lecture 3 Flashcards by megan haugh

points on a regression line

all data points have a residual, we want them to average to zero, does OLS do this?

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what do OLS estimators minimize

the sum of squared residuals
where: uhat i = Y i - Bhat i - Bhat1 Xi

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minimizing the sum of squared residuals gave us

two first order conditions (FOC) (SUM n, i=1)
1. uhat i = 0
2. uhat i X i = 0

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OLS estimators

Bhat 1 = sXY/S^2 X
Bhat o = Ybar - bhat 1 Xbar

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what does necessary conditions mean for FOC

they must be satisfied when we’ve estimated our estimate line via OLS

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SUM n, i=1 uhat i = 0

1st first order condition
- states that the sum of the residuals must be equal to zero

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SUM n, i=1 uhati X i = 0

2nd first order condition

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OLS sample covariance =

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based on OLS first order conditions we have decided

u hat bar = 0
S u caret X = 0

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first order conditions are not

assumptions; they follow from the choice made to use OLS to estimate the regression line

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graphical representations

y = give us the average residuals to sum to zero so the line is through the middle, not higher or lower
u caret = when put onto a residual line we can see that it is not the optimal height for a regression line

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OLS regression lines are

average residual = 0

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what is the height of the regression line

such that the average residuals sum to zero

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what can be said about the OLS estimator of the intercept

Bcaret o + Bcaret 1 X bar = Ybar
OLS sets the height of the regression line at Xbar to Ybar: ensuring average residual will be equal to 0
- plug in mean for X, it should equal Ybar if the average residual is 0

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OLS regressor must go through what

Xbar, Ybar
- slope doesn’t matter, it just needs to hit this point

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even when the average residual is 0 what else is needed

covariance needs to be 0

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what is the 2 step process for OLS

satisfy 2 FOC

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how do the 2 FOC create the best fit line

the first FOC of OLS sets the height of the regression at Xbar to be Ybar, ensuring the average residual is zero
adding the second FOC ensures the slope of the regression results in zero covariance between the X and the residuals

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goal of OLS

estimate a population regression line using a sample

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OLS

estimation method for population regression line

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implications of choosing OLS

1 . OLS must go through (Xbar, Ybar) ensuring residual is zero
2. the OLS slope ensures there is zero covariance between X and u caret

what variance do we prefer for an estimator

the one with more variance
/4 better than /2

least squares assumptions: line written as

there is a population regression line:
Yi = Bo + B1Xi + ui

what can be observed in a population regression line

Y and X
Bo, B1, ui are not observed

least square assumptions #1: zero conditional mean

critical assumption about the error term = E(ui|Xi)

what does the zero conditional mean assumption assume

zero covariance and no correlation between u and X

in the zero conditional mean assumption, u and X are -------

independent

if X and u are correlated then

zero conditional mean assumption cannot be satisfied

what does zero conditional mean assumption do for our line

ensures the true regression line is located in the center of the data

what do we say about u as a function of X in assumption 1

u is not a function of X

two components of the zero conditional mean assumption (about u0

1. expected value of the error term is a constant - it does not depend on x 2. this constant is zero

what graph would OLS not perform well

- with the zero conditional mean assumption slope and intercept surround the true line - data surrounds the true line, center of data is the truth

with zero conditional mean assumption what can we say about true regression line and population conditional mean

true regression line is located at the population conditional mean

what do assumptions 1-3 imply/do

they allow for us to certain of factors concerning OLS estimators

least squares assumption #2: sample is i.i.d.

independently and identically distributed for the joint distribution

joint distribution

Joint: probability that the random variables simultaneously take on certain values, say x and y EX: rain, no rain, long commute, short commute, makes 4 separate options All sum to 1

independent and identically distributed

independent: one observation does not tell you about another, randomly selected identically: drawn from the same population

how to get i.i.d. data

simple random sample from a large population

critical value for 95%

1.96

critical value for 90%

1.64

least squares assumption #: X and Y have non-zero finite moments

- rules out having very large outliers - meant to rule out difficult to work with variables

using these 3 assumptions what 4 things can we say about the OLS estimators; properties we know to be true when sufficient conditions are satisfied

1. OLS estimators are unbiased 2. OLS estimators are consistent 3. we can derive OLS variance estimators 4. OLS estimators are asymptotically normal (only FOCs always hold)

1. OLS estimators are unbiased

E(Bcaret o) = Bo - intercept E(Bcaret o) = B1 - slope this means that the expected values for these are what the OLS estimators turn out to be

2. OLS estimators are consistent

law of large numbers for Bcaret o and Bcaret 1

law of large numbers

When a large number of random variables with the same mean are averaged together, the large values tend to balance the small values, and their sample average is close to their common mean; if you have enough data you can get extremely close

4. OLS estimators are asymptotically normal

approximation works better when there is more data - essentially saying n is sufficiently large that the approximation is good enough

4. OLS estimators are asymptotically normal: infinitely large meaning

normal distribution

properties of OLS that hold ALL the time

1. FOC: u caret bar = 0; OLS regression line passes through (Xbar, Ybar) 2. FOC: s ucaret X = 0; residuals are uncorrelated with regressor

given that 1. FOC holds

then 2. FOC holds

why isn't E(u|X) on the list of properties of OLS that hold all the time

statement is about errors in population not residuals