1. Lecture 19 Preview: Measurement Error and the Instrumental Variables Estimation Procedure
Review: Explanatory Variable/Error Term Correlation and Bias
Introduction to Measurement Error
What Is Measurement Error?
Modeling Measurement Error
The Ordinary Least Squares (OLS) Estimation Procedure and Dependent Variable Measurement Error
The Ordinary Least Squares (OLS) Estimation Procedure and Explanatory Variable Measurement Error
Summary: Explanatory Variable Measurement Error Bias
Explanatory Variable Measurement Error: Attenuation (Dilution) Bias
Might the Ordinary Least Squares (OLS) Estimation Procedure Be Consistent?
Instrumental Variable (IV) Estimation Procedure: A Two Regression Procedure
Mechanics
The Two “Good” Instrument Conditions
Measurement Error Example: Annual, Permanent, and Transitory Income
Definitions and Theory
Might the Ordinary Least Squares (OLS) Estimation Procedure Suffer from a Serious Econometric Problem?
Instrumental Variable (IV) Approach
The Mechanics
Comparison of OLS and IV Approaches
The Two “Good” Instrument Conditions Revisited
Justifying the Instrumental variable (IV) Estimation Procedure

2. Explanatory Variable and Error Term Are Positively Correlated
Review: The Ordinary Least Squares (OLS) Estimation Procedure, Explanatory Variable/Error Term Correlation, and Bias
Esty = bConst + bxx
Esty = bConst + bxx
Explanatory Variable and Error Term Are Positively Correlated
Explanatory Variable and Error Term Are Negatively Correlated
 Estimated Equation More Steeply Sloped Than Actual Equation
 Estimated Equation Less Steeply Sloped Than Actual Equation
 OLS Estimation Procedure for Coefficient Value Is Biased Up
 OLS Estimation Procedure for Coefficient Value Is Biased Down

3. Measurement Error What Is Measurement Error?
Whenever we measure a value we know that we cannot do so perfectly.
Physics Assignment: Measure the amount of time it takes a one pound weight to fall six feet.
Twenty Trials: Use a very accurate stop watch to measure how long it takes the weight to fall six feet.
Question: Will your stop watch report the same amount of time on each trial?
Answer: No. Some times will be lower than other times. Sometimes you will be a little premature in clicking the stop watch button; other times you will be a little late.
It is humanly impossible to measure the actual elapsed time perfectly. No matter how careful you are, sometimes the measured value will be a little low and other times a little high.
How can we model measurement error?
yMeasuredt = yActualt + vt
vt represents measurement error.
vt is a random variable.
We don’t know if the measured value will be low (if vt will be negative) or if the measure value will be high (if vt will be positive) before the trial is conducted.
The mean of vt’s probability distribution equals 0: Mean[vt] = 0.
Question: Why is it important for the mean of vt’s probability distribution to equal 0?
Answer: Since the mean of vt’s probability distribution equals 0, the measured value of yt will not systematically overestimate or underestimate the actual value of yt.

4.  Lab 19.1 Measurement Error: Dependent Variable
yActualt = Const + xActualxActualt + et
et is the model’s error term.
vt is a random variable. vt represents measurement error; the mean of its probability distribution equals 0
yMeasuredt = yActualt + vt
Mean[vt] = 0
yActualt = yMeasuredt – vt
Since the mean of vt’s probability distribution equals 0, the measured value of y will not systematically overestimate or underestimate the actual value.
yActualt = Const xActualxActualt et
yMeasuredt – vt = Const xActualxActualt + et
yMeasuredt = Const xActualxActualt et + vt
yMeasuredt = Const xActualxActualt t where t = et + vt
vt up
Question: Are the explanatory variable, xActualt, and the new error term, t, correlated?
xActualt unaffected
t up
The new error term is not correlated with the explanatory variables.
So, no bias should result.
Type of Actual Mean (Average) Variance of
Measurement YMeas Coef of the Estimated Estimated
Error Var Value Coef Values Coef Values
None 2.0
Dep Vbl
Dep Vbl
Dep Vbl
2.0 1.7
2.0 1.8
2.0 2.0
2.0 2.2
 Lab 19.1
What are the ramifications of dependent variable measurement error?

