Effect of Lag time between explanatory variables and response on linear regression I am curious as to how lag time between explanatory variables and response variables affect linear regression models. 
I am looking at some environmental data, mainly precipitation, temperatures, and change in groundwater levels. I ran a cross-correlation between precipitation and groundwater level, and found out that there is an 11 month lag between precipitation and change in groundwater levels. 
When I try to fit a linear model it gives me a very low t-statistics, a very high P-value and a negligible R-squared. I wonder if these results are a reflection of the lag time? if yes, how do I address that. I would really appreciate your response. 
 A: Unfortunately, I had too many questions to put them all into comments, so I'll ask them in an "Answer".  Once you answer my questions I can edit this response so that it better answers your original question.
First & foremost, a stable 11 month lag between precipitation & water level seems odd (but is possible).  How confident are you about that lag?  How sharp was the peak at 11 months? How did you account the 12-month seasonal cycle?
How many years of data do you have? How many wells? How often are measurements taken? Are they taken at even time intervals? Any missing data? Are the wells ever pumped, or are they just for observation?
How big a region do your wells cover? Is there much geological or topographical variation? If your region is pretty flat and your aquifer has high conduit porosity (like the karstic Edwards aquifer in TX), there's a good chance you can ignore spatial variation.  But if your wells are in an aquifer with low hydraulic conductivity you might have to deal with spatial variation.  If they're in different basins or drilled into different aquifers you will definitely want to take that into account.
Next, consider using the Palmer Hydrological Drought Index - it's a better indicator of recharge than precipitation and temperature.  I'd examine the cross-correlation between water levels and the PHDI.
The main reason I'm suspicious of the 11 month lag is that precipitation experiences lots of high-frequency variation, but recharge doesn't.  Precipitation fuels recharge, but if it takes 11 months for precipitation to make its way from the surface to your well, you'd expect all that time spent percolating would smooth the peaks from rainy months and fill in the troughs for dry months.  E.g. if you experienced a wet month, then a dry month, then another wet month, you'd see a pretty constant recharge to the aquifer.  Eleven months later, you wouldn't see well levels rise for a month, fall for a month, then recover for a month.
Is the study region developed/farmed? Drawdown from nearby pumping might be a factor. In my experience, pumping causes most of the high-frequency changes in water level - recharge is slow-and-steady.  Pumpage (especially agricultural) often depends more directly on precip/temp.  If your wells are in an agricultural area, consider leaving precip/temp in the model, and/or add an indicator of crop moisture like the CMI.  Pumpage in developed areas is also affected by temp and precipitation, though not quite as much as farmland.
Because recharge is usually such a slow process, you will need many, many years of data to detect any patterns in recharge.  The exception to this rule is for aquifers that recharge extremely quickly (e.g. shallow unconfined aquifers overlain with highly permeable sediment), but if that were the case you wouldn't see an 11 month lag between precipitation and water level.
Recharge via precipitation affects aquifer levels on a monthly to yearly time scale.  But precipitation can affect pumpage on the scale of weeks to months (no rain for a couple weeks will spike pumpage for irrigation - especially when the weather is hot).  And pumping affects aquifer levels immediately.  
If your data set covers just a few years, its entirely possible that the 11-month lag you see is actually due to natural variation in the yearly cycle of both precipitation and pumpage.  (This is unlikely if your data set spans many years.)
Also, what's the surface water situation? Does it freeze & melt in your study area? Are there rivers supplied by spring snowmelt? Specifically, how does surface water interact with ground water in your study area? Do you have access to a groundwater model (e.g. MODFLOW)?
Finally (though this should really be your first question): what is your research objective?  Are you trying to find out what is causing changes in well levels?  Do you want to predict well levels?  Do you want to estimate the impact precipitation has on well levels?  This will guide your analysis.
