Linear Regression When E[XU]=0

Assumptions for Linear Regression when E[XU]=0

Let $(Y, X, U)$ be a random vector where $Y$ and $U$ take values in $\mathbf{R}$ and $X$ takes values in $\mathbf{R}^{k+1}$. Assume further that $X=\left(X_0, X_1, \ldots, X_k\right)^{\prime}$ with $X_0=1$ and let $\beta=\left(\beta_0, \beta_1, \ldots, \beta_k\right)^{\prime} \in \mathbf{R}^{k+1}$ be such that

$$Y=X^{\prime} \beta+U$$

We will make following three assumptions for this model.

1. $E[X U]=0$: The justification of 1 varies depending on which of the three preceding interpretations we invoke. This expectation refers to the covariance between the regressor $X$ and the error term $U$ being zero.

   1. For Linear Conditional Expectation: In this interpretation, the regression is viewed as modeling the conditional expectation of $Y$ given $X$. That is, $E[Y \mid X]=\beta_0+\beta_1 X$.

      Here, $U$ represents the deviation of $Y$ from its conditional expectation given $X$. Since $U$ is defined as $Y-E[Y \mid X]$, it inherently means that $E[U \mid X]= 0$ . Consequently, this implies that $E[X U]=0$, because the error term $U$ has no systematic relationship with $X$ when considering the conditional expectation.

   2. For Best Linear Approximation: In the best linear approximation, the regression equation is considered the best linear approximation to the true relationship between $Y$ and $X$, regardless of the true underlying relationship.

      The error term $U$ in this context captures all factors that affect $Y$ but are not captured by $X$. For the linear approximation to be the 'best', it requires that these omitted factors (captured in $U$ ) are uncorrelated with $X$. If $X$ were correlated with $U$, there would be systematic information in $X$ that could improve the approximation, contradicting the premise that the model is the 'best' linear approximation. Hence, $E[X U]=0$ is necessary to ensure that the linear model is indeed the best approximation under the given circumstances.

   3. For Causal Model: the regression is understood as modeling a causal relationship between $X$ and $Y$. Here, $\beta_1$ is interpreted as the causal effect of $X$ on $Y$.

      For $\beta_1$ to represent the causal effect of $X$ on $Y$, it is crucial that all other factors influencing $Y$ are either controlled for or are not correlated with $X$. The error term $U$ includes all these other factors. If $U$ were correlated with $X$, it would mean that there are omitted variables that both affect $Y$ and are related to $X$, which would bias the estimation of the causal effect. Therefore, ensuring $E[X U]=0$ is essential for a valid causal interpretation, as it implies that there are no omitted confounders that are correlated with $X$.

2. $E\left[X X^{\prime}\right]<\infty$, this 2 ensures that $E\left[X X^{\prime}\right]$ exists.

   This matrix $E\left[X X^{\prime}\right]$ is called Design Matrix.

3. There is NO PERFECT COLLINEARITY in $X$ or the matrix $E\left[X X^{\prime}\right]$ is in fact invertible.

   Since $E\left[X X^{\prime}\right]$ is positive semi-definite, invertibility of $E\left[X X^{\prime}\right]$ is equivalent to $E\left[X X^{\prime}\right]$ being positive definite. This ensures there is a unique solution to $\beta$ when solving for it. So, this is also called the identification condition for $\beta$.

We can talk more detail about Invertibility:

Definition:

There is perfect collinearity or multicollinearity in $X$ if there exists nonzero $c \in \mathbf{R}^{k+1}$ such that $P\{c^{\prime} X=0\}=1$, (Here we treat X as a random variable.), i.e., if we can express one component of $X$ as a linear combination of the others.

Lemma:

Let $X$ be such that $E\left[X X^{\prime}\right]<\infty$. Then $E\left[X X^{\prime}\right]$ is invertible iff there is no perfect collinearity in $X$.

