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After looking various sources (1), I have found following equations for system description in kalman filter:

Measurement Equation as: $$ y_t=C_t x_t+r_t \tag{1} $$

and state vector equation as: $$x_{t+1} = A_t x_t + q_t \tag{2}$$

where $r_t \sim N(0, R_t)$ and $q_t \sim N(0, Q_T)$.

I have simple question. Why in equation 2 state vector equation is considered for $x_{t+1}$? Should we have $q_{t+1}$ on the right hand side? What would be the implication for the model if we have considered following equation for state vector equation: $$x_{t} = A_t x_{t-1} + q_t \tag{3}$$

Implication of equation 2 and 3:

If we consider equation 1 and 2, and given the filtration $F_{t-1}$, then in equation 1, only $r_t$ is random variable because $x_t$ is completely determined because $x_t = f(x_{t-1}, q_{t-1})$ but same can not be said about the $x_t$ in equation 1 if we consider equation 1 and 3. In equation 3, $x_t = f(x_{t-1}, q_t)$, it means that given the filtration $F_{t-1}$, $x_t$ is still random variable.

It means conditional expectation of $y_t$ considering (1) and (2) will be: \begin{align} E(y_t|F_{t-1}) =& C_t \, x_t + E(r_t|F_{t-1})\\ =&C_tx_t \tag{4} \end{align} Since, $x_t$ is completely determined given $F_{t-1}$.

If we considered equation (1) and equation (3), then conditional expectation of $y_t$ is :

\begin{align} E(y_t|F_{t-1}) =& C_t \, E(x_t|F_{t-1}) + E(r_t|F_{t-1})\\ =&C_t E(x_t|F_{t-1}) \tag{5} \end{align}

I think both the expression in equation 4 and 5 are different.

Thanks!

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    $\begingroup$ Is there some difference between (2) and the alternative expressed in the final equation? Other than a purely notational issue of whether the process noise and dynamic model matrix from step $s$ to $s+1$ are called $q_s$ and $A_s$ or $q_{s+1}$ and $A_{s+1}$? $\endgroup$
    – Juho Kokkala
    Nov 3, 2017 at 19:54
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    $\begingroup$ Okay, but there is no difference between equation 2 and 3, unless you decide to consider filtrations $F_{t-1}$ as you decided to define them (?) $\endgroup$
    – Juho Kokkala
    Nov 4, 2017 at 13:36
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    $\begingroup$ However, @Taylor's comment indicates that I may have misunderstood something $\endgroup$
    – Juho Kokkala
    Nov 4, 2017 at 13:43
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    $\begingroup$ @Neeraj equation (4) is incorrect, equation (2) is not the same as (5), and there could be a difference between (2) and (3) but it involves things not mentioned in the question (how the noise sequences are correlated cross-sectionally). The Kalman recursions change from these two different ways to write the model. Check out page 354 of the book I mentioned earlier, but note the typos (mentioned in the errata). $\endgroup$
    – Taylor
    Nov 4, 2017 at 18:31
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    $\begingroup$ @Neeraj have you heard of the leverage effect? It refers to the negative correlation between returns and future changes in volatility. Often models that capture this effect will parametrize this with (2) because you can talk about the correlation between $q_t$ and $r_t$. You might be tempted to use (3) instead, and say $q_t$ and $w_t$ are correlated here. However this is probably incorrect because it overstates the predictability of the mean return, and also this mistake has been made in the literature. Just one example, where it's important to talk about the distinction. $\endgroup$
    – Taylor
    Nov 8, 2017 at 17:34

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I believe the subscript does not affect the meaning of the process. It can be treated as $q_{t+1}$ goes through a time delayer but still be identically distributed.

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