Week 4: Simple Regression Model Basics
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Questions and Answers

What is the purpose of a normal probability plot (NPP)?

  • To estimate the parameters of a normal distribution.
  • To determine the significance level for the Jarque-Bera test.
  • To calculate the skewness and kurtosis of a distribution.
  • To visually assess the normality of a distribution by comparing the observed values to expected values under a normal distribution. (correct)

    • What does a straight line on a normal probability plot indicate?

  • The distribution has a high kurtosis.
  • The distribution is approximately normal. (correct)
  • The distribution is skewed to the right.
  • The distribution is skewed to the left.

    • Which of the following is NOT a characteristic of a normal distribution?

  • Skewness of zero.
  • Kurtosis of 3.
  • Mean, median, and mode are equal. (correct)
  • Bell-shaped curve.

    • What is the Jarque-Bera test used for?

    <p>To assess the normality of a distribution. (D)</p> Signup and view all the answers

    What is the decision rule for the Jarque-Bera test?

    <p>Reject the null hypothesis of normality if the computed chi-square value exceeds the critical value. (B)</p> Signup and view all the answers

    What does the central limit theorem state about the distribution of the sum of a large number of independent and identically distributed random variables?

    <p>The distribution tends to be a normal distribution as the number of variables increases indefinitely with a few exceptions. (A)</p> Signup and view all the answers

    Which of these are assumptions related to the OLS estimators?

    <p>All of the above (D)</p> Signup and view all the answers

    What is the main goal of hypothesis testing in this context?

    <p>To determine if there is a relationship between the independent variable and the dependent variable (B)</p> Signup and view all the answers

    What is the typical level of significance used in empirical analysis?

    <p>All of the above (D)</p> Signup and view all the answers

    What is the purpose of the p-value in hypothesis testing?

    <p>To determine the probability of observing the given data if the null hypothesis is true (B)</p> Signup and view all the answers

    What are the degrees of freedom (d.f.) for the two-variable model?

    <p>n - 2 (C)</p> Signup and view all the answers

    Under which circumstances would you reject the null hypothesis based on the p-value?

    <p>When the p-value is less than the level of significance (A)</p> Signup and view all the answers

    What type of test is being used when the alternative hypothesis is H1: B2 ≠ 0?

    <p>Two-tailed test (A)</p> Signup and view all the answers

    What is the critical t-value, for a one-tailed test, with 8 degrees of freedom?

    <p>2.306 (C)</p> Signup and view all the answers

    What does the coefficient of determination (r^2) measure?

    <p>The proportion of the total variation in Y explained by the regression model. (C)</p> Signup and view all the answers

    What is the relationship between the coefficient of determination (r^2) and the total sum of squares (TSS), explained sum of squares (ESS), and residual sum of squares (RSS)?

    <p>r^2 = ESS / TSS (D)</p> Signup and view all the answers

    How are the coefficient of determination (r^2) and the coefficient of correlation (r) related?

    <p>r^2 is the square of r. (D)</p> Signup and view all the answers

    What is a key difference between a one-tailed test and a two-tailed test?

    <p>A one-tailed test is used to determine if the slope coefficient is greater than or less than zero, while a two-tailed test is used to determine if it is simply not equal to zero. (D)</p> Signup and view all the answers

    What does it mean to reject the null hypothesis in this context?

    <p>The data supports the claim that there is a relationship between annual family income and math S.A.T. scores. (D)</p> Signup and view all the answers

    In the math S.A.T. example, what does r^2 = 0.79 indicate?

    <p>79% of the variation in math S.A.T. scores can be explained by the income variable. (D)</p> Signup and view all the answers

    Given a sample coefficient of correlation (r) of -0.85, what can be inferred about the relationship between the two variables?

    <p>There is a strong negative linear relationship between the two variables. (D)</p> Signup and view all the answers

    What does the Gauss-Markov Theorem state about OLS estimators under the assumptions of the classical linear regression model?

