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?


  • 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 $t-1$.


2 Answers 2


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.

  • $\begingroup$ How come I(1) is supposed to imply population growth has a trend? E.g. ARIMA(0,1,0) is a random walk, hence no trend in growth, but a stochastic trend in level. @EconJohn $\endgroup$ Commented Nov 28, 2017 at 21:49
  • $\begingroup$ Use a random walk and calculate the first difference (growth). Check that it has a trend (likely, the mean is 0). $\endgroup$ Commented Nov 30, 2017 at 18:43
  • $\begingroup$ This is incorrect. Check the definition of a random walk in, say, Wikipedia. $\endgroup$ Commented Dec 2, 2017 at 16:15
  • $\begingroup$ Strange that you didn't cite Resnick or Hamilton, but since you seem to like Wikipedia, look here: en.wikipedia.org/wiki/Order_of_integration $\endgroup$ Commented Dec 3, 2017 at 18:57
  • 1
    $\begingroup$ I agree with you (and Wikipedia) completely on this. However, read my comment above carefully. How come I(1) is supposed to imply population growth has a trend? E.g. ARIMA(0,1,0) is a random walk, hence no trend in growth, but a stochastic trend in level. Also, your claim in the next comment is incorrect, because neither a random walk nor its first differences do not have a deterministic trend by the definition of the random walk. $\endgroup$ Commented Dec 3, 2017 at 19:44

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.

  • $\begingroup$ This is a good answer. I just wanted to add the following. You could give the AR component the interpretation you suggest, but that doesn't mean that the AR coefficient you estimate will recover the parameter with this meaning. The act of giving the AR component this interpretation amounts to constructing a structural equation and causal model. The question of whether your estimation procedure recovers parameters with this interpretation is another story. The fact that the AR model has an equivalent MA interpretation makes this point of view relatively easy to see. $\endgroup$
    – jmbejara
    Commented Feb 23, 2018 at 19:55

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