Questions and Answers
 Linear regression is used to explain the relationship between a dependent variable and one or more ______ variables.
independent
 The dependent variable is a ______ variable.
continuous
 The simple linear regression model has ______ independent variable(s).
one
 The multiple linear regression model has ______ or more independent variables.
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 The linear regression model describes how the dependent variable is related to the independent variables and the ______ term.
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 The estimated regression equation shows how to calculate predicted values of the dependent variable using the values of the ______ variables.
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 The least squares method is used to calculate the coefficients so that the errors are as ______ as possible.
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 The coefficient of determination (Rsquared) provides a measure of the ______ of fit for the estimated regression equation.
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 Adjusted Rsquared corrects for the number of independent variables and is ______ to Rsquared.
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 The ttest is used to determine whether the relationship between the dependent variable and ______ independent variable is significant.
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 The ANOVA table shows the total variation, explained variation due to regression, and unexplained variation due to ______.
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 The goal is to find a regression model with coefficients that are ______ significant.
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 An estimator is consistent if it converges in probability to the population parameter as the sample size increases. The probability that the estimator obtained from a sample size will be arbitrarily close to the population parameter goes to 1 as the sample size increases.
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 The OLS estimator is unbiased under assumptions 14 (with the zero conditional mean assumption).
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 Under assumptions 14’ (with the assumption that the regressors are uncorrelated with the error term), the OLS estimator is ______.
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 Unbiasedness is ideal but if it cannot be achieved in a small sample, then ______ can be achieved with a large sample.
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 Omitted variable bias occurs when a relevant variable is omitted from the model and the coefficient will be biased if the omitted variable and the included variable are ______.
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 Under assumptions 15 (Gauss Markov assumptions), the coefficients have asymptotically ______ sampling distribution.
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 In large samples, the normality assumption is not always needed for the OLS estimators to be normal and the ttests and Ftests to be ______.
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 OLS properties hold for any sample, including expected values and unbiasedness under assumptions 14 and variance formulas under ______.
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 GaussMarkov theorem (BLUE) holds under ______ 15.
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 As the sample size increases, standard errors change at a rate of 1/sample size and with larger sample size, standard errors are ______, leading to more significance of the coefficients.
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 The OLS estimator is consistent if the omitted variable is ______ or uncorrelated.
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 The OLS estimator is ______ if assumptions 14’ hold.
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 Linear regression models the relationship between a dependent variable and one or more ______ variables.
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 The regression model can be in linear or ______linear form, and taking logs of variables changes the interpretation of coefficients.
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 Gauss Markov assumptions are standard assumptions for the linear regression model, including linearity in parameters, random sampling, no perfect collinearity, ______, and homoscedasticity.
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 [Blank] means the variance of the error term is constant for each independent variable while heteroscedasticity means the variance differs.
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 The unbiasedness of the OLS estimators is derived from ______ Markov assumptions.
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 The standard errors measure how precisely the regression coefficients are calculated, and lower variance in error term and higher variance in ______ variable is desirable.
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 The variance of the error term can be ______, and the variances of the OLS estimators depend on it.
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 The sample variability in OLS coefficients depends on the variances of the error term and ______ variable.
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 The coefficients are random as the sample is random, and the expected values of the sample coefficients are the ______ parameters.
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 The relationship between y and x is ______ in the population, but the regression model can have logged, squared, or interaction variables.
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 [Blank] or zero conditional mean implies the expected value of the error term given independent variable x is zero.
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 [Blank] is when the variance of the error term is constant for each independent variable.
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 Linear regression models the relationship between a dependent variable and one or more ______ variables.
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 The ______ model can be in linear or loglinear form, and taking logs of variables changes the interpretation of coefficients.
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 Gauss Markov assumptions are standard assumptions for the linear regression model, including linearity in parameters, ______ sampling, no perfect collinearity, exogeneity, and homoscedasticity.
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 ______ means the variance of the error term is constant for each independent variable while heteroscedasticity means the variance differs.
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 The unbiasedness of the OLS estimators is derived from ______ assumptions.
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 The standard errors measure how precisely the regression coefficients are calculated, and lower variance in error term and higher variance in independent variable is ______.
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 The variance of the error term can be estimated, and the variances of the OLS estimators ______ on it.
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 The sample variability in OLS coefficients depends on the variances of the error term and ______ variable.
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 The coefficients are random as the sample is random, and the expected values of the sample coefficients are the ______ parameters.
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 The relationship between y and x is ______ in the population, but the regression model can have logged, squared, or interaction variables.
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 Exogeneity or zero conditional mean implies the expected value of the error term given independent variable x is ______.
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 ______ is when the variance of the error term is constant for each independent variable.
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 Heteroscedasticity refers to a scenario where the variance of the error term differs with the ______ variables.
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 Under heteroscedasticity, OLS estimators are still unbiased and consistent, but the variance formulas for the OLS estimators are not ______.
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 The ttests and Ftests are not valid under heteroscedasticity, and the OLS estimator is not the best linear unbiased estimator (BLUE).
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 Hypothesis testing for heteroscedasticity involves testing whether the expected value of the error term varies with the ______ variables.
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 The BreuschPagan test, White test, and Alternative White test are commonly used tests for ______.