5. Measurement Error: Explanatory Variable
et is the model’s error term
yActualt = Const + xActualxActualt + et
ut is a random variable. ut represents measurement error; the mean of its probability distribution equals 0
xMeasuredt = xActualt + ut
Since the mean of ut’s probability distribution equals 0, the measured value of the explanatory variable, x, will not systematically overestimate or underestimate the actual value.
xActualt = xMeasuredt – ut
yActualt = Const xActual xActualt et
= Const xActual(xMeasuredt – ut) et
= Const xActual xMeasuredt  xActualut et
= Const xActualxMeasuredt et  xActual ut
= Const xActualxMeasuredt t where t = et – xActual ut
Review: Bias and explanatory variable/error term correlation:
Explanatory variable and error term are positively correlated
Explanatory variable and error term are uncorrelated
Explanatory variable and error term are negatively correlated
 OLS estimation procedure for coefficient value is biased upward
 OLS estimation procedure for coefficient value is unbiased
 OLS estimation procedure for coefficient value is biased downward
Question: Are the explanatory variable, xMeasuredt, and the error term, t, correlated?

6. Model: yActualt = Const + xActualxMeasuredt + t
xMeasuredt = xActualt + ut t = et – xActual ut
If xActual < 0
If xActual = 0
If xActual > 0
 ut up
 ut up
 ut up
xMeasuredt up
t up
xMeasuredt up
t unaffected
xMeasuredt up
t down
 The explanatory variable, xMeasuredt, and error term, t, are positively correlated
 The explanatory variable, xMeasuredt, and error term, t, are uncorrelated
 The explanatory variable, xMeasuredt, and error term, t, are negatively correlated
 OLS estimation procedure for coefficient value is biased upward
 OLS estimation procedure for coefficient value is unbiased
 OLS estimation procedure for coefficient value is biased downward
 Biased toward 0
 Biased toward 0

7. Model: yActualt = Const + xActualxMeasuredt + t where t = et – xActual ut
Summary: Explanatory Variable Measurement Error
xActual < 0
xActual = 0
xActual > 0
 OLS estimation procedure for coefficient value is biased upward, toward 0
 OLS estimation procedure for coefficient value is unbiased
 OLS estimation procedure for coefficient value is biased downward, toward 0
 Lab 19.2
Checking Our Logic: Measurement Error Simulation
Type of Actual Mean (Average)
Measurement XMeas Coef of the Estimated Magnitude
Error Var Value Coef Values of Bias
Exp Vbl
Exp Vbl
Exp Vbl 20.0 1.0
Exp Vbl
Explanatory variable measurement error: When the actual coefficient value is positive, downward bias results.
1.11 .89
.56 .44
.56 .44
.00 .00
When the actual coefficient value is negative, upward bias results.
Interesting Observations: Sign of the mean is correct.
When the actual coefficient value is zero there is no bias.
Magnitude of the mean is too low.

8. Explanatory Variable Measurement Error: Attenuation or Dilution Bias
Attenuation or Dilution Bias: The mean of the coefficient estimate’s probability distribution lies between 0 and the actual value of the coefficient:
The sign of the mean of the probability distribution is correct.
The magnitude of the mean of the probability distribution is too low.
As a consequence of explanatory variable measurement error, the estimation procedure underestimates or “attenuates” or “dilutes” the actual effect of the explanatory variable.
Question: How can we explain attenuation bias intuitively?
xMeasuredt = xActualt + ut
yActualt = Const + xActualxActualt + et
xActual = 2
Question: What is the overall average?
Typically yActualt up by 2
xActualt up by 1 ut unchanged
Typically yActualt up by less than 2
xMeasuredt up by 1
Typically yActualt unaffected
xActualt unchanged ut up by 1

9. Least Squares (OLS) Estimation Procedure, Explanatory Variable Measurement Error, Bias, and Consistency
Question: When explanatory variable measurement error is present, might the ordinary least squares (OLS) estimation procedure for the coefficient value be consistent?
If so, it would be good news.
 Lab 19.3
Estimation XMeas Sample Actual Mean of Magnitude Variance of
Procedure Var Size Coef Coef Ests of Bias Coef Ests
OLS
OLS
OLS
1.11 0.89 0.2
1.11 0.89 0.1
Conclusions: When the error terms and the explanatory variables are correlated, the ordinary least squares (OLS) estimation procedure for the coefficient value:
is biased – bad news.
is not consistent – bad news.
Where Do We Stand?
When explanatory variable measurement error is present the ordinary least squares (OLS) estimation procedure is neither unbiased nor consistent.
Question: What should we do when explanatory variable measurement error is present?
Strategy: Consider a different estimation procedure. Ideally, that procedure would be unbiased, but failing that perhaps it will be consistent.

