Merely *a* argument, but one that I like.

I’ve had a discussion with a colleague on the selection of variables in a model. The discussion boils down to the following question:

- supply all the variables that you have into the model and remove the ones that add a risk of overfitting
- start out small and add values to make the model more and more complex?

You should always use a test set to determine the performance of your models and you should apply strategies to prevent it from overfitting, but starting our small and growing the model brings inherit bias into the model. In this document I will provide a mathematical proof of what might be dangerous of this approach.

In this document I’ve assumed that you have some remembrance of college level linear algebra. The following equations should feel readable to you:

\[ \begin{aligned} M_x & = I_n - X(X'X)^{-1}X' \\\ M_x X & = I_nX - X(X'X)^{-1}X'X \\\ & = X - XI_n\\\ & = 0 \end{aligned} \]

From statistics you should hopyfully feel familiar with the following:

\[ \begin{aligned} Y & = X\beta + \epsilon \text{ where } \epsilon \sim N(0,\sigma) \\\ \hat{\beta} & = (X'X)^{-1} X'Y \\\ \mathbb{E}(\hat{\beta} ) & = \mathbb{E}\big((X'X)^{-1} X'Y)\big) \end{aligned} \]

I am going to proove that for linear models you will introduce bias if you use few variables and include more and more as you are building a model and that this will not happen when you start with a lot of variables and reduce. For each case I will show what goes wrong in terms of the expected value of the \(\beta\) variables.

A few things on notation, I will refer to the following linear regression formula:

\[Y = X_1\beta_1 + X_2\beta_2 + \epsilon\]

In this notation, \(X_1,X_2\) are matrices containing data, not vectors, such that \(\beta_1,\beta_2\) are matrices as well.

Suppose that the true model is given through:

\[Y = X_1\beta_1 + X_2\beta_2 + \epsilon\]

If we start out with a smaller model, say by only looking at \(\beta_1\) we would estimate for $ Y = X_1_1 + $ while the whole model should be $ Y = X_1_1 + X_2_2 + $. Then our expected value of \(\beta_1\) can be derived analytically.

\[ \begin{aligned} \mathbb{E}(\beta_1) & = \mathbb{E}\big((X_1'X_1)^{-1} X_1'Y)\big)\\\ & = \mathbb{E}\Big((X_1'X_1)^{-1} X_1'\big(X_1\beta_1 + X_2\beta_2 + \epsilon\big)\Big)\\\ & = \mathbb{E}\Big((X_1'X_1)^{-1} X_1'X_1\beta_1 + (X_1'X_1)^{-1} X_1'X_2\beta_2 + (X_1'X_1)^{-1} X_1'\epsilon\big)\Big)\\\ & = \mathbb{E}\Big(\beta_1 + (X_1'X_1)^{-1} X_1'X_2\beta_2 + (X_1'X_1)^{-1} X_1'\epsilon\big)\Big) \\\ & = \beta_1 + (X_1'X_1)^{-1} X_1'X_2\beta_2 + (X_1'X_1)^{-1} X_1'\mathbb{E}(\epsilon) \\\ & = \beta_1 + (X_1'X_1)^{-1} X_1'X_2\beta_2 \\\ & \ne \beta_1 \end{aligned} \]

So our estimate of \(\beta_1\) is biased. This holds for every subset of variables \(\{\beta_1, \beta_2\}\) that make up \(\beta\).

Suppose that the true model is given through:

\[ Y = X_1\beta_1 + \epsilon \]

If we start out with a larger model, say by including some parameters \(\beta_2\) as well while they do not have any influence on the model then we will initially estimate a wrong model \(Y = X_1\beta_1 + X_2\beta_2 + \epsilon\).

Let’s define a matrix \(M_{X_1} = I_n -X_1(X_1'X_1)^{-1}X_1'\). We can use this matrix to get an estimate of \(\beta_2\).

Start out with the original formula.

\[ \begin{aligned} M_{X_1}Y & = M_{X_1}X_1\beta_1 + M_{X_1}X_2\beta_2 + M_{X_1}\epsilon \\\ M_{X_1}Y & = M_{X_1}X_2\beta_2 + \epsilon \\\ X_2'M_{X_1}Y & = X_2'M_{X_1}X_2\beta_2 + X_2'\epsilon \\\ X_2'M_{X_1}Y & = X_2'M_{X_1}X_2\beta_2 \\\ \beta_2 & = ( X_2'M_{X_1}X_2)^{-1}X_2'M_{X_1}Y \\\ \end{aligned} \]

Notice that \(M_{X_1}X_1 = 0\) and that \(M_{X_1}\epsilon = \epsilon\) because of the definition while \(X_2\epsilon = 0\) because \(\epsilon\) is normally distributed around zero and orthogonal to any of the explanatory variables.

With this definition of \(\beta_2\) we can analyse it to confirm that it should not converge to any other value than zero.

\[ \begin{aligned} \mathbb{E}(\beta_2) & = \mathbb{E}\big(( X_2'M_{X_1}X_2)^{-1}X_2'M_{X_1}Y\big) \\\ & = \mathbb{E}\big(( X_2'M_{X_1}X_2)^{-1}X_2'M_{X_1}\big(X_1\beta_1 + \epsilon\big)\big) \\\ & = \mathbb{E}\big(( X_2'M_{X_1}X_2)^{-1}X_2'M_{X_1}X_1\beta_1 + ( X_2'M_{X_1}X_2)^{-1}X_2'M_{X_1}\epsilon\big) \\\ & = \mathbb{E}\big(( X_2'M_{X_1}X_2)^{-1}X_2'M_{X_1}\epsilon\big) \\\ & = ( X_2'M_{X_1}X_2)^{-1}X_2'M_{X_1}\mathbb{E}(\epsilon) \\\ & = 0 \end{aligned} \]

Notice that \(( X_2'M_{X_1}X_2)^{-1}X_2'M_{X_1}X_1\beta_1 = 0\) because \(M_{X_1}X_1 = 0\). So we see that \(\beta_2\) is correctly estimated, what about \(\beta_1\)?

\[ \begin{aligned} \mathbb{E}(\beta_1) & = \mathbb{E}\big((X_1'X_1)^{-1} X_1'Y)\big)\\\ & = \mathbb{E}\Big((X_1'X_1)^{-1} X_1'\big(X_1\beta_1 + \epsilon\big)\Big)\\\ & = \mathbb{E}\Big(\beta_1 + (X_1'X_1)^{-1} X_1'\epsilon\Big)\\\ & = \beta_1 \end{aligned} \]

So in this case we would remove the variables \(\beta_2\) that are not of influence while our estimate of \(\beta_1\) does not have any bias. This is exactly what we want.

I’ve shown that by starting only a few variables and then adding them to the model has a bias risk in linear models. One can imagine a similar thing happening in other models.