How is regression different from time series?

Question

When you have data $(x_1,y_1), \cdots (x_n,y_n)$, then in regression one tries to find a function $f: X \rightarrow Y$ which explains the dependance between $x_i$ and $y_i$. Here it is assumed that time plays no role (i.e $f$ doesn't change with time). If $f$ dependents on time i.e. $(x_i,y_i)$ is sampled at time $i$, then the regression technique fails and then one studies it as a time series problem. My question is why do we need the concept of time series when we can simply analyze the above situation by having another covariate $t$ along with $x$ and apply regression method to find $f': X \times T \rightarrow Y$.

Answer 1

You can use time as a covariate, but you have to account for several issues.

1. Is time an effect of itself? Although this is a legitimate factor, however, rarely we consider time to be the factor of interest rather than some covariate that changes over time. Hence, time is usually a latent variable that is not of direct interest.
2. By adding explicit $T$ to the regression model, you don't address the dependence between observations. That is, classical regression assumes i.i.d $\{y_i, x_i\}_{i=1}^n$, this assumption is completely violated in time series data. Therefore you need to model this dependence and modify your technique accordingly, without this, just by adding time as a covariate, you will get biased and inconsistent estimators.