Introductory Econometrics: Chapter 2 Slides

7/17/2019 Introductory Econometrics: Chapter 2 Slides

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REVISION MULTIPLE R EGRESSION

Econometrics I - Exercise 2

Lubomır Cingl

March 4, 2014

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REVISION

Simple regression model

y = β 0 + β 1x + u (1)

y dependent / explained

x independent / explanatory / control

u error term

β 0 intercept

β 1 slope

we want to know the relationship

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ASSUMPTIONS

Average value of u in population is 0: E(u) = 0

not very restrictive Zero conditional mean

knowing sth about x does not give any info about u E(u|x) = E(u) = 0 which implies E( y|x) = β 0 + β 1x

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E( y|x) as a linear function of x;distribution of y centered aroundexp. value

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ESTIMATION FROM A SAMPLE

we have a random sample of observations

for each observation holds:

Sample regression line

yi = β 0 + β 1xi + ui (2)

we want ”best” estimates of parameters

β 0, β 1 3 ways how to find them: MoM, OLS, ML

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FORMULAS TO KNOW

interceptβ 0 = ¯ y− β 1x (3)

slope

β 1 =

ni=1(xi − x)( yi − ¯ y)ni=1

(xi − x)2

(4)

if

ni=1(xi − x)

2

> 0

β 1 is sample covar bw x and y div by variance of x

If x and y are positively correlated, the slope positive

R M R

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SAMPLE REGRESSION LINE

e.g. ˆinc = 0.33 + 0.56edu


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R SQUARED

each observation can be made up by explained andunexplained part

yi = ˆ yi + ui we can define following:

( yi − ¯ y)2 is total sum of squares (Var of y)

( ˆ yi − ¯ y)2 explained sum of squares

(ui)

2 residual sum of squares

SST = SSE + SSR

R2 = SSE/SST = 1 − SSR/SST


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PROPERTIES OF OLS ESTIMATOR

Unbiased expected value of estimator is its true value β 1 = β 1 +

(xi−x)ui(xi−x)2 = β 1

Variance Assume homoskedasticity Var(u|x) = σ2

σ2 is the error variance Var β 1 = σ2/

(xi − x) = σ2/sx

2

We also have to estimate σ2

σ2 = 1/(n− 2)u12 = SSR/(n− 2)

Var β 1 = σ2/

(xi − x) = 1/(n− 2)u1

2/sx2


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EXAMPLE 2.7

Consider the savings function

sav = β 0 + β 1inc + u,u =√ inc ∗ e (5)

where e is a random variable with E(e) =

0 and Var(e) = σ2

e .Assume that e is independent of inc.

1. Show that E(u|inc) = 0, so that the key zero conditionalmean assumption is satisfied. [Hint: if e independent of inc, then E(e

|inc) = E(e) and Var(e

|inc) = Var(e)]

2. Show that Var(u|inc) = σ2e ∗ inc.3. Provide a discussion that supports the assumption that the

variance of savings increases with family income.


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INTERPRETATION

linear function y = β 0 + β 1x

marginal effect of x on y

natural log

y = log(x); x > 0

good to know: log(1 + x) ≈ x for x ≈ 0 and

log(x1) − log(x0) ≈ (x1 − x0)/x0 = ∆x/x0 therefore ∆log(x)

∗100

≈%∆x

constant elasticity model

log( y) = β 0 + β 1log(x); y, x > 0

β 1 is elasticity of y w.r.t. x


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EXAMPLE 2.6

Using data from 1988 for houses sold close to garbage mill,following equation relates housing price to distance fromgarbage incinerator:

log( ˆ price) = 9.4 + 0.312log(dist);n = 135;R2 = 0.162 (6)

1. Interpret the coefficient on log(dist). Is the sign of thisestimate what you expect it to be?

2. Do you think simple regression provides an unbiased

estimator of the ceteris paribus elasticity of price withrespect to dist? (Think about the citys decision on where toput the incinerator.)