Proof:

If there is perfect collinearity, such that $\exist c \neq0$ that

$$c^{\prime} \mathbb{E}\left[X X^{\prime}\right] c=\mathbb{E}\left[c^{\prime} X X^{\prime} c\right]=\mathbb{E}\left[\left(c^{\prime} X\right)^2\right]=0 \text { with } P=1$$

In order for $E\left[X X^{\prime}\right]$ to be positive definite, $c^{\prime} \mathbb{E}\left[X X^{\prime}\right] c$ can take value 0, but it cannot have this with P=1. So, $c^{\prime} \mathbb{E}\left[X X^{\prime}\right] c$ is not positive definite, therefore it is not invertible. This is a contradiction.

Note that, based on the calculation rule of the matrix, $(A B)^{\prime}=B^{\prime} A^{\prime}$, we have $(X^{\prime} Z)^{\prime}=Z^{\prime} X$, therefore,

$$\begin{aligned} Z^{\prime} \mathbb{E}\left[X X^{\prime}\right] Z & =\mathbb{E}\left[Z^{\prime} X X^{\prime} Z\right]=\mathbb{E}\left[Z^{\prime} X\left(Z^{\prime} X\right)^{\prime}\right] \\ & =\mathbb{E}\left[\left(Z^{\prime} X\right)^2\right] \geqslant 0 \end{aligned}$$

So, $E\left[X X^{\prime}\right]$ is always positive semi-definite, we need $E\left[X X^{\prime}\right]>0$ with $P>0$ to be positive definite.

Solving for Beta

• $E[U X]=0$ implies that $E\left[X\left(Y-X^{\prime} \beta\right)\right]=0$, which is the FOC for optimization problem $\underset{b \in \mathbb{R}^{k+1}}{\operatorname{argmin}} E\left[\left(Y-X^{\prime} b\right)^2\right]$: $\frac{\partial \mathbb{E}\left[\left(Y-X^{\prime} b\right)^2\right]}{\partial b}=\mathbb{E}\left[\frac{\partial\left(Y-X^{\prime} b\right)^2}{\partial b}\right]=\mathbb{E}\left[\left(Y-X^{\prime} b\right) \cdot X\right]=0$
• As $\beta$ is the solution to this optimization problem. So, we have $\mathbb{E}\left[\left(Y-X^{\prime} \beta\right) \cdot X\right]=0$
• $\mathbb{E}[Y X]-\mathbb{E}\left[X^{\prime} X \beta\right]=0$ $\Rightarrow$ $\mathbb{E}[X Y]=\mathbb{E}[X X^{\prime}]\beta$
• As $E\left[X X^{\prime}\right]<\infty$ and $E\left[X X^{\prime}\right]$ is positive definite. So, it exists and is invertible. We have:

$\beta=\left[\mathbb{E}\left[X X^{\prime}\right]\right]^{-1} \mathbb{E}[X Y]$

Note that, here the $\beta$ we got is the linear projection coefficient.

• And for $E[U X]=0$, we can have that $E[U] = 0$.

Proof:

Since $E[U X]=0$ indicates $X$ is independent with $U$, $\mathbb{E}[x_j u]=0$ for $j = 1 , \cdots, k$.$\Rightarrow$$x_j \perp u$

$Cov(X, U) = \mathbb{E}[XU] - \mathbb{E}[X]\mathbb{E}[U] = 0 - \mathbb{E}[X]\mathbb{E}[U] = 0$

As $\mathbb{E}[X] \neq 0$, so we have $\mathbb{E}[U]=0$.

• If $E\left[X X^{\prime}\right]$ is not invertible (not positive definite), there will be more than one solution to this system of equations. Any two solutions $\beta$ and $\tilde{\beta}$ will necessarily satisfy $X^{\prime} \beta=X^{\prime} \tilde{\beta}$.

Estimating Beta using OLS

Let $(Y, X, U)$ be as described and let $P$ be the marginal distribution of $(Y, X)$. And let $\left(Y_1, X_1\right), \ldots,\left(Y_n, X_n\right)$ be an i.i.d. sequence of random vectors with distribution $P$.

