PS 5052 Mathematical Modeling in Political Science

Fall 2019

This course is an introduction to mathematical techniques used to model phenomena studied in political science, with special attention to the analysis of individual action. Mathematical topics covered include: sets, functions, and graphs; matrix algebra; differential calculus and optimization; probability, mathematical statistics, and decisions under risk; integral calculus; and sequences, series, and limits. All these topics are useful in many settings in political science, including game theory, dynamic modeling, and statistics.

This course website will be updated to reflect any changes in schedule, topics covered, or assignments, as well as to provide relevant links to materials associated with the course.

 

Course Outline and Approximate Schedule

0. Some Preliminaries

Monday and Wednesday, 8/26-8/28

Sets, real numbers, and intervals (reference: Pemberton & Rao sections 3.1-3.2; 31.2)

Areas, sums, and integrals--brief review if needed (P&R 19.1)

Logic and proofs (P&R 31.1)

Sequences, convergence, and limits (P&R 5.1, 5.4; for advanced help, 31.3, 31.4)

Practice problems (not to turn in): 5.1.3 (a-d); 5.2.3 (d only)

Homework problems on sequences and convergence(turn in before class Wednesday 9/4)

 

1. Probability and Statistical Inference

September

Text:   Excerpts (as shown below) from Wasserman, All of Statistics, chapters 1-10

 

1.1 Probability

Wed 9/4 though Mon 9/16

For Wednesday 9/4 read: Wasserman, Chapters 1 and 2.1-2.2

Homework problems to turn in before class Monday 9/9: from Wasserman sec. 1.10,

  • problems 5, 13, 21, 22
  • Note: problems 21 and 22 are "computer exercises". Please turn in your code as well as the relevant output.

Homework problems to turn in before class Monday 9/16: from Wasserman

  • Chapter 1: problems 12, 15, 19
  • Chapter 2: problems 2, 4, 6

 

1.2 Expectations and Moments

Read: sections 3.1-3.5

 

1.3 Convergence of Random Variables

Read: sections 5.1-5.4

 

1.4 Statistical Inference and Hypothesis Testing

Read: sections 6.3, 10.1, 10.2, 13.1

  • 6.3 estimation, confidence sets, and hypothesis testing
    • linear regression as an application: Example 6.6 (p. 89)
  • 10.1 the Wald test: read only through Remark 10.5, plus Example 10.8 (Comparing Two Means) and Theorem 10.10 and the Warning following it.
  • 10.2 p-values
    • linear regression as an application: Section 13.1, especially Example 13.6 (p. 211-212)

Exercises to be assigned eventually (subject to revision); due dates TBA

  • Chapter 1:  Exercises 5, 12, 13, 15, 19; Computer Experiments 21, 22
  • Chapter 2:  Exercises 2, 4, 6, 7, 9, 17, 18
  • Chapter 3:  Exercises 3, 4, 5, 11, 13
  • Chapter 5:
    • Do the following practice problem, which you can label "Problem A":   For each n = 1, 2, ..., let Xn have the uniform distribution on the interval [-1/n, 1 + 1/n]. Let X be distributed uniformly on [0, 1]. Show that Xn converges both in probability and in distribution to X. (You should prove convergence in distribution directly -- that is, without appealing to the fact that convergence in probability implies convergence in distribution.)
    • Do the version of Exercise 4 as we discussed in class: Don't examine convergence in "quadratic mean." Rather, in addition to showing that Xn converges in probability to 0, show (directly, again) that it also converges in distribution to 0.
  • Chapter 6: Exercise 2, 3
  • Chapter 10: Exercise 6 (Hint: n = 1919 is definitely a "large" sample, so the binomial distribution suggested in the exercise is approximately normal. Use the Wald test.)

 

*** First Exam out about W 10/2, in M 10/7 ***

 

(Fall break M 10/14)

 

For all the subjects below, chapter and section numbers refer to Pemberton & Rau, Mathematics for Economists (4th edition)

 

2. Calculus of one variable

2.1 Newton's quotient and differentiation

Read Chapters 6 and 7 (excluding 6.4)

particularly including

  • linear approximation using Newton's quotient
  • composite functions
  • monotone functions
  • inverse functions
Exercises for Chapters 6 and 7 -- due dates TBA
  • 6.1.1
  • 6.2.1 and 6.2.3
  • 7.2.2 and 7.2.3, all parts
  • 7.3.1, all parts
  • 7.4.1

 

*** Second Exam out about W 10/30 in M 11/4 ***

 

 

2.2 Optimization and convexity

Chapter 8

 

2.3 The Exponential and Logarithmic Functions

(Review Sections 4.3, 4.4.) Read Sections 9.1, 9.2

 

2.4 Mean value theorem, l'Hôpital's rule, Taylor's theorem

Sections 10.2 through 10.4

 

