ARIMA modelling and Economic interpretation

Question

ARIMA models, from what I know, are atheoretical models in the sense that they don't provide us with meaningful economic interpretation of a process.

However in population forecasting, can ARIMA models be interpreted in a meaningful way?

i.e.

• can $AR(1)$ process can be interpreted as:

  population growth in period $t$ is a function of population in period $t-1$

• can $MA(1)$ process can be interpreted as:

  population in period $t$ is a function of a policy or ongoing event (like immigration or emigration) which has occured in $t-1$.

Answer 1

You got it quite nicely:

• An AR(1) process would imply population in $t$ depends linearly on the population level at $t-1$
• An MA(1) process would imply population in $t$ depends linearly on the population shock, event or "surprise" at $t-1$.
• An I(1) process would imply population growth (that is, the first difference, $X_t-X_{t-1}$) is a stationary process and doesn't have a trend; the population level itself isn't. In the real world, though, it's strange to find an I(2) model, much less I(>2).

Mix everything up and you can forecast population with a combination of previous population levels, surprising events and population growths.

Answer 2

So we can think of an AR(1) process as:

$y_t = a*y_{t-1} +e_t$

where $e_t$ is some shock / noise process that we can't explain. Note that we can rewrite an AR process as an MA process where by tracing through we get

$y_t = a^2*y_{t-2}+a*e_{t-1}+e_t\approx \sum_{i=0}^{t-1}a^i*e_{t-i}$

So that an AR process is just the weighted sum of all previous shocks where $a$ is what determines the persistence of past shocks.

So then what does it mean to say we have an AR(1) process? Well it means that the outcome today is to some extent dependent on the outcome yesterday (as you yourself suggested). In an extreme case we can set $a=1$ so that we have a random walk process where

$\Delta y_t=y_t-y_{t-1}=e_t$

This implies that the outcome today depends entirely on the outcome yesterday and any additional change in the series is purely due to random noise. Alternatively if $a\geq1$ we would say the process is non-stationary or explosive while if $a<1$ then the process is stationary around the mean of the process or of the shocks themselves.

An MA process has a similar interpretation except we think of it more in terms of unexplainable shocks or noise with some degree of persistence. The important thing is that it is not really something we have a variable for in the MA case but rather this notion that the the process just depends on this noise component. It is more common to find these types of models for forecasting in finance where there's alot of movement but a less clear sense of what might be driving it.

In terms of population forecasting, the AR component could definitely be interpreted as you suggested, while the MA component is less easy to interpret. It's really just a sum of shocks that are otherwise unexplained. If you had an explanation for it (i.e. immigration etc.) then you would likely want to include that as an explanatory variable to model it explicitly rather than approximately through the MA process.