* Explain how it is possible to estimate the partial effect of the exogenous variables, even if ceteris paribus assumption is false.

We can estimate the partial effect of the exogenous variables, even if ceteris paribus assumption is false. It is possible by estimating parameters of the linear model. It let us get results, which we could obtain by comparing observations which do differ in values of one explanatory variable. That way we can estimate the effect on variable yicausedby changes of xkiceteris paribus even though the ceteris paribus assumption is false for all observation in the sample.

* Write down the linear model in conditional expectation form and in the error form and explain why the conditional expectation form of the model is more realistic than the assumption that the regressors are deterministic.

Model in conditional expectation form:

E(a1x1+……….+aKxK|z)=a1E(x1)+……….+aKE(xK)

Model in error form:

y=E(y|x)+u

E(u|x)=0

The conditional expectation form of the model is more realistic than the assumption that the regressors are deterministic, because can be used for nonexperimental data. If we use Clasical Regression model assuming that regressors are deterministic (fixed) it will not be realistic for the nonexperimental data.

* What assumptions about the error term are related to random sample assumption? Give some examples when the random sample assumption can fail in a cross-section. How can we deal with such cases?

Random sample assumption can fail in a cross-section when samples are not representative of underlying population, in fact some data sets are constructed by intentionally oversampling different parts of the population.

2. Ordinary least squares and instrumental variable estimation * In what case the omitted variable can result in the asymptotic bias of an estimator? When the effect of an omitted variable is negligible?

Consider following model, which assumes additive effect of the omitted variable q:

E(y| x1, x2,…, xK, q)= β0 + β1 x1 + β2 x2 +…+ βK xK +γq

We are interested in partial effects of xj (j=1,…,K) on y holding all other variables, including q constant. We can also write a model in error form:

y= β0 + β1 x1 + β2 x2 +…+ βK xK +γq +υ

where υ is structural error and E(υ | x1, x2,…, xK, q)=0

One way to handle the nonobservability of q is to put it into the error term. In doing so, nothing is lost by assuming and E(q)=0 because an intercept is included in equation in error form. Putting q into the error term means we rewrite error term equation as:

y= β0 + β1 x1 + β2 x2 +…+ βK xK +u

where u= γq +υ and E( u)=0, u is uncorrelated with x1, x2,…, xK if and only if q is uncorrelated with x1, x2,…, xK

If q is correlated with any x1, x2,…, xK, then so is u, and we have an endogeneity problem. We cannot expect OLS to consistently estimate any βj (j=1,…,K) and there exists asymptotic bias resulting from omitted variable q.

If q is not correlated with any x1, x2,…, xK then the effect of omitted variable q is negligible.

* Explain what are the necessary conditions for an IV estimator to be consistent. Give an example when the IV estimator can be used to solve the omitted variable problem.

Conditions for an IV estimator to be consistent:

Consider the model:

y= β0 + β1 x1 + β2 x2 +…+ βK xK +u, E( u)=0 and Cov (xj,u)=0 where only xK is correlated with u (xK is endogenous)

The method of instrumental variables (IV) provides a general solution to the problem of an endogenous explanatory variable. To use the IV approach with xK endogenous, we need an observable variable z1 (instrumental variable) that satisfies two conditions:

1) variable z1 is uncorrelated with error term u: Cov( z1, u)=0 2) Coefficient on z1 in linear projection of xK on x1, x2,…, xK-1, z1 is not equal to zero: xK = δ0 + δ 1 x1 + δ 2 x2 +…+ δ K-1 xK-1 +θ1z1+rK, θ1≠0

When we have more than one instrument for endogenous variable...