MULTIPLE REGRESSION ANALYSIS: SPECIFICATION AND DATA ISSUES Chapter 9 1.

MULTIPLE REGRESSION

ANALYSIS: SPECIFICATION AND

DATA ISSUES

Chapter 9

1

I. Introduction2

Failure of zero conditional mean assumption Correlation between error, u, and one or more

explanatory variables. Why variables can be endogenous Possible remedies

Functional Form Misspecification If omitted variable is a function of an

explanatory variable in the model, the model suffers from functional for misspecification

Using proxy variables to address omitted variable bias

Measurement error Not all variables are measured accurately.

II. Functional Form3

Regression model can suffer from misspecification when it doesn’t account for relationship between dependent and explanatory variables.

wage = 0 + 1educ + 2exper + u Omit exper2 or exper*educ

Omitting variable can lead to biased estimates of all regressors

Use wage rather than log(wage) (latter satisfies GM) using wrong variable to relate LHS and RHS can

lead to biased estimates of all regressors.


We can change linear relationship by: using logs on RHS, LHS or both using quadratic forms of x’s Using interactions of x’s

How do we know if we’ve gotten the right functional form for our model? Use F-test for joint exclusion restrictions to detect

misspecification

II. Functional Form Ex: Model of Crime Quadratics or not? Each of sq terms is

individually and jointly signficant (F=31.37, df=3; 2,713

Adding squares makes interpretation more difficult: Before, intuitive (–) sign on

pcnv suggested conviction rate has deterrence on crime.

Now, level is positive, quadratic is negative: for low levels conviction has no deterrent effect, only effective for large levels.

Note: Don’t square qemp86, because it’s a discrete variable taking only few values.

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How do you know what to try? Use economic theory to guide you

Think about the interpretation Does it make more sense for x to affect y in

percentage (use logs) or absolute terms? Does it make more sense for the derivative

of x1 to vary with x1 (quadratic) or with x2 (interactions) or to be fixed?

II. Ramsey’s RESET 7

Know how to test joint exclusion restrictions for higher order terms or interactions. Can be tedious to add and test extra terms May find a square term matters when really

using logs would be even better A test of functional form is Ramsey’s regression

specification error test (RESET) Intuition: If specification okay, no nonlinear functions

of the independent variables should be significant when put in original equation.

Cost: Degrees of freedom

II. Ramsey’s RESET8

RESET relies on a trick similar to the special form of the White test Instead of adding functions of the x’s directly,

we add and test functions of ŷ y = 0 + 1x1 + … + kxk + 1ŷ2 + 1ŷ3 +error

Don’t look at above for parameter estimates, just to test inclusion of extra terms

H0: 1 = 0, 2 = 0 using F~F2,n-k-3 Significant F-stat suggests there’s some sort

of functional for problem

II. Ramsey’s RESET9

Ex: Housing Price Equation (n=88) price = 0 + 1lotsize + 2sqrft +3bdrms +u

RESET statistic (up to yhat3)=4.67 F2,82 and p-value .012 Evidence of functional form misspecification

lprice = 0 + 1llotsize + 2lsqrft +3bdrms +u RESET statistic (up to yhat3)=2.56

F2,82 and p-value .84. No evidence of functional form misspecification

On basis of RESET, log equation is preferred. But just because loq equation “passed” RESET, does

that mean it’s the right specification? Should still use economic theory to determine if

functional form makes sense.

III. Proxy Variables10

Previously, assumed could resolve functional form misspecification because you had the relevant data. What if model is misspecified because no data is

available on an important x variable? Log(wage) = 0 + 1educ +2exper + 3abil + u

Would like to hold ability fixed, but have no measure of it.

Exclusion causes parameter estimates to be biased.

Potential solution: Obtain proxy variable for omitted variable


A proxy variable is something that is related to the unobserved variable that we’d like to control for in our analysis-but can’t. Ex: IQ as proxy for ability x3* = 0 + 3x3 + v3, where * implies unobserved v3 signals that x3 and x3* are not directly related 0 allows different scales to be compared (i.e. IQ

scale may not be how ability measured) just substitute x3 for x3* in y= 0 + 1 x1 +2 x2 +

3 x3* + u


What do we need for this solution to give us unbiased estimates of 1 and 2? Need assumptions on u and v3

1.) u uncorrelated with x1, x2, x3* (standard) Also suggests u uncorrelated with x3…once x1, x2, x3*

included, x3 is irrelevant (i.e. x3 doesn’t directly affect y other than through x3*)

2.) v3 is uncorrelated with x1, x2, x3.

For v3 to be uncorrelated with x1, x2 that means x3* must be good proxy for x3

Formally, this means E(x3* | x1, x2, x3) = E(x3* | x3) = 0 + 3x3 Once x3 controlled for, x3* does not depend on x1, x2


E(abil|educ,exper,IQ)=E(abil|IQ)=0 + 3IQ Implies ability only changes with IQ, and not with

educ and epxer (once include IQ). So are really running: y = (0 + 30) + 1x1+ 2x2 + 33x3 + (u + 3v3) redefined intercept, error term, x3 coefficient

Can rewrite as: y = 0 + 1x1+ 2x2 + 3x3 + e Unbiased estimates of

0 , 1 =12 =2 , 3 Won’t get original 0 or 3.

