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Questions and Answers
What does the partial regression coefficient 𝛽2 represent in a three-variable linear regression model?
What does the partial regression coefficient 𝛽2 represent in a three-variable linear regression model?
- The total change in 𝑌 per unit change in 𝑋2
- The effect of changes in 𝑋3 on the dependent variable 𝑌
- The change in the mean value of 𝑌 per unit change in 𝑋2, holding 𝑋3 constant (correct)
- The overall relationship between 𝑌 and the combination of 𝑋2 and 𝑋3
In the stochastic form of the three-variable regression model, what does the term 𝑢𝑡 represent?
In the stochastic form of the three-variable regression model, what does the term 𝑢𝑡 represent?
- The dependent variable
- The mean value of the dependent variable
- The systematic component of the model
- The stochastic disturbance term (correct)
How are multiple regression models characterized based on the explanation of economic phenomena?
How are multiple regression models characterized based on the explanation of economic phenomena?
- They are developed only from numerous stochastic variables.
- They require at least two explanatory variables.
- They can be explained by a single explanatory variable.
- They often involve three or more explanatory variables. (correct)
What do the equations given in the content indicate about the relationship between 𝑌, 𝑋2, and 𝑋3?
What do the equations given in the content indicate about the relationship between 𝑌, 𝑋2, and 𝑋3?
What does the equation $𝑌𝑡 = 𝛽1 + 𝛽2 X2𝑡 + 𝛽3 X3𝑡 + 𝑢𝑡$ represent?
What does the equation $𝑌𝑡 = 𝛽1 + 𝛽2 X2𝑡 + 𝛽3 X3𝑡 + 𝑢𝑡$ represent?
Which assumption states that there should be no autocorrelation between the error terms?
Which assumption states that there should be no autocorrelation between the error terms?
Which of the following assumptions ensures that the error term has a constant variance?
Which of the following assumptions ensures that the error term has a constant variance?
What is meant by the term 'no exact collinearity' in the context of multiple regression?
What is meant by the term 'no exact collinearity' in the context of multiple regression?
What does the Ordinary Least Squares (OLS) method aim to minimize?
What does the Ordinary Least Squares (OLS) method aim to minimize?
The error term in the multiple regression model follows which distribution according to the assumptions?
The error term in the multiple regression model follows which distribution according to the assumptions?
Which of the following signifies that the error term has a zero mean value?
Which of the following signifies that the error term has a zero mean value?
Which of the following represents the sample regression function (SRF)?
Which of the following represents the sample regression function (SRF)?
What does the notation $𝑦𝑡$ signify in the context of this regression model?
What does the notation $𝑦𝑡$ signify in the context of this regression model?
What distribution do the OLS estimators follow when testing hypotheses in multiple regression?
What distribution do the OLS estimators follow when testing hypotheses in multiple regression?
What is the first step in testing a hypothesis in multiple regression?
What is the first step in testing a hypothesis in multiple regression?
To reject the null hypothesis H0, which of the following conditions must be met regarding the t-statistics?
To reject the null hypothesis H0, which of the following conditions must be met regarding the t-statistics?
What does the null hypothesis H0: β2 = β3 = 0 imply in the context of multiple regression?
What does the null hypothesis H0: β2 = β3 = 0 imply in the context of multiple regression?
Which statistic is used to test the joint hypothesis that β2 and β3 are equal to zero?
Which statistic is used to test the joint hypothesis that β2 and β3 are equal to zero?
What is the relationship between the number of explanatory variables and the R2 value in a regression model?
What is the relationship between the number of explanatory variables and the R2 value in a regression model?
What should be done after calculating the F-statistic in hypothesis testing?
What should be done after calculating the F-statistic in hypothesis testing?
What does the formula for F-statistic in multiple regression assess?
What does the formula for F-statistic in multiple regression assess?
What is the equation used to calculate the variance of the OLS estimator for the coefficient $b_1$?
What is the equation used to calculate the variance of the OLS estimator for the coefficient $b_1$?
How is the standard error of the OLS estimator $b_1$ expressed mathematically?
How is the standard error of the OLS estimator $b_1$ expressed mathematically?
What does the multiple coefficient of determination, $R^2$, represent in a regression model?
What does the multiple coefficient of determination, $R^2$, represent in a regression model?
Which equation correctly represents the relationship among total sum of squares (TSS), explained sum of squares (ESS), and residual sum of squares (RSS)?
Which equation correctly represents the relationship among total sum of squares (TSS), explained sum of squares (ESS), and residual sum of squares (RSS)?
What is the formula for the coefficient of multiple correlation, $R$?
What is the formula for the coefficient of multiple correlation, $R$?
What does $ESS$ represent in the context of regression analysis?
What does $ESS$ represent in the context of regression analysis?
In computing the residual sum of squares (RSS), which formula is correct?
In computing the residual sum of squares (RSS), which formula is correct?
What is the form of the estimated variance of errors in OLS regression?
What is the form of the estimated variance of errors in OLS regression?
What does a higher $R^2$ value signify in a regression model?
What does a higher $R^2$ value signify in a regression model?
Which of the following is true about the standard error of the estimate?
Which of the following is true about the standard error of the estimate?
Flashcards
Multiple Regression Model
Multiple Regression Model
A regression model with more than one explanatory variable.
Dependent Variable (Y)
Dependent Variable (Y)
The outcome or variable being predicted in a regression model.
Explanatory Variables (X2, X3)
Explanatory Variables (X2, X3)
Variables that explain changes in the dependent variable in a regression model.
