LECTURE SIMULTANEOUS EQUATIONS
The whole of this lecture notes will not be done. The essential points are: The nature of the simultaneous equations problem, the concept of identification (but not in detail), the reduced form and structural equations and the method of 2SLS.
The identification problem is dealt with on page labelled 55-57 (top third of 57).. Intuitively it is this. If you have an equation where brand share depends on advertising, price, consumer income:
S = a0 + a1A + a2P + a3Y +u1 (1)
And another where advertising depends upon the age of the product and its brand share
A = b0 + b1S + b2AGE +u2 (2)
Where u1 and u2 are error terms. How do you know when you estimate the brand share equation you are not simply picking up some of the impact of brand share on advertising? The answer to this would appear to be use 2SLS in fully specified equations (i.e. everything which influences the endogenous variable is included for all the equations in the system) and this problem does not arise.
If you use OLS to estimate 1 or 2 you will get biased and inconsistent results. The intuition is easy. In order to use OLS the right hand side variables must be independent of the error term. If they are correlated with the error term, say A is correlated with u1, then in minimising the unexplained residual we will attach too much importance to A. Now in the above case if u1 is positive, S will be greater than it should and if b1 is positive this will lead to A being greater than it should via equation (2). Hence when u1 is positive A will tend to increase, A will be correlated with u1 we cannot use OLS.
The use of 2SLS is described on pages labelled 63-65. Basically you apply OLS twice. The first time to the reduced form equations to get predicted values for the endogenous variables. The second time to the structural equation but on the right hand side of the structural equation instead of the actual value of the endogenous variables you put their predicted value (from the first stage).
By structural equation we mean the original equation system suggested by economic theory.(e.g. at the bottom of page 57). The reduced form equations simply express each endogenous variable as a function of all the predetermined and exogenous variables (i.e. non-endogenous variables) in the system. (as in equation system 6 on page 58).
Now is it always possible to estimate a simultaneous system? No. This will depend upon certain conditions being satisfied we focus on just one of these from these lecture notes: The order condition (found at the bottom of page 60, the top of page 61). In a four equation system, e.g., it is possible that we will be able to use 2SLS to estimate some of the structural equations but not others. We say these equations are identified (slightly different sue of the term than before) and the others under-identified. Note toowe use the term ‘instruments’ to refer to the exogenous and predetermined variables. This appears to me to be slightly unfortunate as this correlates with the instrumental variable technique which can also be used to estimate a simultaneous system but which is different to 2SLS.
You should also know the properties of the 2SLS estimator. These can be found on the last page of these lecture notes, but also you should supplement these by reference to a text book. (NOTE: it is an aim of the course that you should be able to learn to use the text to supplement your knowledge in this way).
Later in the course we will discuss the problem of ‘weak instruments’ and also the technique of 3SLS, which is 2SLS + another technique (seemingly unrelated regressions).
Some have suggested the top of following page is not quite clear. So:
Isolate the systematic element from the random element. Looking at equation 12 (also 9) on page 60 we see that the reduced form of Y gives Y as a function of all the predetermined variables in the model (exogenous, lagged endogenous and constant term). This is also what 2SLS does. An estimate of the systematic component……
Some have suggested the page above is not quite clear. So:
The 2SLS estimates are asymptotically unbiased.
3) The 2SLS estimates are consistent which means that the distribution collapses on the true parameter as n→∞, i.e. it is asymptotically unbiased + the variance of the estimator →0 as n→∞. remarks on 2SLS…………