In exams: Automatic Generation of Exams in R

## data-generating process: linear vs. quadratic, homoscedastic vs. heteroscedastic
type <- sample(c("constant", "linear", "quadratic", "heteroscedastic"), size = 1, prob = c(0.25, 0.35, 0.2, 0.2))
d <- data.frame(x = runif(100, -1, 1))
a <- 0
b <- if(type == "constant") 0 else sample(c(-1, 1), 1) * runif(1, 0.6, 0.9)
c <- if(type != "quadratic") 0 else sample(c(-1, 1), 1) * runif(1, 0.3, 0.6)
s <- if(type != "heteroscedastic") 0.25 else exp(-1.5 + sign(b) * d$x)
d$y <- a + b * d$x + c * d$x^2 + rnorm(100, sd = s)
write.csv(d, "linreg.csv", row.names = FALSE, quote = FALSE)

## model
m <- lm(y ~ x, data = d)
ahat <- coef(m)[1]
bhat <- coef(m)[2]
bpvl <- summary(m)$coefficients[2, 4]
bsol <- c(bpvl >= 0.05, (bpvl < 0.05) & (bhat > 0), (bpvl < 0.05) & (bhat < 0))

## interpretation
bint <- c("`x` and `y` are not significantly correlated", "`y` increases significantly with `x`", "`y` decreases significantly with `x`")
bint <- bint[bsol]
typeint <- switch(type,
  "quadratic" = "the true relationship between `y` and `x` is not linear but quadratic (and hence errors do not have zero expectation)",
  "heteroscedastic" = "the errors are heteroscedastic with increasing variance along with the mean",
  "the assumptions of the Gauss-Markov theorem are reasonably well fulfilled"
)

Question

Theory: Consider a linear regression of y on x. It is usually estimated with which estimation technique (three-letter abbreviation)?

ANSWER1

This estimator yields the best linear unbiased estimator (BLUE) under the assumptions of the Gauss-Markov theorem. Which of the following properties are required for the errors of the linear regression model under these assumptions?

ANSWER2

Application: Using the data provided in linreg.csv estimate a linear regression of y on x. What are the estimated parameters?

Intercept: ##ANSWER3##

Slope: ##ANSWER4##

In terms of significance at 5% level:

ANSWER5

Interpretation: Consider various diagnostic plots for the fitted linear regression model. Do you think the assumptions of the Gauss-Markov theorem are fulfilled? What are the consequences?

ANSWER6

Code: Please upload your code script that reads the data, fits the regression model, extracts the quantities of interest, and generates the diagnostic plots.

ANSWER7

Answerlist

• independent
• zero expectation
• normally distributed
• identically distributed
• homoscedastic
• x and y are not significantly correlated
• y increases significantly with x
• y decreases significantly with x

Solution

Theory: Linear regression models are typically estimated by ordinary least squares (OLS). The Gauss-Markov theorem establishes certain optimality properties: Namely, if the errors have expectation zero, constant variance (homoscedastic), no autocorrelation and the regressors are exogenous and not linearly dependent, the OLS estimator is the best linear unbiased estimator (BLUE).

Application: The estimated coefficients along with their significances are reported in the summary of the fitted regression model, showing that r bint (at 5% level).

summary(m)

Interpretation: Considering the visualization of the data along with the diagnostic plots suggests that r typeint. \

par(mfrow = c(1, 3))
plot(y ~ x, data = d, main = "Data and regression fit")
abline(m)
plot(m, which = 1)
plot(m, which = 3)

Code: The analysis can be replicated in R using the following code.

## data
d <- read.csv("linreg.csv")
## regression
m <- lm(y ~ x, data = d)
summary(m)
## visualization
plot(y ~ x, data = d)
abline(m)
## diagnostic plots
plot(m)

Meta-information

exname: Linear regression extype: cloze exsolution: OLS|01001|r fmt(ahat, 3)|r fmt(bhat, 3)|r mchoice2string(bsol)|nil|nil exclozetype: string|mchoice|num|num|schoice|essay|file extol: 0.01

exams documentation built on Nov. 14, 2022, 3:02 p.m.