Is an element of a vector a scalar or 1x1 vector or 1x1 matrix?

Question

In econometrics, by Hayashi, they defined the error vector of n observations in a $ (n \times K)$ regression funcntion as:

$\epsilon = \begin{bmatrix}\epsilon_{1} \\\epsilon_{2} \\\vdots \\\epsilon_{n}\end{bmatrix}$, where $\epsilon_{i}$ is the ith observation's error term,

and K-dimension x vector of the ith observation as, $x_{i} = \begin{bmatrix}x_{i1} \\x_{i2} \\\vdots \\x_{ik}\end{bmatrix}$

The book says that the cross moment of two random variables E[xy] is zero means that these two random variables are orthogonal. It's not hard to see the point using [0,1] and [1,0] to check their cross product for orthogonality.

But in the book it has a formula for strict exogeneity assumption:

$E[x_{j}\epsilon_{i}] = \begin{bmatrix}x_{j1}\epsilon_{i} \\x_{j2}\epsilon_{i} \\\vdots \\x_{jk}\epsilon_{i}\end{bmatrix} = 0_{(K\times1)}$

So, here the $\epsilon_{i}$ is an element of the $\epsilon$ vector, and the cross moment for a $(k\times1)$ vector and an element of a vector, which are orthogonal, is a $(k\times1)$ 0 vector. My question is that, what is a cross moment of two random variables, is it the expected value of the inner product? And shall I view the $\epsilon_{i}$ as a scalar, or $(1\times1)$ vector, or $(1\times1)$ matrix?

Thank you very much.

Answer 1

Vectors are (in this context) just matrices which have one row or column.

$\def\e{\epsilon}\e_i$ is a scalar, as it is an entry of a matrix, $\e$.

The cross moment of two random variables $X,Y$ is defined as the expected value of their product, where the product being used depends on context.

• If $X$ and $Y$ are scalars, the product is usual multiplication.

• If $X$ is a scalar and $Y$ is a matrix, the product is scalar multiplication.

• If $X$ is and $Y$ are both matrices, then the product is matrix multiplication.

  • If $X$ is an $1\times n$ vector and $Y$ is an $n\times 1$ vector, then the product is a $1\times 1$ matrix, which can be thought of as a scalar with no loss of generality. If so, this scalar is their inner product.

In answer to your second question, there is one special case where the cross moment is the expected value of the inner product.

Answer 2

From what I understand, the n X 1 error vector is a random vector. A random vector of dimension n is a collection of n random variables that are associated to the same event. So in the error vector: $$\epsilon = \begin{bmatrix}\epsilon_{1} \\\epsilon_{2} \\\vdots \\\epsilon_{n}\end{bmatrix}$$ where $\epsilon_i$ is the error in the $i^{th}$ observation, since the error is random every variable is random. Now the cross moment expression should be written like : $$E[x_{j}\epsilon_{i}] = \begin{bmatrix}x_{j1}\epsilon_{i} \\x_{j2}\epsilon_{i} \\\vdots \\x_{jk}\epsilon_{i}\end{bmatrix} = \begin{bmatrix}0 \\0 \\\vdots \\0\end{bmatrix}$$

Now the error $\epsilon_1$ is actually the error observed in the vector $x_1=\begin{bmatrix}x_{11} \\x_{12} \\\vdots \\x_{1k}\end{bmatrix}$ which is again a vector so you are just checking the exogeneity with respect to $\epsilon_1$.

Consider a real number $x$, now this is a scalar because it is plain value. Its physical significance is the magnitude. If we club two such numbers $x$ and $y$ we get a vector$$\left(\begin{array}{c}x\\ y\end{array}\right)$$ Now if we club two such vectors we get a matrix $$\begin{bmatrix}x_1 & x_2 \\y_1 & y_2 \end{bmatrix}$$ The thing is from which space are you viewing the real number. If you are considering only the scalar space then $x$ is a scalar, if you are viewing from a vector space $x$ is a 1X1 vector if you are viewing from matrix space $x$ is a 1X1 matrix. So the hierarchy is like Scalar then Vector then Matrix. Any scalar can be represented as a matrix or a vector but not the other way round.

It is similar to the fact that is $2$ really a complex number. People say it is because in the complex domain it can be represented as $2+0i$,a complex number cant be represented as a real number because it is having real and imaginary parts. Hope this helps...