A natural estimator of $\beta=\left(E\left[X X^{\prime}\right]\right)^{-1} E[X Y]$ is simply:

$$\hat{\beta}=\left(\frac{1}{n} \sum_{1 \leq i \leq n} X_i X_i^{\prime}\right)^{-1}\left(\frac{1}{n} \sum_{1 \leq i \leq n} X_i Y_i\right)=\left(\sum_{1 \leq i \leq n} X_i X_i^{\prime}\right)^{-1}\left(\sum_{1 \leq i \leq n} X_i Y_i\right)$$

This estimator is called the ordinary least squares (OLS) estimator of $\beta$ because it can also be derived as the solution to the following minimization problem:

$$\min _{b \in \mathbf{R}^{k+1}} \frac{1}{n} \sum_{1 \leq i \leq n}\left(Y_i-X_i^{\prime} b\right)^2$$

Here:

• $\hat{\beta}$ is a random variable, since it is $f\left(X_i, Y_i\right)$
• By Law of Large Number: $\frac{1}{n} \sum_{1 \leqslant i \leqslant n} X_i X_i^{\prime} \stackrel{P}{\longrightarrow} \mathbb{E}\left[X_i X_i^{\prime}\right]$
• By Law of Large Number: $\frac{1}{n} \sum_{1 \leqslant i \leqslant n} X_i Y_i^{\prime} \stackrel{P}{\longrightarrow} \mathbb{E}\left[X_i Y_i^{\prime}\right]$
• Note that for more efficient estimator $\hat{\beta}$, its Variance is smaller.

Proof:

Define Objective Function:

$$Q(b)=\frac{1}{n} \sum_{i=1}^n\left(Y_i-X_i^{\prime} b\right)^2$$

We take the derivative of the objective function w.r.t. $b$ and get the FOC:

$$\frac{\partial Q}{\partial b}=-\frac{2}{n} \sum_{i=1}^n X_i\left(Y_i-X_i^{\prime} b\right)=0$$

Then by FOC, we have that:

$$\begin{aligned} & -\frac{2}{n} \sum_{i=1}^n X_i\left(Y_i-X_i^{\prime} b\right)=0 \\ & \sum_{i=1}^n X_i Y_i-\sum_{i=1}^n X_i X_i^{\prime} b=0 \end{aligned}$$

Now, we rearrange to solve for $b$ :

$$\begin{aligned} & \sum_{i=1}^n X_i X_i^{\prime} b=\sum_{i=1}^n X_i Y_i \\ & \left(\sum_{i=1}^n X_i X_i^{\prime}\right) b=\sum_{i=1}^n X_i Y_i \end{aligned}$$

Dividing by $n$ to align with your provided formulation:

$$\left(\frac{1}{n} \sum_{i=1}^n X_i X_i^{\prime}\right) b=\frac{1}{n} \sum_{i=1}^n X_i Y_i$$

Finally, solving for $b$, the result will be our $\hat{\beta}$:

$$\hat{\beta} = b=\left(\frac{1}{n} \sum_{i=1}^n X_i X_i^{\prime}\right)^{-1}\left(\frac{1}{n} \sum_{i=1}^n X_i Y_i\right)$$

Now we finished the proof.

Note that we can also apply weight to the above formula to make this a weighted least squares (WLS) estimator.

$$\min _{b \in \mathbf{R}^{k+1}} \frac{1}{n} \sum_{1 \leq i \leq n}W_i \left(Y_i-X_i^{\prime} b\right)^2$$

So, we have that OLS is a special case of WLS. In this case, $W_i$s are equal to $1$.

Matrix Notation

Define

$$\begin{aligned} \mathbb{Y} & =\left(Y_1, \ldots, Y_n\right)^{\prime} \\ \mathbb{X} & =\left(X_1, \ldots, X_n\right)^{\prime} \\ \hat{\mathbb{Y}} & =\left(\hat{Y}_1, \ldots, \hat{Y}_n\right)^{\prime} =\mathbb{X} \hat{\beta} \\ \mathbb{U} & =\left(U_1, \ldots, \mathbb{U}_n\right)^{\prime} \\ \hat{\mathbb{U}} & =\left(\hat{U}_1, \ldots, \hat{U}_n\right)^{\prime} \\ & =\mathbb{Y}-\hat{\mathbb{Y}} =\mathbb{Y}-\mathbb{X} \hat{\beta} . \end{aligned}$$

