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A typical example of a model for which Two-Stage Least Squares (TSLS) is applicable is the first equation of the following system of equations:
Y2,t = G1X1,t + G2X2,t + U2,t ,
t = 1,2,3,.....,n ,
The equation for
If the error term
Model y1,t = a'X1,t + b'Y2,t + u1,t can be written compactly as
where y is the vector of stacked variables y1,t for
t = 1,2,...,n, X is the matrix with rows
As motivated in the previous section, the error vector u satisfies
Due to the latter, and some further regularity conditions, the parameter vector q can be estimated consistently and asymptotic normally by the Instrumental Variables (IV) approach, using Z as the matrix of instrumental variables. The IV approach is a special case of the Method of Moments (MM) approach. As explained in my lecture notes on the Method of Moments (and in most intermediate econometrics textbooks as well), the IV estimator qn of q takes the form
Of course, we have to require that the matrix X'PZX is nonsingular. A necessary (but not sufficient!) condition for this is that:
The number of variables in
Under regularity conditions, the IV estimator qn is asymptotically normally distributed:
in distribution, where s2 is the variance of u1,t and
The IV estimator qn is also called the TSLS estimator because it can be derived alternatively in the following two steps.
(1) Project linearly the columns of the matrix X on the space spanned by the columns on the
matrix Z. The linear projection involved is the matrix
PZX. Note that linear projection is just regression. Regress
column Xi of X on Z, i.e., estimate the linear regression model
Xi = Zdi + v by OLS.
Then the linear projection of Xi on Z is Z.di, where
(2) Regress y on PZX. Then the OLS estimator of the parameter vector involved is just the IV estimator qn.
The data have been generated artificially, as
Y1 = Y2 + X1 + X2 + U1
Y2 = X1 + X2 + X3 + X4 + U1 + U2
where X1, X2, X3, and X4 have been drawn independently from the N(0,2) distribution, and U1 and U2 have been drawn independently from the N(0,1) distribution. 500 observations on Y1,Y2,X1, X2, X3, and X4 have been generated this way. The data involved is available as file TSLSDATA.CSV in Excel CSV format (US number setting).
The procedure for the selection of the variables in the TSLS model is similar to OLS, except that now also the instrumental variables have to be selected as X variables:
Next, you have to indicate which explanatory variables are endogenous variables. In this case the only endogenous X variable is Y2:
Now you have to remove at least as many exogenous variables from the list as there are endogenous X variables. The variables to be removed are X3 and X4:
Once you click "Exogenous variables OK" the window changes to:
Click "Continue". Then the output appears:
Recall that the actual data generating process is
Click "Continue". Then the NEXTMENU window appears, which provides further options. These options have already been discussed in the guided tour on OLS estimations, and will therefore not be discussed again.