Interpretation of outputs from residuals of regression I am trying to understand a de-trending method and hope someone can explain what it implies. 
Let's say I have rainfall data from 1969 - 2013 for a location. I do this:
    mod <- lm(rainfall ~ year) 

I use the coefficients from mod to predict rainfall for year 2009.
 2009.rain <- mod$coefficients[2]*2009 + mod $coefficients[1]

Now I add 2009.rain to residuals from mod
 detrended.to2009 <- `2009.rain + mod$resid`

The data detrended.to.2009 has been referred to as being "de-trended to 2009 
levels". 
Could anyone explain to me what exactly is happening and how does it affect my results. For example, let' say for the same period, I have crop yield (which has been detrended by taking the first-difference) and I want to regress it to the detrended.to.2009 rainfall data. How will be the coefficients be interpreted from such a model? 
Thank you. 
 A: In the first step you have computed what is the trend (mod$coefficients[2]) and got mod$resid that is the de-trended time series. 
In the second step you have computed what is the value of the rainfall in 2009 according to the model. 
In the third step you have just changed the level of the residual timeseries to adjust it to the 2009 value. IMHO this last step is quite useless.
Going to your specific question: if you regressed the crop yield difference against the de-trended.to.2009 (or against mod$resid, the difference will be just in the inception coefficient), the coefficient of the rainfall, $\beta$, means that for each unit that the rainfall is above of its trended value (mod$coefficients[2]*year + mod$coefficients[1]) the crops yield increases $\beta$ units respect to the last year value.
The difference between regressing against de-trended.to.2009 andmod$resid is in inception / constant coefficient, generally noted $\beta_0$. Think that this coefficient just adjusts to force the residual have zero mean.
In the following model:
$y = \beta_0 + \beta_1 * x+\epsilon$
$\beta_0$ satisfies $\beta_0=E[y]-E[\beta_1x]$ 
thus changing $x$ by $x' = x + c$, just will modify $\beta_0$ by $\beta_1$*c
