and g + 1 columns. A first row contains  the estimated values of the Lorenz curves coordinates from the original sample, (L(0), L(p1), L(p2), . . . , L(pg−1), L(1)) (same values as in (a)). The second and third

row contain respectively the lower and upper bounds 95% confidence interval of these coordinates. Some will use the 2.5 and 97.5 percentiles of the bootstrap. Many pre- fer to use L(p) ± 1.96 × σB(p), where σB(p) is the estimated standard deviation for coordinate L(p) in the bootstrap. Since we are assuming a Gaussian process, the last option is better and easier to use. (10 marks)

c)Inyour R script generate three figures. Each one of the figures displays the line of perfect equality and the Lorenz curves of regions 1 and 2 generated with a grid of 51 points (inclusive of 0) and their 95% confidence bands with 999 bootstrap replications. The list of regional comparisons for these three figures is

(i)OntarioVS Qu´ebec

(ii)Ontario VSAtlantic

(iii)Qu´ebecVS Atlantic

(10 marks)

(d)What are the rankings suggested by visual inspection of these figures. (5marks)

(e)In order to test the rankings obtained by our visual inspection, we have to run a statistical test. This test on Lorenz curves is thefollowing:

H0 :    L1(p) ≥ L2(p) ∀p ∈ [0, 1]

H1 :    L1(p) < L2(p) for some p ∈ [0, 1] 

If L1 and L2 are the estimators of L1 and L2 respectively, it is straightforward to construct a directional KS type test statistic

The bootstrap procedure to implement this test runs as follow:

1)Repeatfor b = 1, . . . , B

  • Draw a sample of size n1with replacement from S1. Compute the nonpara- metric estimator L1b.  
  • Draw a sample of size n2with replacement from S2. Compute the nonpara-metric estimator L2b.
  • Computeτ^b = supp∈[0,1]. N1N2  (L^2b(p) − L^1b(p) − L^2(p) + L^1(p)).

2)Using the sample bτ 1, . . . , bτB, compute the bootstrap p-value

In your MidtermFunctions.R script, create a function named test.Lorenz to run this bootstrap test. This function has four arguments: two matrices and two scalars. The first matrix is the N1 × 2 matrix of the region 1 (data1) and the second matrix is the

N2  × 2 matrix of the region 2 (data2).  The first scalar is the Lorenz curve grid size g

and the second scalar is the number of bootstrap repetitions. (10 marks)

f)As usual, the dominance test has a H0of dominance.  Propose an empirical procedure to tests for Lorenz dominance similar to the empirical procedure for stochastic domi- nance presented in class (don’t forget that in Lorenz dominance, the dominant curve is above and in the stochastic dominance case, it is the opposite). In your Midterm.R script, run this empirical procedure on a grid of 51 points for Lorenz curves and 999 bootstrap replications for the following comparisons:

(i)OntarioVS Qu´ebec

(ii)Ontario VSAtlantic

(iii)Qu´ebecVS Atlantic

What are the empirical conclusion you reach. Interpret their meaning in term of  inequality and compare with the rankings suggested by visual inspection of figures in (d). (10 marks)

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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 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