Loading...

PPT – Research Method PowerPoint presentation | free to view - id: 7ae2c7-MjU2N

The Adobe Flash plugin is needed to view this content

Research Method

- Lecture 9 (Ch9)
- More on specification and Data issues

Using Proxy Variables for Unobserved Explanatory

Variables

- Suppose you are interested in estimating the

return to Education. So you consider the

following model. - Log(Wage)ß0ß1Educ ß2Exp (ß3Abilityu) (1)
- Ability is unobserved, so it is included in the

composite error term. If Ability is correlated

with the year of education, ß1 will be biased. - Question if ability is correlated with Educ,

what is the direction of the bias?

- One way to eliminate the bias is to use a Panel

data then apply the fixed effect or the first

differencing method. - Another method is to use a proxy variable for

ability. This is the topic of this section. - Suppose that IQ is a proxy variable for ability,

and that IQ is available in your data.

- Then, the basic idea is to estimate the

following. - Regress Log(Wage) on Educ, Exp, and IQ (2)
- This is called the plug-in solution to the

omitted variables problem. - The question is under what conditions (2)

produces consistent estimates for the original

regression (1). I will explain these conditions

using the above example (though the arguments can

be easily generalized). - It turns out, the following two conditions ensure

that you get consistent estimates by using the

plug-in solution.

- Condition 1 u is uncorrelated with IQ. In

addition, the original equation should satisfy

the usual conditions (i.e, u is also uncorrelated

with Educ, Exp, and Ability). - Condition 2 E(AbilityEduc, Exp,

IQ)E(AbilityIQ) - Condition 2 means that, once IQ is conditioned,

Educ and Exp does not explain Ability. More

simple way to express condition 2 is that the

ability can be written as - Abilityd0d3IQv3

(3) - where, v3 is a random error which is uncorrelated

with either IQ, Educ or Exp. What it means is

that Ability is a function of IQ only.

Omitted variable

The initial explanatory variables

The proxy variable

- Then, it is clear why these two conditions

guarantee that the plug-in condition produces

consistent estimates. Just plug (3) into (1).

Then you have - Log(Wage)(ß0d0)ß1Educ ß2Exp ß3d3IQ

(uß3v3 ) (4) - Where
- Since u and v3 are uncorrelated with any of the

explanatory variables under condition1 and

condition 2, the slope parameters are consistent.

The intercept has changed, but usually you are

not interested in the intercept. Importantly, you

get consistent estimates for the slope

parameters.

- It is also obvious that, if condition 2 is

violated, then the plug in solution will not

work. If the condition 2 is violated, then

ability will be a function of not only IQ, but

also Educ and Exp. So you will have - Abilityd0 d1Educd2Expd3IQv3 (5)
- If you plug (5) into (1), you have
- Log(Wage)(ß0d0)(ß1ß3d1)Educ (ß2ß3d2)Exp

ß3d3IQ (uß3v3 ) (4) - Thus, the coefficient for Educ is no longer ß1,

but it is ß1ß3d1. Thus, the plug-in solution

produces inconsistent estimates when condition 2

is violated.

If condition 2 is violated then, ability is a

function of all the variables.

Exercise

- Ex.1 Use Wage2.dta to estimate a log wage

equation to examine the return to education.

Include in the equation exper, tenure, married,

south, urban, black. Do you think that the return

to education is unbiased? What do you think is

the direction of the bias - Ex.2 Now, use IQ as a proxy for unobserved

ability. Did the result change? Was your

prediction of the direction of the bias correct?

Answer OLS without IQ

Answer OLS with IQ

Using lagged dependent variable as proxy variables

- Often the lag of the dependent variable is used

as a proxy for the unobserved variables. - First consider the following model.
- (Crime rate) ß0ß1(unemp)

ß2(expenditure) u - If there are omitted factors that directly affect

crime rate and at the same time correlated with

unemployment rate, ß1 will be biased. The omitted

factors may be some pre-existing conditions, like

demographic features (age, race etc). Crime rate

could be different among cities for historical

factors.

- The idea is that, the lag of the dependent

variable may summarize such pre-existing

conditions. - So, estimate the following equation
- (Crime rate)it ß0ß1(unemp)it

ß2(expenditure)it - ß3(Crime rate)it-1

uit - The following slides estimate the model using

CRIME2.dta

Example

- We estimate Crime2.dta to estimate the

regressions. Results are the following.

Without the lag of dependent varriable.

With the lag of dependent variable.

Measurement error

- The existence of important omitted variables

causes endogeneity problem. - Another source of endogeneity is the measurement

error. - This section explains under what circumstance

the measurement error causes endogeneity, and

under what circumstance it does not.

Measurement error in explanatory variable.

- When the explanatory variables are measured with

errors, this causes the endogeneity problem. - This is a common problem. For example, in a

typical survey, the respondents may report their

annual incomes with a lot of errors. Variables

such as GPA or IQ may be reported with errors as

well.

