Exercises Unit C - Applied

Bachelor’s Degree Programme in Philosophy, International and Economic Studies, Ca’ Foscari University of Venice.

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Affiliation

Aldo Solari

Department of Economics, Ca’ Foscari University of Venice

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Chapter 3 Exercise 9.

This question involves the use of multiple linear regression on the Auto data set.

  1. Produce a scatterplot matrix which includes all of the variables in the data set.

  2. Compute the matrix of correlations between the variables using the function cor(). You will need to exclude the name variable, since it is qualitative.

  3. Use the lm() function to perform a multiple linear regression with mpg as the response and all other variables except name as the predictors. Use the summary() function to print the results. Comment on the output. For instance:

    1. Is there a relationship between the predictors and the response?

    2. Which predictors appear to have a statistically significant relationship to the response?

    3. What does the coefficient for the year variable suggest?

  4. Use the plot() function to produce diagnostic plots of the linear regression fit. Comment on any problems you see with the fit. Do the residual plots suggest any unusually large outliers? Does the leverage plot identify any observations with unusually high leverage?

  5. Use the * and : symbols to fit linear regression models with interaction effects. Do any interactions appear to be statistically significant?

  6. Try a few different transformations of the variables, such as \log(X), \sqrt{X}, X^2. Comment on your findings.

Chapter 3 Exercise 10

This question should be answered using the Carseats data set.

  1. Fit a multiple regression model to predict Sales using Price, Urban, and US.

  2. Provide an interpretation of each coefficient in the model. Be careful—some of the variables in the model are qualitative.

  3. Write out the model in equation form, being careful to handle the qualitative variables properly.

  4. For which of the predictors can you reject the null hypothesis H_0 : \beta_j = 0?

  5. On the basis of your response to the previous question, fit a smaller model that only uses the predictors for which there is evidence of association with the outcome.

  6. How well do the models in (a) and (e) fit the data?

Chapter 3 Exercise 15

This problem involves the Boston data set, which we saw in the lab for this chapter. We will now try to predict per capita crime rate using the other variables in this data set. In other words, per capita crime rate is the response, and the other variables are the predictors.

  1. For each predictor, fit a simple linear regression model to predict the response. Describe your results. In which of the models is there a statistically significant association between the predictor and the response? Create some plots to back up your assertions.

  2. Fit a multiple regression model to predict the response using all of the predictors. Describe your results. For which predictors can we reject the null hypothesis H_0 : \beta_j = 0?

  3. How do your results from (a) compare to your results from (b)? Create a plot displaying the univariate regression coefficients from (a) on the x-axis, and the multiple regression coefficients from (b) on the y-axis. That is, each predictor is displayed as a single point in the plot. Its coefficient in a simple linear regression model is shown on the x-axis, and its coefficient estimate in the multiple linear regression model is shown on the y-axis.

  4. Is there evidence of non-linear association between any of the predictors and the response? To answer this question, for each predictor X, fit a model of the form

Y = \beta_0 + \beta_1 X + \beta_2 X^2 + \beta_3 X^3 + \epsilon