ECONOMETRICS I ECON GR5411
Lecture 1 Probability Review by
Seyhan Erden Columbia University MA in Economics
• Professor: Seyhan Erden, PhD seyhan.erden@columbia.edu
• Office hours: Mon and Wed 12:00pm to 12:50pm (If you have questions and want to attend my office hours, please stay at the Zoom meeting at the end of the lecture at 12:00pm)
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• Required Textbook: The topics covered in this course are discussed in a wide variety of textbooks. Below is a list of several recommended textbooks, as well as other references that students will find helpful to supplement the notes from class. Note that earlier editions of many of the textbooks listed below could be used as references for the course material. I will be specific in class if I ever reference a particular section or problem from a specific edition of any of the textbooks.
• Lecture Notes, Problem Sets are your most important resources. Lecture slides will be posted on Courseworks.
• Recommended Textbook:
• Greene, William H., Econometric Analysis, Pearson, (8th edition) 2018
• Angrist, Joshua D. and Pischke, Jorn-Steffen. Mastering Metrics: The Path from Cause to Effect, Princeton University Press, 2014.
• Hayashi, Fumio Econometrics, Princeton University Press, 2000.
• Hansen, Bruce E. Econometrics, University of Wisconsin, 2019
• Wooldridge, Jeffrey M. Econometric Analysis of Cross Section and Panel Data, The MIT Press, Cambridge MA (2nd edition) 2010
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Teaching Assistants and Recitations:
• Martsella Davitaya md3590@columbia.edu
Fridays 8:30 – 10:00am NY time
• Ruoqing Jiang rj2556@columbia.edu
Thursdays 3:00 – 4:30 pm NY time
• Teaching Assistants will hold 90 minute office hours per week, times will be posted on the HOME page on Courseworks.
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COURSE DESCRIPTION:
Econometrics is the application of statistics and economic theory to problems of estimating models of economic behavior. The topics covered in the course will include a review of probability theory and statistics, hypothesis testing, introduction to large sample theory, the single- equation linear regression model, multiple regression, and as time permits extensions such as errors-in- variables, method-of-moments, instrumental variables and the pooling of time-series and cross-section data, panel data analysis, big-data (ridge, lasso) discrete choice models.
PREREQUISITES: Students are expected to have an understanding of multivariate calculus, matrix algebra, probability and statistics, and econometrics at the undergraduate level.
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ASSIGNMENTS: There will be problem sets assigned every other week, each of which involves empirical analysis as well as theoretical questions. The course statistical software is STATA. You may purchase STATA through Columbia at a reduced academic price but this is strictly optional. Stata/IC (Stata for mid-sized datasets) is sufficient for this course. Columbia has discount if you want to “rent” it for 6 months. One can request Stata online or by phone from the information provided at cuit.columbia.edu/content/stata.
If you prefer not to buy the license, there are computer labs on campus with Stata, the closest one is in Lehman Library (the Digital Social Science Center). Of course, most of you are not on campus, so you cannot access the library. CUIT has set up remote access to library computers. You can see the instructions here: https://cuit.columbia.edu/computer-lab- technologies. There are quite a few steps so make sure to set this up well in advance of your problem set deadlines!
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Problem sets and data will be posted on Courseworks. Please hand in your homework assignments at the time (or before) they are due through Gradescope. You will need to sign up to Gradescope ahead of time.
There is a document posted that explains how to submit homework to Gradescope, please read this document ahead of time so that you do not miss the due date due to technical difficulties. Problem sets that are submitted in after the time they are due but before solutions are posted will receive 50% credit. Assignments handed in after solutions are posted on Courseworks will not be graded. Each student must submit their own work. In case of collaboration the grade will be divided among students.
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Piazza: We will be using Piazza for class discussion. The system is highly catered to getting you help fast and efficiently from classmates, the TA, and myself. Rather than emailing questions to the teaching staff, I encourage you to post your questions on Piazza.
Find our class page
at: https://piazza.com/columbia/fall2020/econg r5411_001_2020_3econometricsi/home
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ANY QUESTIONS?