In this notation,

$$\hat{\beta}=\left(\mathbb{X}^{\prime} \mathbb{X}\right)^{-1} \mathbb{X}^{\prime} \mathbb{Y}$$

and may be equivalently described as the solution to

$$\min _{b \in \mathbf{R}^{k+1}}|\mathbb{Y}-\mathbb{X} b|^2 .$$

Hence, $\mathbb{X} \hat{\beta}$ is the vector in the column space of $\mathbb{X}$ that is the closest (in terms of Euclidean distance) to $\mathbb{Y}$.

$$\mathbb{X} \hat{\beta}=\mathbb{X}\left(\mathbb{X}^{\prime} \mathbb{X}\right)^{-1} \mathbb{X}^{\prime} \mathbb{Y}$$

is the orthogonal projection of $Y$ onto the $((k+1)$-dimensional) column space of $\mathbb{X}$.

The matrix

$$\mathbb{P}=\mathbb{X}\left(\mathbb{X}^{\prime} \mathbb{X}\right)^{-1} \mathbb{X}^{\prime}$$

is known as the Projection Matrix. It projects a vector in $\mathbf{R}^n$ (such as $\mathbb{Y}$ ) onto the column space of $\mathbb{X}$.

Note that $\mathbb{P}^2=\mathbb{P}$, which reflects the fact that projecting something that already lies in the column space of $\mathbb{X}$ onto the column space of $\mathbb{X}$ does nothing.

The matrix $\mathbb{P}$ is also symmetric. The matrix

$$\mathbb{M}=\mathbb{I}-\mathbb{P}$$

is also a projection matrix. It projects a vector onto the $((n-k-1)$-dimensional) vector space orthogonal to the column space of $\mathbb{X}$. Hence, $\mathbb{M X}=0$. Note that $\mathbb{M Y}=\hat{\mathbb{U}}$. For this reason, $\mathbb{M}$ is sometimes called the "residual maker" matrix.

Proof:

• $\mathbb{P}\mathbb{X} = \mathbb{X}$:

$$\mathbb{P} \mathbb{X}=\mathbb{X}\left(\mathbb{X}^{\prime} \mathbb{X}\right)^{-1} \mathbb{X}^{\prime} \mathbb{X}=\mathbb{X}\left[\left(\mathbb{X}^{\prime} \mathbb{X}\right)^{-1}\left(\mathbb{X}^{\prime} \mathbb{X}\right)\right]=\mathbb{X} \mathbb{I}=\mathbb{X}$$

• $\mathbb{P}^2 = \mathbb{P}$:

$$\mathbb{P}^2=\left(\mathbb{X}\left(\mathbb{X}^{\prime} \mathbb{X}\right)^{-1} \mathbb{X}^{\prime}\right)\left(\mathbb{X}\left(\mathbb{X}^{\prime} \mathbb{X}\right)^{-1} \mathbb{X}^{\prime}\right)^{\prime} = \mathbb{X}\left(\mathbb{X}^{\prime} \mathbb{X}\right)^{-1} \mathbb{X}^{\prime}\mathbb{X}[\left(\mathbb{X}^{\prime}\mathbb{X}\right)^{-1}]^{\prime} \mathbb{X}^{\prime} = \mathbb{X}\left(\mathbb{X}^{\prime} \mathbb{X}\right)^{-1} \mathbb{X}^{\prime} = \mathbb{P}$$

• $\mathbb{P M}=0$:

$$\mathbb{P M}=\mathbb{P}(\mathbb{I}-\mathbb{P }) = \mathbb{P} - \mathbb{P}^2 = \mathbb{P}-\mathbb{P}=0$$

