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I am having quite a bit of difficulty in reading some economic explanations regarding, e.g., orthogonality, robust standard errors, etc. Often I am thrown off by the use of certain notations. I am left wondering if, e.g., E[X|Y] (the expected value of X conditional on Y) is also sometimes written as (XY), or [XY]. And I am a bit confused as to the meaning of 'X and 'e in the equation

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In addition to clearing up these issues can someone recommend a book that covers various aspects of mathematics applicable to economics? I was thinking to purchase Mathematics for Economists from Blume and Simon.

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    $\begingroup$ They usually mean matrix or vector transpose. $\endgroup$ Nov 21, 2017 at 15:30
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    $\begingroup$ What @CagdasOzgenc said (+1). Economics up to about the Masters level can get by with a superficial understanding of Linear Algebra and some basic theory from Linear Regression. For the former, there are many useful resources -- I first learned it from the (freely available) lectures of Gilbert Strang at MIT OCW and I strongly recommend that. As for regression, there are also many resources of varying quality. Andrew Gelman's books are good intro for a Bayesian perspective, and I've never found one I love for a frequentist POV. Others may be able to chime in. $\endgroup$ Nov 21, 2017 at 23:14

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You have the ' associated with the wrong symbol.

Suppose X is a vector (it is in some cases a matrix here, but the idea is the same). then we usually write X as

$$X=\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}$$

If we want to multiply X times itself (take an inner product), the rules of matrix/vector manipulation require that we make it look like so:

$$\begin{bmatrix}x_1 & x_2 & x_3 \end{bmatrix}\cdot \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix} = x_1^2+x_2^2+x_3^2$$

If $$X=\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}$$ then $$X'=\begin{bmatrix}x_1 & x_2 & x_3 \end{bmatrix}$$ It is just notation for taking the transpose. i.e., making a column vector into a row vector.

So, the way we write "X multiplied by itself" with vectors (or matrices) is not $X\cdot X$, but $X'\cdot X$, so everything has the right shape.

Of course, we don't need the dot in between, so it just comes out as $$X' X$$ The ' goes with the first X, not the second one.

By the way, you might also see the notation $X^T$, which means the same thing. The it would be $$X^T X$$

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