I am reading Chapter 10 Linear Models for Functional Responses of Functional Data Analysis with R and Matlab by Ramsay.
On page 148, the book mentions the model for "Climate Region Effects on Temperature".
$y_i(t)=\beta_0(t)+\sum_{j=1}^4 x_{ij} \beta_j(t) + \epsilon_i(t)$ --- (10.1)
where i refers to the index of 35 Canadian cities; the response variable is the functional response of daily temperature; j refers to the index of four climate zones: Atlantic, Pacific, Prairie and Arctic.
In this case, the values of $x_{ij}$ are either 0 or 1. If the 35 by 5 matrix $\mathrm{Z}$ contains these values, then the first column has all entries equal to 1, which codes the contribution of the Canadian mean temperature; the remaining four columns contain 1 if that weather station in the corresponding climate zone and 0 otherwise. In order to identify the specific effects of the four climate zones, we have to add the constraint
$\sum_{j=1}^4 \beta_j(t)=0$ for all t --- (10.2)
And one of the methods of imposing such constraint is to add the above equation as an additional $36^{th}$ "observation" for which $y_{36}(t)=0$.
I am fine with the model introduction, but I am a little confused why we should take (10.2) into consideration and why adding $y_{36}(t)=0$ is the way to imposing such constraint.
