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Unformatted text preview: n the regression corresponding to equation (14).
3. Run the regression (14) again, but now also including the fitted values
as additional regressors:
4. Use an F-test to test the joint restriction that λ2 = 0, and λ3 = 0. If the
null hypothesis is rejected, Y2 and Y3 should be treated as endogenous. 16 Recursive Systems
• Consider the following system of equations:
(21-23) • Assume that the error terms are not correlated with each other. Can we estimate
the equations individually using OLS? • Equation 21: Contains no endogenous variables, so X1 and X2 are not correlated
with u1. So we can use OLS on (21).
Equation 22: Contains endogenous Y1 together with exogenous X1 and X2. We
can use OLS on (22) if all the RHS variables in (22) are uncorrelated with that
equation’s error term. In fact, Y1 is not correlated with u2 because there is no Y2
term in equation (21). So we can use OLS on (22). • 17 Recursive Systems (cont’d) • Equation 23: Contains both Y1 and Y2; we require these to be
uncorrelated with u3. By similar arguments to the above, equations
(21) and (22) do not contain Y3, so we can use OLS on (23). • This is known as a RECURSIVE or TRIANGULAR system. We do
not have a simultaneity problem here. • But in practice not many systems of equations will be recursive... 18 Indirect Least Squares (ILS) • Cannot use OLS on structural equations, but we can validly apply it to
the reduced form equations. • If the system is just identified, ILS involves estimating the reduced
form equations using OLS, and then using them to substitute back to
obtain the structural parameters. • However, ILS is not used much because
1. Solving back to get the structural parameters can be tedious.
2. Most simultaneous equations systems are over-identified. 19 Estimation of Systems
Using Two-Stage Least Squares • In fact, we can use this technique for just-identified and over-identified
systems. • Two stage least squares (2SLS or TSLS) is done in two stages: Stage 1:
• Obtain and estimate the reduced form equations using OLS. Save the
fitted values for the dependent variables.
• Estimate the structural equations, but replace any RHS endogenous
variables with their stage 1 fitted values.
20 Estimation of Systems
Using Two-Stage Least Squares (cont’d)
Example: Say equations (14)-(16) are required. • • Stage 1:
Estimate the reduced form equations (17)-(19) individually by OLS and
obtain the fitted values,
Replace the RHS endogenous variables with their stage 1 estimated values:
(24)-(26) • Now and will not be correlated with u1,
and will not be correlated with u3 . will not be correlated with u2 ,
21 Estimation of Systems
Using Two-Stage Least Squares (cont’d) • It is still of concern in the context of simultaneous systems whether the
CLRM assumptions are supported by the data. • If the disturbances in the structural equations are autocorrelated, the
2SLS estimator is not even consistent. • The standard error est...
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This note was uploaded on 09/20/2013 for the course FINA 5170 taught by Professor Janebargers during the Summer '13 term at Greenwich School of Management.
- Summer '13
- Financial Markets