For the following regression model, $$ y=\hat{\beta}_0+\hat{\beta}_1\cdot|m(a)-m(b)|+\sum_{k=0}^{12}\hat{\beta}_{k} 1[|m(a)-m(b)=k|] $$ The fitting plot not a linear.
Why the blue line (that is fitted using a regression model that assumes a linear trend) is not a linear line?
"Linear" regression estimate coefficients in order to explain the response of a variable $Y$ to changes in variables $X_k$ using a linear equation of the form $Y=X\beta$. Ex:
$$y_i = \beta_0 + \beta_1 x_{1, i} + \beta_2 x_{2, i} + \epsilon_i$$
"Linear" refers to the fact that $\mathbb{E}(y_i)$ is defined as a linear combination of the parameters $\beta$, not necessarily $X$, which could be modified depending on your objectives/interpretations of the data. Indeed, if some variables $X_k$ are "transformed" variables derived from your actual variable of interest, then the responses will not be linear, but you are still using a linear regression model. Ex:
\begin{split} y_i &= \beta_0 + \beta_1 x_{1, i} + \beta_2 x_{2, i} + \epsilon_i\\ &= \beta_0 + \beta_1 t_{i}^2 + \beta_2 \sqrt{t_{i}} + \epsilon_i \end{split}
With the equation above, $\mathbb{E}(y_i)$ is defined as a linear combination of parameters $\beta$ and of transformed variables $X_k$. But it is not a linear combination of the actual variable of interest, time $t$, which has a non-linear impact on your dependent variable $Y$.
This is what is happening in your case, where your variables $X_k$ are non-linear functions of time $t$, since you introduced absolute values of time differences and binary variables depending on the time of each observation.
Indeed, but in my second equation, the predictors of the linear regression are the variables $X_k$. So these 2 sentences have the same meaning.
– FP0 Jul 23 '22 at 15:31