Three very important topics of econometrics.Multicollinearity,Heteroscedasticity & Autocorrelation

1. Department Of Statistics
Begum Rokeya University, Rangpur
ERROR MINIMIZERS
Remedial Measures
Multicollinearity
Heteroscedasticity
Autocorrelation

2. Multicollinearity
Heteroscedasticity
Autocorrelation
Definition
 Linear relationship between two or more
independent variables.
 Makes overfitting problems.
 Hard to interpret of a model.
 Reduces the precision of the estimated
coefficients.
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3. Multicollinearity
Heteroscedasticity
Autocorrelation
Practical Examples
Physical Activity & Body Fat
1.
Age & Sales Price of A Car
2.
Quality of Product And Network
3.
The Population & Total Gross Income
4.
Years of Education & Annual Income
5.
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4. Multicollinearity
Heteroscedasticity
Autocorrelation
Definition
 Variance of the residuals is unequal over a
range of measured values.
 It arises when 𝐄 𝒖𝒊
𝟐
= 𝝈𝒊
𝟐
 In the presents of outlier it arises.
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5. Multicollinearity
Heteroscedasticity
Autocorrelation
Practical Examples
Food expenditures & income
2.
Explaining student’s test scores & study time
4.
Hours of typing practice & errors per page.
5.
3. Number of populations & flower shops in a city
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Distance of a rocket over time
1.

6. Multicollinearity
Heteroscedasticity
Autocorrelation
Definition
 Relationship between a variable's current
value and its past values.
 Correlation between members of a series
of observations ordered in time or space
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 It arises when,𝑬 𝒖𝒊𝒖𝒋 ≠ 𝟎

7. Multicollinearity
Heteroscedasticity
Autocorrelation
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Practical Examples
Daily stock returns involving regression analysis
using time series data
2.
Temperatures on different days in a month
1.
Similar perform from same class students more
than different classes students
4.
Similar answer from nearby geographic locations
& geographically distant people
3.
Expenditure on households is influenced by the
expenditure of the preceding month
5.

8. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
Detection Methods
High 𝑹𝟐
but few
significan
t ratios
High pair-wise
correlations
among
regressors
Eigen
values &
condition
index
Tolerance
& variance
inflation
factor
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Begum Rokeya University, Rangpur 08
Multicollinearity
Heteroscedasticity
Autocorrelation

9. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
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High 𝑹^𝟐 But Few
Significant T Ratios
 𝑹𝟐
≈ 1, it indicates the variable can be
explained by other predictor variables.
 But t tests show that none or very few
of the predictors are statistically
different from zero.

10. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
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High pair-wise correlations
among regressors
 If the pair-wise correlation between two independent
variable is high (greater than 0.8).
 High zero order correlations may suggest collinearity,
it is not necessary that they be high to have
collinearity in any specific case.

11. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
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Value Eigen and
Condition Index
 Condition number, 𝐤 =
𝑴𝒂𝒙𝒊𝒎𝒖𝒎 𝒆𝒊𝒈𝒆𝒏𝒗𝒂𝒍𝒖𝒆
𝑴𝒊𝒏𝒊𝒎𝒖𝒎 𝒆𝒊𝒈𝒆𝒏𝒗𝒂𝒍𝒖𝒆
 Condition index, CI=
𝑴𝒂𝒙𝒊𝒎𝒖𝒎 𝒆𝒊𝒈𝒆𝒏𝒗𝒂𝒍𝒖𝒆
𝑴𝒊𝒏𝒊𝒎𝒖𝒎 𝒆𝒊𝒈𝒆𝒏𝒗𝒂𝒍𝒖𝒆
= 𝒌
100 1000 ∞
0
0
No
Multicollinearity
Moderate to Strong
Multicollinearity
Severe
Multicollinearity
10 30 ∞
0
0
No
Multicollinearity
Moderate to Strong
Multicollinearity
Severe
Multicollinearity

12. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
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Tolerance & Variance
Inflation Factor (VIF)
 V𝐈𝐅 =
𝟏
𝟏− 𝑹𝟐
 Tolerance, TOL =
𝟏
𝑽𝑰𝑭
5 10 ∞
0
0
No
Multicollinearity
Moderate to Strong
Multicollinearity
Severe
Multicollinearity
0.1 0.2 ∞
0
0
Severe
Multicollinearity
Moderate to Strong
Multicollinearity
No Multicollinearity

13. Park’s
Test
Glejser
Test Spearman's
Rank
Correlation
Test
Goldfeld –
Quandt
Test
Detection Methods
Breusch –
Pagan –
Godfrey
Test
Multicollinearity
Heteroscedasticity
Autocorrelation
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14. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
Park’s Test
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Take the natural log of squared residuals
Take the natural log of Independent variable
suspecting heteroscedastic behavior.
Run OLS for the natural log of regressor against the
natural log of the squared residuals
Run OLS on our data & find squared residuals from it
If the model is insignificant ,there is no
heteroscedasticity in the error variance

15. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
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Glejser Test
15
Run OLS from our data set & find the residuals
Take absolute value of the residuals
Run OLS for regressor against residuals used:
|𝒓𝒊| = 𝜷𝟏 + 𝜷𝟐𝒙𝒊 + 𝒗𝒊 |𝒓𝒊| = 𝜷𝟏 + 𝜷𝟐 𝒙𝒊 + 𝒗𝒊
|𝒓𝒊| = 𝜷𝟏 + 𝜷𝟐
𝟏
𝒙𝒊
+ 𝒗𝒊 |𝒓𝒊| = 𝜷𝟏 + 𝜷𝟐
𝟏
𝒙𝒊
+ 𝒗𝒊
If the model is insignificant ,there is no
heteroscedasticity in the error variance

16. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
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Spearman's Rank Correlation Test
16
Fit the regression model on Y & X and obtain the
residuals 𝒖𝒊
Rank the absolute value of residuals & independent
variable
Find the spearman’s rank correlation coefficient,
𝒓𝒔 = 𝟏 − 𝟔
𝒅𝒊
𝟐
𝒏 𝒏𝟐 − 𝟏
t statistic where, 𝒕 =
𝒓𝒔 𝒏−𝟐
𝟏−𝒓𝒔
𝟐
with n-2 df
If 𝒕𝒄𝒂𝒍 > 𝒕𝒕𝒂𝒃 ,we may reject null hypothesis
and there will be heteroscedasticity in the
error variance

17. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
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Goldfeld – Quandt Test
𝑭 =
𝑹𝑺𝑺𝟐/𝒅𝒇
𝑹𝑺𝑺𝟏/𝒅𝒇
[ df 𝑹𝑺𝑺𝟏 =(n-c)/2 – k & df 𝑹𝑺𝑺𝟐 =(n-c-2k)/2
Rank the observations ascending order according to 𝑿𝒊
Omit c central observations and devide the remaining
(n-c) observations into two groups.
[If n ≈ 30 then c=4, n ≈ 60 then c=8 ]
Fit OLS for the two group in step-2 & find residuals sum
of square 𝑹𝑺𝑺𝟏 & 𝑹𝑺𝑺𝟐
Fit OLS for the two group in step-2 & find residuals sum
of square 𝑹𝑺𝑺𝟏 & 𝑹𝑺𝑺𝟐
If 𝑭𝒄𝒂𝒍 > 𝑭𝒕𝒂𝒃 ,we may reject null
hypothesis and there will be
heteroscedasticity in the error variance.

18. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
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Breusch – Pagan – Godfrey Test
Obtain ESS from constructed model and find Θ = ESS/2
where Θ~ 𝓧𝒎−𝟏
𝟐
&
ESS = total sum of squares – residual sum of squares
Fit a regression model 𝒀𝒊 = 𝜷𝟏 + 𝜷𝟐𝒙𝟐𝒊 + ⋯ + 𝒖𝒊 &
obtain residuals 𝒖𝒊
Obtain 𝝈𝟐 =
𝒖𝒊
𝟐
𝒏
& Construct variables 𝒑𝒊 =
𝒖𝒊
𝟐
𝝈𝟐
Construct the model, 𝒑𝒊 = 𝜶 + 𝜶𝟐𝒙𝟐𝒊 + 𝜶𝟑𝒙𝟑𝒊 + ⋯ + 𝒖𝒊
If𝓧𝒄𝒂𝒍
𝟐
> 𝓧𝒕𝒂𝒃
𝟐
,we may reject null hypothesis
and there will be heteroscedasticity in the
error variance.

