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Questions and Answers
The computed sample regression function takes into account both the mean and variance of x and y.
The computed sample regression function takes into account both the mean and variance of x and y.
True
The null hypothesis states that the true value of beta is less than one.
The null hypothesis states that the true value of beta is less than one.
False
A beta value greater than one indicates that the security is less risky than the market.
A beta value greater than one indicates that the security is less risky than the market.
False
The test statistic is calculated combining beta estimate and its standard error.
The test statistic is calculated combining beta estimate and its standard error.
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In a hypothesis test, a test statistic that falls in the rejection region results in accepting the null hypothesis.
In a hypothesis test, a test statistic that falls in the rejection region results in accepting the null hypothesis.
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The R2 value measures the variability of the predicted values about their mean.
The R2 value measures the variability of the predicted values about their mean.
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The formula for the total sum of squares (TSS) includes all observed values of y relative to the mean ȳ.
The formula for the total sum of squares (TSS) includes all observed values of y relative to the mean ȳ.
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In the formula for β̂, σx,y represents the covariance between x and y.
In the formula for β̂, σx,y represents the covariance between x and y.
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The expression T P (xt − x̄) (yt − ȳ) calculates the total sum of squares (TSS).
The expression T P (xt − x̄) (yt − ȳ) calculates the total sum of squares (TSS).
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The term (yt − ȳ) in the R2 formula is squared to emphasize larger differences.
The term (yt − ȳ) in the R2 formula is squared to emphasize larger differences.
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The formula for R2 can help determine how well a regression model explains variability in y.
The formula for R2 can help determine how well a regression model explains variability in y.
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The total sum of squares (TSS) is calculated using the predicted values of y.
The total sum of squares (TSS) is calculated using the predicted values of y.
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The calculated R2 value can never exceed 1.
The calculated R2 value can never exceed 1.
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The null hypothesis states that β is equal to 0.
The null hypothesis states that β is equal to 0.
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In hypothesis testing, we test the actual values of the coefficients, not their estimated values.
In hypothesis testing, we test the actual values of the coefficients, not their estimated values.
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The rejection region is determined by the test statistic exceeding the critical t-value.
The rejection region is determined by the test statistic exceeding the critical t-value.
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The estimated CAPM beta for the stock must be exactly 1 to not reject the null hypothesis.
The estimated CAPM beta for the stock must be exactly 1 to not reject the null hypothesis.
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A confidence interval is a one-sided interval unless specified otherwise.
A confidence interval is a one-sided interval unless specified otherwise.
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The explained sum of squares (ESS) is represented by the formula $ESS = \sum (ŷ_t - ȳ)^2$.
The explained sum of squares (ESS) is represented by the formula $ESS = \sum (ŷ_t - ȳ)^2$.
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The residual sum of squares (RSS) can be defined as $RSS = \sum (yt - ŷ)^2$.
The residual sum of squares (RSS) can be defined as $RSS = \sum (yt - ŷ)^2$.
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The total sum of squares (TSS) is calculated by $TSS = ESS - RSS$.
The total sum of squares (TSS) is calculated by $TSS = ESS - RSS$.
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The formula for $R^2$ is $R^2 = \frac{ESS}{TSS}$.
The formula for $R^2$ is $R^2 = \frac{ESS}{TSS}$.
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The sum of the explained sum of squares (ESS) and the residual sum of squares (RSS) equals 1000.
The sum of the explained sum of squares (ESS) and the residual sum of squares (RSS) equals 1000.
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If the R-squared value ($R^2$) is 0.9234, it indicates a strong correlation between the independent and dependent variables.
If the R-squared value ($R^2$) is 0.9234, it indicates a strong correlation between the independent and dependent variables.
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The equation $R^2 = (ρ_{x,y})^2$ serves as an alternative interpretation of $R^2$.
The equation $R^2 = (ρ_{x,y})^2$ serves as an alternative interpretation of $R^2$.
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The explained sum of squares (ESS) is always smaller than the total sum of squares (TSS).
The explained sum of squares (ESS) is always smaller than the total sum of squares (TSS).
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Study Notes
Self-Assessment
- The variable on the right-hand side of a linear regression is sometimes called an explanatory variable.
- The variable on the left-hand side of a linear regression is not called a regressor.
- When computing regression parameters using Ordinary Least Squares (OLS), the squared horizontal distances between the model's predictions and the dependent variable values are minimized.
- OLS selects parameters that minimize the sum of squared residuals.
- The sample regression function includes a disturbance term.
- A non-linear model that cannot be transformed into a linear model cannot be estimated using OLS.
- The Classical Linear Regression Model (CLRM) assumes the variance of error terms is not zero.
- The CLRM assumes errors are normally distributed.
- If CLRM assumptions hold, OLS estimators are Best Linear Unbiased Estimators (BLUE).
- Consistency is a weaker condition than unbiasedness.
- The standard error of the slope parameter is the square root of its variance.
Exercises 1
- Calculate arithmetic sample mean and sample variance for x and y data.
- Compute the sample covariance and sample correlation between x and y. Sample covariance = −15.1778.
- Calculate sample correlation coefficient, rxy = −0.9610.
Exercises 1 (cont.)
- Determine the sample regression function using formulas.
- Sample regression function: ŷ = 62.689 - 5.3676x.
- Use the formula involving sample covariance and sample variance. This gives the same beta coefficient as the previous method.
Exercises 2
- Explain the difference between sample and population regression functions using equations.
- Sample regression function describes the relationship between variables estimated from samples.
- Population regression function represents the true but unknown relationship between variables within the entire population.
Exercises 3
- Identify models that can be estimated using OLS (ordinary least squares).
- Models that are linear in parameters are suitable for OLS estimation. Linearization is possible in some cases for models that are not already explicitly linear in the parameters.
Exercises 4
- Null hypothesis (H0): Beta = 1.
- Alternative hypothesis (H1): Beta > 1.
- Evaluate test statistic to determine whether beta equals 1 given the data or whether to reject in favor of beta greater than 1.
- The test statistic (2.682) is greater than the critical t-value (1.671). Reject the null hypothesis.
Exercises 5
- Null hypothesis (H0): Beta = 0.
- Alternative hypothesis (H1): Beta ≠ 0.
- Evaluate test statistic to determine whether beta equals 0 given the data.
- Since test statistic is not in the rejection region, fail to reject the null hypothesis.
Exercises 6
- Form and interpret 95% and 99% confidence intervals for Beta based on calculated data.
Additional Information
- Hypothesis tests are about actual, not estimated coefficients.
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Description
Test your understanding of key concepts in linear regression, particularly focusing on Ordinary Least Squares (OLS) and the Classical Linear Regression Model (CLRM). This quiz covers fundamental principles, assumptions, and properties related to regression analysis, designed for students and enthusiasts of statistics and econometrics.
