An ARIMA(p,d,q) model fitted to a time series implies that the dth difference of the series follows an ARMA(p,q). This means that a current value of the dth difference of the series depends on the last p values and residual values q time points "ago".

You may find your answer in the following studies.

https://www.sciencedirect.com/science/article/pii/016920709290049F

https://arxiv.org/ftp/arxiv/papers/1302/1302.6613.pdf

https://www.bebr.ufl.edu/sites/default/files/Research%20Reports/2004_m_m_projections_.pdf

https://onlinecourses.science.psu.edu/stat510/?q=book/export/html/49

Hi Jacob,

ich don't get why you are asking a question, which answer you already know ;-).

It depends on the variables you use. In general, you got the idea, just some minor tweaks....

can AR(1) process can be interpreted as: "population growth in period t is a function of population in period t−1"

In general, your intuition is correct. An AR(1) just runs an OLS regression of your time series on its last observation. If, e.g., your total population at the end of period t is denoted p(t), then an AR(1) specification might be something like this:

p(t) = phi*p(t-1) + eps_t,

with eps_t being a nuisance parameter. You see, p(t) would be linearly correlated to p(t-1). That's not the growth rate though, but the absolute value of your population.

Your MA(1) example is quite similar. It is correct, that you regress your popolation size on some unobserved variable. Note, the POPULATION SIZE, not the POPULATION GROWTH....your specification would be

p(t) = psi*eps_(t-1) + eps_t.

Hope, this has become clear. Otherwise, I'm happy to answer.

Best regards!

Hello Jacob

I am not a time series expert, but I have used them in my forecasting work. You ask two questions:

• can AR(1) process can be interpreted as: "population growth in period t is a function of population in period t−1"

I would not expect this to be the case, as there is no reason why population size should influence population growth.

• can MA(1)process can be interpreted as: "population growth in period t is a function of some policy or ongoing event (a country's policy on immigration or emigration) which has occurred in t−1"

As a direct effect, a change of policy in period t-1 can be expected to also affect period t. However, a change of policy is not specifically contained in the population data for period t-1. Knowledge of the policy is additional information, which would have to be explicitly incorporated in the model. A substantial change in population size in period t-1 could be temporary (a migration event or change of rules that affects the pace of migrant acceptances - i.e. a tempo effect); this would be noise or could also be modelled by a time-specific dummy variable. So, I would answer 'no': as in economics, time series models involving only one step are atheoretical in population forecasting. However, it may be the case that large-scale in-migration of young adults in period t-1 generates additional population growth in a subsequent range of years through childbearing and the in-migration of fiances, spouses, children and other dependents, and this may possibly be modelled by time series, though forecasting would involve high uncertainty. It may be possible to establish more meaningful interpretations of population size with lags of a generation or so, for obvious reasons, but the age range of childbearing (even in low fertility populations) would weaken the association.

I think it can be applied as long as it is linked to time series events / occurrences of population movements. In this case, you can be able to use ARIMA model to forecast population based on past data. To move things further, you can also use ARIMAX which is simply the application of exogenous variable influences on population.