Chapter 16
Measurement Error Models
A fundamental assumption in all the statistical analysis is that all the observations are correctly measured. In
the context of multiple regression model, it is assumed that the observations on the study and explanatory
variables are observed without any error. In many situations, this basic assumption is violated. There can be
several reasons for such a violation.
For example, the variables may not be measurable, e.g., taste, climatic conditions, intelligence,
education, ability etc. In such cases, the dummy variables are used, and the observations can be
recorded in terms of values of dummy variables.
Sometimes the variables are clearly defined, but it is hard to take correct observations. For example,
the age is generally reported in complete years or in multiple of five.
Sometimes the variable is conceptually well defined, but it is not possible to take a correct
observation on it. Instead, the observations are obtained on closely related proxy variables, e.g., the
level of education is measured by the number of years of schooling.
Sometimes the variable is well understood, but it is qualitative in nature. For example, intelligence is
measured by intelligence quotient (IQ) scores.
In all such cases, the true value of the variable can not be recorded. Instead, it is observed with some error.
The difference between the observed and true values of the variable is called as measurement error or
errors-in-variables.
Difference between disturbances and measurement errors:
The disturbances in the linear regression model arise due to factors like the unpredictable element of
randomness, lack of deterministic relationship, measurement error in study variable etc. The disturbance
term is generally thought of as representing the influence of various explanatory variables that have not
actually been included in the relation. The measurement errors arise due to the use of an imperfect measure
of true variables.
Econometrics | Chapter 16 | Measurement Error Models | Shalabh, IIT Kanpur
1
,Large and small measurement errors
If the magnitude of measurement errors is small, then they can be assumed to be merged in the disturbance
term, and they will not affect the statistical inferences much. On the other hand, if they are large in
magnitude, then they will lead to incorrect and invalid statistical inferences. For example, in the context of
linear regression model, the ordinary least squares estimator (OLSE) is the best linear unbiased estimator of
the regression coefficient when measurement errors are absent. When the measurement errors are present in
the data, the same OLSE becomes biased as well as inconsistent estimator of regression coefficients.
Consequences of measurement errors:
We first describe the measurement error model. Let the true relationship between correctly observed study
and explanatory variables be
y X
where y is a n 1 vector of true observation on study variable, X is a n k matrix of true observations
on explanatory variables and is a k 1 vector of regression coefficients. The value y and X are not
observable due to the presence of measurement errors. Instead, the values of y and X are observed with
additive measurement errors as
y y u
X X V
where y is a n 1 vector of observed values of study variables which are observed with (n 1)
measurement error vector u . Similarly, X is a (n k ) matrix of observed values of explanatory variables
which are observed with n k matrix V of measurement errors in X . In such a case, the usual disturbance
term can be assumed to be subsumed in u without loss of generality. Since our aim is to see the impact of
measurement errors, so it is not considered separately in the present case.
Alternatively, the same setup can be expressed as
y X u
X X V
where it can be assumed that only X is measured with measurement errors V and u can be considered as the
usual disturbance term in the model.
Econometrics | Chapter 16 | Measurement Error Models | Shalabh, IIT Kanpur
2
, In case, some of the explanatory variables are measured without any measurement error then the
corresponding values in V will be set to zero.
We assume that
E (u ) 0, E (uu ') 2 I
E (V ) 0, E (V 'V ) , E (V ' u ) 0.
The following set of equations describes the measurement error model
y X
y y u
X X V
which can be re-expressed as
y y u
X u
X V u
X u V
=X
where u V is called as the composite disturbance term. This model resemble like a usual linear
regression model. A basic assumption in linear regression model is that the explanatory variables and
disturbances are uncorrelated. Let us verify this assumption in the model y X w as follows:
E X E ( X ) ' E ( ) E V '(u V )
E V ' u E V 'V
0
0.
Thus X and are correlated. So OLS will not provide efficient result.
Suppose we ignore the measurement errors and obtain the OLSE. Note that ignoring the measurement errors
in the data does not mean that they are not present. We now observe the properties of such an OLSE under
the setup of measurement error model.