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 Robust standard errors should be used when ______ is found.
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 Weighted Least Squares (WLS) can be used to estimate the model if the heteroskedasticity form is ______.
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 Feasible Generalized Least Squares (FGLS) transforms the variables to get homoscedasticity if the heteroscedasticity form is ______ known.
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 The Rsquared for the regressions of squared residuals on independent variables is used to calculate the test statistics for heteroscedasticity ______.
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 The Ftest and LMtest are commonly used tests for overall significance of ______.
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 The regression model for price needs correction for ______.
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 The Rsquared for the regressions of squared residuals on independent variables is used to calculate the test statistics for heteroscedasticity tests for ______ price.
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Study Notes
Linear Regression Overview
 Linear regression is used to explain the relationship between a dependent variable and one or more independent variables.
 The dependent variable is a continuous variable, while the independent variables can be continuous, discrete, or indicator variables.
 The simple linear regression model has one independent variable, while the multiple linear regression model has two or more independent variables.
 The linear regression model describes how the dependent variable is related to the independent variables and the error term.
 The estimated regression equation shows how to calculate predicted values of the dependent variable using the values of the independent variables.
 The least squares method is used to calculate the coefficients so that the errors are as small as possible.
 The coefficient of determination (Rsquared) provides a measure of the goodness of fit for the estimated regression equation.
 Adjusted Rsquared corrects for the number of independent variables and is preferred to Rsquared.
 The ttest is used to determine whether the relationship between the dependent variable and one independent variable is significant.
 The Ftest is used to test whether the relationship between the dependent variable and all independent variables is significant.
 The ANOVA table shows the total variation, explained variation due to regression, and unexplained variation due to error.
 The goal is to find a regression model with coefficients that are jointly significant.
Linear Regression Overview
 Linear regression is used to explain the relationship between a dependent variable and one or more independent variables.
 The dependent variable is a continuous variable, while the independent variables can be continuous, discrete, or indicator variables.
 The simple linear regression model has one independent variable, while the multiple linear regression model has two or more independent variables.
 The linear regression model describes how the dependent variable is related to the independent variables and the error term.
 The estimated regression equation shows how to calculate predicted values of the dependent variable using the values of the independent variables.
 The least squares method is used to calculate the coefficients so that the errors are as small as possible.
 The coefficient of determination (Rsquared) provides a measure of the goodness of fit for the estimated regression equation.
 Adjusted Rsquared corrects for the number of independent variables and is preferred to Rsquared.
 The ttest is used to determine whether the relationship between the dependent variable and one independent variable is significant.
 The Ftest is used to test whether the relationship between the dependent variable and all independent variables is significant.
 The ANOVA table shows the total variation, explained variation due to regression, and unexplained variation due to error.
 The goal is to find a regression model with coefficients that are jointly significant.
OLS Asymptotics
 An estimator is consistent if it converges in probability to the population parameter as the sample size increases.
 The probability that the estimator obtained from a sample size will be arbitrarily close to the population parameter goes to 1 as the sample size increases.
 The OLS estimator is unbiased under assumptions 14 (with the zero conditional mean assumption).
 Under assumptions 14’ (with the assumption that the regressors are uncorrelated with the error term), the OLS estimator is consistent.
 Unbiasedness is ideal but if it cannot be achieved in a small sample, then consistency can be achieved with a large sample.
 Omitted variable bias occurs when a relevant variable is omitted from the model and the coefficient will be biased if the omitted variable and the included variable are correlated.
 The OLS estimator is consistent if the omitted variable is irrelevant or uncorrelated.
 Under assumptions 15 (Gauss Markov assumptions), the coefficients have asymptotically normal sampling distribution.
 In large samples, the normality assumption is not always needed for the OLS estimators to be normal and the ttests and Ftests to be valid.
 OLS properties hold for any sample, including expected values and unbiasedness under assumptions 14 and variance formulas under assumptions 15.
 GaussMarkov theorem (BLUE) holds under assumptions 15.
 As the sample size increases, standard errors change at a rate of 1/sample size and with larger sample size, standard errors are lower, leading to more significance of the coefficients.
Introduction to Linear Regression
 Linear regression models the relationship between a dependent variable and one or more independent variables.
 The regression model can be in linear or loglinear form, and taking logs of variables changes the interpretation of coefficients.
 Gauss Markov assumptions are standard assumptions for the linear regression model, including linearity in parameters, random sampling, no perfect collinearity, exogeneity, and homoscedasticity.
 Homoscedasticity means the variance of the error term is constant for each independent variable while heteroscedasticity means the variance differs.
 The unbiasedness of the OLS estimators is derived from Gauss Markov assumptions.
 The standard errors measure how precisely the regression coefficients are calculated, and lower variance in error term and higher variance in independent variable is desirable.
 The variance of the error term can be estimated, and the variances of the OLS estimators depend on it.
 The sample variability in OLS coefficients depends on the variances of the error term and independent variable.
 The coefficients are random as the sample is random, and the expected values of the sample coefficients are the population parameters.
 The relationship between y and x is linear in the population, but the regression model can have logged, squared, or interaction variables.
 Exogeneity or zero conditional mean implies the expected value of the error term given independent variable x is zero.
 Homoscedasticity is when the variance of the error term is constant for each independent variable.
Introduction to Linear Regression
 Linear regression models the relationship between a dependent variable and one or more independent variables.
 The regression model can be in linear or loglinear form, and taking logs of variables changes the interpretation of coefficients.
 Gauss Markov assumptions are standard assumptions for the linear regression model, including linearity in parameters, random sampling, no perfect collinearity, exogeneity, and homoscedasticity.
 Homoscedasticity means the variance of the error term is constant for each independent variable while heteroscedasticity means the variance differs.
 The unbiasedness of the OLS estimators is derived from Gauss Markov assumptions.
 The standard errors measure how precisely the regression coefficients are calculated, and lower variance in error term and higher variance in independent variable is desirable.
 The variance of the error term can be estimated, and the variances of the OLS estimators depend on it.
 The sample variability in OLS coefficients depends on the variances of the error term and independent variable.
 The coefficients are random as the sample is random, and the expected values of the sample coefficients are the population parameters.
 The relationship between y and x is linear in the population, but the regression model can have logged, squared, or interaction variables.
 Exogeneity or zero conditional mean implies the expected value of the error term given independent variable x is zero.
 Homoscedasticity is when the variance of the error term is constant for each independent variable.
Heteroscedasticity and its Consequences

