L17 - Simultaneous Equation Models

Question 1

What is a Probability Limit?

Answer 1

Suppose A_T is a random variable based on a sample size T. We say that A_T converges in probability on the value μ_A if the probability that the difference between A_T and μ_A that exceeds any arbitrart small number ε goes to zero as the sample size becomes large.

Question 2

Why are Probability Limits useful?

Answer 2

• Can do things with them that we couldn’t do with expectation operators.

This yields the following rules if two random variables are based on a sample of data of the same size:

1. Product of Probability Limits
2. Ratio of Probability Limits
3. Coefficient of Probability Limits

For Example, E(A_TB_T) does not equal E(A_T)E(B_T) whereas plim(A_TB_T) = plim(A_T)plim(B_T) –> therefore it may be hard to prove unbiasedness it generally easier to prove consistency

Question 3

What are the Sufficient conditions for an estimator to be consistent?

Answer 3

• the arithmetic mean (estimator of population mean based on a sample of size T) is an example of θ_T(hat)
• This is a large sample property

Question 4

What are the Consistency assumptions for the OLS estimator?

Answer 4

Question 5

What is the proof of consistency for the OLS estimator?

Answer 5

• you divide the numerator and denomiator by T first then take plim of both sides.

Question 6

What are the reasons why the X variable may be correlated with the errrors?

Answer 6

• There is correlation between the X variables and omitted ones
• ‘the errors in variables model (EVM)’ - the X variable is measured with error - we sometimes use proxy variables that are related to the X variable we want to use (but dont have observations for) - this leads to measurements that are correlated with the errors
• ‘the simultaneous equation model (SEM)’ - the equation is one equation taken from a set of simultaneous equations e.g. demand and supply models are a both a function of each other
• The RHS variables include lagged endogenous variables e.g. T_t-1 and the errors are autocorrelated

Question 7

Why can you not test for Cov(X,u)=0 to prove there is no correlation and thus consistency?

Answer 7

• Because of the construct of the regression model the equation residual are zero by construct. Thus as they are different to equation errors it cannot be used to prove their correlation with the X variables.

Question 8

What is the Cowles Commission?

Answer 8

• Founded in 1932 as a research institute “t oavance the scientific study and development… of economic theory in its relation to mathematics and statistics”
• Set up and financed by Alfred Cowles - a stock market analyst who was concerned by the failyre to forecast or understand the 1929 crash
• Key figures were Jacob Marschak, Abraham Wald, Tryge Haavelmo and Tjalling Koopmans
• Originally based at University of Chicago but later moved to Yales where it became the Cowles Foundation
• Responsible for the development of the simultaneous equation approach to econometric modelling

Question 9

What is an example of the Cowles Commission approach?

Answer 9

• The simultaneous equation approach to econometric modelling
• There is a lag in the supply curve, in responses to changes in Quantity - this is a frequent assumption in the analysis of agricultural markets - if you have a growing season you cannot respond in a change in price immediately as crops have to grow
• this is an example of the Cowles comission approach as it is a simultaneous equation system and a probabilistic element as each equation has a random shock or disturbance attached to it.
• The shocks of demand and supply are not correlated

Question 10

Will the Demand and Supply example yield a consistent result?

Answer 10

Will estimation by OLS yield consistent results?

In this case yes, because Q does not depend on the current value of P (therefore the lagged variable can be treated as exogenous) and therefore E(Q_tu_1t) = 0.

• This is an example of the ‘Cobweb’ model. What allows us to estimate by OLS is the fact that the current value of P does not appear in the supply curve.

Question 11

What is the Cobweb model?

Answer 11

• its a dynamic model and has lags in the relationship so you dont approach the equilibrium immediately when you are away from it
• in the case of supply and demand to swap between the two curves and iterate until you move toward equilibrium
• it is useful to assume that there is a lag in one of our equations and thus has this cobweb structure as it allows use to estimate these equation by Least Squares.

Question 12

What is the formal definition of exogenous variable in econometrics?

Answer 12

• Koopmans was part of the Cowles Commission body
• This is know as strictly exogenous - however this is not necessary to prove consistency - the X variables just need to be predetermined –> not correlated with future errors but may be correlated with past ones