christopher dougherty ec220 - introduction to econometrics (chapter 9) slideshow: simultaneous...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 9)Slideshow: simultaneous equations bias

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 9). [Teaching Resource]

© 2012 The Author

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SIMULTANEOUS EQUATIONS BIAS

1

This sequence shows why OLS is likely to yield inconsistent estimates in models composed of two or more simultaneous relationships.

wuUpw 321 puwp 21


2

In this example we suppose that we have data on p, the annual rate of price inflation, w, the annual rate of wage inflation, and U, the rate of unemployment, for a sample of countries.

wuUpw 321 puwp 21


3

We hypothesize that increases in wages lead to increases in prices and so p is positively influenced by w.

wuUpw 321 puwp 21


4

We also suppose that workers try to protect their real wages by demanding increases in wages as prices rise, but their ability to so is the weaker, the greater is the rate of unemployment (2 > 0, 3 < 0).

wuUpw 321 puwp 21


5

The model involves some circularity, in that w is a determinant of p, and p is a determinant of w. Variables whose values are determined interactively within the model are described as endogenous variables.

wuUpw 321 puwp 21

p and w are endogenousU is exogenous


6

We will cut through the circularity by expressing p and w in terms of their ultimate determinants, U and the disturbance terms up and uw. Variables such as U whose values are determined outside the model are described as exogenous variables.

wuUpw 321 puwp 21

p and w are endogenousU is exogenous


pw uuUpp )( 32121

7

We will start with p. The first step is to substitute for w from the second equation.

wuUpw 321 puwp 21


22

223211

1

wp uuUp

wp uuUp 22321122 )1(

pw uuUpp )( 32121

8

We bring the terms involving p together on the left side of the equation and thus express p in terms of U, up, and uw.

wuUpw 321 puwp 21


22

223211

1

wp uuUp

wp uuUp 22321122 )1(

wp uUuww 32121 )(

pw uuUpp )( 32121

9

Next we take the equation for w and substitute for p from the first equation.

wuUpw 321 puwp 21


22

223211

1

wp uuUp

wp uuUp 22321122 )1(

wp uUuww 32121 )(

wp uuUw 2312122 )1(

10

pw uuUpp )( 32121

22

23121

1

wp uuUw

We bring the terms involving w together on the left side of the equation and thus express w in terms of U, up, and uw.

wuUpw 321 puwp 21


wuUpw 321

22

223211

1

wp uuUp

wp uuUp 22321122 )1(

wp uUuww 32121 )(

wp uuUw 2312122 )1(

pw uuUpp )( 32121

puwp 21

22

23121

1

wp uuUw

11

The original equations, representing the economic relationships among the variables, are described as the structural equations.

structuralequations


wuUpw 321

22

223211

1

wp uuUp

wp uuUp 22321122 )1(

wp uUuww 32121 )(

wp uuUw 2312122 )1(

pw uuUpp )( 32121

puwp 21

22

23121

1

wp uuUw

12

The equations expressing the endogenous variables in terms of the exogenous variable(s) and the disturbance terms are described as the reduced form equations.

reduced formequation

reduced formequation


wuUpw 321

22

223211

1

wp uuUp

wp uuUp 22321122 )1(

wp uUuww 32121 )(

wp uuUw 2312122 )1(

pw uuUpp )( 32121

puwp 21

22

23121

1

wp uuUw

13

The reduced form equations have two important roles. They can indicate that we have a serious econometric problem, but they may also provide a solution to it.


wuUpw 321

22

223211

1

wp uuUp

wp uuUp 22321122 )1(

wp uUuww 32121 )(

wp uuUw 2312122 )1(

pw uuUpp )( 32121

puwp 21

22

23121

1

wp uuUw

14

The problem is the violation of Assumption B.7 that the disturbance term be distributed independently of the explanatory variable(s). In the first equation, w has a component up. OLS would therefore yield inconsistent estimates if used to fit the equation.

violation of Assumption B.7


wuUpw 321

22

223211

1

wp uuUp

wp uuUp 22321122 )1(

wp uUuww 32121 )(

wp uuUw 2312122 )1(

pw uuUpp )( 32121

puwp 21

22

23121

1

wp uuUw

Likewise, in the second equation, p has a component uw.