10. Instrumental Variable Approach – A Correlated Variable and Two Regressions
Instrument – Correlated Variable:
Choose a “Good” Instrument: A “good” instrument must have two properties:
Correlated with the “problem” explanatory variable.
Uncorrelated with the error term.
Regression 1
Dependent Variable: “Problem” explanatory variable;
Explanatory Variable: Instrument, the correlated variable.
Regression 2
Dependent Variable: Original dependent variable
Explanatory Variable: Estimate of the “problem” explanatory variable based on the results from Regression 1.
Claim: While the instrumental variable estimation procedure does not solve the explanatory variable measurement error bias problem, it mitigates the problem. While the instrumental variable estimation procedure is biased, it is consistent when a good instrument is used.
That is, as the sample size becomes larger:
The magnitude of the bias becomes less.
The variance of the coefficient estimate’s probability distribution becomes less.
You may think of this as a “half a loaf is better than none” strategy. Since we cannot devise an unbiased estimation procedure, we are doing the next best thing. We are devising a consistent estimation procedure.
We shall justify our claim by using a simulation. Before doing so however, we shall illustrate the “nuts and bolts” of the instrumental variable estimation procedure with an example.

11. Measurement Error Example: Annual, Permanent, and Transitory Income
Loosely speaking, permanent income equals what the household earns per year “on average;” permanent income equals the average of annual income.
In some years, the household’s annual income is more than its permanent income, but in other years it is less.
Transitory income equals the difference between annual income and permanent income:
Sometimes transitory income, IncTranst, is positive, sometimes it is negative, on average it is 0.
IncTranst = IncAnnt  IncPermt
or equivalently,
IncAnnt = IncPermt IncTranst
Health Insurance Coverage and Permanent Income
Theory: Additional permanent per capita disposable income in a state increases health insurance coverage within the state.
Model: Coveredt = Const + IncPermIncPermPCt + et
Theory: IncPerm > 0
In reality, permanent income and transitory income cannot be observed.
The only annual income information is available to assess the theory.
Model: Coveredt = Const + IncPermIncAnnPCt + t
Theory: IncPerm > 0
Health Insurance Data: Cross section data of health insurance coverage, education, and income statistics from the 50 states and the District of Columbia in 2007.
Coveredt Adults (25 and older) covered by health insurance in state t (percent)
IncAnnPCt Per capita annual disposable income in state t (thousands of dollars)
HSt Adults (25 and older) who completed high school in state t (percent)

12. Ordinary Least Squares (OLS)
Model: Coveredt = Const + IncPermIncAnnPCt + t
Theory: IncPerm > 0
Dependent Variable: Covered
Explanatory Variable: IncAnnPC
 EViews
Ordinary Least Squares (OLS)
Dependent Variable: Covered
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob
IncAnnPC
0.0352
Const
0.0000
Number of Observations 51
Estimated Equation: EstCovered = IncAnnPC
Interpretation: We estimate that a $1,000 increase in annual per capita disposable income increases the state’s health insurance coverage by .227 percentage points.
Critical Result: The IncAnnPC coefficient estimate equals The positive sign of the coefficient estimate suggests that increases in disposable income increase health insurance coverage. This evidence supports the theory.
H0: IncPerm = 0 Disposable income has no effect on health insurance coverage
H1: IncPerm > 0 Additional disposable income increases health insurance coverage
.0352
Question: Can we use the Prob column? =
Prob[Results IF H0 True] = 2
Answer: Yes.
Question: Might an econometric problem be present here?
Answer: Yes, because annual income can be viewed as permanent income with measurement error.

13. IncAnnPCt = IncPermPCt + IncTransPCt
Sometimes transitory income, IncTranst, is positive, sometimes it is negative, on average it is 0.
 Measurement Error
 IncAnnPCt = IncPermPCt ut
Consequently, we can view annual income as permanent income plus measurement error.
Mean[ut] = 0
IncPermPCt = IncAnnPCt  ut
Coveredt = Const IncPermIncPermPCt et Theory: IncPerm > 0
= Const IncPerm(IncAnnt  ut) et
= Const IncPermIncAnnt  IncPermut et
= Const IncPermIncAnnt et  IncPermut
where t = et  IncPermut
= Const IncPermIncAnnt t
ut up
Theory: IncPerm > 0
t down
IncAnnt up 
 Explanatory varaible, IncAnnt, and the error term, t, are negatively correlated
Attenuation bias: Whenever the explanatory variable suffers from measurement error and the actual coefficient value is positive, the OLS estimation procedure for the coefficient value is biased toward 0.
 OLS estimation procedure for the coefficient value is biased downward

