If adjusted R-squared is zero, can individual coefficients have meaningful interpretation in multiple regression?

Question

It can happen that adjust R-squared metric in multiple regression is zero (or very close to zero), but individual coefficients are statistically significant.

Under these circumstances, can I still interpret the individual coefficient (independent variable) as having an impact on the dependent variable as usual, even if the overall model does not seem to have explanatory power given the zero adjusted R-squared?

Answer 1

Here is a stupid example of a (rightly) significant coefficient but very low R-square value.

> x=1:1000
> y=x+rnorm(1000,0,1000)
> summary(lm(y~x))

Call:
lm(formula = y ~ x)

Residuals:
     Min       1Q   Median       3Q      Max 
-2862.02  -701.14   -18.76   720.95  3088.67 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -40.6993    64.5752   -0.63    0.529    
x             1.1451     0.1118   10.25   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1020 on 998 degrees of freedom
Multiple R-squared:  0.09517,   Adjusted R-squared:  0.09427 
F-statistic:   105 on 1 and 998 DF,  p-value: < 2.2e-16