Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: multicollinearity...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 3)Slideshow: multicollinearity

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 3). [Teaching Resource]

© 2012 The Author

This version available at: http://learningresources.lse.ac.uk/129/

Available in LSE Learning Resources Online: May 2012

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X2 X3 Y

10 19 51

11 21 56

12 23 61

13 25 66

14 27 71

15 29 76

MULTICOLLINEARITY

3232 XXY

12 23 XX

1

Suppose that Y = 2 + 3X2 + X3 and that X3 = 2X2 – 1. There is no disturbance term in the equation for Y, but that is not important. Suppose that we have the six observations shown.

MULTICOLLINEARITY

2

The three variables are plotted as line graphs above. Looking at the data, it is impossible to tell whether the changes in Y are caused by changes in X2, by changes in X3, or jointly by changes in both X2 and X3.

0

10

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1 2 3 4 5 6

Y

X3

X2

Change from previous observation

X2 X3 Y X2 X3 Y

10 19 51 1 2 5

11 21 56 1 2 5

12 23 61 1 2 5

13 25 66 1 2 5

14 27 71 1 2 5

15 29 76 1 2 5

MULTICOLLINEARITY

3

3232 XXY

12 23 XX

Numerically, Y increases by 5 in each observation. X2 changes by 1.

MULTICOLLINEARITY

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Hence the true relationship could have been Y = 1 + 5X2.

0

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1 2 3 4 5 6

Y

X3

X2

Y = 1 + 5X2 ?

MULTICOLLINEARITY

3232 XXY

12 23 XX

5

However, it can also be seen that X3 increases by 2 in each observation.

Change from previous observation

X2 X3 Y X2 X3 Y

10 19 51 1 2 5

11 21 56 1 2 5

12 23 61 1 2 5

13 25 66 1 2 5

14 27 71 1 2 5

15 29 76 1 2 5

MULTICOLLINEARITY

6

Hence the true relationship could have been Y = 3.5 +2.5X3.

0

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Y

X3

X2

Y = 3.5 + 2.5X3 ?

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These two possibilities are special cases of Y = 3.5 – 2.5p + 5pX2 + 2.5(1 – p)X3, which would fit the relationship for any value of p.

0

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Y

X3

X2

Y = 3.5 – 2.5p + 5pX2 + 2.5(1 – p)X3

MULTICOLLINEARITY

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0

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1 2 3 4 5 6

Y

X3

X2

Y = 3.5 – 2.5p + 5pX2 + 2.5(1 – p)X3

There is no way that regression analysis, or any other technique, could determine the true relationship from this infinite set of possibilities, given the sample data.

MULTICOLLINEARITY

uXXY 33221 23 XX

9

What would happen if you tried to run a regression when there is an exact linear relationship among the explanatory variables?

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uXXY 33221 23 XX

10

We will investigate, using the model with two explanatory variables shown above. [Note: A disturbance term has now been included in the true model, but it makes no difference to the analysis.]

MULTICOLLINEARITY

uXXY 33221 23 XX

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23322 XXYYXX iii

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33222

332

22

3322332

XXXXXXXX

XXXXYYXXb

iiii

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The expression for the multiple regression coefficient b2 is shown above. We will substitute for X3 using its relationship with X2.

MULTICOLLINEARITY

uXXY 33221 23 XX

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23322 XXYYXX iii

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33222

332

22

3322332

XXXXXXXX

XXXXYYXXb

iiii

iiii

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2

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2222

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233 ][][

XX

XXXX

XXXX

i

ii

ii

First, we will replace the terms highlighted.

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uXXY 33221 23 XX

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We have made the replacement.

222

222 XXYYXX iii

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33222

2222

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3322332

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XXXXYYXXb

iiii

iiii

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2

222

2222

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233 ][][

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i

ii

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33222

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3322332

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iiii

iiii

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2222

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XX

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i

ii

iiii

Next, the terms highlighted now.

MULTICOLLINEARITY

uXXY 33221 23 XX

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222

222 XXYYXX iii

00

2222

222

2222

22233

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XXYYXXb

iii

iii

222

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22223322 ][][

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i

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We have made the replacement.

MULTICOLLINEARITY

uXXY 33221 23 XX

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222 XXYYXX iii

00

2222

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22233

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XXYYXXb

iii

iii

YYXX

YYXX

YYXXYYXX

ii

ii

iiii

22

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2233 ][][

Finally this term.