III. Proxy Variables

IQ as proxy for ability Want to estimate

return to education 6.5% when run

regression w/o ability proxy

5.4% when include IQ Interact educ*IQ,

allows for possibility that returns to education differ across different ability levels. See that interaction not significant though.

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Proxy variable can still lead to bias if assumptions are not satisfied

Say x3* = 0 + 1x1 + 2x2 + 3x3 + v3 (violation)

Then running: y = (0 + 30) + (1 + 31) x1+ (2 + 32) x2 + 33x3 + (u +

3v3) Bias will depend on signs of 3 and j

Can safely assume 1 >0 and 3 >0, so that return to education is upward biased even when using proxy variable.

This bias may be smaller than omitted variable bias, though (if x3* and x1 correlated less than x3 and x1)

III. Lagged Dependent Variables

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What if there are unobserved variables, and you can’t find reasonable proxy variables?

Can include a lagged dependent variable to account for omitted variables that contribute to both past and current levels of y must think past and current y are related for

this to make sense allows you to account for historical factors that

cause current differences in dependent variables

III. Lagged Dependent Variables Ex: Model of Crime: Effect

of expenditure on crime crime= 0 + 1 unem +2

expend +u Concerned that cities

which have lots of crime react by spending more on crime…biased estimates

Coeff on unem and expend are not intuitive

crime= 0 + 1 unem +2 expend+ 3 crime-1 + u Lagged value controls for

fact that cities with high historical crime rates may spend more on crime prevention

Coefficient estimates now more intuitive

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IV. Properties of OLS under Measurement Error

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Sometimes we have the variable we want, but we think it is measured with error how many hours did you work last year, how

many weeks you used child care when your child was young

When use imprecise measure of variable in our regression, then model contains measurement error.

Consequences of M.E. Model is similar to that of omitted variable bias Often variable with measurement error is the

one we’re interested in measuring There are some conditions under which we still

get unbiased results Measurement error in y different from

measurement error in x

IV. Measurement Error in a Dependent Variable

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Let y* denote variable we’d like to explain, like annual savings. Model: y* = 0 + 1x1 + …+ kxk + u Most often, respondents are not perfect in

their reporting, and so reported savings is denoted y

Define measurement error as observed-actual: e0 = y – y*

Thus, really estimating: y = 0 + 1x1 + …+ kxk + u + e0

IV. Measurement Error in a Dependent Variable

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When will OLS produce unbiased results? Have assumed u has zero mean and that xj and u

are uncorrelated Need to assume

e0 also has zero mean (otherwise just biases 0 ) but more importantly e0 and xj are uncorrelated.

That is, the measurement error in y is statistically independent of each explanatory variable. As result, estimates are unbiased.

Generally find Var(u+ e0 )=u2 +e0

2 >u2

When have m.e. in LHS variable, get larger variances for OLS estimators.

IV. Measurement Error in a Dependent Variable Savings Function sav* = 0 + 1inc +

2size+3educ+ 4age + u e0= sav-sav* Is m.e. correlated with RHS

variables? May think families with higher

incomes or more education more likely to report savings accurately.

Never know if that’s true, so assume there is no systematic relationship: i.e. wealthy or more educated just as likely to mis-report as non-wealthy, uneducated

Scrap Rates Log(scrap*) = 0 + 1grant + u Error assumed to be

multiplicative: y=(y*)*a0 where e0=log(a0) log(scrap)=log(scrap*)+e0

Log(scrap) = 0 + 1grant + u + e0

It’s possible that measurement error more likely to at firms that receive grant underreport scrap rate to make

grant look more effective-so get more in future.

Can’t verify whether true, so assume no relationship: i.e. measurement error not correlated with grant.

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IV. Measurement Error in an Explanatory Variable

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More complicated when measurement error occurs in the explanatory variable(s)

Model: y = 0 + 1 x1* + u x1* is not observed, instead only observe x1 define m.e. as e1 =observed-actual = x1 – x1*

Assume E(e1) = 0 (not strong assumption) E(y| x1*, x1) = E(y| x1*)…means x1 doesn’t affect y

after control for x1*…means u uncorrelated with x1 and x1*….similar to proxy variable assumption.

Now are estimating y = 0 + 1x1 + (u – 1e1)


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What kind of results will OLS give us? depends on our assumption about the

correlation between e1 and x1 Suppose Cov(x1, e1) = 0

OLS remains unbiased Variances larger ( since Var(u-1 e1)=u

2 +1

2 e1 2 )

Assumption that Cov(x1, e1) is analogous to the proxy variable assumption.