Partial Regression Coefficient (β2, β3)
Partial Regression Coefficient (β2, β3)
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Stochastic Disturbance Term (u)
Stochastic Disturbance Term (u)
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Assumption 1
Assumption 1
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Assumption 3
Assumption 3
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Assumption 4
Assumption 4
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Assumption 5
Assumption 5
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Assumption 7
Assumption 7
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Ordinary Least Squares (OLS)
Ordinary Least Squares (OLS)
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Estimation of parameters
Estimation of parameters
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RSS
RSS
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Normal Distribution of u
Normal Distribution of u
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t-Distribution in OLS
t-Distribution in OLS
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Hypothesis Statement
Hypothesis Statement
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Level of Significance (α)
Level of Significance (α)
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t-Statistic Calculation
t-Statistic Calculation
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F-Statistic for Joint Hypothesis
F-Statistic for Joint Hypothesis
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Reject Null Hypothesis
Reject Null Hypothesis
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Adjusted R2
Adjusted R2
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Variance of OLS estimator (β1)
Variance of OLS estimator (β1)
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Standard Error of β1
Standard Error of β1
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Variance of OLS estimator (β2)
Variance of OLS estimator (β2)
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Standard Error of β2
Standard Error of β2
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Variance of OLS estimator (β3)
Variance of OLS estimator (β3)
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Standard Error of β3
Standard Error of β3
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Estimated variance (σ²)
Estimated variance (σ²)
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Total Sum of Squares (TSS)
Total Sum of Squares (TSS)
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Coefficient of Determination (R²)
Coefficient of Determination (R²)
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Coefficient of Multiple Correlation (R)
Coefficient of Multiple Correlation (R)
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Study Notes
Multiple Regression Model
- A regression model with more than one explanatory variable is called a multiple regression model.
- Multiple regression models are useful because many economic phenomena cannot be explained by a single variable.
- This chapter discusses how to estimate multiple regression models, hypothesis testing, and unique features of these models.
Three-Variable Linear Regression Model
- Nonstochastic form: E(Y) = β₁ + β₂X₂t + β₃X₃t
- Stochastic form: Yt = β₁ + β₂X₂t + β₃X₃t + ut
- Yt = Dependent variable
- X₂t and X₃t = Explanatory variables
- ut = Stochastic disturbance term
- t = tth observation
- β₂ and β₃ = Partial regression coefficients
- Equation 4.1 gives the conditional mean value of Y, given fixed values of X₂ and X₃.
- Equation 4.2 is divided into a systematic (deterministic) part (β₁ + β₂X₂t + β₃X₃t) and a nonsystematic (random) part (ut).
Partial Regression Coefficient
- β₂ measures the change in the mean value of Y per unit change in X₂, holding X₃ constant.
- β₃ measures the change in the mean value of Y per unit change in X₃, holding X₂ constant.
- Partial regression coefficients reflect the (partial) effect of one explanatory variable on the mean value of the dependent variable when other variables are held constant.
Assumptions of the Multiple Regression Model
- The regression model is linear in parameters.
- X₂ and X₃ are uncorrelated with the disturbance term u.
- The error term u has a zero mean value (E(u) = 0).
- The variance of u is constant (homoscedastic).
- There is no autocorrelation between error terms (cov(ut,us) = 0 for t ≠ s).
- There is no exact collinearity between the explanatory variables (X₂ and X₃).
- The error term u follows a normal distribution with zero mean and constant variance (u ~ N(0, σ²)).
Estimation of Parameters through Ordinary Least Squares (OLS)
- OLS estimators (b₁, b₂, b₃) are calculated to minimize the sum of squared residuals (RSS).
- The sample counterpart of the model is Yt = b₁ + b₂X₂t + b₃X₃t + et.
- Formulae for calculating b₂, b₃, and b₁ are provided using sums of values of X, Y, X₂ and X₃.
Variance and Standard Errors of OLS Estimators
- Formulas for calculating the variances and standard errors of the OLS estimators (b₁, b₂, b₃) are present, including the formula for the estimator of the unknown variance (σ²).
- These formulas involve sums of the explanatory variables (X₂) and the dependent variable (Y).
Goodness of Fit of the Estimated Multiple Regression
- The multiple coefficient of determination (R²) shows the proportion of the total variation in Y that is explained by X₂ and X₃ jointly.
- TSS (Total Sum of Squares) = ESS (Explained Sum of Squares) + RSS (Residual Sum of Squares).
- R² = ESS / TSS.
- R is the coefficient of multiple correlation (R = ±√R²).
Hypothesis Testing in Multiple Regression: General Comments
- OLS estimators follow a t-distribution with (n-3) degrees of freedom.
- Formulas for calculating t-statistics for b₂, b₃, and b₁ are provided.
- The actual hypothesis testing mechanics have similarities with the two-variable case.
Testing Hypotheses About Individual Partial Regression Coefficients
- Steps for testing individual hypotheses:
- Define the hypotheses.
- Choose a significance level (α).
- Calculate the t-statistics.
- Determine the t-critical value or the p-value.
- Reject the null hypothesis if |t| > t-critical or if p-value < α.
Testing Joint Hypothesis
- Steps for testing joint hypotheses:
- Define the joint null hypothesis (e.g., β₂ = β₃ = 0).
- Choose a significance level (α).
- Calculate the F-statistic.
- Determine the F-critical value from an F-distribution table.
- Reject the null hypothesis if F > F-critical.
Comparing Two R² values: The Adjusted R²
- The adjusted R² is a measure of goodness of fit that accounts for the number of explanatory variables.
- R² will tend to increase as more explanatory variables are added to the model.
- Adjusted R² is calculated to account for this and allow comparison of models with different numbers of explanatory variables.
- Adjusted R² will be less than or equal to the unadjusted R².
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