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Random Variables
Data generating process (DGP) – underlying statistical model that represents the random variation in observations. Sometimes DGP is referred to underlying common distribution of the population.
Random Variable (RV) – the outcome of a random process.
Discrete RV – countable, finite, X is discrete if X can take only a finite number of different values, ex: sum of two dice.
Continuous RV – infinitely divisible, hence not countable, X is continuous if X can take every value in an interval, ex: age, weight
Once recorded it is discrete.
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Probability Distribution:
A listing of the values 𝑥 taken by a random variable 𝑋 and their associated probabilities is a probability distribution, 𝑓𝑥
For Discrete RV, probability
𝑓 𝑥 =𝑃𝑟𝑜𝑏(𝑋=𝑥)
The axioms of probability require
1. 0≤𝑃𝑟𝑜𝑏𝑋=𝑥 ≤1 2. ∑/𝑓𝑥=1
For discrete RV, 𝑋, the probability that 𝑋 is less than or equal to 𝑎 is denoted as 𝐹(𝑎). 𝐹(𝑥) is the cumulative distribution function (cdf) and it follows
𝐹𝑥 =3𝑓𝑋 =𝑃𝑟𝑜𝑏(𝑋≤𝑥)
45/
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Simple Example to Discrete Probability Distribution:
Ex: 𝑋 = getting Heads when a coin flipped twice
𝑿
𝒇(𝒙)
Prob. distr. (PD)
𝑭(𝒙)
Cumulative PD
TT
0
.25
.25
TH, HT
1
.50
.75
HH
2
.25
1
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Probability Density Function (pdf) and Cumulative Density (cdf):
For Continuous RV,
the probability associated with any particular point is zero,
we can assign positive probabilities to intervals in the range
of 𝑥.
The probability density function (pdf) is <
𝑃𝑟𝑜𝑏 𝑎≤𝑥≤𝑏 =:𝑓 𝑥 𝑑𝑥≥0 ;
The area under 𝑓 𝑥 in the range from 𝑎 to 𝑏.
For continuous RV cdf is
A@
𝐹 𝑥 = : 𝑓 𝑥 𝑑𝑥 = 1
?@
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pdf and cdf for continuous RV:
If the range of 𝑥 is not infinite, then 𝑓 𝑥 = 0 anywhere outside the range. Because the probability associated with any individual point is 0. Since Pr 𝑋 = 𝑥 = 0
𝑃𝑟𝑜𝑏 𝑎≤𝑥≤𝑏 =𝑃𝑟𝑜𝑏 𝑎≤𝑥<𝑏 =𝑃𝑟𝑜𝑏 𝑎<𝑥≤𝑏
= 𝑃𝑟𝑜𝑏(𝑎 < 𝑥 < 𝑏)
and for a continuous RV, pdf is the derivative of cdf
𝑓𝑥 =𝑑𝐹(𝑥) 𝑑𝑥
Convention: an upper case letter for the cdf, where as a lowercase letter for the pdf.
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Expectations of RV (measure of central tendency) :
The mean, or expected value, of a RV is:
3 𝑥𝑓 𝑥 𝑖𝑓 𝑥 𝑖𝑠 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒,
/
: 𝑥𝑓 𝑥 𝑑𝑥 𝑖𝑓 𝑥 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
/
The mean is usually denoted as 𝝁. It is not necessarily a value taken by the RV. For example the expected value of Heads in one toss of a coin is 1⁄2.
Other measures of central tendency are the median and the mode.