Sub-Vectors of Beta

Let $(Y, X, U)$ be a random vector where $Y$ and $U$ take values in $\mathbf{R}$ and $X$ takes values in $\mathbf{R}^{k+1}$. Let $\beta=\left(\beta_0, \beta_1, \ldots, \beta_k\right)^{\prime} \in \mathbf{R}^{k+1}$ such that

$$Y=X^{\prime} \beta+U$$

Partition $X$ into $X_1$ and $X_2$, where $X_1$ takes values in $\mathbf{R}^{k_1}$ and $X_2$ takes values in $\mathbf{R}^{k_2}$. Partition $\beta$ into $\beta_1$ and $\beta_2$ analogously. In this notation,

$$Y=X_1^{\prime} \beta_1+X_2^{\prime} \beta_2+U$$

Our preceding results imply that:

$$\left(\begin{array}{l} \beta_1 \\ \beta_2 \end{array}\right)=\left[E\left(\begin{array}{l} X_1 \\ X_2 \end{array}\right)\left(X_1^{\prime} X_2^{\prime}\right)\right]^{\prime} E\left(\begin{array}{l} X_1 \\ X_2 \end{array}\right) Y$$

$$\left(\begin{array}{l} \beta_1 \\ \beta_2 \end{array}\right)=\left(\begin{array}{ll} E\left[X_1 X_1^{\prime}\right] & E\left[X_1 X_2^{\prime}\right] \\ E\left[X_2 X_1^{\prime}\right] & E\left[X_2 X_2^{\prime}\right] \end{array}\right)^{-1}\left(\begin{array}{l} E\left[X_1 Y\right] \\ E\left[X_2 Y\right] \end{array}\right)$$

This is derived from the general linear model estimation method, specifically the Ordinary Least Squares (OLS) method, under the assumption that:

• The expected value of the error term $U$ is zero and that $U$ is uncorrelated with $X$.
• Existence and invertibility of matrix:

$$\left(\begin{array}{ll} E\left[X_1 X_1^{\prime}\right] & E\left[X_1 X_2^{\prime}\right] \\ E\left[X_2 X_1^{\prime}\right] & E\left[X_2 X_2^{\prime}\right] \end{array}\right)$$

Question: Can we derive formulae for $\beta_1$ and $\beta_2$ that admit some interesting interpretations?

1. Partial Effects: The coefficients in $\beta_1$ and $\beta_2$ can be interpreted as the partial effects of the variables in $X_1$ and $X_2$ on $Y$, controlling for the other variables. This means that $\beta_1$ captures the impact of $X_1$ on $Y$ while holding $X_2$ constant, and vice versa for $\beta_2$.
2. Multicollinearity Consideration: In cases where there is multicollinearity (i.e., high correlation) between some variables in $X_1$ and $X_2$, partitioning the regression can help understand how each group of variables uniquely contributes to explaining the variation in $Y$.
3. Group-Specific Analysis: This partitioning is particularly useful when $X_1$ and $X_2$ represent distinct groups of variables, such as demographic factors versus economic indicators. It allows for the isolation of the effects of each group on $Y$.
4. Interpreting in Context: The interpretation of $\beta_1$ and $\beta_2$ will highly depend on the specific context of the study. For instance, in an economic growth model, $X_1$ might include labor and capital variables, while $X_2$ includes policy variables. The coefficients would then tell us about the distinct impacts of these groups on economic growth.
5. Statistical Significance: It's important to evaluate the statistical significance of the estimated coefficients in $\beta_1$ and $\beta_2$ to determine if the observed relationships are not due to random chance.
6. Estimation Challenges: If $X_1$ and $X_2$ are highly correlated, the inverse of the matrix in the formula may be difficult to compute accurately, leading to estimation problems. This is a common issue in models with multicollinearity.

Result Based on BLP

BLP: for a random variable $A$ and a random vector $B$, denote by $\operatorname{BLP}(A \mid B)$ the best linear predictor of $A$ given $B$, i.e.

$$\operatorname{BLP}(A \mid B) \equiv B^{\prime}\left(E\left[B B^{\prime}\right]\right)^{-1} E[B A] .$$

If $A$ is a random vector, then define $\mathrm{BLP}(A \mid B)$ component-wise.

And we also define $\tilde{A}=A-\operatorname{BLP}(A \mid B)$.