15

violation of Assumption B.7


22

22

22121

2OLS2

][

][][

ww

uuww

ww

uuwwww

ww

uwuwww

ww

ppwwb

i

ppii

i

ppiii

i

ppiii

i

ii

16

wuUpw 321 puwp 21

We will investigate the sign of the bias in the slope coefficient if OLS is used to fit the price inflation equation.


22

22

22121

2OLS2

][

][][

ww

uuww

ww

uuwwww

ww

uwuwww

ww

ppwwb

i

ppii

i

ppiii

i

ppiii

i

ii

17

wuUpw 321 puwp 21

As usual we start by substituting for the dependent variable using the true model. For this purpose, we could use either the structural equation or the reduced form equation for p.

22

223211

1

wp uuUp


22

22

22121

2OLS2

][

][][

ww

uuww

ww

uuwwww

ww

uwuwww

ww

ppwwb

i

ppii

i

ppiii

i

ppiii

i

ii

18

wuUpw 321 puwp 21

The algebra is simpler if we use the structural equation.

22

223211

1

wp uuUp


The 1 terms cancel. We rearrange the rest of the second factor in the numerator.

22

22

22121

2OLS2

][

][][

ww

uuww

ww

uuwwww

ww

uwuwww

ww

ppwwb

i

ppii

i

ppiii

i

ppiii

i

ii

19

wuUpw 321 puwp 21


Hence we obtain the usual decomposition into true value and error term.

22

22

22121

2OLS2

][

][][

ww

uuww

ww

uuwwww

ww

uwuwww

ww

ppwwb

i

ppii

i

ppiii

i

ppiii

i

ii

20

wuUpw 321 puwp 21


22

OLS2

ww

uuwwb

i

ppii

21

wuUpw 321 puwp 21

We will now investigate the properties of the error term. Of course, we would like it to have expected value 0, making the estimator unbiased.


22

OLS2

ww

uuwwb

i

ppii

22

wuUpw 321 puwp 21

22

23121

1

wp uuUw

However, the error term is a nonlinear function of both up and uw because both are components of w. As a consequence, it is not possible to obtain a closed-form analytical expression for its expected value.


22

OLS2

ww

uuwwb

i

ppii

23

wuUpw 321 puwp 21

22

23121

1

wp uuUw

We will investigate the large-sample properties instead. We will demonstrate that the estimator is inconsistent, and this will imply that it has undesirable finite-sample properties.


22

OLS2

ww

uuwwb

i

ppii

24

wuUpw 321 puwp 21

To investigate the plim of the error term, we need to divide both the numerator and the denominator by n. Otherwise they would increase indefinitely with n.

)var(

),cov(1

plim

1 plim

1

1

plimplim

2

22

w

uw

wwn

uuwwn

wwn

uuwwn

ww

uuww

p

i

ppii

i

ppii

i

ppii


22

OLS2

ww

uuwwb

i

ppii

25

wuUpw 321 puwp 21

Having divided by n, they have plims and we can make use of the plim quotient rule.

)var(

),cov(1

plim

1 plim

1

1

plimplim

2

22

w

uw

wwn

uuwwn

wwn

uuwwn

ww

uuww

p

i

ppii

i

ppii

i

ppii


22

OLS2

ww

uuwwb

i

ppii

26

wuUpw 321 puwp 21

)var(

),cov(1

plim

1 plim

1

1

plimplim

2

22

w

uw

wwn

uuwwn

wwn

uuwwn

ww

uuww

p

i

ppii

i

ppii

i

ppii

It can then be shown that the plim of the numerator is the covariance between w and u, and the plim of the denominator is the variance of w.


22

22

222

2

3121

22

22

23121

10)(var00

11

,cov,cov

,cov][,cov

11

1,cov,cov

pu

p

wppp

pp

wppp

u

uuuu

Uuu

uuUuwu

wuUpw 321 puwp 21

27

)var(

),cov( plim 2

OLS2 w

wub p

We will start by deriving the limiting value of the numerator, the first step being to substitute for w from its reduced form equation. (Note: Here we must use the reduced form equation. If we use the structural equation, we will find ourselves going round in circles.)

22

23121

1

wp uuUw


22

22

222

2

3121

22

22

23121

10)(var00

11

,cov,cov

,cov][,cov

11

1,cov,cov

pu

p

wppp

pp

wppp

u

uuuu

Uuu

uuUuwu

wuUpw 321 puwp 21

28

)var(

),cov( plim 2

OLS2 w

wub p

We use Covariance Rule 1 to decompose the expression.


