I am reading a paper related to timeseries forecasting in which I have a question regarding the seasonal AR model described in equation (1.2) namely:
$log(y_t)$~$log(y_{t-1}) + log(y_{t-12}) + x^{(1)}_t + e_t$
My understanding of the equation is that to predict the $log$ of the value of the variable $y$ at timestamp $t$ , the author is considering its value at timestamp $t-1$ and it's value during the same month (since the data is split up by month), the previous year. The $x^{(1)}_t$ indicates a google trends parameter included as side information.
My question is however related to the error term $e_t$. I have never quite intuitively understood how these error terms are generated and this is tripping me up. I understand that this is required to account for the residual effect, I have read in a couple of places that these error terms are assumed to be i.i.d (which I'm assuming is normally distributed with $\mu$ (mean) zero and constant $\sigma^2$ (variance) . $\mathcal{N}(0,\sigma^2)$ and this requirement comes from the fact that they are assumed to be uncorrelated. But What I get confused about, is, how are these error terms drawn from the distribution , meaning how are they scaled so as to not dominate over the other terms in the equation? The result of a least-squares regression in the same paper is :
$log(y_t)$~$0.114 \cdot log(y_{t-1}) + 0.709 \cdot log(y_{t-12}) + 0.006 \cdot x^{(1)}_t + 2.312$
How was the value 2.312 generated, so an insight into the process that generated this value would be nice?
My second question is more to check my interpretation of the regression parameters, am I correct in assuming that in this case, the regression parameters indicate that the most of the variance in the data is explained by the value of $y_{t-12}$, the next most important value is $y_{t-1}$ as far as explaining the variance is concerned and finally the $x_{t}^{(1)}$ which although has a positive correlation, is weakly correlated?
1) You are confusing the notation of R programming language with the mathematical meaning. Viz, the theoretical model is $$ \log(y_t) = b_0 + b_1\log(y_{t-1}) + b_2\log(y_{t-12}) + b_3x_1 + e_t, $$ and in R notation is written as $$ \log(y_t) \sim \log(y_{t-1}) + log(y_{t-12}) + x_1. $$ Hence, $2.312$ is simply the OLS estimator of the intercept term $b_0$.
2) No, the absolute value of $\hat{b}$ may depend on the scale of the corresponding variables as well. If you want to check "importance", you can perform model selection by the $R^2$ index.
(Surely, this not the best possible model for "importance" selections, however, without some advance statistical knowledge, it's probably the simplest one. It gives you the marginal contribution of each of the variables to the "explained variance": $ESS/TSS$.)