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Heteroskedasticity, Tests for homoskedasticity, Remedies for the heteroskedasticity, Assumption of OLS, Unbiasedness, Eiker White Test, Partitioning the data are points you can learn about Econometric in this lecture.
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As we have studied in the previous lecture, the desirable properties of OLS are conditional on the validity of assumption. Among these assumptions, one assumption is the homoskedasticity. In this chapter we will study
What is homoskedasticity?
Recall from the previous lecture that an assumption of OLS is
The variance of error term should be fixed i.e. ๐ฃ๐๐(๐๐) = ๐ 2 which is independent of the index i. This property is also called homoskedasticity (antonym: heteroskedasticity)
In the previous lecture we have shown two figures to clears the concept of homoskedasticity. However it is not always possible to judge the existence of homoskedasticity in such a straight forward way especially when we have multiple regressors. We need formal testing to investigate the incidence of homoskedasticity in a model.
What happens if errors are not homoskedastic?
We have learned that OLS has some desirable properties and these properties including
a. Unbiasedness b. Efficiency c. Consistency
In absence of homoskedasticity OLS remain unbiased and consistent but it is no longer efficient.
The loss of efficiency means that there exist some other estimators which give more accurate result than OLS and those estimators should get preference.
Testing For Homoskedasticity
There are many tests for the heteroskedasticity. Two of these tests are:
a. Goldfeld Quandt Test b. Eiker White Test (or White Test)
Dr. Asad zaman alongwith his co-authors have proposed more than one tests. One test was jointly proposed by Dr. Asad Zaman and Mumtaz Ahmed which they call as MZ test.
We will not go into detail of these tests because of the computational difficulty involved. We will learn only the two tests mentioned above.
Goldfeld Quandt Test
The procedure of GQ test is as follows:
Suppose you have data ๐ฆ 1 , ๐ฆ 2 , โฆ. ๐ฆ๐ on the dependent variable and ๐ฅ 1 , ๐ฅ 2 , โฆ. ๐ฅ๐ on the independent variable. We have to test null of homoskedasticity against the alternative of heteroskedasticity i.e.
๐ป0: ๐๐^2 = ๐ 2 ๐๐๐ ๐๐๐ ๐
๐ป0: ๐๐^2 โ ๐ 2 ๐๐๐ ๐๐๐ ๐
2
Example:
Consider the following data set, which has data on consumption and income for India
INCOME; GDP per capita constant US$ 2000
CONS: Household Final Consumption per capita constant US$ 2000
Multiple R 0. R Square 0. Adjusted R Square 0. Standard Error 4. Observations 11 Coefficients Standard Error t Stat P-value Intercept 9.395835 22.10532 0.425049 0. X Variable 1 0.745186 0.101469 7.344001 4.36E- 05
In the above output ๐๏ฟฝ 1 and ๐๏ฟฝ 2 are highlighted which are 4.005 and 1.235 respectively. The GQ statistics
is thus: ๐บ๐ = ๏ฟฝ ๐๏ฟฝ๐๏ฟฝ^21 ๏ฟฝ
2 = ๏ฟฝ 41 ..^005235 ๏ฟฝ
2 = 10.
Here n=23, n1=12 thus
Df1=12-2=10 and df2=11-2=
The critical value is found by writing =FINV(.05,9,10) which is 3.
Decision: since the calculated test statistics is greater than 5% critical value H0 is rejected.
Note about partitioning the data
GQ test requires making two parts of data. These parts may be equal or unequal. If there is some natural breakpoint in the economy, than we can take this point as breakpoint, regardless of whether it is in the middle of data or not. But if we cannot identify any natural break point, than we can make two halves of the data as we have done in the above example (part I with 12 and Part 2 with 11 observations, almost equal parts).
1n 1972, due to the oil price shock, the dynamics of most of economies suddenly changed so it can be taken as break point.
Eiker White Test
The procedure of this test is as follows:
Let Y be the dependent variable and X1, X2, โฆ.Xk are the independent variables;
Example
Consider the following data set on consumption, GDP and CPI for Australia
CONS GDP CPI 9.8 16.3 10. 10.0 17.1 10. 10.2 17.2 10. 10.3 17.3 10. 10.2 17.3 11. 10.2 16.9 11. 10.3 17.0 11. 10.4 17.1 11. 10.7 17.6 11. 10.9 18.0 11. 11.0 18.3 11. 11.2 18.7 11. 11.4 19.3 11. 11.5 19.2 11. 12.0 20.3 11. 12.1 20.6 11. 12.3 20.8 11.
Run the following regression ๐ถ๐ก = ๐ผ + ๐ฝ 1 ๐ฆ๐ก + ๐ฝ 2 ๐๐ก + ๐๐ก and test the residual for heteroskedasticity using EW test
Solution
The output of regression is as follows:
SUMMARY OUTPUT
Regression Statistics Multiple R 0. R Square 0. Adjusted R Square 0.
Taking first column as dependent variable, the regression output is
Therefore EW stat is 18*0.226=4.
This EW statistics has Chi-square distribution with df=6 (number of variables in the auxiliary regression)
The critical Value can be found by writing =CHIINV(0.05,6) in MS excel which is 12.
EW statistics is smaller than the 5% critical value therefore the null hypothesis of homoskedasticity cannot be rejected.
We have learned that in presence of heteroskedasticity the OLS is not efficient. Therefore to get efficient estimate of the model, following remedies can be suggested.
Here we will show how the first remedy works
Consider the data on Indian Cons-GDP where we found the problem of heteroskedasticity. We will take log transform of the data the results are as follows:
GDP CONS LogGDP LogCONS 1960 186 194.7 5.2257 5. 1961 192 185 5.2575 5. 1962 195 181.7 5.2730 5. 1963 206 188.2 5.3279 5. 1964 216 193.8 5.3753 5. 1965 232 210.6 5.4467 5. 1966 239 218.2 5.4765 5. 1967 245 232.3 5.5013 5. 1968 255 240.5 5.5413 5. 1969 262 236.7 5.5683 5. 1970 283 283.1 5.6454 5.
2 = 1.
Now the GQ statistics is much smaller than 5% critical value which was 3.02, so the null of heteroskedasticity is not rejected.