What is a 'State'? I've been working with State-Space modelling in R using the KFAS package and I'm a bit confused as to what exactly a 'State' is.. 
I know that in state space modelling, we have an observation equation and a state equation, I have a general understanding of the observation equation but not exactly of the state equation. 
To put some context in what i'm trying to do using this modelling...
I've got a time series of 2160 data points, and about 5 different predictors. Using the KFAS package, I wanted to build a state space model so that I could plot the change in the beta coefficients of the predictors over time. 
To do this I used a Kalman Smoother to get the 'filtered' estimates of the states and then plotted those. However, I'm not sure if what I've done is what I was trying to obtain.. Which brings me back to being confused about what exactly a 'state' is and what i'm really plotting here.. Any help would be appreciated.
 A: A state can be or represent anything. There are too many examples to list to do the generality of this any justice. For you, it just needs to be a continuous state space Markov Chain. That should sound pretty abstract. 
It sounds like you're using dynamic linear models. If that's true, your state will represent a time-varying coefficient. It will be like a linear regression model, only your coefficients will be allowed to vary over time randomly. 

To do this I used a Kalman Smoother to get the 'filtered' estimates of
  the states and then plotted those.

This isn't correct. A smoother gives you smoothed estimates of the states, while a filter gives you filtered estimates of your states. More on that in the next paragraph. 

However, I'm not sure if what I've done is what I was trying to obtain

It depends on your application. If you want to use an entire window of observed data $y_1, \ldots, y_T$ to estimate your states $x_1, \ldots, x_T$, then you use a smoother. This is more accurate than filtering, however, it is using data you wouldn't have had at certain times. A smoother will give you the probability distribution $p(x_{1:T}|y_{1:T})$ or in the marginal case $p(x_t|y_{1:T})$ for $t=1,\ldots,T$. If you plot the means over time of this distribution, you will see how your coefficients/states change.
However, if you're using a filter, you get the sequence of distributions $p(x_{1:t}|y_{1:t})$ or $p(x_t|y_{1:t})$ for $t=1,\ldots,T$. Notice that these sequences only use information up to the most recent time period. There is no looking forward with filtering distributions. However, they are less accurate than smoothing distributions typically, in the sense that they have higher variance. 
And by the way, all of this assumes that you know the variance parameters of your model. 
A: State variables and the Markov property
A stochastic process satisfies the Markov property if the past is independent from the future given the present state of the system.
In dynamic models, in time series models, what's referred to as state variables is generally some minimal set of variables that capture enough information so that the Markov property is satisfied. The Markov property is a highly desirable property for modeling!
Examples:


*

*Rocket tracking


*

*Our state variables here may be the rocket position (x, y, pitch, yaw...), velocity vector, fuel level, etc...


*A simple AR(2):


*

*Imagine we have the simple AR(2) model $y_{t+1} = b_0 + b_1 Y_{t} + b_2 Y_{t-1} + \epsilon_t$. You could define $X_t = \begin{bmatrix} Y_t \\ Y_{t-1} \end{bmatrix}$ as a state variable. 


*Simple macro-economics, neoclassical growth model


*

*You might have the level of capital $K_t$ and productivity $A_t$ as state variables.



The state variables are generally some minimal set of variables such that additional information about the past or the present would be irrelevant. They capture the current state of the system.