22

22

222

2

3121

22

22

23121

10)(var00

11

,cov,cov

,cov][,cov

11

1,cov,cov

pu

p

wppp

pp

wppp

u

uuuu

Uuu

uuUuwu

wuUpw 321 puwp 21

The first term is 0 because (1 + 21) is a constant. The second term is 0 because U is exogenous and so distributed independently of up. The fourth term is 0 if the disturbance terms are distributed independently of each other.

29

)var(

),cov( plim 2

OLS2 w

wub p


22

22

222

2

3121

22

22

23121

10)(var00

11

,cov,cov

,cov][,cov

11

1,cov,cov

pu

p

wppp

pp

wppp

u

uuuu

Uuu

uuUuwu

wuUpw 321 puwp 21

30

)var(

),cov( plim 2

OLS2 w

wub p

However, the third term is nonzero because the limiting value of a sample variance is the corresponding population variance.


222

2222

223

2

233

23

222

22

23

22

23

22

121

1

,cov2

,cov2,cov2

varvarvar

1

1

1var

11var)var(

wp uuU

wp

pw

wp

wpwp

uu

uUuU

uuU

uuUuuUw

31

wuUpw 321 puwp 21

)var(

),cov( plim 2

OLS2 w

wub p

Next we will derive the limiting value of var(w). Variances are unaffected by additive constants, so the first part of the expression may be dropped.

22

23121

1

wp uuUw


Using Variance Rule 1, the numerator decomposes into three variances and three covariances. The denominator is a constant common factor and may be taken, squared, outside the expression using Variance Rule 2.

222

2222

223

2

233

23

222

22

23

22

23

22

121

1

,cov2

,cov2,cov2

varvarvar

1

1

1var

11var)var(

wp uuU

wp

pw

wp

wpwp

uu

uUuU

uuU

uuUuuUw

32

wuUpw 321 puwp 21

)var(

),cov( plim 2

OLS2 w

wub p


222

2222

223

2

233

23

222

22

23

22

23

22

121

1

,cov2

,cov2,cov2

varvarvar

1

1

1var

11var)var(

wp uuU

wp

pw

wp

wpwp

uu

uUuU

uuU

uuUuuUw

33

wuUpw 321 puwp 21

)var(

),cov( plim 2

OLS2 w

wub p

The covariances are all 0 on the assumption that U, up, and uw are distributed independently of each other. Thus the numerator consists of the variance expressions. Remember that we have to square 3 and 2 when we take them out of the variance expressions.


Hence we obtain an expression for the limiting value of the OLS estimator of the slope coefficient. We can see that the estimator is inconsistent. Can we determine the sign of the bias?

22

22

1),cov(

pu

p wu

2

22

2222

223

1)var(

wp uuU

w

34

2222

223

22

222OLS2 )1( plim

wp

p

uuU

ub

wuUpw 321 puwp 21

)var(

),cov( plim 2

OLS2 w

wub p


22

22

1),cov(

pu

p wu

2

22

2222

223

1)var(

wp uuU

w

35

2222

223

22

222OLS2 )1( plim

wp

p

uuU

ub

wuUpw 321 puwp 21

)var(

),cov( plim 2

OLS2 w

wub p

The sign of the bias will depend on the sign of the term (1 – 22), since 2 must be positive and all the variance components are positive.


wuUpw 321

22

223211

1

wp uuUp

wp uuUp 22321122 )1(

wp uUuww 32121 )(

wp uuUw 2312122 )1(

pw uuUpp )( 32121

puwp 21

22

23121

1

wp uuUw

36

Looking at the reduced form equation for w, w should be a decreasing function of U. 3should be negative. So (1 – 22) must be positive.


Uw 3

wuUpw 321 puwp 21

37

In fact, it is a condition for the existence of equilibrium in this model. Suppose that the exogenous variable U changed by an amount U. The immediate effect on w would be to change it by 3U (in the opposite direction, since 3 < 0).


Uw 3Uwp 322

wuUpw 321 puwp 21

38

This would cause p to change by 23U.


Uw 3

U

pUw

322

23

)1(

wuUpw 321 puwp 21

39

This would cause a secondary change in w equal to 223U.

