Rsquared Dependence

> set.seed(1977)
> n <- 1500
> beta.actual <- matrix(c(2, 3), ncol = 1)
> beta.sample <- cbind(rnorm(n, beta.actual[1]), rnorm(n, beta.actual[2]))
> error <- rnorm(n)
> income <- cbind(rep(1, n), seq(from = 1, to = 5, length.out = n))
> consumption <- income[, 1] * beta.sample[, 1] + income[, 2] *
+     beta.sample[, 2] + error
> savings <- income - consumption

> summary(lm(consumption ~ income + 0))
Call:
lm(formula = consumption ~ income + 0)

Residuals:
      Min        1Q    Median        3Q       Max
-15.19239  -1.97241   0.02677   2.07748  13.28453

Coefficients:
        Estimate Std. Error t value Pr(>|t|)
income1   2.0679     0.2511   8.236 3.83e-16 ***
income2   2.9242     0.0781  37.442  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.495 on 1498 degrees of freedom
Multiple R-squared: 0.9136,     Adjusted R-squared: 0.9134
F-statistic:  7916 on 2 and 1498 DF,  p-value: < 2.2e-16

> summary(lm(savings ~ income + 0))
Response Y1 :

Call:
lm(formula = Y1 ~ income + 0)

Residuals:
      Min        1Q    Median        3Q       Max
-13.28453  -2.07748  -0.02677   1.97241  15.19239

Coefficients:
        Estimate Std. Error t value Pr(>|t|)
income1  -1.0679     0.2511  -4.253 2.24e-05 ***
income2  -2.9242     0.0781 -37.442  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.495 on 1498 degrees of freedom
Multiple R-squared: 0.8987,     Adjusted R-squared: 0.8986
F-statistic:  6646 on 2 and 1498 DF,  p-value: < 2.2e-16

Response Y2 :

Call:
lm(formula = Y2 ~ income + 0)

Residuals:
      Min        1Q    Median        3Q       Max
-13.28453  -2.07748  -0.02677   1.97241  15.19239

Coefficients:
        Estimate Std. Error t value Pr(>|t|)
income1  -2.0679     0.2511  -8.236 3.83e-16 ***
income2  -1.9242     0.0781 -24.638  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.495 on 1498 degrees of freedom
Multiple R-squared: 0.8448,     Adjusted R-squared: 0.8446
F-statistic:  4078 on 2 and 1498 DF,  p-value: < 2.2e-16