CS代考 OLSX 01 - PowCoder代写

OLS and the Conditional Expectation Function

Empirical Finance: Methods and Applications Imperial College Business School
January 10th and 11th, 2022

􏰀 Course Details
􏰀 Basic housekeeping
􏰀 Course tools: Menti, R, and R-Studio 􏰀 Introduction to tidy data
􏰀 OLS and the Conditional Expectation Function 􏰀 Review and properties of the CEF
􏰀 Review, implementation, and value of OLS

Course Details: Contact
􏰀 Lecturer:
􏰀 Office: 53 Prince’s Gate, 5.01b 􏰀 Phone: +44 (0)20 7594 1044

Course Details: Assessment
􏰀 Two assignments (proposed schedule) 􏰀 Assignment 1 (25%)
􏰀 Assigned Tuesday of Week 3
􏰀 Due by 4pm on Tuesday of Week 5 􏰀 Assignment 2 (25%)
􏰀 Assigned Tuesday of Week 6
􏰀 Due by 5:30pm Tuesday of Week 8
􏰀 Final Exam (50%)

Course Details: Tentative Office Hours and Tutorials
􏰀 Tentative office hours
􏰀 Tuesdays from 14:00-15:00
􏰀 Or by appointment
􏰀 Formal tutorials will begin in Week 2
􏰀 Joe will be available online this week to help with R/RStudio

Course Details: Mentimeter
􏰀 On your phone (or computer) go to Menti.com

Course Details: R and R-Studio
􏰀 Make sure you have the most up-to-date version of R: 􏰀 https://cloud.r-project.org/
􏰀 And an up-to-date version of RStudio:
􏰀 https://www.rstudio.com/products/rstudio/download/

Course Details: In Class Exercises
􏰀 Throughout the module we’ll regularly do hands on exercises 􏰀 Lets start with a quick example:
􏰀 On the insendi course page find the data: ols basic.csv
􏰀 5variables: Y,X,Y sin,Y 2,Y nl
􏰀 Load the data into R, and run an OLS regression of Y on X. 􏰀 What is the coefficient on X?

Course Details: Projects in R-Studio
􏰀 For those with R-Studio set up:
􏰀 Open R-Studio and select File ⇒ ⇒ ⇒

􏰀 Name the directory “EF lecture 1” and locate it somewhere convenient
􏰀 Each coursework should be completed in a unique project folder

Course Details: R set up
􏰀 Download all data files from the hub and place them in EF lecture 1 􏰀 s p price.csv
􏰀 ols basics.csv
􏰀 ames testing.csv 􏰀 ames training.csv

Course Details: The Tidyverse
􏰀 The majority of the coding we do will utilize the tidyverse
The tidyverse is an opinionated collection of R packages designed for data science.
All packages share an underlying design philosophy, grammar, and data structures.
􏰀 For an excellent introduction and overview:
􏰀 ’s R for Data Science: https://r4ds.had.co.nz/
install.packages(“tidyverse”)
library(tidyverse)

Course Details: Tidy Data
􏰀 The tidyverse is structured around tidy datasets
􏰀 There are three interrelated rules which make a dataset tidy:
1. Each variable must have its own column 2. Each observation must have its own row 3. Each value must have its own cell
􏰀 For the theory underlying tidy data:
􏰀 http://www.jstatsoft.org/v59/i10/paper

An Example of Tidy Data

An Example of Non-Tidy Data

Fixing An Observation Scattered Across Rows
tidy2 <- table2 %>%
pivot_wider(names_from=type, values_from=count)

Another Example of Non-Tidy Data

Fixing Columns as Values: gather()
tidy4a <- table4a %>%
pivot_longer(c(‘1999‘, ‘2000‘), names_to = “year”, values_to =

Introducing the Pipe: %≥%
􏰀 You’ll notice that both of these operations utilize a “pipe”: %≥%
􏰀 A tool for clearly expressing a sequence of multiple operations 􏰀 Can help make code easy to read and understand
􏰀 Consider evaluating the following: x = 􏰶(log(e9)) 􏰀 Could write it as:
x <-sqrt(log(exp(9))) 􏰀 Or with pipes: x <- 9 %>%

This Week: Two Parts
(1) Introduction to the conditional expectation function (CEF) 􏰀 Why is the CEF a useful (and widely used) summary of the
relationship between variables Y and X
(2) Ordinary Least Squares and the CEF
􏰀 Review, implementation, and the utility of OLS

Part 1: The Conditional Expectation Function
􏰀 Overview
􏰀 Key takeaway: useful tool for describing the relationship between
variables Y and X
􏰀 Why: (at least) three nice properties:
1. Law of iterated expections 2. CEF decomposition property 3. CEF prediction property