Econometrics | Chapter 16 | Measurement Error Models | Shalabh, IIT Kanpur
3
Measurement Error Models
A fundamental assumption in all the statistical analysis is that all the observations are correctly measured. In
the context of multiple regression model, it is assumed that the observations on the study and explanatory
variables are observed without any error. In many situations, this basic assumption is violated. There can be
several reasons for such a violation.
For example, the variables may not be measurable, e.g., taste, climatic conditions, intelligence,
education, ability etc. In such cases, the dummy variables are used, and the observations can be
recorded in terms of values of dummy variables.
Sometimes the variables are clearly defined, but it is hard to take correct observations. For example,
the age is generally reported in complete years or in multiple of five.
Sometimes the variable is conceptually well defined, but it is not possible to take a correct
observation on it. Instead, the observations are obtained on closely related proxy variables, e.g., the
level of education is measured by the number of years of schooling.
Sometimes the variable is well understood, but it is qualitative in nature. For example, intelligence is
measured by intelligence quotient (IQ) scores.
In all such cases, the true value of the variable can not be recorded. Instead, it is observed with some error.
The difference between the observed and true values of the variable is called as measurement error or
errors-in-variables.
Difference between disturbances and measurement errors:
The disturbances in the linear regression model arise due to factors like the unpredictable element of
randomness, lack of deterministic relationship, measurement error in study variable etc. The disturbance
term is generally thought of as representing the influence of various explanatory variables that have not
actually been included in the relation. The measurement errors arise due to the use of an imperfect measure
of true variables.
Econometrics | Chapter 16 | Measurement Error Models | Shalabh, IIT Kanpur
1
,Large and small measurement errors
If the magnitude of measurement errors is small, then they can be assumed to be merged in the disturbance
term, and they will not affect the statistical inferences much. On the other hand, if they are large in
magnitude, then they will lead to incorrect and invalid statistical inferences. For example, in the context of
linear regression model, the ordinary least squares estimator (OLSE) is the best linear unbiased estimator of
the regression coefficient when measurement errors are absent. When the measurement errors are present in
the data, the same OLSE becomes biased as well as inconsistent estimator of regression coefficients.
Consequences of measurement errors:
We first describe the measurement error model. Let the true relationship between correctly observed study
and explanatory variables be
y X
where y is a n 1 vector of true observation on study variable, X is a n k matrix of true observations
on explanatory variables and is a k 1 vector of regression coefficients. The value y and X are not
observable due to the presence of measurement errors. Instead, the values of y and X are observed with
additive measurement errors as
y y u
X X V
where y is a n 1 vector of observed values of study variables which are observed with (n 1)
measurement error vector u . Similarly, X is a (n k ) matrix of observed values of explanatory variables
which are observed with n k matrix V of measurement errors in X . In such a case, the usual disturbance
term can be assumed to be subsumed in u without loss of generality. Since our aim is to see the impact of
measurement errors, so it is not considered separately in the present case.
Alternatively, the same setup can be expressed as
y X u
X X V
where it can be assumed that only X is measured with measurement errors V and u can be considered as the
usual disturbance term in the model.
Econometrics | Chapter 16 | Measurement Error Models | Shalabh, IIT Kanpur
2
, In case, some of the explanatory variables are measured without any measurement error then the
corresponding values in V will be set to zero.
We assume that
E (u ) 0, E (uu ') 2 I
E (V ) 0, E (V 'V ) , E (V ' u ) 0.
The following set of equations describes the measurement error model
y X
y y u
X X V
which can be re-expressed as
y y u
X u
X V u
X u V
=X
where u V is called as the composite disturbance term. This model resemble like a usual linear
regression model. A basic assumption in linear regression model is that the explanatory variables and
disturbances are uncorrelated. Let us verify this assumption in the model y X w as follows:
E X E ( X ) ' E ( ) E V '(u V )
E V ' u E V 'V
0
0.
Thus X and are correlated. So OLS will not provide efficient result.
Suppose we ignore the measurement errors and obtain the OLSE. Note that ignoring the measurement errors
in the data does not mean that they are not present. We now observe the properties of such an OLSE under
the setup of measurement error model.
Econometrics | Chapter 16 | Measurement Error Models | Shalabh, IIT Kanpur
3