14. Ordinary Least Squares (OLS)
Instrumental Variables: An Instrument (a Correlated Variable) and Two Regressions
Instrument – Correlated Variable:
Choose a variable that is correlated with the “problem”
explanatory variable (the explanatory variable suffering from measurement error that creates the ordinary least squares (OLS) bias problem).
Adults completing high school (percent), HS
Regression 1:
Dependent variable: Problem explanatory variable.
IncAnnPC
Explanatory variable: Instrument, the correlated variable. HS
Question: Would you expect education and income to be correlated?
Ordinary Least Squares (OLS)
Dependent Variable: IncAnnPC
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob HS
0.0230
Const 
0.7543
Number of Observations 51
 EViews
Estimated Equation: EstIncAnnPC =  HS

15. Ordinary Least Squares (OLS)
Regression 1:
Estimated Equation: EstIncAnnPC =  HS
Regression 2:
Dependent variable: Original dependent variable
Covered
Explanatory variable: Estimate of the “problem” explanatory variable based on the results from Regression 1.
EstIncAnnPC
Ordinary Least Squares (OLS)
Dependent Variable: Covered
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob
EstIncAnnPC
0.0000
Const
0.0002
Number of Observations 51
 EViews
Estimated Equation: EstCovered = EstIncAnnPC
Interpretation: We estimate that a $1,000 increase in permanent per capita disposable income increases the state’s health coverage by 1.39 percentage points.
Critical Result: The EstIncAnnPC coefficient estimate equals The positive sign of the coefficient estimate suggests that increases in permanent disposable income increase health insurance coverage. This evidence supports the theory.
The Ordinary Least Squares (OLS) and Instrumental Variables (IV) Estimates
IncPerm Estimate SE t-Statistic Prob
Ordinary Least Squares (OLS)
Instrumental Variables (IV) <.0001

16. Ordinary Least Squares (OLS)
Good Instrument Conditions Revisited
Instrument/”Problem” Explanatory Variable Correlation: The instrument, HSt, must be correlated with the “problem” explanatory variable, IncAnnPCt .
Instrumental Variables (IV) Regression 1
Ordinary Least Squares (OLS)
Dependent Variable: IncAnnPC
Explanatory Variable(s):
Estimate SE
t-Statistic
Prob HS
0.0230
Const 
0.7543
Number of Observations 51
EstIncAnnPC =  HS
The estimate, EstIncAnnPCt , will be a “good” surrogate only if the instrument, HSt, is correlated with the “problem” explanatory variable, IncAnnPCt; that is, only if the estimate is a good predictor of the “problem” explanatory variable.
The sign of the HSt coefficient is positive supporting our view that annual income and high school education are positively correlated.
Furthermore, the coefficient is significant at the 5 percent level and nearly significant at the 1 percent level.
Consequently, it is not unreasonable to judge that the instrument meets the first condition.

17. Instrument/Error Term Independence: The instrument, HSt, and the error term, t, must be independent.
Coveredt = Const + IncPermEstAnnIncPC + t
Question: Are EstIncPCt and t independent?
t = et  IncPermut
EstIncAnnPCt =  HSt
Answer: Only if HSt and t are independent.
The explanatory variable/error term independence premise will be satisfied only if the instrument, HSt, and the error term, t, are independent.
Question: Is there a good reason to believe that they are correlated? No
Unfortunately, there is no way to confirm this empirically with our data.
NB: This can be the “Achilles heel” of the instrumental variable (IV) estimation procedure, however. Finding a good instrument can be very tricky.

18. Justifying the Instrumental Variable (IV) Estimation Procedure
Claim: When explanatory variable measurement error is present the instrumental variables estimation procedure for the coefficient value is biased but consistent.
 Lab 19.4
Estimation XMeas Sample Actual Mean of Magnitude Variance of
Procedure Var Size Coef Coef Ests of Bias Coef Ests IV IV IV
2.24 0.24 8.8
2.21 0.21 5.4
2.17 0.17 3.4
Conclusions: When measurement error is present, the instrumental variable estimation procedure for the coefficient value
is biased – bad news;
is consistent – good news.
Question: What have we learned about explanatory variable measurement error when the coefficient is nonzero?
Answer: Both the ordinary least squares (OLS) estimation procedure and the instrumental variables (IV) estimation procedure for the coefficient value are biased.
The ordinary least squares (OLS) estimation procedure is also not consistent.
The instrumental variables (IV) estimation procedure is consistent when the good instrument conditions are satisfied.