3. What other factors about a house affect its price? Mightthese be correlated with distance from the incinerator?


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REGRESSION THROUGH ORIGIN

restriction: x = 0 if y = 0 then we estimate y = β 1x

we know that

β 1 =

ni=1

(xi − x)( yi − ¯ y)

ni=1

(xi − x)2(7)

if we set x = 0, ¯ y = 0 we get possibly biased estimator:

β 1 =

ni=1

(xi)( yi)

ni=1

(xi)2

(8)


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EXAMPLE 2.8

Consider standard simple regression model with standardassumptions met. Let β 1 be the estimator of β 1 obtained byassuming the intercept is zero.

1. Find E( β 1) in terms of the xi, β 0, and β 1. Verify that β 1 is

unbiased for β 1 when the population intercept β 0 is zero.Are there other cases where β 1 is unbiased?

2. Find the variance of β 1. Hint: It does not depend on β 0.

3. Show that Var( β 1) ≤ Var( β 1). Hint:

(xi)2 ≥(xi − x)2

with strict inequality unless x = 0.4. Comment on the trade-off between bias and variance

when choosing between β 1 and β 1.


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EXAMPLE 2.9

1. Let β 0 and β 1 be the intercept and slope from the regressionof yi on xi, using n observations. Let c1 and c2, with c2 = 0, be constants. Let β 0 and β 1 be the intercept and slope from

the regression c1 yi on c2xi. Show that β 1 = (c1/c2) β 1 andβ 0 = c1 β 0, thereby verifying the claims on units of measurement in Section 2.4. [Hint: To obtain β 1, β 0 plugthe scaled versions of x and y into the OLS formulas.]

2. Now let and β 0 and β 1 be from the regression (c1 + yi) on(c2 + xi) (with no restriction on c1 or c2). Show that β 1 = β 1and β 0 = β 0 + c1 − c2 β 1.


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EXAMPLE 2.9

3. Now let and β 0 and β 1 be the OLS estimates from theregression log( yi) on (xi), y > 0. For c1 > 0 let β 0 and β 1 be the

intercept and slope from the regression of log(c1 yi) on (xi).Show that β 1 = β 1 and β 0 = β 0 + log(c1).4. Now, assuming that xi > 0, let β 0 and β 1 be the intercept andslope from the regression of yi on log(c2xi). How do β 0 and β 1compare with the intercept and slope from the regression of yi

on log(xi)?


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Multiple regression


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MULTIPLE REGRESSION MODEL

paralels with simple reg model

y = β 0 + β 1x1 + β 2x2 + ... + u (9)

y dependent / explained

x independent / explanatory / control u error term

similar assumptions about the error term

β 0 intercept

linear - linear in parameters β 1,β 2, ... slopes

Interpretation: ∆ˆ y = ∆ β 1x1 ceteris paribus: holding other factors constant


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EXAMPLE I

effect of education on wage

wage = β 0 + β 1educ + β 2exper + u (10)

we take exper out from error term

ceteris paribus: holding experience constant, we canmeasure the effect of education on wage

partial effect


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EXAMPLE II

effect of income on consumption

cons = β 0 + β 1inc + β 2(inc)2 + u (11)

same way of getting parameters interpretation different: alone inc non-sense

∆cons

∆inc ≈ β 1 + 2β 2inc

marginal effect of income on consumption depends on both terms


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ASSUMPTIONS

Expected value of u is 0: E(u) = 0 Zero conditional mean (ASS3)

knowing sth about x does not give any info about u

E(u|x1, x2,..., xk ) = 0 all unobserved factors in error term uncorrelated with all xi

No perfect collinearity (ASS4) no independent variable is constant no exact linear relationship among them holds

x’s can be correlated, but not perfectly we need more observations than parameters: n ≥ k + 1


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ESTIMATION FROM A SAMPLE

we have a random sample of observations (ASS2)

for each observation holds (ASS1):

yi = β 0 + β 1xi1 + β 2xi2 + ... + β k xik + ui (12)

we want ”best” estimates of parameters

β 0, β 1, ...