Heteroscedasticity refers to a scenario where the variance of the error term differs with the independent variables.

Under heteroscedasticity, OLS estimators are still unbiased and consistent, but the variance formulas for the OLS estimators are not valid.

The ttests and Ftests are not valid under heteroscedasticity, and the OLS estimator is not the best linear unbiased estimator (BLUE).

Hypothesis testing for heteroscedasticity involves testing whether the expected value of the error term varies with the independent variables.

The BreuschPagan test, White test, and Alternative White test are commonly used tests for heteroscedasticity.

Robust standard errors should be used when heteroscedasticity is found.

Weighted Least Squares (WLS) can be used to estimate the model if the heteroskedasticity form is known.

Feasible Generalized Least Squares (FGLS) transforms the variables to get homoscedasticity if the heteroscedasticity form is not known.

The Rsquared for the regressions of squared residuals on independent variables is used to calculate the test statistics for heteroscedasticity tests.

The Ftest and LMtest are commonly used tests for overall significance of heteroscedasticity.

The regression model for price needs correction for heteroscedasticity.

The Rsquared for the regressions of squared residuals on independent variables is used to calculate the test statistics for heteroscedasticity tests for log price.Heteroscedasticity and Regression Models

Heteroscedasticity is a condition where the variance of the errors is not constant across the range of values of the independent variable.

Heteroscedasticity can lead to biased coefficients, incorrect variance for the coefficients, and invalid ttests and Ftests.

Three tests are used to identify heteroscedasticity: BreuschPagan test, White test, and Alternative White test.

The BreuschPagan test and White test are based on regressing the squared residuals on the independent variables.

The Alternative White test is based on regressing the squared residuals on the fitted values and squared fitted values.

If the heteroscedasticity form is known, weighted least squares (WLS) can be used to correct for it.

WLS assigns weights to each observation based on the inverse of the variance of the error term.

If the heteroscedasticity form is unknown, feasible generalized least squares (FGLS) can be used.

FGLS assigns weights to each observation based on the inverse of the estimated variance of the error term.

The coefficients are the same for ordinary least squares (OLS) and OLS with robust standard errors, but the standard errors and significance can differ.

The coefficients are different for OLS as compared to WLS and FGLS because of the use of weights.

After correcting for heteroscedasticity, the results of the various regression models (OLS, OLS with robust standard errors, WLS, and FGLS) are similar, except for the loss of significance of one coefficient in some models.
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Description
Test your understanding of linear regression with this quiz! Learn about the basics of linear regression, including the differences between simple and multiple regression models, how to calculate predicted values using the estimated regression equation, and the importance of the coefficient of determination. This quiz will also cover the least squares method, ttests, Ftests, and the ANOVA table. Sharpen your skills and see how well you understand this fundamental statistical tool.