    <p>OLS estimators are unbiased and have the minimum variance among all linear estimators. (A)</p> Signup and view all the answers

    Which assumption of the classical linear regression model ensures that the variance of each error term is constant?

    <p>Assumption 4: The variance of each $u_i$ is constant or homoscedastic. (A)</p> Signup and view all the answers

    What does it mean to say that OLS estimators are unbiased?

    <p>The expected value of the estimators is equal to the true value of the population parameters. (D)</p> Signup and view all the answers

    What is the standard error of the regression (SER) used for?

    <p>To measure the goodness of fit of the estimated regression line. (A)</p> Signup and view all the answers

    Which of the following terms is NOT a property of OLS estimators under the assumptions of the classical linear regression model?

    <p>Consistent. (D)</p> Signup and view all the answers

    What assumption of the classical linear regression model relates to the absence of systematic relationships between error terms?

    <p>Assumption 5: There is no correlation between two error terms. (A)</p> Signup and view all the answers

    How does the assumption of homoscedasticity affect the estimation of the variance of the OLS estimators?

    <p>It decreases the estimated variance of the OLS estimators. (A)</p> Signup and view all the answers

    Which of the following scenarios would violate the assumption that the explanatory variable is uncorrelated with the disturbance term?

    <p>A study analyzing the relationship between education and earnings, where individuals with higher education levels also tend to come from families with higher incomes. (C)</p> Signup and view all the answers

    Flashcards

    Classical Linear Regression Model

    A statistical method to model the relationship between a dependent variable and one or more independent variables assuming specific assumptions.

    Assumption 1

    The regression model is linear in parameters, meaning the relationship is expressed as a straight line with respect to coefficients.

    Assumption 2

    The explanatory variable is uncorrelated with the disturbance term and is non-stochastic, ensuring unbiased estimates.

    Assumption 3

    The mean value of the disturbance term is zero, implying errors do not systematically overestimate or underestimate the model.

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    Assumption 4

    The variance of the disturbance term is constant (homoscedastic), indicating that error variability is stable across all levels of the independent variable.

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    Gauss-Markov Theorem

    States that OLS estimators are BLUE (Best Linear Unbiased Estimators) under the classical linear regression model assumptions.

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    Property 1 of OLS

    The OLS estimators (b1, b2) are linear, meaning they can be expressed as a linear combination of the observed data.

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    Error Variance in OLS

    The estimator of error variance is unbiased, indicating that it accurately reflects the true variability of error terms across samples.

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    Efficient estimators

    Estimators that have the smallest variance among linear unbiased estimators.

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    Normal distribution of error terms

    Error terms that follow a normal distribution with mean zero and variance σ².

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    Central Limit Theorem

    States that the distribution of sums of many independent random variables tends to a normal distribution.

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    Null hypothesis (H0)

    A statement suggesting no relationship or effect, often implying a parameter equals zero.

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    Alternative hypothesis (H1)

    A statement suggesting there is a relationship, often indicating the parameter is not zero.

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    Test statistic

    A calculated value used to decide whether to reject the null hypothesis.

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    Degrees of freedom (d.f.)

    The number of independent values in a calculation, typically (n - 2) for simple regression.

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    P-value

    The probability of obtaining test results at least as extreme as the observed results, under H0.

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    Histogram of Residuals

    A graphical representation showing the PDF of a variable based on interval observations.

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    Normal Probability Plot (NPP)

    A plot comparing observed values against expected values for a normal distribution.

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    Coefficient of Skewness

    A measure indicating the asymmetry of a probability density function (PDF).

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    Kurtosis

    A measure of how peaked or flat a PDF is compared to the normal distribution.

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    Jarque-Bera Test

    Statistical test to check for normality using skewness and kurtosis.

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    Null Hypothesis

    A statement that there is no effect or relationship in a population.

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    Critical t value

    The threshold value that the t statistic must exceed to reject the null hypothesis.

