# More models

We cover a couple of additional models.

## Threat to identification the linear model

We want to think about what is the estimand that a linear model would recover in a couple of interesting cases.

### Omitted variable bias

We consider the following model:

together with E[\epsilon_i | X_i, Z_i] = 0 and we ask what does the regression coefficient of Y_i on X_i alone recovers?

### Measurment error bias

We consider the following model:

where we only observe X_i = X^*_{i} + u_i

together with E[\epsilon_i,u_i | X_i] = 0 and we ask what does the regression coefficient of Y_i on X_i alone recovers?

## Instrumental variable

Perhaps we are willing to assume that for a given Z_i variable we have that E[U_i | Z_i] = 0 even if it is not true for X_i. In this case we can identify \beta from

assuming that E[Z_i X'_i] is square and invertible. We then define the instrumental variable estimator as

This can address both earlier mentioned issues. Two important assumptions:

- exclusion restriction: E[U_i | Z_i] = 0
- relevance: E[Z_i X'_i] is invertible

We can show consistency and asymptotic normality. But what about unbiasedness?

TBD in class!

### 2SLS

In more general case, when for instance one has more instruments than regressors. We introduce the 2SLS estimator:

$$ \beta_n^{2SLS} = (X'_n P_n X_n)^{-1} X'_n P_n Y_n $$

where P_n = Z_n ( Z'_n Z_n)^{-1} Z'_n. We see that P_n P_n = P_n and so we can write

and so this is like regressing Y_n on P_n X_n where we note that P_n X_n = Z_n ( Z'_n Z_n)^{-1} Z'_n X_n, is the predicted value from the regression of X_n on Z_n.