Difference between $y_t = \alpha + \beta t$ and $y_t = y_{t-1} + \beta$ Would someone mind walking me through the differences between:
\begin{align}
y_t &= \alpha + \beta t  \\
    &\&  \\
y_t &= y_{t-1} + \beta
\end{align}
as well as between
\begin{align}
y_t &= \alpha + \beta t + a_t  \\
    &\&  \\
y_t &= y_{t-1} + \beta + a_t
\end{align}
where I believe $a_t$ is a sequence of independent normal random variables, $\mathcal N(0,1)$?
Basically all I can draw from this is that $y_t = \alpha + \beta t$ is a regression model where there's a dependent variable and independent variables and $y_t = y_{t-1} + \beta$ is a time series model that depends on the past and some variable $\beta$. Clearly that isn't a very good explanation of what the difference between the equations are. 
 A: $$y_t = \alpha + \beta t$$ 
and 
$$y_t = y_{t-1} + \beta$$ 
are the same up to a constant: take the second equation and iteratively substitute for $y_{t-1}$ to get 
$$y_t = y_{t-1} + \beta = y_{t-2} + 2 \beta = \dotsc = y_0 + \beta t.$$
So if $\alpha = y_0$, the two coincide. If not, the difference is $\alpha - y_0$, which is a constant.

Meanwhile,
$$y_t = \alpha + \beta t + a_t$$ 
and 
$$y_t = y_{t-1} + \beta + a_t$$
are quite different by nature: take the second of the latter two equations and iteratively substitute for $y_{t-1}$ to get 
$$y_t = y_{t-1} + \beta + a_t = y_{t-2} + 2 \beta + a_{t-1} + a_t = \dotsc = y_0 + \beta t + \sum_{\tau=1}^t a_{\tau}.$$
The difference is $\alpha - y_0 - \sum_{\tau=1}^{t-1} a_{\tau}$ which involves a constant component $\alpha - y_0$ and a random walk component $\sum_{\tau=1}^{t-1} a_{\tau}$. When $t$ grows larger, the random walk component will dominate the constant component and the difference will essentially be a random walk.
A: Can you be more specific on $\beta$, is it a variable or a constant? if constant, what values can it take? 
I believe, it is a constant. These are time series models.
The equations with time trend is a model of deterministic trend, however, it could easily be shown that $y_t$ is stationary if $\beta$ approaches $0$. The second model is an AR(1) model where the coefficient of $y_{t-1}$ is 1. The DGP in both cases is sensitive to the values $\beta$ takes. 
To see this, run simulations in excel varying the values that $\beta$ can take.
A: If $\alpha + \beta t$ are not random then $y_t$ is a completely deterministic function (i.e. this is just some line with intercept $\alpha$ and slope $\beta$. 
The next equation is also not random (as long as $\beta$ and the initial value $y_0$ is not random)
The next set of equations are what people might more typically refer to as regression and time series regression respectively. 
