Why is sum of squared residuals non-increasing when adding explanatory variable? In my econometric textbook(Introductory Econometrics) covering OLS, the author write, "SSR must fall when another explanatory variable is added." 
Why is it?
 A: Assuming you have a linear regression model, for easy notation consider first one then two covariables.  This generalizes to two sets of covariables.
The first model is
$$
  I \colon y_i=\beta_0 + \beta_1 x_{1i}+\epsilon_{i}
$$
the second model is 
$$
  II \colon  y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \epsilon_i
$$
This is solved by minimizing sum of squared residuals, for model one we want to minimize $\text{SSR}_1 = \sum_i (y_i-\beta_0-\beta_1 x_{1i})^2$
and for model two you want to minimize 
$\text{SSR}_2 = \sum_i (y_i-\beta_0-\beta_1 x_{1i}-\beta_2 x_{2i})^2$. 
Lets say you have found the correct estimators for model 1, then you can obtain that exact same residual sum squares in model two by choosing the same values for $\beta_0, \beta_1$ and letting $\beta_2=0$.  Now you can find, possibly, a lower sum squares residual by searching for a better value for $\beta_2$. 
To summarize, the models are nested, in the sense that everything we can model with model 1 can be matched by model two, model two is more general than model 1.  So, in the optimization, we have larger freedom with model two so can always find a better solution.
This has really nothing to do with statistics but is a general fact about optimization.
A: SSR is a measure of the discrepancy between the data and an estimation model.
If you have the option to take into account another variable , then if this variable contains more information, the fit would naturally be tighter, which means a lower SSR.
