How to derive the least square estimator for multiple linear regression?

Question

In the simple linear regression case $y=\beta_0+\beta_1x$, you can derive the least square estimator $\hat\beta_1=\frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum(x_i-\bar x)^2}$ such that you don't have to know $\hat\beta_0$ to estimate $\hat\beta_1$

Suppose I have $y=\beta_1x_1+\beta_2x_2$, how do I derive $\hat\beta_1$ without estimating $\hat\beta_2$? or is this not possible?

Answer 1

The derivation in matrix notation

Starting from $y= Xb +\epsilon $, which really is just the same as

$\begin{bmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{N} \end{bmatrix} = \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1K} \\ x_{21} & x_{22} & \cdots & x_{2K} \\ \vdots & \ddots & \ddots & \vdots \\ x_{N1} & x_{N2} & \cdots & x_{NK} \end{bmatrix} * \begin{bmatrix} b_{1} \\ b_{2} \\ \vdots \\ b_{K} \end{bmatrix} + \begin{bmatrix} \epsilon_{1} \\ \epsilon_{2} \\ \vdots \\ \epsilon_{N} \end{bmatrix} $

it all comes down to minimzing $e'e$:

$\epsilon'\epsilon = \begin{bmatrix} e_{1} & e_{2} & \cdots & e_{N} \\ \end{bmatrix} \begin{bmatrix} e_{1} \\ e_{2} \\ \vdots \\ e_{N} \end{bmatrix} = \sum_{i=1}^{N}e_{i}^{2} $

So minimizing $e'e'$ gives us:

$min_{b}$ $e'e = (y-Xb)'(y-Xb)$

$min_{b}$ $e'e = y'y - 2b'X'y + b'X'Xb$

$\frac{\partial(e'e)}{\partial b} = -2X'y + 2X'Xb \stackrel{!}{=} 0$

$X'Xb=X'y$

$b=(X'X)^{-1}X'y$

One last mathematical thing, the second order condition for a minimum requires that the matrix $X'X$ is positive definite. This requirement is fulfilled in case $X$ has full rank.

The more accurate derivation which goes trough all the steps in greater dept can be found under http://economictheoryblog.com/2015/02/19/ols_estimator/

Answer 2

It is possible to estimate just one coefficient in a multiple regression without estimating the others.

The estimate of $\beta_1$ is obtained by removing the effects of $x_2$ from the other variables and then regressing the residuals of $y$ against the residuals of $x_1$. This is explained and illustrated How exactly does one control for other variables? and How to normalize (a) regression coefficient?. The beauty of this approach is that it requires no calculus, no linear algebra, can be visualized using just two-dimensional geometry, is numerically stable, and exploits just one fundamental idea of multiple regression: that of taking out (or "controlling for") the effects of a single variable.

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In the present case the multiple regression can be done using three ordinary regression steps:

1. Regress $y$ on $x_2$ (without a constant term!). Let the fit be $y = \alpha_{y,2}x_2 + \delta$. The estimate is $$\alpha_{y,2} = \frac{\sum_i y_i x_{2i}}{\sum_i x_{2i}^2}.$$ Therefore the residuals are $$\delta = y - \alpha_{y,2}x_2.$$ Geometrically, $\delta$ is what is left of $y$ after its projection onto $x_2$ is subtracted.

2. Regress $x_1$ on $x_2$ (without a constant term). Let the fit be $x_1 = \alpha_{1,2}x_2 + \gamma$. The estimate is $$\alpha_{1,2} = \frac{\sum_i x_{1i} x_{2i}}{\sum_i x_{2i}^2}.$$ The residuals are $$\gamma = x_1 - \alpha_{1,2}x_2.$$ Geometrically, $\gamma$ is what is left of $x_1$ after its projection onto $x_2$ is subtracted.

3. Regress $\delta$ on $\gamma$ (without a constant term). The estimate is $$\hat\beta_1 = \frac{\sum_i \delta_i \gamma_i}{\sum_i \gamma_i^2}.$$ The fit will be $\delta = \hat\beta_1 \gamma + \varepsilon$. Geometrically, $\hat\beta_1$ is the component of $\delta$ (which represents $y$ with $x_2$ taken out) in the $\gamma$ direction (which represents $x_1$ with $x_2$ taken out).

Notice that $\beta_2$ has not been estimated. It easily can be recovered from what has been obtained so far (just as $\hat\beta_0$ in the ordinary regression case is easily obtained from the slope estimate $\hat\beta_1$). The $\varepsilon$ are the residuals for the bivariate regression of $y$ on $x_1$ and $x_2$.

The parallel with ordinary regression is strong: steps (1) and (2) are analogs of subtracting the means in the usual formula. If you let $x_2$ be a vector of ones, you will in fact recover the usual formula.

