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I often see the concepts Time Series Regression and Time Series Forecasting refering to something similar but I don't see clearly what's the difference among these two concepts. By now, the idea I have for each concpet is the next one:

  • Time Series Forecasting: The action of predicting future values using previously observed values.

  • Time Series Regression: This is more a method to infer a model to use it later for predicting values.

But so many times, I see people use the second concept to refer to the first one. I don't know if this is correct or if I am missing something. If anyone can help me make myself clear I would appreciate it

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In principle what you have said captures the difference. But may be people refer TS Forecasting as TS Regression because technically, TS forecasting involves TS regression.

Another point (concerning your second bullet point) is that every TS regression is not necessarily used for predicting. It may simply be for understanding the relationship between two variables.

Consider a very naive example:

$$C_t = \beta_0+\beta_1Y_t + \epsilon_t$$ where $C_t$ is consumption, $Y_t$ is income.

In the above example, an economist may simply be interested in estimating, $\beta_1$ which is the marginal propensity to consume. In theory, the same model can be used for predicting $C_{t+h}$ given $Y_{t+h}$. But it is the practice (rather than theory) that is more interesting and complicated. Let me mention just 2 differences between TS regression and forecasting that one experiences in practice:

  1. Take above equation. At a given time, you are likely to have published figures for $Y_{t+h}$ but not $C_{t+h}$. This is where first practical different in TS Regression: Choice of Variables. Forecasting requires leading indicators, i.e. those explanatory variables that are available before the the response variable value is known. Here there is less interest discovering whether the given explanatory variable explains the response variable. Interest is more in how well it can predict the response variable. For example, for forecasting $C_t$ you may want to use volume of credit card transactions. We know that it is obviously going to explain $C_t$ (so not of interest to an economist) but very helpful in forecasting if the data is available in advance.

  2. Diagnostics: In TS regression the diagnostics of regression output usually involves checking for significance of explanatory variables - to ensure how well it explains changes in response variables. In forecasting it may not be so important (Rob Hyndman has given some example somewhere in his blog, I don't remember, however, exact topic). Another example is $R^2$ - important in forecasting, but not as much in general regression. There are many more (frankly, I have been thinking of posting this question for a list of separate diagnostic list for regression and forecasting).

I am sure others here can give some more interesting inputs.

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  • $\begingroup$ In theory, the same model can be used for predicting $C_{t+h}$ given $Y_{t+h}$. I guess this would not be forecasting since $Y_{t+h}$ is not known before $C_{t+h}$ is known, while forecasting should be about the future given the present and the past. Also, $R^2$ might be more relevant for regression (indicating how well the dependent variable is explained) than forecasting (where in-sample measures are of limited use). $\endgroup$
    – Richard Hardy
    Oct 25, 2019 at 12:33
  • $\begingroup$ Correct. But if $Y_{t+h}$ is know before $C_{t+h}$, then the equation can be used. I meant from data availability point of view. For example, most governments publish initial estimates of output before they publish consumption estimates. But I agree with your comment that in strict sense even this isn't forecasting. I forgot there's a separate term for this: nowcasting. $\endgroup$
    – Dayne
    Oct 25, 2019 at 12:49
  • $\begingroup$ About $R^2$: I am not sure but let me take an example. Say the equation used in the answer is the real data generating process. Now if variance of error term is large, the $R^2$ is bound to be small. However, the model is true and the estimate of $\beta_1$ would be BLUE. So the estimate of $E(C_t)$ is excellent but not of $C_t$. The latter is of more interest in forecasting while the former in understanding relationships. $\endgroup$
    – Dayne
    Oct 25, 2019 at 12:58
  • $\begingroup$ Thanks. I think our comments are not contradictory; your $R^2$ example is a counterexample, but it is not difficult to find pro-examples as well (it depends on what the goal of regression analysis is: measuring effect sizes or explaining the variation in the dependent variable; in-sample $R^2$ is not necessarily a good measure of predictive performance either). $\endgroup$
    – Richard Hardy
    Oct 25, 2019 at 13:15
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Time Series Forecasting (to my way of thinking ) purely uses the past of the endogenous series as the basis of the model (ARIMA or Box-Jenkins) .

Time Series Regression also uses causal (exogenous)series and their lags in addition to the history of the endogenous series. These models are generally called Transfer Functions or Dynamic Regression Models or more frequently SARMAX models https://autobox.com/pdfs/SARMAX.pdf.

Both approaches are best employed while detecting and incorporating latent deterministic structure such as pulses, level shifts and local time trends and possible transient effects of parameters or model error variance over time..

Please see http://www.autobox.com/pdfs/regvsbox-old.pdf which I wrote a number of years ago to provide more details/contrasts on "Regression vs Box_Jenkins"

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Well, regression is about estimating/fitting a model and the concept expands beyond time series data. A model is fit to establish the relationship between the endogenous and exogenous variables.

Forecasting/Prediction of the endogenous variable is one use case of a fitted model, but not all fitted models are good for forecasting purposes.

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