QuantEcon / QuantEcon/lecture-python-advanced.myst

Update terms for ESS, TSS, and SSR to specify uncentered vs centered?

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Description

I think a more commonly seen definitions of (centered) TSS, ESS and SSR are given here where the decomposition only holds when $X$ includes intercept term.

So, I think it might be clearer if we replace

Here are some more standard definitions:

* The **total sum of squares** is $:=  \| y \|^2$.
* The **sum of squared residuals** is $:= \| \hat u \|^2$.
* The **explained sum of squares** is $:= \| \hat y \|^2$.

> TSS = ESS + SSR

with

Define:

* The (uncentered) **total sum of squares** (TSS) is $:=  \| y \|^2$.
* The (uncentered) **sum of squared residuals** (SSR) is $:= \| \hat u \|^2$.
* The **explained sum of squares** (ESS) is $:= \| \hat y \|^2$.

 We have the relationship:

$$
\text{TSS} = \text{ESS} + \text{SSR}
$$

```{note}
For the centered case, see [here](https://en.wikipedia.org/wiki/Explained_sum_of_squares).
```

Please let me know your thoughts on this!

Originally posted by @HumphreyYang in https://github.com/QuantEcon/lecture-python-advanced.myst/pull/217#discussion_r2250749514

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Research direction

Locate the lecture text containing the displayed TSS, SSR, and ESS definitions, using the wording from this issue and the original discussion in pull request #217. Update the terminology as proposed, including the centered-case note, then verify that the rendered equations and note are correct.

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tex
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