Page 1

Course Overview

• Title: Data Science Exercises

• Institution: OLS HOCHSCHULE Prof. Diego d'Andria SCHMALKALDEN

• Semester: WS 2024-2025

• Context: University of Applied Sciences

Page 2

Instructions

• Software used: R

• Built-in datasets accessed via data() command in R console.

• Uses standard R commands with exceptions:

  • Exercise 4: Uses sandwich and lmtest packages for robust standard errors.

  • Exercise 6: Uses dataset from the ggplot2 package.

Page 3

Exercise 1: Describing Saving Rates Data

• Dataset: LifeCycleSavings

  • Time period: 1960–1970

• Tasks:

  1. Load the data and summarize each variable (min, mean, max).

  2. Find the variable with the largest range of values.

  3. Obtain a correlation matrix: identify pairs with highest and lowest correlations.

Page 4

Exercise 2: OLS on Saving Rates

• Dataset: LifeCycleSavings

• Tasks:

  1. Run OLS model with saving rate (sr) as dependent variable.

  2. Identify statistically significant regressors (p < 0.10).

  3. Create reduced model excluding insignificant regressors: assess if explanatory power improves.

Page 5

Exercise 3: Diagnosing Errors from OLS Model

• Dataset: LifeCycleSavings

• Tasks:

  1. Run full model from Exercise 2:

  • Plot histogram of residuals: analyze normal distribution.

  2. Conduct Shapiro-Wilk normality test on residuals: evaluate reliability of the hypothesis test.

Page 6

Exercise 4: Obtaining Heteroskedasticity-Robust Standard Errors

• Dataset: LifeCycleSavings

• Tasks:

  1. Execute full model from Exercise 2.

  2. Obtain robust standard errors: investigate changes in p-values for variables pop15 and ddpi.

Page 7

Exercise 5: The Mass of Wood from Fallen Trees

• Dataset: trees

  • Variables: Diameter (Girth), Height, Volume of cherry trees.

• Tasks:

  1. Run OLS model with Volume ~ Girth + Height.

  2. Assess how well the regressors predict Volume.

  3. Evaluate normality of residuals.

  4. Run log-transformed model: ln(Volume) ~ ln(Girth) + ln(Height. Compare significance and R² with first OLS model.

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Exercise 6: Poverty in the US Midwest

• Dataset: midwest from ggplot2

• Tasks:

  1. OLS model: percbelowpoverty ~ poptotal + popdensity + perchsd + percollege + percprof + percadults.

  2. Compute percadults as popadults/poptotal.

  3. Test necessity of all six regressors: omit one at a time and verify model fit.

Page 9

Solutions

• Overview of the solutions provided for exercises 1-6.

Page 10

Exercise 1 - Solution

• Load dataset:

  data(LifeCycleSavings)

• Descriptive statistics:

  summary(LifeCycleSavings)

• Correlation matrix:

  cor(LifeCycleSavings)

Page 11

Exercise 2 - Solution

• Load dataset:

  data(LifeCycleSavings)

• OLS model:

  model1 <- lm(sr ~ pop15 + pop75 + dpi + ddpi, data=LifeCycleSavings)
  summary(model1)

• Significant variables: pop15 and ddpi.

• Reduced model:

  model2 <- lm(sr ~ pop15 + ddpi, data=LifeCycleSavings)
  summary(model2)

• Compare Adjusted R²:

  • Full model: 0.2797

  • Reduced model: 0.2575

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Exercise 3 - Solution

• Load dataset and run regression:

  data(LifeCycleSavings)
  model1 <- lm(sr ~ pop15 + pop75 + dpi + ddpi, data=LifeCycleSavings)
  summary(model1)

• Histogram of residuals:

  hist(model1$residuals)

• Shapiro-Wilk test:

  shapiro.test(rstandard(model1))

• Interpretation: p-value = 0.91, not rejecting normality.

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Exercise 4 - Solution

• Load dataset and run regression:

  data(LifeCycleSavings)
  model1 <- lm(sr ~ pop15 + pop75 + dpi + ddpi, data=LifeCycleSavings)
  summary(model1)

• Obtain robust errors:

  library(sandwich)
  library(lmtest)
  model1_robust <- coeftest(model1, vcov = vcovHC)
  print(model1_robust)

• Interpretation: pop15 remains significant while ddpi loses significance.

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Exercise 5 - Solution

• Load dataset:

  data(trees)
  trees <- data.frame(trees)

• OLS model:

  reg <- lm(Volume ~ Girth + Height, data=trees)
  summary(reg)

• Model R² = 0.948, F-statistic p-value < 0.001.

