# Multiple Linear Regression Simulation

I’m new to the R language. I would like to know how to simulate from a multiple linear regression model that fulfills all four assumptions of the regression.

ok.. thank you.

Let say i want to simulate the data based on this data set:

y<-c(18.73,14.52,17.43,14.54,13.44,24.39,13.34,22.71,12.68,19.32,30.16,27.09,25.40,26.05,33.49,35.62,26.07,36.78,34.95,43.67)
x1<-c(610,950,720,840,980,530,680,540,890,730,670,770,880,1000,760,590,910,650,810,500)
x2<-c(1,1,3,2,1,1,3,3,2,2,1,3,3,2,2,2,3,3,1,2)

fit<-lm(y~x1+x2)
summary(fit)


then i get the output:

Call:
lm(formula = y ~ x1 + x2)

Residuals:
Min       1Q   Median       3Q      Max
-13.2805  -7.5169  -0.9231   7.2556  12.8209

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 42.85352   11.33229   3.782  0.00149 **
x1          -0.02534    0.01293  -1.960  0.06662 .
x2           0.33188    2.41657   0.137  0.89238
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 8.679 on 17 degrees of freedom
Multiple R-squared:  0.1869,    Adjusted R-squared:  0.09127
F-statistic: 1.954 on 2 and 17 DF,  p-value: 0.1722


My question is how to simulate a new data that mimic the original data above?

1. If you don’t have them already, start by setting up some predictors, $x_1$, $x_2$, …

2. Choose the population (‘true’) coefficients of your predictors, the $\beta_i$’s, including $\beta_0$, the intercept.

3. Choose the error variance, $\sigma^2$ or equivalently its square root, $\sigma$

4. generate the error term, $\varepsilon$, as an independent random normal vector, with mean 0 and variance $\sigma^2$

5. Let $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + … + \beta_k x_k + \varepsilon$

then you can regress the $y$ on your $x$’s

e.g. in R you could do something like:

x1 <- 11:30
x2 <- runif(20,5,95)
x3 <- rbinom(20,1,.5)

b0 <- 17
b1 <- 0.5
b2 <- 0.037
b3 <- -5.2
sigma <- 1.4

eps <- rnorm(x1,0,sigma)
y <- b0 + b1*x1  + b2*x2  + b3*x3 + eps


produces a single simulation of $y$ from the model. Then running

 summary(lm(y~x1+x2+x3))


gives

Call:
lm(formula = y ~ x1 + x2 + x3)

Residuals:
Min      1Q  Median      3Q     Max
-2.6967 -0.4970  0.1152  0.7536  1.6511

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 16.28141    1.32102  12.325 1.40e-09 ***
x1           0.55939    0.04850  11.533 3.65e-09 ***
x2           0.01715    0.01578   1.087    0.293
x3          -4.91783    0.66547  -7.390 1.53e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.241 on 16 degrees of freedom
Multiple R-squared:  0.9343,    Adjusted R-squared:  0.9219
F-statistic: 75.79 on 3 and 16 DF,  p-value: 1.131e-09


You can simplify this procedure in several ways, but I figured spelling it out would help to begin with.

If you want to simulate a new random $y$ but with the same population coefficients, just rerun the last two lines of the procedure above (generate a new random eps and y), corresponding to steps 3 and 4 of the algorithm.