This generalizes in the obvious way to regression with more than two variables: to estimate $\hat\beta_1$, regress $y$ and $x_1$ separately against all the other variables, then regress their residuals against each other. At that point none of the other coefficients in the multiple regression of $y$ have yet been estimated.

Answer 3

A simple derivation can be done just by using the geometric interpretation of LR.

Linear regression can be interpreted as the projection of $Y$ onto the column space $X$. Thus, the error, $\hat{\epsilon}$ is orthogonal to the column space of $X$.

Therefore, the inner product between $X'$ and the error must be 0, i.e.,

$<X', y-X\hat{\beta}> = 0$

$X'y - X'X\hat{\beta} = 0$

$X'y = X'X\hat{\beta}$

Which implies that,

$(X'X)^{-1}X'y = \hat{\beta}$.

Now the same can be done by:

(1) Projecting $Y$ onto $X_2$ (error $\delta = Y-X_2 \hat{D}$), $\hat{D} = (X_2'X_2)^{-1}X_2'y$,

(2) Projecting $X_1$ onto $X_2$ (error $\gamma = X_1 - X_2 \hat{G}$), $\hat{G} = (X_2'X_2)^{-1}X_2X_1$,

and finally,

(3) Projecting $\delta$ onto $\gamma$, $\hat{\beta}_1$

Answer 4

The ordinary least squares estimate of $\beta$ is a linear function of the response variable. Simply put, the OLS estimate of the coefficients, the $\beta$'s, can be written using only the dependent variable ($Y_i$'s) and the independent variables ($X_{ki}$'s).

To explain this fact for a general regression model, you need to understand a little linear algebra. Suppose you would like to estimate the coefficients $(\beta_0, \beta_1, ...,\beta_k)$ in a multiple regression model,

$$ Y_i = \beta_0+\beta_1X_{1i}+...+\beta_kX_{ki}+\epsilon_i $$

where $\epsilon_i \overset{iid}{\sim} N(0,\sigma^2)$ for $i=1,...,n$. The design matrix $\mathbf{X}$ is a $n\times k$ matrix where each column contains the $n$ observations of the $k^{th}$ dependent variable $X_k$. You can find many explanations and derivations here of the formula used to calculate the estimated coefficients $\boldsymbol{\hat{\beta}}=(\hat{\beta}_0, \hat{\beta}_1, ..., \hat{\beta}_k)$, which is

$$ \boldsymbol{\hat{\beta}}=(\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{X}^\prime \mathbf{Y} $$

assuming that the inverse $(\mathbf{X}^\prime \mathbf{X})^{-1}$ exists. The estimated coefficients are functions of the data, not of the other estimated coefficients.

Answer 5

One small minor note on theory vs. practice. Mathematically $\beta_0, \beta_1, \beta_2 ... \beta_n$ can be estimated with the following formula:

$$ \hat{\beta} = (X'X)^{-1} X'Y$$

where $X$ is the original input data and $Y$ is the variable that we want to estimate. This follows from minimizing the error. I will proove this before making a small practical point.

Let $e_i$ be the error the linear regression makes at point $i$. Then:

$$ e_i = y_i - \hat{y_i} $$

The total squared error we make is now:

$$ \sum_{i=1}^n e_i^2 = \sum_{i=1}^n (y_i - \hat{y_i})^2$$

Because we have a linear model we know that:

$$ \hat{y_i} = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + ... + \beta_n x_{n,i} $$

Which can be rewritten in matrix notation as:

$$ \hat{Y} = X\beta $$

We know that

$$ \sum_{i=1}^n e_i^2 = E'E $$

We want to minimize the total square error, such that the following expression should be as small as possible

$$ E'E = (Y-\hat{Y})' (Y-\hat{Y}) $$

This is equal to:

$$ E'E = (Y-X\beta)' (Y-X\beta)$$

The rewriting might seem confusing but it follows from linear algebra. Notice that the matrices behave similar to variables when we are multiplying them in some regards.

We want to find the values of $\beta$ such that this expression is as small as possible. We will need to differentiate and set the derivative equal to zero. We use the chain rule here.

$$ \frac{dE'E}{d\beta} = - 2 X'Y + 2 X'X\beta = 0$$

This gives:

$$ X'X\beta = X'Y $$

Such that finally: $$ \beta = (X'X)^{-1} X'Y $$

So mathematically we seem to have found a solution. There is one problem though, and that is that $(X'X)^{-1}$ is very hard to calculate if the matrix $X$ is very very large. This might give numerical accuracy issues. Another way to find the optimal values for $\beta$ in this situation is to use a gradient descent type of method. The function that we want to optimize is unbounded and convex so we would also use a gradient method in practice if need be.