Assignment:
Terms and Concepts
Explain the zero condition mean assumption E(u|x) = 0
Define an unbiased estimator
Explain the zero mean and zero covariance assumption E(u) = 0 and Cov(u, x) = 0
Define an exogenous explanatory variable
Define an endogenous explanatory variable
List the three main causes of endogeneity
Omitted variables
Measurement Error
Simultaneity
Describe omitted variable bias - our example was ability
Define an instrumental variable
Know that an instrumental variable must satisfy:
Cov (z, u) = 0 - z should have no partial effect on y after x and omitted variables are controlled for. z should be uncorrelated with omitted variables. (instrument exogeneity).
Cov (z, x) ≠ 0 - z should be related to x. (instrument relevance)
Write the reduced form for an endogenous variable
Show how to write the IV estimator in terms of the population moments Cov(z, x) and Cov (z, y). What assumption is needed?
Write the formula for the IV estimator using sample analogs of the Cov (z, y) and Cov (z, x)
Define a consistent estimator - write in terms of probability limits (plims).
Sketch a short proof that the instrumental variables estimator is consistent. I am thinking about Problem #1 on your first homework here. You may rely on the law of large numbers and you may rely on the result that the sample variance and sample covariance are consistent estimators of the population variance and the population covariance.
Be able to write the variance of the IV estimator in the simple case of one explanatory variable. Be able to compare the variance of the IV estimator to the variance of the OLS estimator. Illustrate the cost of estimating a model by IV.
Review Lab # 2.
Describe what happens if (z, u) are moderately correlated and the instruments are weak.
Describe and explain 2 consequences of weak instruments
Describe how you would test the relationship between z and x in both the simple regression model with one explanatory variable and the multiple regression model.
What do we do if we have one endogenous explanatory variable and more than one instrument? Describe the two parts of Two-Stage Least Squares (2SLS).
Describe the Hausman test. What is the null hypothesis? Be able to explain the steps you would take to carry out the Hausman test.
Review Lab # 3
Be able to prove that the OLS estimator for the coefficient for the variable X is biased towards zero (attenuation bias) when X is measured with error and we have the classical errors-in-variables assumption.
Be able to discuss simultaneous equations bias using a simple supply and demand model.
Be able to illustrate the problem using a supply and demand graph.
In the context of a simultaneous equations model, be able to illustrate simultaneous equations bias. I am not looking for a formal proof. Suppose we have two equations.
Focus on the first equation and write the endogenous right-hand side variable as a reduced form. Then discuss why the right-hand side endogenous variable is correlated with the error term in the first structural equation. The trick is to look at the reduced form error term and note that it is a function of the error term in the first structural equation.
Discuss the order condition for identification. How would you show that the order condition is satisfied?
Discuss the rank condition for identification. How would you show that the rank condition is satisfied in the simple two-equation model?
Review Lab #4.
What is a binary dependent variable?
What is a latent variable model for a binary dependent variable?
List three reasons why we should not estimate the binary dependent variable model as a linear probability model.
What is the underlying cumulative distribution function for the logit model?
What is the underlying cumulative distribution function for the probit model?
Write the expression for the marginal effect of a continuous variable on the probability
that Y = 1 for a binary dependent variable model. Simply use the g( ) function. I just want the general expression.
Write the expression for the marginal effect of a dummy variable on the probability that Y = 1 for the binary dependent variable model. Simply use the G ( ). I just want the general expression.
Be able to write the expressions for the average marginal effects for both continuous and binary explanatory variables.
Be able to interpret the marginal effects.
Review Lab # 5 (Parts I and II).
Be able to write down the log likelihood function (using the G( ) function) for the simple binary dependent variable model.
Be able to use log likelihoods (Likelihood Ratio Test) to test general exclusion restrictions.
Be able to calculate the Pseudo R-Square
Review Lab # 6
Describe when we would use a multinomial logit model.
Be able to write the odds ratio for two alternatives, say Pi3/Pi1
Write the expression for the change in the log odds with respect to a change in the explanatory variable.
Be able to interpret marginal effects from the multinomial logit model.
What is an ordered probit model?
Be able to construct the relevant probabilities for an ordered probit model.
Be able to interpret the marginal effects from an ordered probit.
Review Lab 7, Parts I and II and III.