L18 - Identification and Indirect Least Squares

Question 1

What is the difference between the structural and reduced form of an economic model?

Answer 1

• The structural form of a model is the set of equations determined by economic theory. Each equation will typically contain more than one endogenous variable.
  
  • alpha and beta are the structural parameter –> they describe the economic model - the ones we are interested in
• The reduced form of a model is obtained by solving it so that each equation contains only one endogenous variable.
  
  • we can get the reduced form by subbing Q₁ into the demand equation

Question 2

Why do we want to derive the reduced form?

Answer 2

• As all the RHS variable are predetermined and not endogenous we can solve the equation using OLS
  
  • thus they will not be correlated with the error terms of their equations - not effected by biasedness and consistency issues
• We can then find the parameter of the reduced form to then find the parameters of the structural form - the ones we are interested in
  
  • this is the method of indirect least squares

Question 3

What happens when their is a unique mapping from the reduced form coefficients to the structural form parameters?

Answer 3

• The system is just identified
  
  • there is just enough information to calculate the unknown structural parameter
    
    • this is not always the case there may be too much information to determine them or there may be multiple answers

the errors is the reduced forms will usually be correlated with each other even if the errors of the structural form are not

Question 4

How do you generally calculate Indirect Least Squares?

Answer 4

• Substitute so that you have a pair of equation that have been adapted from a structural to a reduced form
• Estimate their parameters using OLS
• Sub these into the implicit structural form estimate equations to determine the unknown structural form parameters

ILS is possible when a system (or equation) is just identified

• the number of structural parameter = the number of reduced form parameter

Question 5

When is a model under/over identified?

Answer 5

• underidentified - structural parameter > reduced form parameters
• overidentified structural parameter < reduced form parameters

Exogenous variables are important as they help us generate extra equations in the reduced form –> in this case there are none therefore the system is underidentified

Question 6

What is the General Structural Form equation?

Answer 6

Question 7

How do you derive the reduced form from the general structural form?

Answer 7

• We can always obtain consistent estimates of the reduced form by OLS.
• If there is possible to use the reduced form parameters to solve for the structural parameters then we say that a model (or an equation) is identified.
• It is possible for some equations in a model to be identified while other are not. In our example the model as a whole cannot be identified because there are three structural parameters but only two reduced form parameters.
• However, it is possible to show that the first equation is identified because:

β₁₁=π₁₁/π₂₁

Question 8

What is the general feature of simultaneous equation models?

Answer 8

Question 9

What are the General Coniditions for Equation Identification?

Answer 9

• K-k –> the number of excluded exogenous variables (total number of exogenous variables - those actual present in the equation)
• the minus 1 is a normalisation restriction - as one variable is always set equation to 1? (8.30)

Question 10

What are the Order condition for identification?

Answer 10