To get a neat proof of the equivalence I was trying to prove today (and save time!) I asked ChatGPT-4 (with the *WolframAlpha* extention) for help.
If we are using the derivations in an appendix (which I think we should), I would proudly mention/cite ChatGPT, as I would have done if the proof was taken from a book or even a Q&A forums like CrossValidated (stats.StackExchange.com).

The following is the proof for the SAR model, and it is possible to make similar proofs for the SMA and SARMA models as well.
Although those proofs become pretty messy, they really work---trust me!
Since ChatGPT's response needs qiute a bit of cleaning, which I am not going to present them here.

So, here is how you can prove the structural time series modeling framework ($y_t = a_t + A_t$ with the independence assumption) is equivalent to the SAR model (without the need for distinguiching $a_t$ from $A_t$).

# Proof for the SAR model

## My prompt

Autoregressive seasonal time series model for $y_t$ with weekly (period $7$) seasonality can be modeled in two ways:

1. Based on structural time series (STS) modeling framework, that decomposes $y_t$ into two independent latent components, namely a daily component $a_t$ and a weekly component $A_t$:

$$y_t = a_t + A_t$$

and then modeling each component with an AR model

$$a_t = \phi a_{t-1} + u_t \\
A_t = \Phi A_{t-7} + U_t$$

wherein $\phi$ and $\Phi$ are the AR parameters of the daily and weekly processes, and $u_t$ and $U_t$ are the processes innovation terms (which, following the independence of $a_t$ and $A_t$, are themselves independent).

2. Using a SAR model:

$$y_t = \phi y_{t-1} + \Phi y_{t-7} - \phi \Phi y_{t-8} + \epsilon_t$$

Prove, step-by-step, that these two approaches are erquivalent.

## ChatGPT-4's answer

To prove that the two approaches are equivalent, we'll start by expressing each model and then manipulating the equations to show their equivalence.

**Approach 1: Decomposition into Daily and Weekly Components**

Given:
1. $y_t = a_t + A_t$
2. $a_t = \phi a_{t-1} + u_t$
3. $A_t = \Phi A_{t-7} + U_t$

**Approach 2: SAR Model**

Given:

4. $y_t = \phi y_{t-1} + \Phi y_{t-7} - \phi \Phi y_{t-8} + \epsilon_t$

**Step-by-Step Proof:**

**Step 1:** Express $y_{t-1}$ and $y_{t-7}$ using the decomposition model:

From (1):

$$y_{t-1} = a_{t-1} + A_{t-1} \\
y_{t-7} = a_{t-7} + A_{t-7}$$

**Step 2:** Substitute the AR models for $a_t$ and $A_t$ from (2) and (3) into (1):

$$y_t = \phi a_{t-1} + u_t + \Phi A_{t-7} + U_t$$

**Step 3:** Substitute the expressions for $y_{t-1}$ and $y_{t-7}$ from Step 1 into the above equation:

$$y_t = \phi (y_{t-1} - A_{t-1}) + u_t + \Phi (y_{t-7} - a_{t-7}) + U_t$$

**Step 4:** Now, express $A_{t-1}$ and $a_{t-7}$ using their respective AR models:

From (2):

$$a_{t-7} = \phi a_{t-8} + u_{t-7}$$

From (3):

$$A_{t-1} = \Phi A_{t-8} + U_{t-1}$$

**Step 5:** Substitute the above expressions into the equation from Step 3:

$$y_t = \phi y_{t-1} - \phi \Phi A_{t-8} - \phi U_{t-1} + u_t + \Phi y_{t-7} - \Phi \phi a_{t-8} - \Phi u_{t-7} + U_t$$

**Step 6:** Group the terms:

$$y_t = \phi y_{t-1} + \Phi y_{t-7} - \phi \Phi y_{t-8} + \epsilon_t$$

Where
$$\epsilon_t = u_t + U_t - \phi U_{t-1} - \Phi u_{t-7}$$

This equation matches the SAR model given in (4), proving that the two approaches are equivalent.

**Note:**
The term $\epsilon_t$ in the SAR model is a composite error term that encompasses the innovations from both the daily and weekly processes, as well as the lagged effects of these innovations.

# My own note

The reason the indepenednce assumption is not needed in the SAR formulation is that the latent components $a_t$ and $A_t$ eventually get together and can be simplified into $y_t$.
In doing so, the weird $-\phi \Phi y_{t-8}$ term appears---which was not in the separate daily and weekly AR processes (Equations 2 and 3 above)---essentially takes care of the equivalence.
In other words, the reason that $SAR \neq dailyAR + weeklyAR$ lies in the assumption of $a_t \mathrel{\unicode{x2AEB}} A_t$
**Furthermore**, the independent $u_t$ and $U_t$ are *grinded and mixed* together into $\epsilon_t$ as expressed in Step 6 above.

***So, in sum***, the independence assumption is needed if you *really* want to **separately** model the **latent processes** $a_t$ and $A_t$---that would also require explicitely modeling $u_t$ and $U_t$ and separately estimating their variances---which can be done using the structural time series modeling framework using state-space formulations different from that of the SAR model.
The *elegance* of the SAR(MA) model is that it grinds the latent processes under the hood and it only concerns the observed $y_t$ and **a single** residual variance term $\epsilon$.