Review: Expectation of a Random Variable Y
􏰀 Suppose Y is a random variable with a finite number of outcomes y1,y2,···yk occurring with probability p1,p2,···pk:
􏰀 The expectation of Y is:
k E[Y]= ∑yipi
􏰀 For example: if Y is the value of a (fair) dice roll:
E[Y]=1×61+2×16+3×16+4×16+5×16+6×16 =3.5
􏰀 Suppose Y is a (continuous) random variable whose CDF F(y) admits density f (y )
􏰀 The expectation of Y is:
􏰀 This is just a number!
E[Y]= yf(y)dy

The Conditional Expectation Function (CEF)
􏰀 We are often interested in the relationship between some outcome Y and a variable (or set of variables) X
􏰀 A useful summary is the conditional expectation function: E[Y|X] 􏰀 Gives the expectation of Y when X takes any particular value
􏰀 Formally, if fy(·|X) is the conditional density of Y|X: 􏰊
E[Y|X]= zfy(z|X)dz
􏰀 E[Y|X] is a random variable itself: a function of the random X
􏰀 Can think of it as E[Y|X]=h(X)
􏰀 Alternatively, evaluate it at particular values: for example X = 0.5
E[Y|X =0.5] is just a number!

Unconditional Expectation of Height for Adults: E[H]
Adult Height (Inches)
54 60 66 72 78

Unconditional Expectation of Height for Adults: E[H]
E[H]=67.5 In.
Adult Height (Inches)
54 60 66 72 78

Why the Conditional Expectation Function?
􏰀 E[Y|X] is not the only function that relates Y to X
􏰀 For example, consider 95th Percentile of Y given X: P95(Y|X)
P95[H|G=Male]
P95[H|G=Female]
􏰀 But E[Y|X] has a bunch of nice properties
Adult Height (Inches)
54 60 66 72 78

Property 1: The Law of Iterated Expectations
EX[E[Y|X]]=E[Y]
􏰀 Example: let Y be yearly wages for MSc graduates
􏰀 E[Y]=£1,000,900
􏰀 Two values for X : {RMFE, Other}
􏰀 Say 10% of MSc students are RMFE, 90% in other programs 􏰀 E[Y|X=RMFE]=£10,000,000
􏰀 E[Y|X=Other]=£1000
􏰀 The expectation works like always (just over E[Y|X] instead of X): E[E[Y|X]]=E[Y|X =RMFE]×P[X =RMFE]+E[Y|X =Other]×P[X =Other]
􏰎 􏰍􏰌 􏰏􏰎 􏰍􏰌 􏰏
£10,000,000×0.1 £1000×0.9 = £1, 000, 900

Property 1: The Law of Iterated Expectations
E[E[Y|X]]=E[Y]
􏰀 Not true, for example, for the 95th percentile: E[P95[Y|X]]̸=P95[Y]

Property 2: The CEF Decomposition Property
􏰀 Any random variable Y can be broken down into two pieces Y =E[Y|X]+ε
􏰀 Where the residual ε has the following properties: (i) E[ε|X] = 0 (“mean independence”)
(ii) ε uncorrelated with any function of X
􏰀 Intuitively this property says we can break down Y into two parts: (i) The part of Y “explained by” X: E[Y|X]
􏰀 This is the (potentially) useful part when predicting Y with X (ii) The part of Y unrelated to X: ε

Property 3: The CEF Prediction Property
􏰀 Out of any function of X, E[Y|X] is the best predictor of Y
􏰀 In other words, E[Y|X] is the “closest” function to Y on average
􏰀 What do we mean by closest?
􏰀 Consider any function of X, say m(X)
􏰀 m(X) is close to Y if the difference (or “error”) is small: Y −m(x) 􏰀 Close is about magnitude, treat positive/negative the same…
􏰀 m(X) is also close to Y if the squared error is small: (Y −m(x))2 􏰀 E[Y|X] is the closest, in this sense, in expectation:
E [Y |X ] = arg min E [(Y − m(X ))2 ] m(X)
􏰀 “Minimum mean squared error”

Summary: Why We Care About Conditional Expectation Functions
􏰀 Useful tool for describing relationship between Y and X
􏰀 Several nice properties
􏰀 Most statistical tests come down to comparing E[Y|X] at certain X 􏰀 Classic example: experiments

Part 2: Ordinary Least Squares
􏰀 Linear regression is arguably the most popular modeling approach across every field in the social sciences
􏰀 Transparent, robust, relatively easy to understand
􏰀 Provides a basis for more advanced empirical methods 􏰀 Extremely useful when summarizing data
􏰀 Plenty of focus on the technical aspects of OLS last term 􏰀 Focus today on an applied perspective

Review of OLS in Three Parts
1. Overview
􏰀 Intuition and Review of Population and Sample Regression Algebra
􏰀 Connection With Conditional Expectation Function 􏰀 Estimating a Linear Regression in R
2. An Example: Predicting Home Prices
3. Rounding Out Some Details
􏰀 Scaling and Implementation

OLS Part 1: Overview

OLS Estimator Fits a Line Through the Data
βOLS +βOLSX 01

A Line Through the Data: Example in R
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How Do We Choose Which Line?

