4.2 Linear Regression Models

SW 4.1

In order to get around the curse of dimensionality that we discussed in the previous section, we will often an impose a linear model for the conditional expectation. For example,

\[ \mathbb{E}[Y|X] = \beta_0 + \beta_1 X \] or

\[ \mathbb{E}[Y|X_1,X_2,X_3] = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 \] If we know the values of \(\beta_0\), \(\beta_1\), \(\beta_2\), and \(\beta_3\), then it is straighforward for us to make predictions. In particular, suppose that we want to predict the outcome for a new observation with characteristics \(x_1\), \(x_2\), and \(x_3\). Our prediction would be

\[ \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 \]

Example 4.1 Suppose that you are studying intergenerational income mobility and that you are interested in predicting a child’s income whose parents’ income was $50,000 and whose mother had 12 years of education. Let \(Y\) denote child’s income, \(X_1\) denote parents’ income, and \(X_2\) denote mother’s education. Further, suppose that \(\mathbb{E}[Y|X_1,X_2] = 20,000 + 0.5 X_1 + 1000 X_2\).

In this case, you would predict child’s income to be

\[ 20,000 + 0.5 (50,000) + 1000(12) = 57,000 \]


The above model can be equivalently written as \[\begin{align*} Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + U \end{align*}\] where \(U\) is called the error term and satisfies \(\mathbb{E}[U|X_1,X_2,X_3] = 0\). There will be a few times where this formulation will be useful for us.