Variable Subset Selection

Variable Subset Selection is a technique for estimating parameters of a Linear Model, where we identify a subset of independent variables that are most predictive of the dependent variable.

Consider the following linear model:

$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \ldots + \beta_px_p + \epsilon$

In this context, $y$ represents the dependent variable, $\beta_0$ is the intercept, $\beta_1, \beta_2, \ldots, \beta_p$ are the coefficients, $x_1, x_2, \ldots, x_p$ are the independent (predictor) variables, and $\epsilon$ denotes the error term.

Variable subset selection aims to identify a subset of the $p$ predictor variables $x_1, x_2, \ldots, x_p$ of size $d$ that are most strongly associated with the dependent variable. By narrowing down the predictors, we simplify the model while retaining its predictive power.

Once we determine this smaller subset of predictors, we fit a least squares linear regression model using only these variables. For example, if we believe that only $x_1$ and $x_2$ are significantly related to $y$, we set $d = 2$ and fit a model as follows:

$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon$

This approach assumes that the remaining variables $x_3, \ldots, x_p$ do not significantly contribute to explaining the variability in the response $y$.

But how do we determine which variables are important? In some cases, prior data analysis or domain expertise can guide the selection process. More commonly, however, a systematic approach is needed to assess the relevance of each variable.

The number of potential models that can be formed depends on the total number of predictors. Including the intercept term $\beta_0$, each of the $p$ predictors $x_1, \ldots, x_p$ can either be included in or excluded from the model, resulting in $2^p$ possible combinations to consider.