Purpose
Simulate Multiple Reg and test the Hypothesis

> set.seed(1977)
> n <- 15000
> beta.actual <- matrix(c(2, 3, 4, 5), ncol = 1)
> beta.sample <- cbind(rnorm(n, beta.actual[1]), rnorm(n, beta.actual[2]),
+     rnorm(n, beta.actual[3]), rnorm(n, beta.actual[4]))
> error <- rnorm(n)
> x <- cbind(rep(1, n), runif(n), runif(n), runif(n))
> y <- x[, 1] * beta.sample[, 1] + x[, 2] * beta.sample[, 2] +
+     x[, 3] * beta.sample[, 3] + x[, 4] * beta.sample[, 4] + error
> summary(lm(y ~ x + 0))
Call:
lm(formula = y ~ x + 0)


Residuals:
      Min        1Q    Median        3Q       Max
-6.877817 -1.153599 -0.001764  1.160492  6.736600


Coefficients:
   Estimate Std. Error t value Pr(>|t|)
x1  1.95455    0.04478   43.65   <2e-16 ***
x2  3.02972    0.04889   61.97   <2e-16 ***
x3  3.98549    0.04898   81.37   <2e-16 ***
x4  5.09754    0.04900  104.03   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 1.737 on 14996 degrees of freedom
Multiple R-squared: 0.9578,     Adjusted R-squared: 0.9578
F-statistic: 8.515e+04 on 4 and 14996 DF,  p-value: < 2.2e-16

Test this Hypothesis b2 = 0 My mind says that there are atleast 3 ways to do it.. Firstly the restricted and unrestricted case

> library(faraway)
> g <- lm(sr ~ pop15 + pop75 + dpi + ddpi, savings)
> RSS0 <- sum(resid(g)^2)
> g.res <- lm(sr ~ pop75 + dpi + ddpi, savings)
> RSS1 <- sum(resid(g.res)^2)
> fstat <- ((RSS1 - RSS0)/3)/(RSS0/46)
> fstat
[1] 3.464173
> qf(0.95, 3, 46)
[1] 2.806845

Since the value is greater than 95 percent level, reject the hypothesis that the coefficient of pop15 =0

Second case is anova

> g1 <- lm(sr ~ pop15 + pop75 + dpi + ddpi, savings)
> g2 <- lm(sr ~ pop75 + dpi + ddpi, savings)
> anova(g1, g2)
Analysis of Variance Table


Model 1: sr ~ pop15 + pop75 + dpi + ddpi
Model 2: sr ~ pop75 + dpi + ddpi
  Res.Df     RSS Df Sum of Sq      F   Pr(>F)
1     45  650.71
2     46  797.72 -1   -147.01 10.167 0.002603 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Null Hypo is that pop15 = 0 which is obviously rejected if you look at ftest results Third is the complicated procedure of actually computing variance of Rb-r Actually the results will produce the same fstat as procedure 1