One Data Point

vi: Observation i’s Deviation from β0 +β1xi

One Data Point
yi=β0 + β1xi+vi vi

Choosing the Regression Line
􏰀 For any line β0 +β1X, the data point (yi,xi) may be written as: yi =β0+β1xi+vi
􏰀 vi will be big if β0 +β1xi is “far” from yi 􏰀 vi willbesmallifβ0+β1xi is“close”toyi 􏰀 We refer to vi as the residual

Choosing the (Population) Regression Line
yi =β0+β1xi+vi
􏰀 An OLS regression is simply choosing the βOLS,βOLS that make vi
as “small” as possible on average 􏰀 How do we define “small”?
􏰀 Want to treat positive/negative the same: consider vi2
􏰀 Choose βOLS,βOLS to minimize: 01
E[vi2] = E[(yi −β0 −β1xi)2]

(Population) Regression Anatomy
{βOLS,βOLS} = arg min E[(y −β −β x )2] 0 1 {β0,β1} i 0 1i
􏰀 In this simple case with only one xi , β OLS has an intuitive definition: 1
βOLS = Cov(yi,xi) 1 Var(xi)
βOLS =y ̄−βOLSx ̄ 01

Regression Anatomy (Matrix Notation)
yi =β0+β1xi+vi
􏰀 You will often see more concise matrix notation:
β=β Xi=x 1i
􏰎 􏰍􏰌 􏰏 􏰎􏰍􏰌􏰏
2×1 2×1 y =X′β+v
􏰀 This lets us write the OLS Coefficients as:
βOLS = argminE[(y −X′β)2]
⇒ βOLS = E[X X′]−1E[X y ] ii ii

(Sample) Regression Anatomy
βOLS = argminE[(y −X′β)2] {β} i i
βOLS = E[X X′]−1E[X y ] ii ii
􏰀 Usually do not explicitly know these expectations, so compute sample analogues:
⇒βˆOLS =(X′X)−1(X′Y)
1 x1 y1
1 x2 y2 X=. . Y=.
. . . 1 xN yn
􏰎 􏰍􏰌 􏰏 􏰎􏰍􏰌􏰏
ˆOLS N (yi −Xi′β)2
β =argmin∑ {β} i=1

