FN6809 Notes 2

26 Questions

According to the text, why do we only need three dummy variables in a model?

To avoid linear dependence in X

In the given example, what does the $\beta_0$ parameter represent in the model for executive salary?

The mean salary for females

Based on the given data, is there evidence that Harris Bank discriminated against female employees?

Yes, there is evidence of discrimination

What is the forecasted average salary for males with 12 years of education, 10 years of experience, and with 'months' equal to 15?

$5692.903

According to the text, which variable is considered the most important independent variable in the regression model?

Living area

Which of the following is true about extra sums of squares?

They measure the marginal reduction in the error sum of squares when predictor variables are added to the regression model.

What does the coefficient of partial determination measure?

The marginal contribution of one X variable when all others are already included in the model.

What problems arise when multicollinearity exists in a regression analysis?

Increased likelihood of rounding errors and confusing and misleading regression results.

Which of the following is NOT a method for detecting multicollinearity?

Principal Components Analysis

Which of the following statements is true about the sampling variances of the estimated coefficients in the presence of collinearity?

The sampling variances increase sharply with increasing collinearity between the independent variables.

What is the formula for Variance Inflation Factor (VIF)?

VIFi = (1 - Ri^2)^-1

Which type of regression model includes more than one independent variable?

Multiple linear regression model

What is the general linear regression model equation for a first-order model with k predictor variables?

$Y_i = \beta_0 + \beta_1X_{i1} + \beta_2X_{i2} + \ldots + \beta_kX_{ik} + \epsilon_i$

What does the general linear regression model assume about the error term?

The error term is normally distributed

Which of the following is the formula for the coefficient of determination R2 in multiple regression analysis?

$R2 = SSR/SST = 1 - SSE/SST$

What is the purpose of using adjusted R Square (Ra2) in regression analysis?

To measure how well the model fits the data after adjusting for the number of independent variables and sample size

What is the formula for calculating the adjusted R Square (Ra2) value in multiple regression analysis?

$Ra2 = 1 - (181.95/410.4)(14/12)$

Why does the coefficient of determination R2 generally increase when more independent variables are included in a multiple regression equation?

Because SSR generally increases and SSE decreases

Which of the following is true about the interaction term in the interaction model with two independent variables?

The interaction term represents the cross-product of X1 and X2.

Which of the following is true about nested models?

Two models are nested if one model contains all the terms of the second model and at least one additional term.

Which of the following is the correct formula for the partial F-test statistic for comparing nested models?

F = [(SSER - SSEC)/(k-g)]/MSEC

Which of the following is true about beta coefficients in multiple regression?

Beta coefficients are the coefficients of the independent variables when all variables are expressed in standardized form.

Which of the following is the correct equation for the general linear regression model in matrix terms?

$Y = X\beta + \epsilon$

What is the formula for estimating the regression coefficients in the least squares method?

$b = (X'X)^{-1}X'Y$

What is the formula for the variance-covariance matrix of the regression coefficients?

$Var(b) = \sigma^2 (X'X)^{-1}$

What is the null hypothesis for the F-test associated with the ANOVA table in regression analysis?

$H_0: \beta_1 = \beta_2 = ... = \beta_k = 0$

Study Notes

Multiple Regression Model

In a model with k categorical variables, only k-1 dummy variables are needed to represent the categories.
The β0 parameter represents the intercept or the average value of the response variable when all independent variables are equal to zero.

Interpretation of Regression Coefficients

In a model for executive salary, the coefficient of a dummy variable represents the difference in average salary between two groups (e.g., males and females).

Hypothesis Testing

The presence of multicollinearity can lead to unstable and unreliable estimates of the regression coefficients.
Multicollinearity can be detected using methods such as variance inflation factor (VIF), tolerance, and condition index.

Variance Inflation Factor (VIF)

VIF is calculated as 1 / (1 - R^2) where R^2 is the coefficient of determination of the regression of the independent variable on the remaining independent variables.

Multiple Regression Model Equation

The general linear regression model equation for a first-order model with k predictor variables is Y = β0 + β1X1 + β2X2 + … + βkXk + ε.

Assumptions of the Linear Regression Model

The error term is assumed to be normally distributed with a mean of zero and a constant variance.

Coefficient of Determination (R2)

R2 measures the proportion of the total variation in the response variable that is explained by the independent variables.
R2 is calculated as 1 - (SSE / SST) where SSE is the sum of the squared errors and SST is the total sum of squares.

Adjusted R Square (Ra2)

Ra2 is a measure of the proportion of the variation in the response variable that is explained by the independent variables, adjusted for the number of independent variables.
Ra2 is calculated as 1 - ((n-1) / (n-k-1)) * (1 - R2) where n is the sample size and k is the number of independent variables.

Interaction Model

The interaction term in the interaction model with two independent variables represents the change in the effect of one independent variable on the response variable when the other independent variable changes.

Nested Models

Nested models are models where one model is a subset of the other model.
The partial F-test can be used to compare nested models.

Partial F-test

The partial F-test is used to compare the fit of two nested models.
The partial F-test statistic is calculated as ((R2_full - R2_reduced) / (k_full - k_reduced)) / ((1 - R2_full) / (n - k_full - 1)) where R2_full is the R2 of the full model, R2_reduced is the R2 of the reduced model, k_full is the number of independent variables in the full model, k_reduced is the number of independent variables in the reduced model, and n is the sample size.

Beta Coefficients

Beta coefficients are standardized regression coefficients that measure the change in the response variable for a one-unit change in the independent variable, while controlling for the other independent variables.

Matrix Form of the Linear Regression Model

The general linear regression model can be written in matrix form as Y = Xβ + ε.

Least Squares Method

The least squares method is used to estimate the regression coefficients.
The estimated regression coefficients are calculated as β = (X^T X)^-1 X^T Y.

Variance-Covariance Matrix of Regression Coefficients

The variance-covariance matrix of the regression coefficients is calculated as σ^2 (X^T X)^-1.

F-test

The null hypothesis for the F-test associated with the ANOVA table in regression analysis is that all the regression coefficients are equal to zero.

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