𝐸(𝑥)
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Expectations of a function of RV:
Let 𝑔 𝑥 𝑔 𝑥 is:
be a function of 𝑥 then expected value of
3𝑔 𝑥 𝑃𝑟𝑜𝑏(𝑋 = 𝑥) /
: 𝑔(𝑥)𝑓 𝑥 𝑑𝑥
= 𝑎 + 𝑏𝑥 for constants 𝑎 and 𝑏, then 𝐸 𝑔(𝑥) =𝐸 𝑎+𝑏𝑥 =𝑎+𝑏𝐸(𝑥)
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𝐸 𝑔(𝑥)
/
𝑖𝑓 𝑥 𝑖𝑠 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒, 𝑖𝑓 𝑥 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠
If 𝑔 𝑥 9/10/20
Variance of RV (measure of dispersion):
The variance of a RV is: 𝑉𝑎𝑟𝑥 =𝐸(𝑥−𝜇)U
3(𝑥 − 𝜇)U𝑓(𝑥) 𝑖𝑓 𝑥 𝑖𝑠 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒, /
:(𝑥 − 𝜇)U𝑓 𝑥 𝑑𝑥 𝑖𝑓 𝑥 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 /
=
𝑉𝑎𝑟 𝑥
must be positive, is usually denoted by 𝜎U 𝑉𝑎𝑟𝑔(𝑥) =𝑉𝑎𝑟𝑎+𝑏𝑥 =𝑏U𝑉𝑎𝑟(𝑥)
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Standard Deviation and Chebychev Inequality:
The standard deviation is the positive square root of the variance and usually denoted by 𝜎, hence it has the same unit as 𝑥 and 𝜇.
The Chebychev inequality states that, for any positive constant 𝑘, 1
𝑃𝑟𝑜𝑏 𝜇−𝑘𝜎≤𝑥≤𝜇+𝑘𝜎 ≥1−𝑘U
Which basically means “at least 1 − 1/𝑘2 of the distribution's values are within k standard deviations of the mean” (or equivalently Chebychev inequality guaranties that “no more than 1/k2 of the distribution's values can be more than k standard deviations away from the mean”)
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Markov Inequality⇒Chebychev Inequality:
The Markov Inequality: if 𝑥 is nonnegative RV, that is, if Pr(𝑥 < 0) = 0, and 𝑘 is any positive constant, then Pr 𝑥 ≥ 𝑘 ≤ 𝐸(𝑥)/𝑘
Proof:
[[@\
Now, 𝑎 ≥ 0, so 𝐸(𝑥) ≥ 𝑏. Also 𝑏 ≥ 𝑘 ∫\ 𝑓 𝑥 𝑑𝑥 = 𝑘 Pr 𝑥 ≥ 𝑘
So, 𝐸(𝑥) ≥ 𝑘 Pr 𝑥 ≥ 𝑘 [from Goldberger page 31]
The Chebychev inequality states that, replacing 𝑘𝜎 by 𝜖, if 𝑥 is RV with 𝐸 𝑥 = 𝜇 and variance 𝑣𝑎𝑟 𝑥 = 𝜎U, and 𝜖 is a positive constant, then
@\@
𝐸 𝑥 =:𝑥𝑓 𝑥 𝑑𝑥=:𝑥𝑓 𝑥 𝑑𝑥+:𝑥𝑓 𝑥 𝑑𝑥=𝑎+𝑏
or the equivalently
𝑃𝑟𝑜𝑏 |𝑥−𝜇|≥𝜖 ≤𝜎U 𝜖U
𝑃𝑟𝑜𝑏 𝑥−𝜇 <𝜖 >1−𝜎U 𝜖U
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Skewness and Kurtosis:
Skewness is a measure of asymmetry of a distribution: 𝐸 (𝑥 − 𝜇)b
For symmetric distributions 𝑓 𝜇 − 𝑥 = 𝑓(𝜇 + 𝑥) and skewness = 0
Kurtosis is a measure of thickness of the tails of the distribution
𝐸 (𝑥 − 𝜇)c
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Joint Distributions:
The joint density function for two RVs 𝑋 and 𝑌 denoted by 𝑓(𝑥, 𝑦) is defined as
𝑃𝑟𝑜𝑏 𝑎≤𝑥≤𝑏,𝑐≤𝑦≤𝑑
3 3 𝑓 𝑥,𝑦 𝑖𝑓𝑥𝑎𝑛𝑑𝑦𝑎𝑟𝑒𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒
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