How to tackle the heteroscedasticity
? Our ability to tackle the problem will depend upon the
assumptions about error variance. The following situations may emerge.
i)
When
is known
In this situation the CLRM:
can be transformed, dividing each value by
corresponding
thus,
(
)
This effectively transforms error terms to
which is homoscedastic and therefore, the OLS
estimators would be free of heteroscedasticity. The estimates of
and
in this case is called the
Weighted Least Squares (WLS) estimators.
i)
When
is unknown
: We make assumption about the error variance: a) error variance proportional to
the
such as
. Then the transformed regression model is
(
)

Research Methodology (ECO 484): Bachelor of Social Sciences, Level 4 Semester 2
Dr. Mohammad Sadiqunnabi Choudhury
Here, the coefficient on
becomes the constant term. The term
gives homoscedastic variances.
b) when error variance proportional to the
such as
. Then we divide by
√
to get the
transformed regression model
√
√
(
√
)
√
Here, the coefficient on
becomes the constant term. The term
√
will be free of
heteroscedasticity.
The problem of autocorrelation:
The classical regression model also assumes that disturbance terms
s do not have any serial correlation. But, in many situations this assumption may not hold. The
consequences of the presence of serial or auto correlation are similar to those of heteroscedasticity: the
OLS are no longer BLUE.
Symbolically,
no
autocorrelation means
. Autocorrelation can arise in
economic data on account of many factors:
i)
Business cycle: cyclical ups and downs in economic time series continue till something happens
which reverse the situation.
ii)
Misspecification of model: fewer
variables in the model leaving large systematic components to
be clubbed with errors.
iii)
Cobweb phenomenon: certain types of economic time series (especially agricultural output) in
which supply demand interaction either converges to equilibrium or diverges from it.
The consequences of autocorrelation are not different from those of heteroscedasticity listed in the
previous section. Here too OLS estimators are biased or are not BLUE,
&
tests are no longer
reliable. Therefore, computed value of
is not reliable estimate of true goodness of fit.
There are many tests for detecting autocorrelation
–
ranging from visual inspection of error plots, the
Runs Test, Swed-Eisenhart critical runs test. But most commonly used in Durbin-Watson
d
test defined
as:
∑
∑
However, again, we are holding back information on practical detections and avoidance of problem of
autocorrelation for the reasons of limitation of space here.
6.4.3 Maximum likelihood estimations
Let
be n-vector of sample values, dependent on some k-vector of unknown
parameters,
. Let the joint density function be
which indicates the
dependence on
. This density may be interpreted in two ways. For a given
it indicates the probability
of a set of sample outcomes. Alternatively it may be interpreted as a function of
conditional on a set of
sample outcomes. The latter interpretation is referred to as a likelihood function:

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