Fuel Usage Regression Analysis

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Questions and Answers

According to the linear model summary, which variable has the most statistically significant association with fuel usage?

  • Income
  • logMiles
  • Dlic (correct)
  • Tax

Based on the linear model summary, a higher gasoline tax (Tax) is associated with increased fuel usage per capita.

False (B)

In the linear model, what does the 'Residual standard error' represent?

the average distance that the observed values fall from the regression line

In the context of the baseball salaries data, the training set contains approximately _____ % of observations.

<p>70</p>
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Match the baseball statistic with its description:

<p>CRuns = Cumulative Runs Hits = Number of successful hits PutOuts = Number of times a fielder puts a batter or runner out AtBat = Number of times a player has been at bat</p>
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Based on the training set regression summary, which variable is NOT statistically significant at a 0.05 significance level?

<p>DivisionW (A)</p>
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In the test set regression summary, the intercept is statistically significant at a 0.05 level.

<p>True (A)</p>
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What is the purpose of splitting a dataset into training and test sets when building a predictive model?

<p>to evaluate the model's performance on unseen data</p>
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In Problem 3, the regression being performed is described as 'regression through the _____' signifying the absence of an intercept term.

<p>origin</p>
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Match the term with its definition within the context of linear models without an intercept:

<p>Weighted Least Squares (WLS) = A method that uses a weight matrix to account for unequal variances. Ordinary Least Squares (OLS) = A method that minimizes the sum of squared differences between observed and predicted values. Weight Matrix (W) = A matrix used in WLS to adjust for different levels of precision or reliability in the observations.</p>
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According to the problem, what is assumed about the predictor variable x in Problem 3?

<p>It is assumed to be fixed/constant. (D)</p>
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In problem 4, it is assumed that the weights (wii) are equal to each other for all individuals.

<p>False (B)</p>
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In problem 4, what is the potential consequence of incorrectly assuming that Var(ε) = σ²Ω⁻¹ when in fact Var(ε) = σ²W⁻¹ ?

<p>biased estimates and incorrect standard errors</p>
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In Problem 5, a matrix A is said to have orthonormal columns if Ā'A = _____

<p>Ipxp</p>
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Match the term to the description:

<p>Orthonormal Columns = Columns are orthogonal to one another and have a length of 1 Ridge Regression = Regression technique that adds a penalty term to the OLS function to prevent overfitting OLS = Finds the parameters that minimize the sum of the squares of the errors</p>
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In problem 5, what is assumed about the columns of X?

<p>They are orthonormal and have a mean of zero. (D)</p>
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In Problem 6, using more knots in cubic regression spline always improve out-of-sample R².

<p>False (B)</p>
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In Problem 6, what is the effect of increased knots on in-sample R²?

<p>It will increase</p>
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In the model specified in page 1, Fuel = Tax + Dlic + Income + logMiles, data = datafuel, 'Fuel' is the ______ variable.

<p>dependent</p>
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Match

<p>AIC = A method for model selection that seeks to find the model that best explains the data with a minimum number of parameters. Forward Stepwise Regression = Starts with no predictors and adds variables one at a time. Lasso = A method for model selection that adds a penalty term to shrink the coefficients</p>
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Flashcards

Variables in Fuel Usage Prediction

Predict fuel usage based on gasoline tax (Tax), driver's license proportion (Dlic), per capita income (Income), and log of highway miles (logMiles).

Estimated Variance

The estimated variance for the estimated difference calculated.

Orthonormal Matrix Definition

A matrix where columns are orthogonal to each other and normalized.

Ridge Regression Coefficients

Estimated intercept and slopes from ridge regression for a fixed λ.

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Cubic Regression Spline Polynomials

Polynomials providing predictions for x ≤ ξ and x > ξ in a cubic regression spline.

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Free Parameters in Cubic Spline.

The number of free parameters when fitting a cubic spline with one knot, including the intercept.

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In-Sample R²

The phenomenon where increasing the number of knots in a cubic regression spline leads to a higher R-squared value when evaluated on the training data.

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Out-of-Sample R²

The phenomenon where increasing the number of knots in a cubic regression spline may lead to a lower or overfitting dataset when evaluated on the testing data

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Study Notes

  • The data includes average fuel usage per capita in the 50 US states plus Washington, D.C. (Fuel).
  • It uses variables like gasoline tax (Tax, in cents), proportion of residents with a driver's license (Dlic), per capita income (Income), and base-e logarithm of highway miles (logMiles) to predict fuel usage.
  • The regression model predicts fuel usage based on Tax, Dlic, Income, and logMiles.

