1. EC220 - Introduction to econometrics (chapter 11)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 11)
Slideshow: simultaneous equations models
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Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 11). [Teaching Resource]
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Available in LSE Learning Resources Online: May 2012
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2. SIMULTANEOUS EQUATIONS MODELS
Most of the issues relating to the fitting of simultaneous equations models with time series data are similar to those that arise when using cross-sectional data. 1

3. SIMULTANEOUS EQUATIONS MODELS
One needs to make a distinction between endogenous and exogenous variables, and between structural and reduced form equations, and find valid instruments, when necessary, for the endogenous variables. 2

4. SIMULTANEOUS EQUATIONS MODELS
The main difference is the potential use of lagged endogenous variables as instruments.
We will use a simple macroeconomic model for a closed economy to illustrate the discussion. 3

5. SIMULTANEOUS EQUATIONS MODELS
Private sector consumption, Ct, is determined by income, Yt . Private sector investment, It, is determined by the rate of interest, rt . Income is defined to be the sum of consumption, investment, and government expenditure, Gt . 4

6. SIMULTANEOUS EQUATIONS MODELS
ut and vt are disturbance terms assumed to have zero mean and constant variance. Realistically, since both may be affected by economic sentiment, we should not assume that they are independent. 5

7. SIMULTANEOUS EQUATIONS MODELS
As the model stands, we have three endogenous variables, Ct , It , and Yt , and two exogenous variables, rt and Gt . 6

8. SIMULTANEOUS EQUATIONS MODELS
The consumption function is overidentified and we would use TSLS. The investment function has no endogenous variables on the right side and so we would use OLS. The third equation is an identity and does not need to be fitted. So far, so good. 7

9. SIMULTANEOUS EQUATIONS MODELS
However, even in the simplest macroeconomic models, it is usually recognized that the rate of interest should not be treated as exogenous. 8

10. SIMULTANEOUS EQUATIONS MODELS
We extend the model to the IS–LM model of conventional introductory macroeconomics by adding a relationship for the demand for money, Mtd, relating it to income (the transactions demand for cash) and the interest rate (the speculative demand). 9

11. SIMULTANEOUS EQUATIONS MODELS
We assume that the money market clears with the demand for money equal to the supply, Mts, which initially we will assume to be exogenous. 10

12. SIMULTANEOUS EQUATIONS MODELS
We now have five endogenous variables (the previous three, plus rt and Mtd) and two exogenous variables, Gt and Mts. On the face of it, the consumption and investment functions are both overidentified and the demand for money function exactly identified. 11

13. SIMULTANEOUS EQUATIONS MODELS
However, we have hardly started with the development of the model. Government expenditure will be influenced by budgetary policy that takes account of government revenues from taxation. Tax revenues will be influenced by the level of income. 12

14. SIMULTANEOUS EQUATIONS MODELS
The money supply will respond to various pressures, and so on. Ultimately, it has to be acknowledged that all important macroeconomic variables are likely to be endogenous and consequently that none is able to act as a valid instrument. 13

15. SIMULTANEOUS EQUATIONS MODELS
The solution is to take advantage of a feature of time-series models that is absent in cross-sectional ones: the inclusion of lagged variables as explanatory variables. Suppose, as seems realistic, the static consumption function is replaced by the ADL(1,0) model shown. 14

16. SIMULTANEOUS EQUATIONS MODELS
Ct–1 has already been fixed by time t and is described as a predetermined variable. Subject to an important condition, predetermined variables can be treated as exogenous variables at time t and can therefore be used as instruments. 15

17. SIMULTANEOUS EQUATIONS MODELS
Hence, in the investment equation, Ct–1 could be used as an instrument for rt. To serve as an instrument, it must be correlated with rt, but this is assured by the fact that it is a determinant of Ct, and thus Yt, and so with rt via the demand for money relationship. 16

18. SIMULTANEOUS EQUATIONS MODELS
To give another example, many models include the level of the capital stock, Kt–1, as a determinant of current investment (usually negatively: the greater the previous stock, the smaller the immediate need for investment, controlling for other factors). So the investment function becomes as shown. 17

19. SIMULTANEOUS EQUATIONS MODELS
Then, since Kt–1 is a determinant of It, and It is a component of Yt, Kt–1 can serve as an instrument for Yt in the consumption function. 18

20. SIMULTANEOUS EQUATIONS MODELS
On the whole, it is reasonable to expect relationships to be dynamic, rather than static, and that lagged variables will feature among the regressors. Suddenly, a model that lacked valid instruments has become full of them. 19

21. SIMULTANEOUS EQUATIONS MODELS
It was mentioned above that there is an important condition attached to the use of predetermined variables as instruments. This concerns the properties of the disturbance terms in the model. 20

22. SIMULTANEOUS EQUATIONS MODELS
We will discuss the point in the context of the use of Ct–1 as an instrument for rt in the investment equation. For Ct–1 to be a valid instrument, it must be distributed independently of the disturbance term vt . We acknowledge that Ct–1 will be influenced by vt–1. 21

23. SIMULTANEOUS EQUATIONS MODELS
(Considering the interactions between the relationships at time t – 1, vt–1 influences It–1 by definition and hence Yt–1 because It–1 is a component of it. Thus vt–1 also influences Ct–1, by virtue of the consumption function.) 22

24. SIMULTANEOUS EQUATIONS MODELS
Now, the fact that Ct–1 depends on vt–1 does not, in itself, matter. The requirement is that
Ct–1 should be distributed independently of vt . So, provided that vt and vt–1 are distributed independently of each other, Ct–1 remains a valid instrument. 23

25. SIMULTANEOUS EQUATIONS MODELS
But suppose that there is time persistence in the values of the disturbance term and vt is correlated with vt–1. Then Ct–1 will not be distributed independently of vt and will not be valid instrument. 24

26. SIMULTANEOUS EQUATIONS MODELS
We will return to the issue of time persistence in the disturbance term when we discuss autocorrelation (serial correlation) in Chapter 12. Unfortunately, autocorrelation is a common problem with time series data and it limits the use of predetermined variables. 25

27. 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 11.6 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
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
or the University of London International Programmes distance learning course
20 Elements of Econometrics