The Method Of Instrumental Variables And Its Application

The method of Instrumental Variables is one of the most important estimation methods in econometrics, which has made a huge contribution to the development of econometrics. The method of Instrumental Variables is used to define a series of estimation methods. It means that the estimation of an equation can be realised by choosing an appropriate instrumental variable. In Model Estimation , Instrumental Variables are used, as a instrument, to substitute the random explanatory variables which are related to the random error terms. There is a misunderstanding that the original model will be changed in this method since the random explanatory variables in the original model is considered as a instrument. However, instrumental variable does not change the original model. It only substitutes the random explanatory variables in the process of parameter estimates actually. There are several ways to select instrumental variables, and different estimation results will be led to as different methods being used. In the dissertation, the Methods of Instrumental Variables and its application will be introduced and discussed in four parts, as followed.In Part One, the limitation of ordinary least square estimator is discussed firstly. For model:When E(ut|Xt) = 0 is not true, generally, ordinary least square estimator has bias unconsistent.. There are two situations. The first one is variable error, which means that errors exist in the measure of independent variables. It is the differences between the measure data and the real data that lead to correlation of regressors with measure errors and error terms. This is called OLS estimating unconsistently. The second one is simultaneous equations. In the economy theories, it is believed that two and more endogenous variable are determined simultaneously, which means that all the endogenous variables are absolutely related to all the error terms in the equations. In this situation, all the endogenous variables in the regression function estimated by ordinary least square are not the answer for the question. Therefore, the correlation of all the regressors with measure errors and all the error terms exists as long as simultaneous equations model, and results in the OLS estimating unconsistently. The definition of Instrumental Variables is given in Section Two: the Instrumental Variables must satisfy three requirements simultaneously, highly related to the random explanatory variables substitute; not related to the random error terms; and not related to the other explanatory variables in the model. Take model (1.1.1) as an example,In Part Two, the application of Methods of Instrumental Variables in estimation of single equation linear model and simultaneous equations model is introduced briefly. Take estimation of single equation linear model (2.1.1) as an example, Z is the instrumental variable to substitute x, the parameter estimates is:The parameter estimates gained by using Methods of Instrumental Variables have the following characteristics:It is biased in small-sample,it is asymptotic unbiased in large-sample,furthermore the distribution of estimator focus on true structural parameter. In estimation of simultaneous equations model,make application to national income obtain the result,In small-sample,namelyα|^is not unbiased ofα.But in large-sample,α₁|^is consistent estimator ofα₁. after that,we introduce how to choose the best instrumental variables.Theorem 2.3.1 Letβ|^_IV1,β|^_IV2,...β|^_IVk the k instrumental variables estimator of parameter modelβ, and then is the best instrumental variables which isconsisted ofβ|^_IV1,β|^_IV2,...,β|^_IVkTheorem 2.3.2 If axe the q groups instrumental variables estimator ofβ的q,and Var(β|^_ij=σ_i² If then is the best instrumental variables which is consisted by q groups instrumental vari-ables,where In Part There,the two important kinds of Methods of Instrumental Variables, Indirect Least Square and Two Stage Least Squares are introduced in P art Three. Indirect Least Square is a kind of single equation estimator method in simultaneous equations econometric model. It should be applied to the just-identified equation in structural model. The main procedures are: expressing all the endogenous variables in the structural equation estimated as the function of all the predetermined explanatory variable and random variables, leading to the relative reduced equation ; making all the reduced equations satisfy the requirements of hypothesis of ordinary least square estimator, so that OLS method can be used to get the estimation of reduced parameters; applying the results of first two steps to the reduced form, since the equation is the best realised one, the estimation of structural equation parameter can be gained indirectly. The parameter estimation gained from the reduced form is linear, unbiased and least variance.Two Stage Least Squares is a kind of single equation method which is suitable for the estimation of structural equation parameter in best realised equations as well as oyeridentified equations, and it is also most frequently used estimation method of simultaneous equations model parameter.2SLS is brought forward to eliminate errors from the existing of endogenous variable in explanatory variables in the simultaneous equations model. Its first phase is to get the estimation of endogenous variable through applying OLS to reduced equation in the model. In the second phase, the endogenous variable from the right side of the structural equation are applied to the estimation results of the first phase, so that the random variables are determined. Then the estimation results of structural parameter are gained by applying OLS to the structural equation again. 2SLS is asymptotic unbias.ILS and 2SLS are both Methods of Instrumental Variables, and ILS, 2SLS, IV are equivalence when structural equation are best realised.From It can be concluded that ISL is actually a method of Instrumental Variables. Choose k predetermined explanatory variable one by one as the instrumental variables of g₁-1 endogenous variable and k₁ predetermined explanatory variable in the structural equation. The predetermined explanatory variable which are originally chosen as explanatory variables also choose other predetermined explanatory variable as their instrumental variables other that using themselves as their instrumental variables.Formandare totally equivalence. It can be concluded that 2SLS is a method of instrumental variables, which uses the linear combination of all the predetermined explanatory variable as the instrumental variables of the endogenous explanatory variables in the structural equation.Use the way of IV to express the results of ILS, 2SLS, and IV, like (3.3.3), (3.3.6), (3.3.7). On the surface, there are differences among them. Actually, they are equivalence when the structural equation are best realised.In the final part of the dissertation, the equivalence of ILS, 2SLS, and IV, when structural equation are best realised, has been proved according to simulation results of National Income Model through the China macro-economics data by State statistical bureau.