Now, come back to our problem: $Y=X_1^{\prime} \beta_1+X_2^{\prime} \beta_2+U$

Define $\tilde{Y}=Y-\operatorname{BLP}\left(Y \mid X_2\right)$ and $\tilde{X}_1=X_1-\operatorname{BLP}\left(X_1 \mid X_2\right)$. Consider the linear regression

$$\tilde{Y}=\tilde{X}_1^{\prime} \tilde{\beta}_1+\tilde{U} \text { where } E\left[\tilde{X}_1 \tilde{U}\right]=0$$

as for example, in the second interpretation of the linear regression model: Best Linear Approximation that we described before, then we can have that:

$$\tilde{\beta}_1=\left(E\left[\tilde{X}_1 \tilde{X}_1^{\prime}\right]\right)^{-1} E\left[\tilde{X}_1 \tilde{Y}\right]=\beta_1$$

Proof:

First we can have the following result based on the definition of $\tilde{Y}$ and $\tilde{X}_1$:

• $Y=X_2^{\prime} \gamma+U_2$ $\Rightarrow$ $\mathbb{E}\left[X_2 U_2\right]=0$ $\Rightarrow$ $U_2=\tilde{Y}$$=Y-X_2^{\prime} \gamma$
• $X_1 = X_2^{\prime} \eta+e$ $\Rightarrow$ $\mathbb{E}\left[X_2 e\right]=0$ $\Rightarrow$ $e=\tilde{X}_1$$\Rightarrow$ $\mathbb{E}\left[X_2 \tilde{X}_1^{\prime}\right]=0$

Above can be got because $U_2$ and $e$ are projections from linear regression models.

For formula $\tilde{\beta}_1=\left(E\left[\tilde{X}_1 \tilde{X}_1^{\prime}\right]\right)^{-1} E\left[\tilde{X}_1 \tilde{Y}\right]$, we will first separate the right-hand side $E\left[\tilde{X}_1 \tilde{Y}\right]$

$E\left[\tilde{X}_1 \tilde{Y}\right] = E\left[\tilde{X}_1 (Y-X_2^{\prime} \gamma)\right]$$= E\left[\tilde{X}_1 Y\right]-E\left[\tilde{X}_1X_2^{\prime} \gamma\right]$

Since $\tilde{X}_1$ can be viewed as a scaler, we can have that as $\mathbb{E}\left[X_2 \tilde{X}_1^{\prime}\right]=0$, then $E\left[\tilde{X}_1 X_2^{\prime} \gamma\right] = E\left[X_2\tilde{X}_1^{\prime} \gamma\right] = E\left[X_2\tilde{X}_1^{\prime} \right]\gamma = 0$

Now we have that: $E\left[\tilde{X}_1 \tilde{Y}\right]=E\left[\tilde{X}_1 Y\right]$, take the original formula $Y=X_1^{\prime} \beta_1+X_2^{\prime} \beta_2+U$ inside, we can have that:

• $U \perp X_1$, and $U \perp X_2$, since we are under the interpretation of the Best Linear Approximation.
• $E[\tilde{X}_1 Y] = E[\tilde{X}_1 (X_1^{\prime} \beta_1+X_2^{\prime} \beta_2+U)]$$=E\left[\tilde{X}_1X_1^{\prime} \right]\beta_1+E\left[\tilde{X}_1X_2^{\prime}\right] \beta_2+E\left[\tilde{X}_1U\right]$

  • As we have shown $E\left[\tilde{X}_1 X_2^{\prime}\right] =0$ $\Rightarrow$ $E\left[\tilde{X}_1 X_2^{\prime}\right] \beta_2=0$
  • As $U \perp X_1$ $\Rightarrow$ $E\left[\tilde{X}_1 U\right]=0$

Now we got that $E\left[\tilde{X}_1 \tilde{Y}\right]=E\left[\tilde{X}_1 Y\right]=E\left[\tilde{X}_1 X_1^{\prime}\right] \beta_1$, now for $E\left[\tilde{X}_1 X_1^{\prime}\right]$, we can have that:

$E\left[\tilde{X}_1 X_1^{\prime}\right] = E\left[\tilde{X}_1 (X_2^{\prime} \eta+e)^{\prime}\right] = E\left[\tilde{X}_1 X_2^{\prime} \eta\right]+E\left[\tilde{X}_1e^{\prime}\right]$