Summary of the Model

  • Formula: Fuel ~ Tax + Dlic + Income + logMiles
  • Residuals range from -163.145 to 183.499

Coefficients:

  • Intercept: 154.192845
  • Tax: -4.227983
  • Dlic: 0.471871
  • Income: -0.006135
  • logMiles: 26.755176

Significance Codes:

  • 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1
  • Residual standard error: 64.89 on 46 degrees of freedom
  • Multiple R-squared: 0.5105
  • Adjusted R-squared: 0.4679
  • F-statistic: 11.99 on 4 and 46 DF, p-value: 9.331e-07

Predicting Fuel Usage Difference Between Two States

  • State 1 has a gas tax that is ten cents higher than State 2.
  • State 1 has 10% more highway miles than State 2.
  • A prediction is formed for the difference in per capita fuel usage between these two states.

Regression with Base-10 Logarithm of Highway Miles

  • The regression uses the base-10 logarithm of highway miles instead of the base-e logarithm.
  • State 1 has a gas tax that is ten cents higher than State 2
  • State 1 has 10% more highway miles than State 2.
  • Additional information may be needed if a prediction cannot be formed based on provided data.

Variance Calculation for Estimated Difference

  • The matrix X contains the design matrix.
  • The function signif rounds to a specified number of significant figures.
  • The function solve conducts matrix inversion.

95% Confidence Intervals for logMiles Slope Coefficient

  • Method 1: Using quantiles from the t-distribution and conventional standard errors.
  • Method 2: Using quantiles from the t-distribution and heteroskedasticity-consistent standard errors.
  • Method 3: Using the pairs bootstrap.
  • Testing the null hypothesis that the true slope coefficient on log(Miles) equals zero with a two-sided alternative.

Histogram and Normal Quantile Plot

  • Shows the estimated distribution for the quantity based on the pairs bootstrap.
  • The x-axis is (βlog(Miles) - βlog(Miles))/SeHC2(βlog(Miles))

Salaries Data

  • Includes data on Salaries for 263 Major League Baseball fielders.
  • Comprises 19 predictor variables related to offensive/defensive performance and team.
  • Objective: Use this model to predict a given player's salary.
  • Data is split into training (70%) and test sets.

Model Selection

  • Forward stepwise selection and AIC were used.
  • Summary tables created when fitting regressions using selected variables in the training and test sets.

Training Set Summary

  • Formula: Salary ~ CRuns + Hits + PutOuts + AtBat + Walks + CWalks + Division + CRBI
  • Residuals range from -733.10 to 918.47

Coefficients:

  • Intercept: -0.24802
  • CRuns: 0.92374
  • Hits: 7.14723
  • PutOuts: 0.31852
  • AtBat: -1.83699
  • Walks: 5.25774
  • CWalks: -0.89600
  • DivisionW: -76.05482
  • CRBI: 0.40357

Significance Codes:

  • 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1
  • Residual standard error: 275.4 on 175 degrees of freedom
  • Multiple R-squared: 0.6065
  • Adjusted R-squared: 0.5885
  • F-statistic: 33.72 on 8 and 175 DF, p-value: < 2.2e-16

Test Set Summary

  • Formula uses forwardstep$terms
  • Residuals range from -592.36 to 1812.87

Test Set Coefficients:

  • Intercept: 353.0238
  • CRuns: 0.1244
  • Hits: 8.2062
  • PutOuts: 0.1230
  • AtBat: -3.0819
  • Walks: 9.3373
  • CWalks: -0.3766
  • DivisionW: -186.4704
  • CRBI: 0.8619

Significance Codes:

  • 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1
  • Residual standard error: 385.3 on 70 degrees of freedom
  • Multiple R-squared: 0.466
  • Adjusted R-squared: 0.405
  • F-statistic: 7.636 on 8 and 70 DF, p-value: 2.735e-07
  • A model including Putouts substantially improves the predictive performance of the model.

Tuning the Penalty Parameter

  • Instead of AIC, it optimizes the information criterion penalty parameter A based on out-of-sample R² on the test set.

Lasso Regression Tuning

  • Chooses the tuning parameter A based on the value of A that minimizes the sum of squared errors in the training set.

Regression Through the Origin

  • Regression of y on a single predictor x without an intercept term.

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