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I'm studying ML estimation and I have a silly question that I'm not able to see its solution. Suppose I have an AR(1) process:

$$y_t = c+ \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim}\hbox{Normal}(0,\sigma^2)$$

The first step is find the density of $y_1$: $$y_1 = c+ \phi y_{0}+ u_1, \quad u_1 \sim \hbox{Normal} (0,\sigma^2)$$ According to this, I have that $y_1 \sim \hbox{Normal}(c+ \phi y_{0}, \sigma^2)$, since a normal with mean zero plus a constant gives a mean translation by the constant and the variance remains unchanged. (This is my reasoning)

But the Hamilton book (Time series Analysis) says that $y_1 \sim \hbox{Normal}\left( \frac{c}{1-\phi}, \frac{\sigma^2}{1-\phi^2}\right)$.

The justification that the book gives is that the AR(1) given above is such that its mean is $\frac{c}{1-\phi}$ and its variance is $\frac{\sigma^2}{1-\phi^2}$, regardless of $t$. Ok, this I understand by the stationarity of AR(1), but this is inconsistent with my reasoning.

Well, I'm probably making a huge mistake in my reasoning and I'd like to know where I'm going wrong.

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    $\begingroup$ stats.stackexchange.com/q/603410/20519 $\endgroup$
    – Zhanxiong
    Commented Apr 20, 2023 at 11:11
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    $\begingroup$ The problem of your reasoning is that $y_0$ is also a random variable, not a constant, which makes the translation property of normal does not apply. $\endgroup$
    – Zhanxiong
    Commented Apr 20, 2023 at 11:14
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    $\begingroup$ You condition on $y_0$ while the unconditional distribution of $y_1$ assumes stationarity, i.e., that the chain has been running since time $t=-\infty$. $\endgroup$
    – Xi'an
    Commented Apr 20, 2023 at 14:37

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