What if that’s not the case? Suppose only that Cov(x1

*, e1) = 0 Called classical errors-in-variables assumption More realistic assumption than assuming Cov(x1, e1) =0

This means: Cov(x1, e1) = E(x1e1)-E(x1 )E(e1 ) =E[(x1

*+e1)(e1)]= E(x1*e1) + E(e1

2) = 0 + e2 ≠0.

This means x1 is correlated with the error so estimate is biased and inconsistent

)(

*)(1

)*,cov(2

),(),(,ˆplim

1

1122

*

2*

122*

2

1

1122

*

21

11

11111

1

11111

xVar

xVar

exxVar

exCovuxCov

xVar

euxCov

ex

x

ex

e

ex

e

24


Economics 20 - Prof. Anderson

25

Notice that the multiplicative portion Var(x1*)/Var(x1)< 1 Means the estimate is biased toward zero – called

attenuation bias True regardless of if 1 is (+) or (-) Larger Var(x1*)/Var(x1) suggests inconsistency with

OLS is small, because variation in “noise” (a.k.a. m.e.) is small relative to variation in true value.

It’s more complicated with a multiple regression, but can still expect attenuation bias when assume classical errors in variables.



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y = 0 + 1x*1 + 2x2 + 3x3 +u Assume u uncorrelated with x*1,x1,x2,x3

If assume e1 uncorrelated with x1,x2,x3 then get y = 0 + 1x1 + 2x2 + 3x3 +u -1e1

get consistent estimates

But, if e1 uncorrelated with x2,x3 but not necessarily x1, get

If x*1 uncorrelated with x2,x3 get consistent estimates of 2, 3

If this doesn’t hold, then other estimates will be inconsistent (size and direction are indeterminate)

*

132210*1

*1

21

*21

*21

11

xfrom is r

ˆplim

rxxwhere

er

r



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Ex: GPA with measurement error colGPA = 0 + 1faminc* + 2hsGPA+3SAT +4smoke + u

faminc* is actual annual family income faminc=faminc*+e1

Assuming CEV holds, get OLS estimator of 1 that is attenuated (biased toward zero).

colGPA = 0 + 1faminc + 2hsGPA+3SAT +4smoke* + u smoke=smoke*+e1

CEV unlikely to hold, because those who don’t smoke are really unlikely to mis-report. Those that do smoke can mis-report, such that error and actual number of times smoked (smoked*) are correlated.

Deriving the implications of measurement error when CEV doesn’t hold is difficult and out of scope of text.

V. Missing Data, Nonrandom Samples, Outlying Observations


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Introduction into data problems that can violated MLR.2 of G-M assumptions Cases when data problems have no effect

on OLS estimates Other cases when get biased estimates

Missing Data Generally collect data from random sample

of observations (people, schools, firms) Discover that information from these

observations on key variables are missing

V. Missing Data – Is it a Problem?


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Consequences If any observation is missing data on one of

the variables in the model, it can’t be used Data missing at Random

If data is missing at random, using a sample restricted to observations with no missing values will be fine

Simply reduces sample size, thus reducing precision of estimates

V. Missing Data – Is it a Problem?


30

Data not missing at random A problem can arise if the data is missing

systematically High income individuals refuse to provide income data Low education people generally don’t report education People with high IQ more likely to report IQ

When missing data does not lead to bias Sample chosen on basis of independent variables Ex: Savings, income, age, size for population of

people 35 years and older No bias because E(savings|income, age, size) is same

for any subset of population described by income, age, size in this data.

V. Nonrandom Samples


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When missing data leads to bias If the sample is chosen on the basis of the y

variable, then we have sample selection bias

Ex: estimating wealth based on education, experience, and age. Only those with wealth below 250k included OLS gives biased estimates because E(wealth|

educ, exper, age) not same as expected value conditional on wealth being less than 250k.

V. Outliers /Influential Observations


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Sometimes an individual observation can be very different from the others “Influential” for estimates if dropping that

observation(s) from the analysis changes the key OLS estimates by a lot

Particularly important with small data sets OLS susceptible to outliers because by definition,

minimizes sum of squared residual, and this outlier will have “large” residual.

Causes of outliers errors in data entry – one reason why looking at

summary statistics is important sometimes the observation will just truly be very

different from the others

V. Outliers /Influential Observations Example: R& D

Intensity & Firm Size

Sales more than triples, and now statistically significant.

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1137.R ,1728.R 31,n

(.0445) (.000084) (.592)

arg0478.000186.297.2int

0124.R ,0761.R 32,n

(.0462) (.000044) (.586)

arg0446.000053.625.2int

argint

2_2

^

2_2

^

210

profmsalesensrd

profmsalesensrd

uprofmsalesensrd

V. Outliers


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Not unreasonable to fix observations where it’s clear there was just an extra zero entered or left off, etc.

Not unreasonable to drop observations that appear to be extreme outliers, although readers may prefer to see estimates with and without the outliers

Can use Stata to investigate outliers graphicall

MULTIPLE REGRESSION ANALYSIS: SPECIFICATION AND DATA ISSUES Chapter 9 1.

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