> 1. "n = 1" is misleading. In a time series,
> the n is the number of time samples; thus n
> is large here, not 1.
In fact, the extraction or consumption rate you use as a data point *for a given period* is a sample *of one* drawn from the population random variable, i.e. the random variable that includes all possible extraction or consumption rates *for that period in that economy*.
Ex post, that rate is just one realization of that population random variable, like the winning lottery number is just one realization of all the possible lotto numbers. But ex ante, the rate can be any among a bunch of possible ones.
And your extraction or consumption rate for the next period will be another sample *of one*. It will not be drawn from *the same* population random variable, the one including *last year's* possible extraction or consumption rates in *last year's* economic conditions. The economy from which you draw your size-one samples is shifting. And it may be shifting in response to those very changes.
Think of a bucket with rubber balls of different colors. You can easily draw inferences about the distribution of colored balls in that bucket, as long as the content of the bucket doesn't change, the proportion of colored balls remains fixed, etc. But if the colors and their proportion are mixed up every time you draw a sample (one rubber ball), you're in trouble. And if the mix-up of colors happens in response (or in anticipation!) to the kind of colors you (may) get in your previous (or future) draws, then you're in deep trouble.
It's not hard to imagine that the mathematical conditions for you to be able to draw any meaningful inference about the population random variable from *a sequence of size-one samples* are demanding. They boil down to having the probability distribution of your population random variable staying (basically) fixed over time. In plain words, it requires that the economy (its basic structure) remains "stationary" or fixed over time. If you assume this, you're excluding a priori the very result that can falsify your model. How is that scientific?
Not to be nihilistic about time-series regression, you can take differences of higher and higher order to your data -- e.g. instead of using extraction/consumption growth rates, you can use *the change* in those growth rates, or the change in the change, etc.). If you keep doing that, at some point, your data will start to look more and more *as if* drawn from a stationary probability distribution.
But you can never tell for sure that the distribution of higher-order differences is truly stationary, since you cannot preclude the underlying distribution from shifting in ways you can't predict and messing up the assumptions of your model. *That can happen* because of the self-referential nature of the economy -- its chicken-and-egg properties. Self-referential dynamic systems are undetermined: things can go every which way. And, believe me, interpreting the results of a regression on higher-order differences won't be as clear cut as predicting future oil rates.