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Hayden Economics. Source: Economic Report of the President, To give some numerical results, we obtained the data shown in Table First we estimate the reduced-form equations, regressing separately price and quantity on per capital real consumption expenditure. The results are as follows:. These results show how OLS can distort the "true" picture when it is applied in inappropriate situations.

We have seen that the estimators of the reduced-form coefficients are consistent and under appropriate assumptions also best unbiased or asymptotically efficient see footnote 9.

Do these properties carry over to the ILS estimators? It can be shown that the ILS estimators inherit all the asymptotic properties of the reduced-form estimators, such as consistency and asymptotic efficiency.

But the small sample properties such as unbiasedness do not generally hold true. It is shown in Appendix 20A, Section 20A. For large-sample size, however, standard errors of the structural coefficients can be obtained approximately.

However, as shown in App. The income equation, a hybrid of quantity-theory-Keynesian approaches to income determination, states that income is determined by money supply, investment expenditure, and government expenditure.

The money supply function postulates that the stock of money is determined by the Federal Reserve System on the basis of the level of income. Obviously, we have a simultaneous-equation problem, which can be checked by the simultaneity test discussed in Chapter Applying the order condition of identification, we can see that the income equation is underidentified whereas the money supply equation is over-identified.

There is not much that can be done about the income equation short of changing the model specification. The overidentified money supply function may not be estimated by ILS because there are two estimates of p21 the reader should verify this via the reduced-form coefficients. As a matter of practice, one may apply OLS to the money supply equation, but the estimates thus obtained will be inconsistent in view of the likely correlation between the stochastic explanatory variable Y1 and the stochastic disturbance term u2.

Suppose, however, we find a "proxy" for the stochastic explanatory variable Y1 such that, although "resembling" Y1 in the sense that it is highly correlated with Y1 , it is uncorrelated with u2.

Such a proxy is also known as an instrumental variable see Chapter If one can find such a proxy, OLS can be used straightforwardly to estimate the money supply function. But how does one obtain such an instrumental variable? The process is as follows:. Stage 1. To get rid of the likely correlation between Y1 and u2, regress first Y1 on all the predetermined variables in the whole system, not just that equation.

In the present case, this means regressing Y1 on X1 and X2 as follows:. From Eq. Note that Comparing What is the advantage of It can be shown that although Y1 in the original money supply equation is correlated or likely to be correlated with the disturbance term U2 hence rendering OLS inappropriate , Y1t in As a result, OLS can be applied to As this two-stage procedure indicates, the basic idea behind 2SLS is to "purify" the stochastic explanatory variable Y1 of the influence of the stochastic disturbance U2.

This goal is accomplished by performing the reduced-form regression of Y1 on all the predetermined variables in the system Stage 1 , obtaining the estimates Y1t and replacing Y1t in the original equation by the estimated Y1t, and then applying OLS to the equation thus transformed Stage 2.

The estimators thus obtained are consistent; that is, they converge to their true values as the sample size increases indefinitely. The reason is as follows: From Now even if the predetermined variables are truly nonstochastic, the ft's, being estimators, are stochastic.

Therefore, Yjt is stochastic too. Now from our discussion of the reduced-form equations and indirect least-squares estimation, it is clear that the reduced-coefficients, the n's, are functions of the stochastic disturbances, such as U2.

But as noted previously, this correlation disappears as the sample size tends to infinity. The upshot of all this is that in small samples the 2SLS procedure may lead to biased estimation. Both X3 and X4 are predetermined.

It can be readily verified that both Eqs. To apply 2SLS, we proceed as follows: In Stage 1 we regress the endogenous variables on all the predetermined variables in the system. In Stage 2 we replace Y1 and Y2 in the original structural equations by their estimated values from the preceding two regressions and then run the OLS regressions as follows:.

The estimates thus obtained will be consistent. It can be applied to an individual equation in the system without directly taking into account any other equation s in the system. Hence, for solving econometric models involving a large number of equations, 2SLS offers an economical method.

For this reason the method has been used extensively in practice. Unlike ILS, which provides multiple estimates of parameters in the overidentified equations, 2SLS provides only one estimate per parameter. It is easy to apply because all one needs to know is the total number of exogenous or predetermined variables in the system without knowing any other variables in the system.

Although specially designed to handle overidentified equations, the method can also be applied to exactly identified equations. If the R2 values in the reduced-form regressions that is, Stage 1 regressions are very high, say, in excess of 0.

But this result should not be surprising because if the R2 value in the first stage is very high, it means that the estimated values of the endogenous variables are very close to their actual values, and hence the latter are less likely to be correlated with the stochastic disturbances in the original structural equations. In other words, in this case, the Ys will be very poor proxies for the original Y's. Notice that in reporting the ILS regression in But we can do this for the 2SLS estimates because the structural coefficients are directly estimated from the second-stage OLS regressions.

There is, however, a caution to be exercised. The estimated standard errors in the second-stage regressions need to be modified because, as can be seen from Eq. However, the modification required can be easily effected by the formula given in Appendix 20A, Section 20A. The statistical justification of the 2SLS is of the large-sample type. When there are no lagged endogenous variables,. If these two conditions are satisfied, the sampling distribution of 2SLS coefficient estimators becomes approximately normal for large samples.

When the equation system contains lagged endogenous variables, the consistency and large-sample normality of the 2SLS coefficient estimators require an additional condition,. If the [disturbances appearing in the various structural equations are] not independently distributed, lagged endogenous variables are not independent of the current operation of the equation system.

If these variables are nevertheless treated as predetermined in the 2SLS procedure, the resulting estimators are not consistent. To illustrate the 2SLS method, consider the income-money supply model given previously in Eqs.

As shown, the money supply equation is overidentified. To estimate the parameters of this equation, we resort to the two-stage least-squares method. The data required for analysis are given in Table Stage 1 Regression.

We first regress the stochastic explanatory variable income Y1, represented by GDP, on the predetermined variables private. Discover the hidden rules and beat the rich at their own game.

They resent those who have it, but spend their total lives attempting to get it for themselves. The reason an immense majority of individuals never accumulate a substantial savings is because they don't comprehend the nature of money or how it works.

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## Elements of Econometrics: Second Edition

Hayden Economics. Source: Economic Report of the President, To give some numerical results, we obtained the data shown in Table First we estimate the reduced-form equations, regressing separately price and quantity on per capital real consumption expenditure. The results are as follows:. These results show how OLS can distort the "true" picture when it is applied in inappropriate situations.

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## Elements of Econometrics

After earning his PhD in Economics with a minor in Statistics from Stanford under Kenneth Arrow in , [2] Kmenta held academic positions at the University of Wisconsin —65, Michigan State University —73, and the University of Michigan emeritus and was a visiting faculty member at universities in five countries. Kmenta received 24 academic honors, awards, and prizes during his career, beginning with being made a fellow of the American Statistical Association in and a fellow of the Econometric Society in , and stretching through when he received the NEURON Award for Lifetime Achievement in Economics. Kmenta wrote extensively on econometric model building as well as econometric methods. He edited two books with James B. A wide-ranging econometrician, his papers analyze topics as disparate as small sample properties of estimators, missing observations, estimation of production function parameters, and ridge regression among many others. As a result, his work is referenced in publications in medicine, political science, insurance underwriting, antitrust litigation, and energy issues, to list but a few.