Chapter 8
Heteroskedasticity
In the multiple regression model
y X ,
it is assumed that
V ( ) 2 I ,
i.e.,
Var ( i2 ) 2 ,
Cov( i j ) 0, i j 1, 2,..., n.
In this case, the diagonal elements of the covariance matrix of are the same indicating that the variance of
each i is same and off-diagonal elements of the covariance matrix of are zero indicating that all
disturbances are pairwise uncorrelated. This property of constancy of variance is termed as homoskedasticity
and disturbances are called as homoskedastic disturbances.
In many situations, this assumption may not be plausible, and the variances may not remain the same. The
disturbances whose variances are not constant across the observations are called heteroskedastic disturbance,
and this property is termed as heteroskedasticity. In this case
Var ( i ) i2 , i 1, 2,..., n
and disturbances are pairwise uncorrelated.
The covariance matrix of disturbances is
12 0 0
0 22 0
V ( ) diag ( 1 , 2 ,..., n )
2 2 2 .
0 0 n2
Econometrics | Chapter 8 | Heteroskedasticity | Shalabh, IIT Kanpur
1
, Graphically, the following pictures depict homoskedasticity and heteroskedasticity.
Homoskedasticity
Heteroskedasticity (Var(y) increases with x) Heteroskedasticity (Var(y) decreases with x)
Examples: Suppose in a simple linear regression model, x denote the income and y denotes the expenditure
on food. It is observed that as the income increases, the expenditure on food increases because of the choice
and varieties in food increase, in general, up to a certain extent. So the variance of observations on y will not
remain constant as income changes. The assumption of homoscedasticity implies that the consumption pattern
of food will remain the same irrespective of the income of the person. This may not generally be a correct
assumption in real situations. Instead, the consumption pattern changes and hence the variance of y and so the
variances of disturbances will not remain constant. In general, it and will be increasing as income increases.
Econometrics | Chapter 8 | Heteroskedasticity | Shalabh, IIT Kanpur
2
Heteroskedasticity
In the multiple regression model
y X ,
it is assumed that
V ( ) 2 I ,
i.e.,
Var ( i2 ) 2 ,
Cov( i j ) 0, i j 1, 2,..., n.
In this case, the diagonal elements of the covariance matrix of are the same indicating that the variance of
each i is same and off-diagonal elements of the covariance matrix of are zero indicating that all
disturbances are pairwise uncorrelated. This property of constancy of variance is termed as homoskedasticity
and disturbances are called as homoskedastic disturbances.
In many situations, this assumption may not be plausible, and the variances may not remain the same. The
disturbances whose variances are not constant across the observations are called heteroskedastic disturbance,
and this property is termed as heteroskedasticity. In this case
Var ( i ) i2 , i 1, 2,..., n
and disturbances are pairwise uncorrelated.
The covariance matrix of disturbances is
12 0 0
0 22 0
V ( ) diag ( 1 , 2 ,..., n )
2 2 2 .
0 0 n2
Econometrics | Chapter 8 | Heteroskedasticity | Shalabh, IIT Kanpur
1
, Graphically, the following pictures depict homoskedasticity and heteroskedasticity.
Homoskedasticity
Heteroskedasticity (Var(y) increases with x) Heteroskedasticity (Var(y) decreases with x)
Examples: Suppose in a simple linear regression model, x denote the income and y denotes the expenditure
on food. It is observed that as the income increases, the expenditure on food increases because of the choice
and varieties in food increase, in general, up to a certain extent. So the variance of observations on y will not
remain constant as income changes. The assumption of homoscedasticity implies that the consumption pattern
of food will remain the same irrespective of the income of the person. This may not generally be a correct
assumption in real situations. Instead, the consumption pattern changes and hence the variance of y and so the
variances of disturbances will not remain constant. In general, it and will be increasing as income increases.
Econometrics | Chapter 8 | Heteroskedasticity | Shalabh, IIT Kanpur
2