3 ways how to find them: MoM, OLS, ML

we will get the estimates ⇒ Sample regression ft

ˆ yi = β 0 + β 1xi2 + β 2xi2 + ... + β k xik (13)


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PROPERTIES

if ASS1 thru ASS4 hold, then our OLS estimator isunbiased

i.e. the procedure of getting the estimate is unbiased

if we add irrelevant variable no effect on parameters of relevant variables - still unbiased slight increase in R2

if we omit important variable OLS will be biased, usually ”omitted variable bias”


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OMMITED VARIABLE BIAS

y = β 0 + β 1x1 + β 2x2 + u

˜ y = β 0 + β 1x1

E( β 1) = β 1 + β 2

(xi1 − x1)xi2(xi1 − x1)2 = β 1 + β 2 δ 1


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VARIANCE

Assume homoskedasticity Var(u|x1, x2...xk ) = σ2

variance of u same for all combinations of outcomes of x’s

Var(β j) = σ2

(xij−

x j)(1−R2

j

) =

σ2

SST j(1−R2

j

) (14)

R2 j is R2 from the regression of x j on all other x’s size important

We also have to estimate σ2

σ2 = 1/(n

−k

−1) u1

2= SSR/df

Varβ j = σ

2

SST j(1−R2 j )

seβ j = σ SST j(1−R2 j )


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EXAMPLE 3.9

The following equation describes the median housing price in acommunity in terms of amount of pollution (nox for nitrousoxide) and the average number of rooms in houses in thecommunity (rooms):

log( price) = β 0 + β 1log(nox) + β 2rooms + u

1. What are the probable signs of the regression slopes?Interpret β 1, explain.

2. Why might nox [more precisely, log(nox)] and rooms benegatively correlated? If this is the case, does the simpleregression of log( price) on log(nox) produce an upward ordownward biased estimator of β 1?


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EXAMPLE 3.9

3. Using the data in HPRICE2.RAW, the following equationswere estimated:

log( ˆ price) = 11.71

−1.043log(nox),n = 506,R2 = 0.264

log( ˆ price) = 9.23−0.718log(nox)+0.306rooms,n = 506,R2 = 0.514

Is the relationship between the simple and multiple regressionestimates of the elasticity of price with respect to nox what you

would have predicted, given your answer in part 2.? Does thismean that -0.718 is definitely closer to the true elasticity than-1.043?


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EXAMPLE 3.12

1. Consider the simple regression model y = β 0 + β 1x+ u underthe first four Gauss-Markov assumptions. For some functiong(x), for example g(x) = x2 or g(x) = log(1 + x2) define zi = g(xi). Define a slope estimator as

β 1 =

( zi − ¯ z) yi/

( zi − z)xi

Show that β 1 is linear and unbiased. Remember, becauseE(u|x) = 0, you can treat both xi and zi as nonrandom in yourderivation.2. Add the homoskedasticity assumption, MLR.5. Show that

Var( β 1) = σ(

( zi − ¯ z)2)/(

( zi − z)xi)2


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EXAMPLE 3.12

3. Show directly that, under the Gauss-Markovassumptions,Var( β 1) ≤ Var( β 1), where β 1 is the OLS estimator.

[Hint: The Cauchy-Schwartz inequality in Appendix B impliesthat

(n−1

( zi − z)(xi − x))2 ≤ σ(n−1

( zi − ¯ z)2)(n−1

(xi − x)2)

notice that we can drop ¯x from the sample covariance.]


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EXAMPLE 3.15

The file CEOSAL2.RAW contains data on 177 chief executiveofficers, which can be used to examine the effects of firmperformance on CEO salary.

1. Estimate a model relating annual salary to firm sales and

market value. Make the model of the constant elasticityvariety for both independent variables. Write the resultsout in equation form.

2. Add profits to the model from part 1. Why can this

variable not be included in logarithmic form? Would yousay that these firm performance variables explain most of the variation in CEO salaries?


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EXAMPLE 3.15

3. Add the variable ceoten to the model in part 2. What is theestimated percentage return for another year of CEO tenure,

holding other factors fixed?4. Find the sample correlation coefficient between the variableslog(mktval) and profits. Are these variables highly correlated?What does this say about the OLS estimators?

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Introductory Econometrics: Chapter 2 Slides

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