Regression modelling on time series data Consider a time series dataset, such as the daily average temperature of Ohio city. Typically, we can employ the ARIMA or SARIMA modelling approach to analyse the data. May I ask is it possible for us not to use the ARIMA or SARIMA models; instead, we use a regression approach to analyse the dataset. To be precise,

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*The outcome variable, Y: today's (T)'s temperature

*Predictor 1, X1: yesterday's (T-1)'s temperature

*Predictor 2, X2: ereyesterday's (T-2)'s temperature

*Predictor 3, X3: (T-3)'s temperature

So, we run an OLS regression to obtain the relationship for Y=f(X1, X2, X3), such that the estimated coefficients beta0, beta1, beta2, and bets3 are obtained.
Is this something correct? And may I get any previous study that focused on this kind of statistical modelling approach?
 A: Ad hoc and for very short term forecasting, yes, you can do that. IIRC, you lose some desirable statistical properties, but you would need to consult a textbook on mathematical time series modeling to learn more, or hope for a real time series modeler to come along - I'm just a forecaster, and as such would happily do this. (After all, "yesterday's weather" is often very hard to beat as a forecast.)
Long-range forecasts would be problematical, because you would need to include much longer lags to capture the seasonality. Because of the length of the seasonal cycle (365), you would presumably need to include multiple such long lags, and suddenly your model is very large.
SARIMA does not work well with daily data and yearly seasonality, at least when you do automatic modeling (e.g., the forecast and fable packages in R), which runs for a long time to determine the model. It may be better to look at BATS/TBATS models, which model seasonality using Fourier terms.
A: This is a very popular approach to forecasting. Actually, most deep neural network approaches to forecasting work like this. But this method is also successfully used with other models like random forest or gradient boosting machines.
Those data-driven methods are often quite successful, provided there is sufficient data available. The great disadvantage relative to model-driven methods like ARIMA is that they are rather black boxes, which can do forecasting but don't really give you more information about the time series itself, i.e. there is no real model with interpretable parameters.
