# 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.

• If you assume there is a lag (it seems you do) and design a model that allows for a lagged effect, then you cannot blame the lag for a low $R^2$ and the like. If you design a model where only contemporaneous effects are allowed, what else would you expect than a low $R^2$? So what is your model? Anyway, if your cross-correlation analysis has a significant spike at 11 months, you should likely find a corresponding regressor statistically significant; so do you? (Although statistical significance does not guarantee large effect size.) Commented Jan 27, 2015 at 18:27
• How many years of data do you have? How many wells? How often are measurements taken? Commented Jan 28, 2015 at 3:02

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.

• This is a lot of questions. But first of all thanks for your answer.
– Baks
Commented Jan 28, 2015 at 21:15
• This is a lot of questions. But first of all thanks for your answer. My objective is to find out what proporting of change in groundwater level can be attributed to climate ( precip + Temp). I have data varying in legnth from 5 to over 20 years, but of course with missing data in the longer series. the 5 year data is continuous (level loggers, hourly). I have 100s of wells over a 35,000 km2 area. But I was trying with a few wells ( 5 years of record) to see if it was altogether possible to estimate the proportion of variability related to climate. I will explore the PHDI option.
– Baks
Commented Jan 28, 2015 at 21:29
• @Baks - I hadn't considered how US-specific my advice was, but if you're measuring area in km2 then PHDI & CMI might not apply. I'm sure other indices are computed elsewhere. Commented Jan 29, 2015 at 3:40
• @Baks - How are you defining climate? If you define climate as weather patterns averaged over several years (5+ at a minimum, 20-30 seems more common), then you probably don't have records long enough to detect the effect of climate change. Commented Jan 29, 2015 at 3:50
• @Baks - You can use the wells with 5 years of hourly measurements for characterizing the relationship between weather & aquifer level. If you have any records of pumping from the same aquifer, you can analyze that too. Understanding the relationship between weather and aquifer level is an important first step for modeling the effects of climate change. Commented Jan 29, 2015 at 4:01