Uwp 322


Uw 3

U

pUw

322

23

)1(

Uwp 32222 )1(

wuUpw 321 puwp 21

40

This in turn would cause p to change by a secondary amount 223U.

Uwp 322

2


Uw 3

U

pUw

322

23

)1(

Uwp 32222 )1(

U

U

pUw

332

32

22

2222

322

2222

23

...)1(

)1(

wuUpw 321 puwp 21

41

This would cause w to change by a further amount 223U.2 2

Uwp 322


42

And so on and so forth. The total change will be finite only if 22 < 1. Otherwise the process would be explosive, which is implausible.

Uw 3

U

pUw

322

23

)1(

Uwp 32222 )1(

U

U

pUw

332

32

22

2222

322

2222

23

...)1(

)1(

122

wuUpw 321 puwp 21

Uwp 322


2222

223

222

2222

222

3

2223

2

2222

223

222

2222

222

3

222

2

2222

223

222

22

2

2222

223

22

222OLS2

1

11

1

)1( plim

wp

p

wp

w

wp

p

wp

p

wp

p

wp

p

uuU

u

uuU

uU

uuU

u

uuU

u

uuU

u

uuU

ub

wuUpw 321 puwp 21

Next we will look at the bias graphically.

43


2222

223

222

2222

222

3

2223

2

2222

223

222

2222

222

3

222

2

2222

223

222

22

2

2222

223

22

222OLS2

1

11

1

)1( plim

wp

p

wp

w

wp

p

wp

p

wp

p

wp

p

uuU

u

uuU

uU

uuU

u

uuU

u

uuU

u

uuU

ub

wuUpw 321 puwp 21

44

To do this, it is helpful to rearrange the expression.


2222

223

222

2222

222

3

2223

2

2222

223

222

2222

222

3

222

2

2222

223

222

22

2

2222

223

22

222OLS2

1

11

1

)1( plim

wp

p

wp

w

wp

p

wp

p

wp

p

wp

p

uuU

u

uuU

uU

uuU

u

uuU

u

uuU

u

uuU

ub

wuUpw 321 puwp 21

45

We have now gathered the terms involving 2 together.


46

Thus we see that the limiting value of the OLS estimator is a weighted average of the true value 2 and 1/2.

2222

223

222

2222

222

3

2223

2

2222

223

222

2222

222

3

222

2

2222

223

222

22

2

2222

223

22

222OLS2

1

11

1

)1( plim

wp

p

wp

w

wp

p

wp

p

wp

p

wp

p

uuU

u

uuU

uU

uuU

u

uuU

u

uuU

u

uuU

ub

wuUpw 321 puwp 21


47

Variations in U cause variations in p and w, the change in p being 2 times the change in w. Such movements trace out the true relationship.

wuUpw 321 puwp 21

2222

223

222

2222

222

3

2223

2OLS2

1 plim

wp

p

wp

w

uuU

u

uuU

uUb

22

23121

1

wp uuUw

22

223211

1

wp uuUp


wuUpw 321 puwp 21

2222

223

222

2222

222

3

2223

2OLS2

1 plim

wp

p

wp

w

uuU

u

uuU

uUb

22

23121

1

wp uuUw

22

223211

1

wp uuUp

puwp 5.05.1 wuUpw 4.05.05.2

48

We will illustrate this with a Monte Carlo model, choosing parameter values as shown above. Note, in particular, that 2 is 0.5.

0

1

2

3

4

5

0 1 2 3 4


49

The values chosen for U were 2, 2.25, rising by steps of 0.25 to 6.75. This is what we would see if there were no disturbance terms in the model. The slope is 0.5.

p

w

wp 5.05.1


50

To simplify the graphics, we will drop uw from the model, so that we can see the effect of up more clearly.

w

p

0

1

2

3

4

5

0 1 2 3 4

wp 5.05.1


wuUpw 321 puwp 21

2222

223

222

2222

222

3

2223

2OLS2

1 plim

wp

p

wp

w

uuU

u

uuU

uUb

22

23121

1

wp uuUw

22

223211

1

wp uuUp

puwp 5.05.1 wuUpw 4.05.05.2

51

p and w are both affected by variations in up, the change in p being 1/2 times the change in w.