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    One-Tailed Test

    A hypothesis test that assesses the direction of an effect or relationship (either increase or decrease).

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    Coefficient of Determination (r²)

    A measure that explains the proportion of variance in the dependent variable that's explained by the independent variable.

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    Total Sum of Squares (TSS)

    The total variation in the dependent variable around its mean.

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    Residual Sum of Squares (RSS)

    The variation in the dependent variable that is not explained by the regression model.

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    Coefficient of Correlation (r)

    Quantifies the strength and direction of the linear relationship between two variables.

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    Study Notes

    Week 4: Simple Regression Model (Part I)

    • Model: A linear regression model is used to understand the relationship between a dependent variable (Y) and one or more independent variables (X).
    • Assumptions of the Classic Linear Regression Model:
      • Assumption 1 (Linearity): The relationship between variables is linear in the parameters. This means that the dependent variable can be described as a straight line (or plane) when graphed against the independent variable(s).
      • Assumption 2 (Exogeneity): The independent variable(s) is not correlated with the error term. In simpler terms, the independent variable is unrelated to factors that influence the dependent variable through the error term.
      • Assumption 3 (Zero Conditional Mean): The expected value of the error term (u) is zero, given the value of the independent variable (X). Simply put, the error term's mean is zero. This ensures that factors not explained by X are not systematically linked to the independent variables.
      • Assumption 4 (Homoscedasticity): The variance of the error term is constant for all values of the independent variable(s). The spread of the dependent variable's values around the regression line is consistent across different values of the independent variables.
      • Assumption 5 (No Autocorrelation): There is no correlation between the error terms at different observations. This condition prevents the errors from being correlated. In simpler terms, the error at one observation does not influence the error at another observation.
      • Assumption 6 (Correct Specification): The regression model accurately represents the relationship between the variables.
      • Assumption 7 (Normality): The error terms follow a normal distribution with a mean of zero and constant variance (homoscedasticity).

    Variances and Standard Errors of OLS Estimators

    • The variances and standard errors of Ordinary Least Squares (OLS) estimators for b1 and b2 are essential for assessing the variability or uncertainty in these estimates.

    Standard Error of Regression (SER)

    • SER is a measure of the goodness of fit of the estimated regression line. It estimates the error in prediction. It is calculated using the formula: σ^2 = Σei^2 / (n - 2) and the standard error of the regression is its square root (ô = √σ^2)). This represents the average vertical distance between the observed data points and the fitted regression line.

    Hypothesis Testing

    • Testing hypotheses about the relationship between variables is a crucial aspect of regression analysis. Statistical tests are used to determine if the relationship is significant. A null hypothesis (e.g., no relationship between variables) is posited, and evaluated against an alternative hypothesis. Results are drawn based on either a confidence interval or significance test approach.

    Test of Significance Approach

    • The test of significance approach is a procedure for hypothesis testing, which depends on calculating a t-value and using critical values on a t-table.

    One-Tailed Test

    • A one-tailed test considers only a specific direction of the effect.

    Two-Tailed Test

    • A two-tailed test considers both directions of the effect.

    Coefficient of Determination (r²)

    • It measures how much of the variation in the dependent variable is explained by the regression model. It has values between 0 and 1. A higher value of r² indicates a better fit of the model.

    Coefficient of Correlation (r)

    • It measures the strength and direction of the linear relationship between the two variables. The value is between -1 and 1. A value of +1 or –1 signifies a perfect linear relationship, zero represents no relationship.

    Normality Tests

    • Various normality tests are utilized to verify that the errors or residuals in the regression model follow a normal distribution.
      • Histogram of residuals: Visualizes the distribution of residuals.
      • Normal Probability Plot (NPP): Plots the observed residuals against their expected values under a normal distribution. A straight line in the plot suggests a normal distribution.
      • Jarque-Bera Test: Tests the skewness and kurtosis of the residuals; a low p-value suggests deviations from normality.

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