Contemporary design research: emerging multisensorality

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MODULE 5: SIMULTANEOUS EQUATION, BINARY CHOICE, AND

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5.1.2.0 OBJECTIVE

The main objective of this unit is to demonstrate to the students that in practice most economic relationships interact with others in a system of simultaneous equations and when this is the case the application of OLS to a single relationship in isolation yields biased estimates.

5.1.3.0 MAIN CONTENTS

5.1.3.1 Simultaneous Equations Models: Structural and Reduced Form Equations

As explained earlier in other modules, measurement error is not the only probable cause why the fourth Gauss–Markov condition may not be satisfied. Simultaneous equations bias is another. To illustrate this; suppose there is an investigation on the determinants of price inflation and wage inflation. For ease, it would be better to start with a very simple model that supposes that p, the annual rate of growth of prices, is related to w, the annual rate of growth of wages, it being assumed that increases in wage costs force prices upwards:

That is;

…[5.01]

Here, w is related to pand U, the rate of unemployment, workers protecting their real wages by demanding increases in wages as prices rise, but their ability to do so being the weaker, the higher the rate of unemployment ( ). Which is stated as:

…[5.02]

where, are disturbance terms

Clearly, this simultaneous equations model involves a certain amount ofcomplexity:wdetermines p in the first equation [5.01], and in turn,p helps to determine w in the second [5.02]. For better clarity in resolving this complexity, we need to make a distinction between endogenousand exogenous variables. Endogenous variables are variables whose values are determined by the interaction of the relationships in the model. Exogenous ones are those whose values are determined

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externally. Thus in the present case,p and ware both endogenous, and U is exogenous.

The exogenous variables and the disturbance terms ultimately determine the values of the endogenous variables, once the complexity is cleared. The mathematical relationships expressing the endogenous variables regarding the exogenous variables and disturbance terms are known as the reduced form equations. The original equations that we wrote down when specifying the model are

described as the structural equations. We will derive the reduced form equations for p and w. To obtain that for p, we take the structural equation for p and substitute for w from the second equation:

( ) …[5.03]

Hence,

( ) …[5.04]

and so we have the reduced form equation for p;

( ) …[5.05]

Similarly we obtain the reduced form equation for w:

( )

…[5.06]

Hence

( ) …[5.07]

and so

…[5.08]

5.1.3.2 Simultaneous Equations Bias

In almost all simultaneous equations models, the reduced form equations express the endogenous variables regarding all of the exogenous variables and all of the

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disturbance terms. You can see that this is the case with the price inflation/wage inflation model. In this model, there is only one exogenous variable, U.

wdepends on it directly; p does not depend on it directly but does so indirectly because w determines it. Similarly, both p and wdepend on , p directly and w indirectly. And both depend on , w directly and p indirectly.

The dependence of w on means that OLS would yield inconsistent estimates if used

to fit

equation [5.01], the structural equation for p. w is a stochastic regressor and its random component is not distributed independently of the disturbance term . Similarly the dependence of p on

means that OLS would yield inconsistent estimates if used to fit [5.02]. Since [5.01] is a simple regression equation, it is easy to analyze the large-sample bias in the OLS estimator of .

5.1.5.0 SUMMARY

In this unit, we started by explaining structural and reduced form of equations which was illustrated by showing that measurement error is not the only probable cause why the fourth Gauss–Markov condition may not be satisfied for which the biasness of simultaneous equations is an example. We then went further to explain the simultaneous equations bias.

5.1.4.0 CONCLUSION

In this unit, Simultaneous equations estimation is discussed. The structural and reduced form of equations as it relates to Simultaneous equations model bias is also explained. Clearly, for deeper understanding of the equation models, the students should make a distinction between endogenousand exogenous variables in Simultaneous equations estimation.

5.1.6.0 TUTOR-MARKED ASSIGNMENT

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1.) Simple macroeconomic model consists of a consumption function and an income identity:

whereC is aggregate consumption, I isaggregate investment, Y is aggregate income, and u is a disturbance term. On the assumption that I is exogenous, derive the reduced form equations for C and Y.

2.) From the model above, demonstrate that OLS would yield inconsistent results if used to fit the consumption function, and investigate the direction of the bias in the slope coefficient.

5.1.7.0 REFERENCES /FURTHER READING

Maddala, G. S., &Lahiri, K. (1992).Introduction to econometrics (Vol. 2). New York.

Dougherty, C. (2007). Introduction to econometrics. Oxford University Press, USA.

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Unit 2: Binary Choice and LimitedDependent Models andMaximum Likelihood