L19 - Instrumental Variables

Question 1

What is one case in which we can still use OLS to give consistent parameter estimates?

Answer 1

• In the case of recursive models for example the cobweb model
• These are models in which the endogenous variable can be solved in sequence
• If the errros in the individual equations are not correlated with each other then OLS will give consistent estimates of any individual equation taken from a recursive system.

Question 2

What is an example of a recursive model?

Answer 2

• Always use OLS when can e.g. we have proven its still consistent as it will have the lower variance

Question 3

What are the two problems with the Indirect Least Squares Estimator?

Answer 3

• the estimator might not be unique - if the equation is overidentified
• hard to work through algebra and compute the answers

Question 4

What is the instrumental variables estimator?

Answer 4

Question 5

What is the equation for β(hat) as defined by the instrumental variable estimator?

Answer 5

Question 6

Is the IV estimator consistent?

Answer 6

Question 7

What is the Variance of the IV estimator?

Answer 7

• The Variance of IV is high that OLS because it is divided by the correlation coefficient of X and Z
  
  • as it is squared it must be between 0 and 1, thus the denominator of the IV will always be smaller than that in the OLS equations thus IV is less efficient
• The Closer Z and X are correlated the more efficent the IV estimator is - instrument strength
  
  • ​But a good instrument should not be correlated with the error terms, thus the more it is correlated with the X variables, it must then be more correlated with the errors thus there is a trade off between Instrument strength and instrument validity

Question 8

Where would we get an instrument?

Answer 8

• in a system of simultaneous equations, instruments are often provided to us in the form of the exogenous variable

Question 9

When does IV and ILS give indentical parameter estimates?

Answer 9

When the equation is just identified

• IV is better however because:
  
  • dont have to worry about computed difficult algebra
  • we have formulas for the SE of the coefficient and can then compute hypothesis tests of them

Question 10

Do we always want to go for a consistent estimator at all costs?

Answer 10

• no there may be times where we pick a biased or inconsistent estimator because of it trade-off with efficiency - Mean Square error