• Shapiro-Wilk test:

  shapiro.test(rstandard(reg))

• Normality implication: Cannot reject normality (p-value ~ 0.64).

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Exercise 5 - Solution (cont.)

• Log transformed model:

  reg_ln <- lm(log(Volume) ~ log(Girth) + log(Height), data=trees)
  summary(reg_ln)

• Outcomes: Significance for Height improved, R² increased to 0.977.

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Exercise 6 - Solution

• Load dataset:

  library(ggplot2)
  data(midwest)
  midwest <- data.frame(midwest)
  midwest$percadults <- midwest$popadults/midwest$poptotal

• Run full OLS model:

  reg <- lm(percbelowpoverty ~ poptotal + popdensity + perchsd + percollege + percprof + percadults, data=midwest)
  summary(reg)

Page 17

Exercise 6 - Solution (cont.)

• Test each regressor's necessity:

  reg1 <- lm(percbelowpoverty ~ popdensity + perchsd + percollege + percprof + percadults, data=midwest)
  reg2 <- lm(percbelowpoverty ~ poptotal + perchsd + percollege + percprof + percadults, data=midwest)
  reg3 <- lm(percbelowpoverty ~ poptotal + popdensity + percollege + percprof + percadults, data=midwest)
  reg4 <- lm(percbelowpoverty ~ poptotal + popdensity + perchsd + percprof + percadults, data=midwest)
  reg5 <- lm(percbelowpoverty ~ poptotal + popdensity + perchsd + percollege + percadults, data=midwest)
  reg6 <- lm(percbelowpoverty ~ poptotal + popdensity + perchsd + percollege + percprof, data=midwest)

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Exercise 6 - Solution (cont.)

• Compare Adjusted R² for models:

  summary(reg)$adj.r.squared
  summary(reg1)$adj.r.squared
  summary(reg2)$adj.r.squared
  summary(reg3)$adj.r.squared
  summary(reg4)$adj.r.squared
  summary(reg5)$adj.r.squared
  summary(reg6)$adj.r.squared

• Findings: Omitting poptotal yields highest R² (0.441); omitting perchsd results in lowest R² (0.129).

Page 19

R Code for Exercises

• Exercise 1:

  data(LifeCycleSavings)
  summary(LifeCycleSavings)
  cor(LifeCycleSavings)

• Exercise 2:

  model1 <- lm(sr ~ pop15 + pop75 + dpi + ddpi, data=LifeCycleSavings)
  summary(model1)
  model2 <- lm(sr ~ pop15 + ddpi, data=LifeCycleSavings)
  summary(model2)

• Exercise 3:

  hist(model1$residuals)
  shapiro.test(rstandard(model1))

• Exercise 4:

  library(sandwich)
  library(lmtest)
  model1_robust <- coeftest(model1, vcov = vcovHC)
  print(model1_robust)

Page 20

R Code for Exercises (cont.)

• Exercise 5:

  trees <- data.frame(trees)
  reg <- lm(Volume ~ Girth + Height, data=trees)
  summary(reg)
  shapiro.test(rstandard(reg))
  reg_ln <- lm(log(Volume) ~ log(Girth) + log(Height), data=trees)
  summary(reg_ln)

• Exercise 6:

  library(ggplot2)
  data(midwest)
  midwest <- data.frame(midwest)
  midwest$percadults <- midwest$popadults/midwest$poptotal
  reg <- lm(percbelowpoverty ~ poptotal + popdensity + perchsd + percollege + percprof + percadults, data=midwest)
  reg1 <- lm(percbelowpoverty ~ popdensity + perchsd + percollege + percprof + percadults, data=midwest)
  reg2 <- lm(percbelowpoverty ~ poptotal + perchsd + percollege + percprof + percadults, data=midwest)
  reg3 <- lm(percbelowpoverty ~ poptotal + popdensity + percollege + percprof + percadults, data=midwest)
  reg4 <- lm(percbelowpoverty ~ poptotal + popdensity + perchsd + percprof + percadults, data=midwest)
  reg5 <- lm(percbelowpoverty ~ poptotal + popdensity + perchsd + percollege + percadults, data=midwest)
  reg6 <- lm(percbelowpoverty ~ poptotal + popdensity + perchsd + percollege + percprof, data=midwest)
  summary(reg)$adj.r.squared
  summary(reg1)$adj.r.squared
  summary(reg2)$adj.r.squared
  summary(reg3)$adj.r.squared
  summary(reg4)$adj.r.squared
  summary(reg5)$adj.r.squared
  summary(reg6)$adj.r.squared