- Now, let us understand the nature of the problem.
- Suppose that you want to estimate the following

simple regression. - yß0ß1x1 u .(1)
- where x1 is the measurement-error free variable.

Suppose that this regression satisfies MLR.1

through MLR.4. - Now, suppose that you only observe the

error-ridden variable x1. That is - x1x1e1
- where e1 is a random error uncorrelated with x1.

- To be more precise, the measurement error is such

that - x1x1e1 .(2)
- and
- Cov(x1, e1)0 .(3)
- (2) and (3) is called the classical

errors-in-variables (CEV) assumption. - Note that the above assumption has nothing to do

with u. We maintain the assumption that u is

uncorrelated with both x1 and x1. This also

means that u is uncorrelated with e1.

- Because we only observe the error-ridden variable

x1, we can only estimate the following model. - yß0ß1x1v.(4)
- Under the CEV assumption, the observed

(error-ridden) variable in regression (4) is

endogenous. - To see this, plug x1x1-e1 into the original

regression (1) to get - yß0ß1x1(u- ß1e1).(5)

- So, we have vu- ß1e1
- Now, notice that
- Cov(x1, v)Cov(x1, u- ß1e1) ?0
- See the front board for the proof.
- Therefore, x1 is correlated with the error term.

Therefore, x1 is endogenous. Thus, OLS will be

biased.

- Under the CEV assumption, we can predict the

direction of the inconsistency (characterization

of the bias is difficult). Let be the

estimated coefficient from the error-ridden

variable regression (4). Then, we have

- Proof see the front board
- Since the term inside the parenthesis is always

smaller than 1, there is a bias towards zero.

This is called the attenuation bias.

Error in variable (more general case)

- Suppose you want to estimate the following model.
- yß0ß1x1ß2x2.ßkxku
- where x1 is measurement free variable.
- However, you only observe error-ridden variable

x1. So you can only estimate the above regression

by replacing x1 with x1.

- Assume that other variables are measurement error

free. - Then the probability limit of is given by

where is the population error from the

following regression. x1d0d1x2

dk-1xk r1

Measurement error in the dependent variable

- When the measurement error is in the dependent

variable, but explanatory variables have no

measurement-errors, there will be no bias in OLS. - Consider the following model.
- yß0ß1x1 u .(1)
- where y is the measurement free variable.
- But, you only observe the error-ridden variable y.

- Assume the following
- yye ..(2)
- and
- Cov(y, e)0 ...(3)
- Again, we maintain the assumption that u is

uncorrelated with both x1 and x1. This also

means that u is uncorrelated with e1. - By plugging yy-e into (1), we have the

following OLS. - yß0ß1x1 (ue) (5)
- Since e and u are not correlated with the

explanatory variables, (5) causes no bias in the

estimation.

Non random sampling 1 Exogenous sampling

- Consider the following regression
- Savingß0ß1(income)ß2(age)u
- Suppose that the survey is conducted for people

over 35 years old. This is non-random sampling,

but the sampling criteria is based on the

independent variable. This is called the sample

selection based on the independent variables, and

is an example of exogenous sample selection. - In this case, OLS regression of the above model

has no bias.

Non random sampling 2 Enogenous sampling

- Consider the following regression.
- Wealthß0ß1(Educ)ß2(Exper)u
- However, suppose that only people with wealth

below 250,000 are included in the sample. Then

the sample selection criteria is based on the

dependent variable. This is called the sample

selection based on dependent variable, and is an

example of endogenous sample selection. - In this case, OLS estimate of the above

regression are always biased.

Stratified sampling

- This is a common survey method, in which the

population is divided into non-overlapping

groups, or strata. The sampling is random within

each group. - However, some groups are often oversampled in

order to increase observations for that group.

Whether this causes the bias depends on whether

the selection is exogenous or endogenous.

- If females are oversampled, and you are

interested in the gender differences in savings,

then this is the exogenous sample selection.

Thus, this causes no bias. - If people with low wealth are oversampled, and if

you are interested in the wealth regression, then

this is endogenous sample selection. This causes

a bias in the regression.

More subtle form of sample selection.

- Suppose that you are interested in estimating the

wage offer regression. - Low(wage offer) ß0ß1(Educ)ß2(Exper)u
- When the wage offer is too low for a particular

person, the person may decide not to work. Thus,

this person will not be included in the sample.

This is the case where sample selection is caused

by the persons decision to work or not.

- When the decision is based on unobserved factors,

then the OLS regression causes a bias. This is

called the sample selection bias. - This is typically a problem for the study of the

wage offer for women. - This course does not cover the method to correct

for this type of bias. In the fall semester, I

will cover this type of issues in a new course

the Cross Section and Panel Data Analysis.