0

1

2

3

4

5

0 1 2 3 4


52

p

w

uwp 5.05.1

As a consequence the actual observations are shifted away from the true relationship up or down along lines with slope 1/2, the dotted lines in the graph. Since 2 is 0.5, 1/2 is 2.0.


53

The regression line thus overestimates 2.

wp 11.136.0ˆ

p

w

uwp 5.05.1

0

1

2

3

4

5

0 1 2 3 4


54

We will calculate the large-sample bias in the slope coefficient. U and up were chosen so that they had population variances 2.08 and 0.64, respectively.

wuUpw 321 puwp 21

puwp 5.05.1 wuUpw 4.05.05.2

99.0

64.025.008.216.064.025.0

0.2

64.025.008.216.008.216.0

5.0

1 plim 222

222

3

222

2222

222

3

2223

2OLS2

wp

p

wp

w

uuU

u

uuU

uUb


55

The large sample bias is 0.99. Our regression estimate, 1.11, was a little higher.

99.0

64.025.008.216.064.025.0

0.2

64.025.008.216.008.216.0

5.0

1 plim 222

222

3

222

2222

222

3

2223

2OLS2

wp

p

wp

w

uuU

u

uuU

uUb

wuUpw 321 puwp 21

puwp 5.05.1 wuUpw 4.05.05.2


56

Here are the results for 10 samples in this simulation. The slope coefficient was overestimated every time and does appear to be distributed around its plim, 0.99.

wuUpw 321 puwp 21

puwp 5.05.1 wuUpw 4.05.05.2

b1 s.e.(b1) b2 s.e.(b2)

1 0.36 0.49 1.11 0.22

2 0.45 0.38 1.06 0.17

3 0.65 0.27 0.94 0.12

4 0.41 0.39 0.98 0.19

5 0.46 0.92 0.77 0.22

6 0.26 0.35 1.09 0.16

7 0.31 0.39 1.00 0.19

8 1.06 0.38 0.82 0.16

9 –0.08 0.36 1.16 0.18

10 1.12 0.43 0.69 0.20

b1 s.e.(b1) b2 s.e.(b2)

1 0.36 0.49 1.11 0.22

2 0.45 0.38 1.06 0.17

3 0.65 0.27 0.94 0.12

4 0.41 0.39 0.98 0.19

5 0.46 0.92 0.77 0.22

6 0.26 0.35 1.09 0.16

7 0.31 0.39 1.00 0.19

8 1.06 0.38 0.82 0.16

9 –0.08 0.36 1.16 0.18

10 1.12 0.43 0.69 0.20


wuUpw 321 puwp 21

puwp 5.05.1 wuUpw 4.05.05.2

57

Because the slope coefficient was overestimated, the intercept was underestimated every time.


wuUpw 321 puwp 21

puwp 5.05.1 wuUpw 4.05.05.2

58

No attention should be paid to the standard errors because they are invalidated by the simultaneous equations bias.

b1 s.e.(b1) b2 s.e.(b2)

1 0.36 0.49 1.11 0.22

2 0.45 0.38 1.06 0.17

3 0.65 0.27 0.94 0.12

4 0.41 0.39 0.98 0.19

5 0.46 0.92 0.77 0.22

6 0.26 0.35 1.09 0.16

7 0.31 0.39 1.00 0.19

8 1.06 0.38 0.82 0.16

9 –0.08 0.36 1.16 0.18

10 1.12 0.43 0.69 0.20


The chart plots the distribution of the slope coefficient for 1 million samples. Almost all the estimates are above the true value of 0.5, confirming the large-sample analysis.

0

1

2

3

4

0 0.25 0.5 0.75 1 1.25 1.5

59


The plim (0.99) in this case provides a good guide to the size of the bias since the mean of the distribution is 0.95, fairly close to the plim even though, with only 20 observations, each sample was quite small.

0

1

2

3

4

0 0.25 0.5 0.75 1 1.25 1.5

60

plim b2 = 0.99mean of b2 = 0.95

Copyright Christopher Dougherty 2011.

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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 9.2 of C. Dougherty,

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

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

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or the University of London International Programmes distance learning course

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11.07.25

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

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

christopher dougherty ec220 - introduction to econometrics (chapter 9) slideshow: simultaneous...

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simultaneous equations

terms of u

original equations

structural equations

disturbance terms u

exogenous slide

endogenous u

simultaneous relationships