Transcript Christopher Dougherty EC220 - Introduction to econometrics (chapter 11) Slideshow: simultaneous equations models Original citation: Dougherty, C.

Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 11)
Slideshow: simultaneous equations models
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 11). [Teaching Resource]
© 2012 The Author
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Available in LSE Learning Resources Online: May 2012
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
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
SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
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
SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
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
SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
As the model stands, we have three endogenous variables, Ct , It , and Yt , and two
exogenous variables, rt and Gt .
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
However, even in the simplest macroeconomic models, it is usually recognized that the rate
of interest should not be treated as exogenous.
8
SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
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).
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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
SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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
SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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
SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
The solution is to take advantage of a feature of time-series models that is absent in crosssectional 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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt   3 K t 1  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt   3 K t 1  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt   3 K t 1  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt   3 K t 1  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt   3 K t 1  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt   3 K t 1  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
(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.)
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt   3 K t 1  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt   3 K t 1  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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.
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SIMULTANEOUS EQUATIONS MODELS
C t   1   2Yt   3C t 1  ut
I t   1   2 rt   3 K t 1  v t
Yt  C t  I t  Gt
M td   1   2Yt   3 rt  wt
M td  M ts
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
Copyright Christopher Dougherty 2011.
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Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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11.07.25