19. Detection Methods
Durbin
Watson
Test
Run
Test
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Multicollinearity
Heteroscedasticity
Autocorrelation

20. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
Durbin Watson test
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Fit a model, 𝒀𝒕 = 𝜷𝟏 + 𝜷𝟐𝑿𝟐𝒕 + 𝜷𝟑𝑿𝟑𝒕+ 𝒖𝒕
Find the residuals 𝒖𝒊 and take a lag 𝒖𝒕−𝟏
Calculate, 𝒅 = 𝒕=𝟐
𝒕=𝒏
(𝒖𝒕−𝒖𝒕−𝟏)𝟐
𝒕=𝟏
𝒕=𝒏 𝒖𝒕
𝟐 ; d lies between 0 to 4
For the given sample size & number of regressor find
out the critical 𝒅𝑳 & 𝒅𝒖

21. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
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Decision Rules
Durbin Watson Test
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Reject 𝑯𝟎:
Positive
autocorrelation
Inconclusive
Do not Reject
𝑯𝟎:No evidence
autocorrelation
Inconclusive
Reject
𝑯𝟎:Negative
autocorrelation
𝒅𝑳 𝒅𝒖 2 4-𝒅𝑳
4-𝒅𝒖
4
0 𝒅𝑳 𝒅𝒖
0

22. Find Prob [E (R) – 1.96𝝈𝑹 ≤ 𝑹 ≤ E (R) + 1.96𝝈𝑹] = 0.95
Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
Run test
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Mean: 𝑬 𝑹 =
𝟐𝑵𝟏𝑵𝟐
𝑵
+ 𝟏
Variance: 𝝈𝑹
𝟐
=
𝟐𝑵𝟏𝑵𝟐(𝟐𝑵𝟏𝑵𝟐 − 𝑵)
𝑵𝟐(𝑵 −𝟏)
Here, 𝑁1 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 " + “, and 𝑁2 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 " − “
𝑁 = 𝑁1 + 𝑁2 and R = Numbers of Runs
Note the signs (+ or -) of the residuals
Count the number of runs & define the length of a run
Fit a regression model and find the residuals

23. Detection of
Multicollinearity
Detection of
Heteroscedasticity
Detection of
Autocorrelation
𝑯𝟎: No autocorrelation
𝑯𝟏: There is autocorrelation
Decision Rule: Run Test
Decision Rules
Run Test
Accept 𝑯𝟎 if R lies in the confidence interval
otherwise reject
Positive autocorrelation when R will be few
Negative autocorrelation when r will be many
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24. Multicollinearity
Heteroscedasticity
Autocorrelation
Remedial Measures
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A priori information
1
Combining cross sectional & time series data
2
Dropping a variable(s) and specification bias
3
Transformation of variables
4
Additional or new data
5

25. Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
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A Priori Information
 Consider a regression model:
 Supose a priori believe that β𝟑 = 𝟎. 𝟏𝟎𝛃𝟐
𝒀𝒊 = 𝜷𝟏 + 𝜷𝟐𝑿𝟐𝒊 + 𝜷𝟑𝑿𝟑𝒊+ 𝒖𝒊
 Run the model:
𝒀𝒊 = 𝜷𝟏 + 𝜷𝟐𝑿𝟐𝒊 + 𝟎. 𝟏𝟎 𝜷𝟐𝑿𝟑𝒊+ 𝒖𝒊
= 𝜷𝟏 + 𝜷𝟐𝑿𝟐𝒊+ 𝒖𝒊
Where 𝑿𝒊 = 𝑿𝟐𝒊 + 𝟎. 𝟏𝟎 𝑿𝟑𝒊

26. Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
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Combining Cross-sectional
And Time Series Data
 Fit a model having cross section & time
series data:
𝒍𝒏𝒀𝒕 = 𝜷𝟏 + 𝜷𝟐𝒍𝒏𝑷𝒕 + 𝜷𝟑𝒍𝒏𝑰𝒕 + 𝒖𝒕
where P and I are highly correlated
 We have to fit:
𝒀𝒕
∗
= 𝜷𝟏 + 𝜷𝟐𝒍𝒏𝑷𝒕 + 𝒖𝒕
Where 𝒀∗ = 𝒍𝒏𝒀 − 𝜷𝟑𝒍𝒏𝑰
𝐘𝐭
∗
represent that value of 𝐘 after removing Multicollinearity problem

27. Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
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Dropping A Variable(s) And
Specification Bias
 Consider a regression model:
𝒀𝒊 = 𝜷𝟏 + 𝜷𝟐𝑿𝟐𝒊 + 𝜷𝟑𝑿𝟑𝒊+ 𝒖𝒊
• If in the model , the regressor are highly
correlated then drop a variable and fit a model
having no multicollinearity problem.

28. Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
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Transformation of Variables
 Regression model for time series data:
𝒀𝒕= 𝜷𝟏 + 𝜷𝟐𝑿𝟐𝒕 + 𝜷𝟑𝑿𝟑𝒕+ 𝒖𝒕
 Subtract above two model we get:
𝒀𝒕 - 𝒀𝒕−𝟏= 𝜷𝟐(𝑿𝟐𝒕 − 𝑿𝟐,𝒕−𝟏+ 𝜷𝟑(𝑿𝟑𝒕 −𝑿𝟑,𝒕−𝟏)+ 𝒗𝒕
Where, 𝒗𝒕 = 𝒖𝒕 − 𝒖𝒕−𝟏
 Fit a model at time t-1:
𝒀𝒕−𝟏 = 𝜷𝟏 + 𝜷𝟐𝑿𝟐,𝒕−𝟏 + 𝜷𝟑𝑿𝟑,𝒕−𝟏+ 𝒖𝒕−𝟏
Which model will reduce the multicollinearity problem.

29. Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
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Additional or New Data
 Multicollinearity is a sample feature. So, we can
add another sample to the same variable.
 Increasing sample size, multicollinearity problem
may reduce.

30. Remedial Measures
when
𝝈𝟐 is
known
when
𝝈𝟐 is
unknown
There are two approaches to remediation
Multicollinearity
Heteroscedasticity
Autocorrelation
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31. Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
When 𝝈𝟐 is Known: Method of
Weighted Least Square
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 Consider the simple linear regression model:
𝒀𝒊 = 𝜶 + 𝜷𝑿𝒊 + 𝒖𝒊
Then the transformed error will have a constant variance,
𝑽 𝝁𝒊
∗
= 𝑽
𝝁𝒊
𝝈𝒊
=
𝟏
𝝈𝒊
𝟐
𝑽 𝝁𝒊 +
𝟏
𝝈𝒊
𝟐
𝝈𝒊
𝟐
= 𝟏
 Run a model:
𝒀𝒊
𝝈𝒊
= 𝜷𝟏
∗
𝟏
𝝈𝒊
+ 𝜷𝟐
∗
𝑿𝟐𝒊
𝝈𝒊
+ 𝜷𝟑
∗
𝑿𝟑𝒊
𝝈𝒊
+
𝑼𝒊
𝝈𝒊
If 𝐕(𝒖𝒊) = 𝝈𝒊
𝟐
then heteroscedasticity is present

32. Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
When 𝝈𝟐 is Unknown
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 The error variance is proportional to 𝑿𝒊
𝟐
:
𝑬(𝑼𝒊
𝟐
) = 𝝈𝟐 𝑿𝒊
𝟐
 The error variance is proportional to 𝑿𝒊:
𝑬(𝑼𝒊
𝟐
) = 𝝈𝟐𝑿𝒊
 The error variance is proportional to the square of the
mean value of Y :
𝑬(𝑼𝒊
𝟐
) = 𝝈𝟐
[𝑬(𝒀𝒊)]𝟐
 A log transformation such as:
𝒍𝒏𝒀𝒊 = 𝜷𝟏 + 𝜷𝟐 𝒍𝒏𝑿𝒊 + 𝑼𝒊

