# Semi-parametric regression

## The Robinson estimator

### The Robinson estimator

#### Partialling out

$$y_i=\mathbf x_i\theta +g(\mathbf z_i) +\epsilon_i$$

Consider:

$$E(y_i|\mathbf z_i)=E(\mathbf x_i\theta +g(\mathbf z_i) + \epsilon_i|\mathbf z_i)$$

$$E(y_i|\mathbf z_i)=E(\mathbf x_i\theta|\mathbf z_i)+E(g(\mathbf z_i)|\mathbf z_i) + E(\epsilon_i|\mathbf z_i)$$

$$E(y_i|\mathbf z_i)=E(\mathbf x_i|\mathbf z_i)\theta+g(\mathbf z_i)$$

We can now remove the parametric part:

$$y_i-E(y_i|\mathbf z_i)=\mathbf x_i\theta +g(\mathbf z_i) + \epsilon_i - E(\mathbf x_i|\mathbf z_i)\theta -g(\mathbf z_i)$$

$$y_i-E(y_i|\mathbf z_i)=(\mathbf x_i- E(\mathbf x_i|\mathbf z_i))\theta +\epsilon_i$$

We define:

• $$\bar y_i = y_i-E(y_i|\mathbf z_i)$$

• $$\bar x_i = \mathbf x_i- E(\mathbf x_i|\mathbf z_i)$$

$$\bar y_i =\bar x_i \theta +\epsilon_i$$

#### Estimating $$\bar y_i$$ and $$\bar x_i$$

So we can use OLS if we can estimate.

• $$E(y_i|\mathbf z_i)$$

• $$E(\mathbf x_i|\mathbf z_i)$$

We can do this with non-parametric methods.

### Bias and variance of the Robinson estimator

#### Moments of the Robinson estimator

If IID then

$$Var (\hat \theta) =\dfrac{\sigma^2_\epsilon }{\sum_i(x_i-\hat X_i)^2}$$

Otherwise, can use GLM

What are the properties of the estimator?

$$E[\hat \theta ]=E[\dfrac{\sum_i (X_i-\hat X_i)(y_i-\hat y_i)}{\sum_i(x_i-\hat X_i)^2}]$$