1. Simply said, when a line is not going to be a really great fit to your data set;

2. BIC (Bayes information criterion) will assist you: you take the model that has the smallest BIC as being the optimal model order. Hence, compute BIC quantities for every model order, then make a plot of them (BIC as a function of the polynomial order), and finally what you want to look for is a minimum point. i.e. the minimum point is the optimal model order.

Polynomial Regression is a form of Linear regression known as a special case of Multiple linear regression which estimates the relationship as an nth degree polynomial. Polynomial Regression is sensitive to outliers so the presence of one or two outliers can also badly affect the performance.

Degree 1 – linear. Degree 2 – quadratic. Degree 3 – cubic. Degree 4 – quartic (or, if all terms have even degree, biquadratic) Degree 5 – quintic.

Multivariate polynomial regression was used to generate polynomial iterators for time series exhibiting autocorrelations. A stepwise technique was used to add and remove polynomial terms to ensure the model contained only those terms that produce a statistically significant contribution to the fit.

Polynomial models are a great tool for determining which input factors drive responses and in what direction. ... A quadratic (second-order) polynomial model for two explanatory variables has the form of the equation below. The single x-terms are called the main effects.

https://methods.sagepub.com/dataset/howtoguide/age-polynomial-in-wtms-2009-stata

Preprint Progression of COVID19 Pandemic in India: A Concurrent Linea...

Chapter Lineare Regression