• For $E\left[\tilde{X}_1 X_2^{\prime} \eta\right]$, as $e \perp X_2^{\prime} \eta$ because of the projection, we have $\tilde{X}_1 = e \perp X_2^{\prime} \eta$, so $E\left[\tilde{X}_1 X_2^{\prime} \eta\right]=0$
• For $E\left[\tilde{X}_1 e^{\prime}\right]$, as $e=\tilde{X}_1$, we have that $E\left[\tilde{X}_1 e^{\prime}\right] = E\left[\tilde{X}_1 \tilde{X}_1^{\prime}\right]$

Now we got that $E\left[\tilde{X}_1 \tilde{Y}\right]=E\left[\tilde{X}_1 Y\right]=E\left[\tilde{X}_1 X_1^{\prime}\right] \beta_1 = E\left[\tilde{X}_1 \tilde{X}_1^{\prime}\right]\beta_1$

Take this into the formula of $\tilde{\beta}_1$, we can have that:

$$\tilde{\beta}_1=\left(E\left[\tilde{X}_1 \tilde{X}_1^{\prime}\right]\right)^{-1} E\left[\tilde{X}_1 \tilde{Y}\right] = \left(E\left[\tilde{X}_1 \tilde{X}_1^{\prime}\right]\right)^{-1}E\left[\tilde{X}_1 \tilde{X}_1^{\prime}\right]\beta_1＝\mathbb{I}\beta_1=\beta_1$$

Now we successfully proved the above equation.

Interpretation

Above equaation can be interpreted like this: $\beta_1$ in the linear regression of $Y$ on $X_1$ and $X_2$ is equal to the coefficient in a linear regression of the error term from a linear regression of $Y$ on $X_2$ on the error terms from a linear regression of the components of $X_1$ on $X_2$.

This formalizes the common description of $\beta_1$ as the "effect" of $X_1$ on $Y$ after "controlling for $X_2$. "

Take $X_2=$ constant and $X_1 \in \mathbf{R}$. Then $\tilde{Y}=Y-E[Y]$ and $\tilde{X}_1=X_1-E\left[X_1\right]$. Hence,

$$\beta_1=\left(E\left[\left(X_1-E\left[X_1\right]\right)^2\right]\right)^{-1} E\left[\left(X_1-E X_1\right)(Y-E[Y])\right]=\frac{\operatorname{Cov}\left[X_1, Y\right]}{\operatorname{Var}\left[X_1\right]}$$

Proof:

As $X_2$ is a constant, we define $X_2=1$ for simplicity. In this way, we have $Y=\beta_2+\beta_1 X_1+U$.

Therefore, we use $X_2$ to estimate $X_1$ and $Y$, we can have that:

• $X_1 = C + e$, to get the BLP of $X_1$, we minimize over $\min _C E\left(X_1-C\right)^2$ $\Rightarrow$ FOC: $-2 E\left(X_1-C\right)=0$ $\Rightarrow$ $C=E[X_1]$
• $Y=D+e$, to get the BLP of $Y$, we minimize over $\min _D E\left(Y-D\right)^2 \Rightarrow$ FOC: $-2 E\left(Y-D\right)=0 \Rightarrow D=E\left[Y\right]$

In this way, we can get that $\tilde{Y}=Y-E[Y]$ and $\tilde{X}_1=X_1-E\left[X_1\right]$.

Now, based on the interpretation we shown above, we can have that:

$$\tilde{Y}=\beta_ 1\tilde{X}_1+\epsilon$$

So, in order to solve the $\beta_1$, we have that we need to minimize over $E_{\beta}(\tilde{Y}-\beta_1 \tilde{X}_1)^2$

The FOC for this problem is $E(\tilde{Y}-\beta _1\tilde{X}_1) \tilde{X}_1=0$ $\Rightarrow$ $E(\tilde{Y} \tilde{X}_1)=\beta _1E(\tilde{X}_1 \tilde{X}_1)$

So, we got

$$\beta_1=\frac{E\left(\tilde{Y} \tilde{X}_1\right)}{E\left(\tilde{X}_1^2\right)}$$

Take $\tilde{Y}=Y-E[Y]$ and $\tilde{X}_1=X_1-E\left[X_1\right]$ into this equation, we have $\beta_1=\left(E\left[\left(X_1-E\left[X_1\right]\right)^2\right]\right)^{-1} E\left[\left(X_1-E X_1\right)(Y-E[Y])\right]=\frac{\operatorname{Cov}\left[X_1, Y\right]}{\operatorname{Var}\left[X_1\right]}$ being proved.