Exercises for Chapters 8, 9, 10 (differential calculus) -- due dates TBA

  • 8.2.2, 8.2.3, 8.2.4 finding and assessing critical points, using the second derivative; and curve-sketching
    • refers back to 8.1.2, 8.1.3
  • 8.3.1, 8.3.2 global max & min
  • 8.4.2, 8.4.3, 8.4.4 concavity, convexity.
    • For optional extra practice do 8.4.1 (don't turn in).
  • 9.1.4
  • 9.2.2, 9.2.4, 9.2.5
  • 10.2.3 all parts (a)-(c)
  • 10.2.1, 10.3.2; 10.4.1 (a) and (b)

 

2.5 Integration

Chapters 19, 20 (excluding 19.3)

Exercises for Chapters 19, 20 (integral calculus) -- due dates TBA

  • area: 19.1.3
  • integration: 19.2.3, 19.2.4 (all 3 parts); 19.2.6
  • integration by parts & by substitution: 20.1.1, 20.1.3, 20.1.4, 20.1.5
  • improper integrals: 20.2.1

 

(Thanksgiving break W 11/27)

 

3. Linear algebra

  • vectors, linear dependence, length: Section 11.1, 13.3
  • matrices as linear mappings; inverses: Section 11.2, 12.3
  • linear dependence and rank: Section 12.4
  • determinants and quadratic forms: Chapter 13
  • eigenvalues and eigenvectors: Chapter 27

Exercises for Chapters 11-13 (linear algebra) -- due dates TBA

  • vector arithmetic: 11.1.4
  • vector length; norm: 13.3.1, 13.3.2, 13.3.7
  • matrices as linear mappings: 11.2.3
  • matrix rank: 12.4.3 (refers to 11.1.4)
  • determinant: 13.1.1
  • quadratic forms: 13.4.7

 

A nice supplemental FREE linear algebra textbook: Pemberton and Rau occasionally make the connection between matrix operations and the corresponding properties of linear transformations, but their treatment of linear transformations is not complete. For your future reference, a nice-looking accessible supplementary treatment can be found online:

  • David Cherney, Tom Denton, Rohit Thomas, and Andrew Watson, Linear Algebra, online (1.4 megabytes, 460 pages), University of California, Davis (2013).
We have covered material corresponding to some of that in Chapters 1, 2, 4-12, and 16.

 

A succinct statement of the test for positive and negative (semi-)definiteness of a symmetric matrix based on its principal minors (by economist Nathan Barczi) can be found at MIT's website.

Another alternative, and interesting, definition of the determinant, in terms of permutations of the columns is given here.

 

4. Multivariate calculus

material from Chapters 14, 15, 16

  • partial derivatives: Read Section 14.1 up through p. 271 only.
  • the chain rule: Read Section 14.2, especially begining from top of p. 276
  • if time: implicit functions, section 15.1
  • optimization: Read sections 16.1, 16.2

Exercises for Chapters 14-16 (multivariate calculus) -- due dates TBA

  • critical points and second-order conditions: 16.1.2 (a, b, c), 16.1.3 (a, b, c), 16.1.5
    • Hint for 16.1.3: consider movements along the line y = kx for any constant k (whether positive or negative); then
      • in (a), f(x,y) = x4 + k4x4 = (1 + k4) x4;
      • In (c), it becomes (1 - k4) x4, and whether f reaches a max or min at (0, 0) depends on whether k is greater than or less than 1.
  • global optima, concavity, and convexity: 16.2.1 (for a, b, c only); 16.2.2

 

 

Third Exam

  • Fri. 12/6 regular help session; Mon. 12/9 optional review session
  • Exam distributed by Wed. 12/11
  • Exam due M 12/16

 


 

 

This page written by Randall Calvert ©2019
Email comments and questions to calvert at wustl.edu
Monday & Wednesday 10:00-11:20
classroom: Seigle 205

Jump directly to CURRENT topics & assignments

Instructor: Randall Calvert
Seigle 238; WU email: calvert
Office hours: Tu-Th 3:00-4:00; Fri. 1:30-2:30; and by appointment

Assistant to Instructor: Ryan Johnson
WU email: thomas.johnson
Office Hours:  TBA

Optional Help Sessions: Fridays 10:00-11:20 in Seigle 205

Textbooks

  • Malcolm Pemberton and Nicholas Rau, Mathematics For Economists: An Introductory Textbook, 4th ed. Book website includes solutions.
  • Larry Wasserman, All of Statistics. Textbook website offers some data and desiultory R code.

Course Requirements

All assignments are to be turned in as LaTeX documents; hand-drawn illustrations OK.

  • Three exams, 75%.:   non-cumulative, closed-book, take-home, un-timed. Dates indicated below; yoiu will get at least two weeks warning of any date changes.
  • Weekly exercises, 25%.:   collaboration encouraged. Grading will be based heavily on (1) completion and (2) effort.
  • Possible short quizzes.  I retain the option of administering a short quiz from time to time; to count as part of the 25% for homework.
  • Attendance.  Provided you're not contagious, I expect your attendance at every class meeting. Please let me know, in advance if possible, if problems arise that will require you to be absent.