1 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.
2 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.
3 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.
4 SIMULTANEOUS EQUATIONS MODELS Private sector consumption, C t, is determined by income, Y t. Private sector investment, I t, is determined by the rate of interest, r t. Income is defined to be the sum of consumption, investment, and government expenditure, G t.
5 SIMULTANEOUS EQUATIONS MODELS u t and v t 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.
6 SIMULTANEOUS EQUATIONS MODELS As the model stands, we have three endogenous variables, C t, I t, and Y t, and two exogenous variables, r t and G t.
7 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.
8 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.
9 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, M t d, relating it to income (the transactions demand for cash) and the interest rate (the speculative demand).
10 SIMULTANEOUS EQUATIONS MODELS We assume that the money market clears with the demand for money equal to the supply, M t s, which initially we will assume to be exogenous.
11 SIMULTANEOUS EQUATIONS MODELS We now have five endogenous variables (the previous three, plus r t and M t d ) and two exogenous variables, G t and M t s. On the face of it, the consumption and investment functions are both overidentified and the demand for money function exactly identified.
12 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.
13 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.
14 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.
15 SIMULTANEOUS EQUATIONS MODELS C t–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.
16 SIMULTANEOUS EQUATIONS MODELS Hence, in the investment equation, C t–1 could be used as an instrument for r t. To serve as an instrument, it must be correlated with r t, but this is assured by the fact that it is a determinant of C t, and thus Y t, and so with r t via the demand for money relationship.
17 SIMULTANEOUS EQUATIONS MODELS To give another example, many models include the level of the capital stock, K t–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.
18 SIMULTANEOUS EQUATIONS MODELS Then, since K t–1 is a determinant of I t, and I t is a component of Y t, K t–1 can serve as an instrument for Y t in the consumption function.
19 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.
20 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.
21 SIMULTANEOUS EQUATIONS MODELS We will discuss the point in the context of the use of C t–1 as an instrument for r t in the investment equation. For C t–1 to be a valid instrument, it must be distributed independently of the disturbance term v t. We acknowledge that C t–1 will be influenced by v t–1.
22 SIMULTANEOUS EQUATIONS MODELS (Considering the interactions between the relationships at time t – 1, v t–1 influences I t–1 by definition and hence Y t–1 because I t–1 is a component of it. Thus v t–1 also influences C t–1, by virtue of the consumption function.)
23 SIMULTANEOUS EQUATIONS MODELS Now, the fact that C t–1 depends on v t–1 does not, in itself, matter. The requirement is that C t–1 should be distributed independently of v t. So, provided that v t and v t–1 are distributed independently of each other, C t–1 remains a valid instrument.
24 SIMULTANEOUS EQUATIONS MODELS But suppose that there is time persistence in the values of the disturbance term and v t is correlated with v t–1. Then C t–1 will not be distributed independently of v t and will not be valid instrument.
25 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.
Copyright Christopher Dougherty 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