This Should (Hopefully) Look Familiar
RSS(b)= ∑(yi −Xi′b)2 i=1

Estimating a Linear Regression in R
􏰀 Simple command to estimate OLS
􏰀 ols v1<-lm(y∼ x, data= ols basics) 􏰀 And to display results: 􏰀 summary(ols v1) A Line Through the Data: Example in R ●●●● ● ●● ● ●●●● ●● ● ● ● ● ● ●● ● ● ●●● ● ●●●●● ● ● ●● ● ● ●●●● ● ● ●●● ●●●● ●● ●●● ●●●●● ●●● ● ●●● ● ●● ●● ●● ● ●●●●●●●●● ●● ●● ● ●● ●●●●●● ●● ● ● ●●● ● ● ● ● ● ●● ●● ● ●● ● ●● ● ●●● ● ● ● ●● ● ●● ● ●●● ●● ●●● ● ● ●●●● ● ● ● ● ●● ●● ●● ● ● ● ●● ●●●●●● ●● ●● ● ●● ●●● ● ●● ● ● ●●● ● ●●●● ● ●●● ● ●● ●●●●● ● ● ● ●●● ●● ●●●●●● ●●●● ●● ● ● ● ●●●●●● ● ●●●●●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ●● ● ● ● ● ● ●● ● ● ●●● ● ●● ● ● ● ● ● ●●● ● ● ● ● ●● ●●● ● ● ● ● ●● ●●●● ●●●● ●● ● ● ● ●●●●●● ●●●●● ●● ●●●●● ● ●● ●●●●●● ●●● ● ● ●●● ● ● ● ●●● ●●● ● ● ●● ●● ● ●● ● ● ●● ● ● ●● ● ●●●●● ●● ● ● ●● ●● ●●●●●● ●●●●● ● ● ●● ● ● ● ●● ● ●● ●●● ●●●● ● ●●●● ● ●●●●●●●● ●●●● ● ●●● ●● ●●● ●● ● ●● ●●● ● ●●●●●●●●●● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ●●●● ●● ● ●● ●● ● ● ●●● ● ● ●●● ● ●●● ●●●● ●●●● ● ● ●●●● ●●● ● ● ● ●● ● ●●● 􏰀 Intercept looks something like 1 􏰀 Slope, approximately 2? Recall, for comparison Regression and the Conditional Expectation Function 􏰀 Why is linear regression so popular? 􏰀 Simplest way to estimate (or approximate) conditional expectations! 􏰀 Three simple results 􏰀 OLS perfectly captures CEF if CEF is Linear 􏰀 OLS generates best linear approximation to the CEF if not 􏰀 OLS perfectly captures CEF with binary (dummy) regressors Regression captures CEF if CEF is Linear 􏰀 Take the special case of a linear conditional expectation function: E[y |X ] = X′β iii 􏰀 Then OLS captures E[yi|Xi] ●● ● ●● ●● ●● ●● ●●●●● ●● ● ● ●●● ● ●● ● ● ●●●● ●●●● ●● ● ● ● ● ● ● ●●● ● ● ● ●● ● ●●●● ●● ● ●● ●● ● ●●● ●●●●● ●●● ● ●●● ● ●● ●● ●● ● ● ●●●●●●●●●● ●● ●●●●● ● ●●●● ● ●●●●●●●● ● ● ●●● ●●● ● ● ●●●● ● ● ● ● ●●●●●●● ● ●● ● ●● ●●●●●●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●●●●●●●●● ●● ● ●●●● ●●●●● ● ● ●●●● ●●●●● ●● ●●●●● ● ●●● ●●●● ●●●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ●● ● ●● ●●● ● ●● ● ●● ● ● ●● ● ● ●● ● ●●●●● ●● ● ● ●● ●● ●● ●● ● ●● ● ● ●●● ● ● ● ● ● ● ●● ● ●● ●●● ●●●● ● ●●●● ● ●●●●●●●●● ●● ●● ●●● ● ●●●●● ●●● ● ●●● ● ● ● ●●●●●●●● ●●● ● ●●●●●●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●●●● ●●● ● ●● ● ● ●●●●● ●● ●● ●● ● ● ● ● ● ●●●● ●●●● ● ●● ● ●●● ●●●●●● ●●●●●●● ● ●● ●●● ●●●● ● ●●●●● ● ●●● ●● ●● ● ● ●●● ● ●●●● ● ● ●●● ● ● ●● ●●● ●●●● ●● ●●●● ●●●● ● ● ●●●●● ●●● ● Regression captures CEF if CEF is Linear 􏰀 Why? Recall from the CEF decomposition property: E[Xiεi] = E[Xi(Yi −E[Yi|Xi])] = 0 􏰀 Replacing 􏰀 So Xi′β =Xi′βOLS =E[Yi|X] E[Y |X ] = X′β iii ⇒E[X(Y −X′β)]=0 iii ⇒β =E[X X′]−1E[X Y ]=βOLS ii ii Conditional Expectation Function Often Non-Linear ●●●● ●● ● ●●●●● ● ●●● ●●● ●● ● ● ●●●● ●●● ● ● ●● ● ● ●●●● ● ● ●● ● ●●● ●●● ● ●● ● ●● ● ● ●●● ● ●● ●● ●●● ●● ● ● ● ●● ● ● ●● ● ● ●●●●● ● ● ● ●● ● ●●● ● ● ● ●● ● ●●●●● ●●●● ● ●● ●●● ●● ●● ●●● ● ● ●●● ●● ●●●●● ●●●● ●●●●●●●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ●● ● ●●●● ● ●● ●●●●● ● ●●●●● ●●●●●● ● ●● ●● ●● ●● ●●●● ●●● ●● ●●● ●●●●●● ●● ● ●●● ●●●●●● ● ● ●● ● ● ● ●●● ● ● ● ● ● ●● ●●●● ●●● ● ●●● ● ●● ●● ●● ●●● ● ●●● ●●●●● ● ●●●● ●●● ●● ● ●● ●●●●● ●●● ●● ●● ● ● ●●● ● ●●●●● ●●●● ● ●● ●●● ●● ● ● ●● ● ●●● ● ● ●●●●● ● ● ●● ● ● ●●● ●● ●●●● ● ● ●●●●●●● ● ●● ●●●●● ●● ● ● ● ● ● ● ●● ● ● ●●●● ●● ●●●● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● OLS Provides Approximation to CEF ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ●● ● ●● ● ● ●● ●●● ●● ● ●●● ●● ● ●● ●●●● ●●●●●● ●● ●●●● ● ●●●● ● ● ● ● ●● ● ● ● ●● ● ● ● ●●● ●● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ●● ● ● ●● ● ●● ●● ●● ●●●●●●● ● ● ●●●●● ●● ●● ●● ● ● ●● ●● ●●●●●●● ●●●●●● ● ● ●●● ●●● ●●●●●● ●● ● ● ●● ●●● ● ● ●● ●● ● ● ●● ●●●● ● ● ●● ● ● ●●●●● ● ●● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ●● ●●●● ●● 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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