Now, we start to check the individual elements in vector $\beta$:

If we use our formula to interpret the coefficient $\beta_j$, we obtain:

$$\beta_j=\frac{\operatorname{Cov}\left[\tilde{X}_j, Y\right]}{\operatorname{Var}\left[\tilde{X}_j\right]} .$$

$\Rightarrow$ each coefficient in a multivariate regression is the bivariate slope coefficient for the corresponding regressor, after "partialling out" all the other variables in the model.

Estimating Sub-Vectors of Beta

Partition $X$ and $\beta$ as before and consider

$$Y=X_1^{\prime} \beta_1+X_2^{\prime} \beta_2+U$$

We use $\hat{\beta}=\left(\hat{\beta}_1^{\prime}, \hat{\beta}_2^{\prime}\right)^{\prime}$ to denote the LS estimator of $\beta$ in a regression of $Y$ on $X$.

We now derive estimation counterparts to the previous results about solving for sub-vectors of $\beta$. That is, $\hat{\beta}_1$ can also be obtained from a "residualized" regression.

As we can have that $\tilde{Y}=\tilde{X}_1^{\prime}\beta_1 +\epsilon$, just treat this residualized regression as an ordinary regression we have shown before. Using OLS to get the best estimator of $\beta_1$

$$\hat{\beta}_1=\left(\frac{1}{n} \sum_{1 \leq i \leq n} \tilde{X}_{1 i} \tilde{X}_{1 i}{ }^{\prime}\right)^{-1}\left(\frac{1}{n} \sum_{1 \leq i \leq n} \tilde{X}_{1 i} \tilde{Y}_i\right)=\left(\sum_{1 \leq i \leq n} \tilde{X}_{1 i} \tilde{X}_{1 i}{ }^{\prime}\right)^{-1}\left(\sum_{1 \leq i \leq n} \tilde{X}_{1 i} \tilde{Y}_i\right)$$

Matrix Form

Let $\mathbb{X}_1=\left(X_{1,1}, \ldots, X_{1, n}\right)^{\prime}$ and $\mathbb{X}_2=\left(X_{2,1}, \ldots, X_{2, n}\right)^{\prime}$.

Denote by $\mathbb{P}_1$ the projection matrix onto the column space of $\mathbb{X}_1$ and $\mathbb{P}_2$ the projection matrix onto the column space of $\mathbb{X}_2$.

Define Residual Makers $\mathbb{M}_1=\mathbb{I}-\mathbb{P}_1$ and $\mathbb{M}_2=\mathbb{I}-\mathbb{P}_2$.

• Apply $\mathbb{M}_2$ to $\mathbb{X}_1$ to get the residualized version of $\mathbb{X}_1$: $\tilde{\mathbb{X}}_1=\mathbb{M}_2 \mathbb{X}_1$
• Apply $\mathbb{M}_2$ to $\mathbb{Y}$ to get the residualized version of $\mathbb{Y}$: $\tilde{\mathbb{Y}}=\mathbb{M}_2 \mathbb{Y}$

The regression model now becomes $\widetilde{\mathbb{Y}}=\beta_1 \widetilde{\mathbb{X}}_1+\epsilon$, where $\epsilon$ is the error term.

Then use Ordinary Least Squares (OLS) to estimate $\beta_1$. The formula for $\hat{\beta}_1$ is similar to before:

$$\hat{\beta}_1=\left(\tilde{\mathbb{X}}_1^{\prime} \tilde{\mathbb{X}}_1\right)^{-1} \tilde{\mathbb{X}}_1^{\prime} \widetilde{\mathbb{Y}}$$