1. EC220 - Introduction to econometrics (chapter 9)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 9)
Slideshow: simultaneous equations bias
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Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 9). [Teaching Resource]
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2. SIMULTANEOUS EQUATIONS BIAS
This sequence shows why OLS is likely to yield inconsistent estimates in models composed of two or more simultaneous relationships. 1

3. SIMULTANEOUS EQUATIONS BIAS
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. 2

4. SIMULTANEOUS EQUATIONS BIAS
We hypothesize that increases in wages lead to increases in prices and so p is positively influenced by w. 3

5. SIMULTANEOUS EQUATIONS BIAS
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 (a2 > 0, a3 < 0). 4

6. SIMULTANEOUS EQUATIONS BIAS
p and w are endogenous
U is exogenous
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. 5

7. SIMULTANEOUS EQUATIONS BIAS
p and w are endogenous
U is exogenous
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. 6

8. SIMULTANEOUS EQUATIONS BIAS
We will start with p. The first step is to substitute for w from the second equation. 7

9. SIMULTANEOUS EQUATIONS BIAS
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. 8

10. SIMULTANEOUS EQUATIONS BIAS
Next we take the equation for w and substitute for p from the first equation. 9

11. SIMULTANEOUS EQUATIONS BIAS
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. 10

12. SIMULTANEOUS EQUATIONS BIAS
structural
equations
The original equations, representing the economic relationships among the variables, are described as the structural equations. 11

13. SIMULTANEOUS EQUATIONS BIAS
reduced form
equation
reduced form
equation
The equations expressing the endogenous variables in terms of the exogenous variable(s) and the disturbance terms are described as the reduced form equations. 12

14. SIMULTANEOUS EQUATIONS BIAS
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. 13

15. SIMULTANEOUS EQUATIONS BIAS
violation of Assumption B.7
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. 14

16. SIMULTANEOUS EQUATIONS BIAS
violation of Assumption B.7
Likewise, in the second equation, p has a component uw. 15

17. SIMULTANEOUS EQUATIONS BIAS
We will investigate the sign of the bias in the slope coefficient if OLS is used to fit the price inflation equation. 16

18. SIMULTANEOUS EQUATIONS BIAS
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. 17

19. SIMULTANEOUS EQUATIONS BIAS
The algebra is simpler if we use the structural equation. 18

20. SIMULTANEOUS EQUATIONS BIAS
The b1 terms cancel. We rearrange the rest of the second factor in the numerator. 19

21. SIMULTANEOUS EQUATIONS BIAS
Hence we obtain the usual decomposition into true value and error term. 20

22. SIMULTANEOUS EQUATIONS BIAS
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. 21

23. SIMULTANEOUS EQUATIONS BIAS
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

24. SIMULTANEOUS EQUATIONS BIAS
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. 23

25. SIMULTANEOUS EQUATIONS BIAS
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. 24

26. SIMULTANEOUS EQUATIONS BIAS
Having divided by n, they have plims and we can make use of the plim quotient rule. 25

27. SIMULTANEOUS EQUATIONS BIAS
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. 26

28. SIMULTANEOUS EQUATIONS BIAS
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.) 27

29. SIMULTANEOUS EQUATIONS BIAS
We use Covariance Rule 1 to decompose the expression. 28

30. SIMULTANEOUS EQUATIONS BIAS
The first term is 0 because (a1 + a2b1) 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

31. SIMULTANEOUS EQUATIONS BIAS
However, the third term is nonzero because the limiting value of a sample variance is the corresponding population variance. 30

32. SIMULTANEOUS EQUATIONS BIAS
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. 31

33. SIMULTANEOUS EQUATIONS BIAS
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. 32

34. SIMULTANEOUS EQUATIONS BIAS
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 a3 and a2 when we take them out of the variance expressions. 33

35. SIMULTANEOUS EQUATIONS BIAS
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? 34

36. SIMULTANEOUS EQUATIONS BIAS
The sign of the bias will depend on the sign of the term (1 – a2b2), since a2 must be positive and all the variance components are positive. 35

