1 simultaneous equations models most of the issues relating to the fitting of simultaneous equations...

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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.

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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.


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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.


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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 .


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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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As the model stands, we have three endogenous variables, Ct , It , and Yt , and two exogenous variables, rt and Gt .


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


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We extend the model to the IS–LM model of conventional introductory macroeconomics by adding a relationship for the demand for money, Mt

d, relating it to income (the transactions demand for cash) and the interest rate (the speculative demand).


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We assume that the money market clears with the demand for money equal to the supply, Mt

s, which initially we will assume to be exogenous.


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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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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.


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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.


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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.


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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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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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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).


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So the investment function becomes as shown.


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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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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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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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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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(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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Now, the fact that Ct–1 depends on vt–1 does not, in itself, matter. The requirement is thatCt–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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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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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.

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Copyright Christopher Dougherty 2013.

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

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2013.01.27

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1 simultaneous equations models most of the issues relating to the fitting of simultaneous equations...

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