A 'Pure' Time Series Model

Question

Can anyone explain to me what is meant by a 'pure' time series model?

I believe it might have something to do with the exclusion of external factors but I'm really not too sure.

For example, the term is used here[1] (Sec 4.8),

a pure time series based approach can be used to generate forecasts for the predictor variable

And also in a post here on Cross Validated:

consider a pure time series model with AR(1) DGP: $x_t=ϕx_{t−1}+c$

[1]: Hyndman, Rob J and George Athana­sopou­los,
Forecasting: principles and practice
https://www.otexts.org/fpp

Answer 1

If you place the first quote in the broader context of what's being discussed, the meaning can be discerned.

It's in a section titled Regression with time series data; the term is mentioned here:

An alternative approach is to use genuine forecasts for the predictor variable. For example, a pure time series based approach can be used to generate forecasts for the predictor variable

In that context a "pure time series model" is one not including external predictors.

That is, the model for $y_t$ depends on $x_t$, but when it comes to forecasting $y_{n+1}$ (say), you need a value for $x_{n+1}$; so it, to must be predicted. An ARMA model for $x$ would be an example of a 'pure time series' model.

Looking at your second link, again, the context in which it occurs gives us the clue. The question itself discusses trying to measure the impact of external variables ("the effectiveness of my sales campaign" and "the impact of competitors"). A pure time series model doesn't have those additional variables, just the series itself, as with the AR(1) model mentioned there.

Answer 2

What is probably meant is a "univariate" time series model (e.g. ARIMA or exponential smoothing). A univariate model only looks at the variable in question and usually tries to estimate some kind of auto-regression, trend, and seasonality in the time series.

Contrast this with a "multivariate" time series model that takes into account external factors.

For example, pretend you're trying to forecast electricity usage for a large utility. You have a historic dataset of electric usage (in megawatt hours) for each month for the past several years. You might build a simple univariate model by predicting that load this month will be the same as load last month (this is known as a "naive" model). You might improve this model by averaging usage for each month and forecasting each future month to be it's historic mean (this is known as a "seasonally naive" model). These models are known as "univariate" models because they only take into account a single variable: historic electric usage data.

Now lets pretend you're getting a little more creative, and you realize that on hot days people use a lot more electricity (to run air conditioners). You add a second variable to your model: average monthly temperature, and build a regression where electricity load is a function of what month it is and how hot it is. Since this model takes into account 2 variables, it is a "multivariate" time series model.

Finally, one small problem with multivariate models is that you need to forecast each of the component variables (e.g. temperature) before forecasting. This is why univariate models remain useful.