Difference between $y_t = \alpha + \beta t$ and $y_t = y_{t-1} + \beta$

Question

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.

Answer 1

$$y_t = \alpha + \beta t$$

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.

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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.

Answer 2

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.

Answer 3

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.