37. SIMULTANEOUS EQUATIONS BIAS
Looking at the reduced form equation for w, w should be a decreasing function of U. a3 should be negative. So (1 – a2b2) must be positive. 36

38. SIMULTANEOUS EQUATIONS BIAS
In fact, it is a condition for the existence of equilibrium in this model. Suppose that the exogenous variable U changed by an amount DU. The immediate effect on w would be to change it by a3DU (in the opposite direction, since a3 < 0). 37

39. SIMULTANEOUS EQUATIONS BIAS
This would cause p to change by b2a3DU. 38

40. SIMULTANEOUS EQUATIONS BIAS
This would cause a secondary change in w equal to a2b2a3DU. 39

41. SIMULTANEOUS EQUATIONS BIAS
This in turn would cause p to change by a secondary amount a2b2a3DU. 2 40

42. SIMULTANEOUS EQUATIONS BIAS
This would cause w to change by a further amount a2b2a3DU. 2 2 41

43. SIMULTANEOUS EQUATIONS BIAS
And so on and so forth. The total change will be finite only if a2b2 < 1. Otherwise the process would be explosive, which is implausible. 42

44. SIMULTANEOUS EQUATIONS BIAS
Next we will look at the bias graphically. 43

45. SIMULTANEOUS EQUATIONS BIAS
To do this, it is helpful to rearrange the expression. 44

46. SIMULTANEOUS EQUATIONS BIAS
We have now gathered the terms involving b2 together. 45

47. SIMULTANEOUS EQUATIONS BIAS
Thus we see that the limiting value of the OLS estimator is a weighted average of the true value b2 and 1/a2. 46

48. SIMULTANEOUS EQUATIONS BIAS
Variations in U cause variations in p and w, the change in p being b2 times the change in w. Such movements trace out the true relationship. 47

49. SIMULTANEOUS EQUATIONS BIAS
We will illustrate this with a Monte Carlo model, choosing parameter values as shown above. Note, in particular, that b2 is 0.5. 48

50. SIMULTANEOUS EQUATIONS BIASp w
The values chosen for U were 2, 2.25, rising by steps of 0.25 to This is what we would see if there were no disturbance terms in the model. The slope is 0.5. 49

51. SIMULTANEOUS EQUATIONS BIASp w
To simplify the graphics, we will drop uw from the model, so that we can see the effect of up more clearly. 50

52. SIMULTANEOUS EQUATIONS BIAS
p and w are both affected by variations in up, the change in p being 1/a2 times the change in w. 51

53. SIMULTANEOUS EQUATIONS BIASp w
As a consequence the actual observations are shifted away from the true relationship up or down along lines with slope 1/a2, the dotted lines in the graph. Since a2 is 0.5, 1/a2 is 2.0. 52

54. SIMULTANEOUS EQUATIONS BIASp w
The regression line thus overestimates b2. 53

55. SIMULTANEOUS EQUATIONS BIAS
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. 54

56. SIMULTANEOUS EQUATIONS BIAS
The large sample bias is Our regression estimate, 1.11, was a little higher. 55

57. SIMULTANEOUS EQUATIONS BIAS
b s.e.(b1) b s.e.(b2)
9 –
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. 56

58. SIMULTANEOUS EQUATIONS BIAS
b s.e.(b1) b s.e.(b2)
9 –
Because the slope coefficient was overestimated, the intercept was underestimated every time. 57

59. SIMULTANEOUS EQUATIONS BIAS
b s.e.(b1) b s.e.(b2)
9 –
No attention should be paid to the standard errors because they are invalidated by the simultaneous equations bias. 58

60. SIMULTANEOUS EQUATIONS BIAS
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. 59

61. SIMULTANEOUS EQUATIONS BIAS
plim b2 = 0.99
mean of b2 = 0.95
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. 60

62. Copyright Christopher Dougherty 2011.
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The content of this slideshow comes from Section 9.2 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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