P value of multiple linear regression I have extremely large number of observations (8524152) of soil moisture, precipitation, evapotranspiration, delta precipitation, and delta evapotranspiration. I ran a multiple linear regression model and my result looks like 
Call:
lm(formula = SMDI ~ ET + delta_ET + PRCP + delta_PRCP, data = regData)

Residuals:
 Min  vvvv     1Q   Median       3Q      Max 
-10414.0     67.1    133.9    192.2   8737.3 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -87.508196   0.797889 -109.67   <2e-16 ***
ET            0.083853   0.001225   68.46   <2e-16 ***
delta_ET      0.267973   0.001270  211.04   <2e-16 ***
PRCP          0.237649   0.003255   73.02   <2e-16 ***
delta_PRCP    0.257458   0.003250   79.23   <2e-16 ***



Residual standard error: 1705 on 8524147 degrees of freedom
Multiple R-squared:  0.4424,    Adjusted R-squared:  0.4424 
F-statistic: 1.691e+06 on 4 and 8524147 DF,  p-value: < 2.2e-16

The t-stat for evapotranspiration (ET), Precipitation (PRCP), delta_PRCP, and delta_ET are same, and the combined p-value is also extremely small. allmost < 2.2e-16. is this possible?
Juvin
 A: What you are getting here is actually not a coincidence in your data - it is just the numerical limits of statistical computing.  If you check the characteristics of the machine you will see that R represents numerical values in binary form, and there is an imposed limit on the number of binary digits it uses to store a number.  The value you are seeing here the smallest positive number that can be represented with fifty-two binary digits, which is presumably the present limit setting of your machine.  You can check this in R by running the following code:
.Machine$double.ulp.digits
[1] -52

.Machine$double.eps
[1] 2.220446e-16

2^(-52)
[1] 2.220446e-16

identical(2^(-52), .Machine$double.eps)
[1] TRUE

So, all that you are seeing in your regression output is that all your p-values are below the smallest positive floating-point number in the present setting in R.  This does not imply any amazing co-incidence of your p-values; it just means they are being expressed below the same upper bound.  If you would like to get more accuracy in your p-values, you will need to look at how to change the calculation limits in your machine (see e.g., here).
A: The t-statistics are similar, but different.  The p-values are very small and appear similar, but in fact, are likely very small, but outside the range of numerical accuracy (so small they can't be reliably computed, so you see the values you are seeing).  They are extremely small, so the p-values are simply indicating strong significance, assuming a properly constructed model that meets model assumptions.