00

2222

222

2222

22222

2

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XXYYXXb

iii

iii

MULTICOLLINEARITY

uXXY 33221 23 XX

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222

222 XXYYXX iii

YYXX

YYXX

YYXXYYXX

ii

ii

iiii

22

22

2233 ][][

Again, we have made the replacement.

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222 XXYYXX iii

00

2222

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2222

22222

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XXXXXX

XXYYXXb

iii

iii

It turns out that the numerator and the denominator are both equal to zero. The regression coefficient is not defined.

MULTICOLLINEARITY

uXXY 33221 23 XX

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222

222 XXYYXX iii

00

2222

222

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22222

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XXYYXXb

iii

iii

It is unusual for there to be an exact relationship among the explanatory variables in a regression. When this occurs, it s typically because there is a logical error in the specification.

. reg EARNINGS S EXP EXPSQ

Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 45.57 Model | 22762.4472 3 7587.48241 Prob > F = 0.0000 Residual | 89247.7839 536 166.507059 R-squared = 0.2032-------------+------------------------------ Adj R-squared = 0.1988 Total | 112010.231 539 207.811189 Root MSE = 12.904

------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.754372 .2417286 11.39 0.000 2.279521 3.229224 EXP | -.2353907 .665197 -0.35 0.724 -1.542103 1.071322 EXPSQ | .0267843 .0219115 1.22 0.222 -.0162586 .0698272 _cons | -22.21964 5.514827 -4.03 0.000 -33.05297 -11.38632------------------------------------------------------------------------------

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However, it often happens that there is an approximate relationship. For example, when relating earnings to schooling and work experience, it if often reasonable to suppose that the effect of work experience is subject to diminishing returns.

uEXPSQEXPSEARNINGS 4321




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A standard way of allowing for this is to include EXPSQ, the square of EXP, in the specification. According to the hypothesis of diminishing returns, 4 should be negative.





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We fit this specification using Data Set 21. The schooling component of the regression results is not much affected by the inclusion of the EXPSQ term. The coefficient of S indicates that an extra year of schooling increases hourly earnings by $2.75.


. reg EARNINGS S EXP


------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.678125 .2336497 11.46 0.000 2.219146 3.137105 EXP | .5624326 .1285136 4.38 0.000 .3099816 .8148837 _cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213------------------------------------------------------------------------------

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In the specification without EXPSQ it was 2.68, not much different.





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The standard error, 0.23 in the specification without EXPSQ, is also little changed and the coefficient remains highly significant.




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By contrast, the inclusion of the new term has had a dramatic effect on the coefficient of EXP. Now it is negative, which makes little sense, and insignificant.

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Previously it had been positive and highly significant.







MULTICOLLINEARITY

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The coefficient of EXPSQ is also strange. It is positive, suggesting increasing returns to experience. However, it is not significant.




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The reason for these problems is that EXPSQ is highly correlated with EXP. This makes it difficult to discriminate between the individual effects of EXP and EXPSQ, and the regression estimates tend to be erratic.

. cor EXP EXPSQ(obs=540)

| EXP EXPSQ------+------------------ EXP | 1.0000EXPSQ | 0.9812 1.0000





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The high correlation causes the standard error of EXP to be larger than it would have been if EXP and EXPSQ had been less highly correlated, warning us that the point estimate is unreliable.





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When high correlations among the explanatory variables lead to erratic point estimates of the coefficients, large standard errors and unsatisfactorily low t statistics, the regression is said to said to be suffering from multicollinearity.





MULTICOLLINEARITY

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Note that the coefficients remain unbiased and the standard errors remain valid.





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Multicollinearity may also be caused by an approximate linear relationship among the explanatory variables. When there are only 2, an approximate linear relationship means there will be a high correlation, but this is not always the case when there are more than 2.

Copyright Christopher Dougherty 2011.

These slideshows may be downloaded by anyone, anywhere for personal use.

Subject to respect for copyright and, where appropriate, attribution, they may be

used as a resource for teaching an econometrics course. There is no need to

refer to the author.

The content of this slideshow comes from Section 3.4 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

Additional (free) resources for both students and instructors may be

downloaded from the OUP Online Resource Centre

http://www.oup.com/uk/orc/bin/9780199567089/.

Individuals studying econometrics on their own and who feel that they might

benefit from participation in a formal course should consider the London School

of Economics summer school course

EC212 Introduction to Econometrics

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

11.07.25

Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: multicollinearity...

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