33. Remedial Measures
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33
First-Difference Transformation
1
Generalized Transformation
2
Newey-West Method
3
Multicollinearity
Heteroscedasticity
Autocorrelation

34. First-difference Transformation
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34
Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
 If autocorrelation is of AR(1) type, we have:
𝒖𝒕 − 𝝆𝒖𝒕−𝟏 = 𝒗𝒕
• Assume ρ=1 and run first-difference model
(taking first difference of dependent variable
and all regressors)
𝒀𝒕 − 𝒀𝒕−𝟏 = 𝜷𝟐(𝑿𝒕 − 𝑿𝒕−𝟏)+ (𝒖𝒕−𝒖𝒕−𝟏)

35. Generalized Transformation
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35
Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
 Estimate value of ρ through regression of residual
on lagged residual and use value to run transformed
regression
 𝒀𝒕 = 𝜷𝟏 + 𝜷𝟐𝑿𝒕+ 𝒖𝒕
 𝒀𝒕−𝟏 = 𝜷𝟏 + 𝜷𝟐𝑿𝒕−𝟏+ 𝒖𝒕−𝟏
 ρ𝒀𝒕−𝟏 = ρ𝜷𝟏 + ρ𝜷𝟐𝑿𝒕−𝟏+ ρ𝒖𝒕−𝟏
 𝒀𝒕 − ρ𝒀𝒕−𝟏 = 𝜷𝟏(𝟏 − ρ) + 𝜷𝟐(𝑿𝒕 − 𝝆𝑿𝒕−𝟏)+ (𝒖𝒕 − ρ𝒖𝒕−𝟏)

36. Newey-West Method
Remedials of
Multicollinearity
Remedials of
Heteroscedasticity
Remedials of
Autocorrelation
 Generates HAC (heteroscedasticity and
autocorrelation consistent) standard errors.
Department Of Statistics
Begum Rokeya University, Rangpur
36

37. Multicollinearity
Heteroscedasticity
Autocorrelation
Plot Explanation
Department Of Statistics
Begum Rokeya University, Rangpur
37
From the matrix of scatter plot, we suspect
Multicollinearity problem between first & second
independent variable because they are highly correlated.

38. Multicollinearity
Heteroscedasticity
Autocorrelation
Plot Explanation
Department Of Statistics
Begum Rokeya University, Rangpur
38
The following figure illustrates the
typical pattern of the residuals if
the error term is homoscedastic

39. Multicollinearity
Heteroscedasticity
Autocorrelation
Plot Explanation
Department Of Statistics
Begum Rokeya University, Rangpur
39
In this figure, each plot exhibits
the potential existence of
heteroscedasticity with various
relationships between the residual
variance (squared residuals) and
the values of the independent
variable X

40. Multicollinearity
Heteroscedasticity
Autocorrelation
Plot Explanation
Department Of Statistics
Begum Rokeya University, Rangpur
40
The graph of residuals
against fitted value
shows funnel shape,
which indicate presence
of heteroscedasticity.

41. Multicollinearity
Heteroscedasticity
Autocorrelation
Plot Explanation
Department Of Statistics
Begum Rokeya University, Rangpur
41
The plot of residuals against time shows a cyclic
pattern. So, we can suspect positive autocorrelation.

42. Multicollinearity
Heteroscedasticity
Autocorrelation
Plot Explanation
Department Of Statistics
Begum Rokeya University, Rangpur
42
The plot of residuals against time shows upward and
downward trend in the disturbance. So, we can suspect
negative autocorrelation.

43. Multicollinearity
Heteroscedasticity
Autocorrelation
Plot Explanation
Department Of Statistics
Begum Rokeya University, Rangpur
43
The plot of residuals against time shows random scatter
plot. So, we can suspect no autocorrelation.

44. Department Of Statistics
Begum Rokeya University, Rangpur
43