Question

 

If we could draw many random samples from the same population, and each time we ran the exact same regression, then we would get the same regression coefficients but different standard errors.

Answer 1

False

coefficients can also change

x <- rnorm(100)
e <- rnorm(100)
y <- 2*x + 5 + e
model <- lm(y~x)
summary(model)

running above code again, coefficients are different

Call:
lm(formula = y ~ x)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.90224 -0.60599  0.06986  0.57724  2.61120 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  5.09043    0.09851   51.67   <2e-16 ***
x            2.05151    0.09796   20.94   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9848 on 98 degrees of freedom
Multiple R-squared:  0.8174,    Adjusted R-squared:  0.8155 
F-statistic: 438.5 on 1 and 98 DF,  p-value: < 2.2e-16

> x <- rnorm(100)
> e <- rnorm(100)
> y <- 2*x + 5 + e
> model <- lm(y~x)
> summary(model)

Call:
lm(formula = y ~ x)

Residuals:
     Min       1Q   Median       3Q      Max 
-3.08156 -0.64059 -0.04581  0.66655  2.32896 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  4.97905    0.09979   49.90   <2e-16 ***
x            1.79899    0.09973   18.04   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9977 on 98 degrees of freedom
Multiple R-squared:  0.7685,    Adjusted R-squared:  0.7662 
F-statistic: 325.4 on 1 and 98 DF,  p-value: < 2.2e-16