# Insignificant input variable but high R2

Can a linear regression model with a single input variable x with x not significant but the model has a high R2?

Can this exist? If yes, what are the reasons?

This can happen, but only with very small amounts of data. An example (in R) with three data points:

x <- c(0,1,2)
y <- c(-0.1,0.7,1.2)
summary(lm(y ~ x))

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.0500     0.1118  -0.447   0.7323
x             0.6500     0.0866   7.506   0.0843 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1225 on 1 degrees of freedom
Multiple R-squared:  0.9826,    Adjusted R-squared:  0.9651
F-statistic: 56.33 on 1 and 1 DF,  p-value: 0.08432


Notice the R-squared is very high, but the x parameter is not 95%-significant.

With lots of data, one cannot have a very high R-squared parameter without the corresponding parameter being significant.