Question 1

The regression R² is a measure of

Accepted Answer

The regression R² is a measure of the goodness of fit of your regression line. It represents the proportion of variance in the dependent variable (Y) explained by the independent variable (X) in a linear regression model. The higher the value of the regression coefficient, the better the fit of the regression line.

Question 2

(Requires Appendix)The sample regression line estimated by OLS

Accepted Answer

The sample regression line estimated by Ordinary Least Squares (OLS) methodology will always pass through the point where $\bar{X}$ is the mean of the independent variable and $\bar{Y}$ is the mean of the dependent variable. This is a property of the OLS regression line, ensuring that the line is the best fit in terms of minimizing the sum of squared residuals.

Question 3

The variance of Y_i is given by

Accepted Answer

The variance of Y_i in a simple linear regression model is given by $\beta _ { 1 } ^ { 2 }$ var(X_i) + var(u_i), where $\beta _ { 1 }$ is the coefficient of X_i and u_i is the error term.

Question 4

When the estimated slope coefficient in the simple regression model, $\hat { \beta }$ ₁, is zero, then

Accepted Answer

When the estimated slope coefficient in a simple regression model is zero, it implies that there is no linear relationship between the independent variable and the dependent variable. This situation leads to a coefficient of determination, $R^2$, equal to 0, indicating that the model explains none of the variability of the response data around its mean.

Question 5

The normal approximation to the sampling distribution of $\hat { \beta }$ ₁ is powerful because

Accepted Answer

The normal approximation to the sampling distribution of $\hat{\beta}$ is powerful because it enables econometricians to use normal distribution properties to develop methods for statistical inference, such as hypothesis testing and confidence intervals. This is not necessarily because real-life explanatory variables are normally distributed (A), other distributions are not symmetric (C), or it implies OLS is the BLUE estimator (D).

Question 6

The slope estimator, β₁, has a smaller standard error, other things equal, if

Accepted Answer

The standard error of the slope estimator, β₁, is inversely proportional to the standard deviation of the explanatory variable, X. Therefore, if there is more variation in X, the standard error of the slope estimator will be smaller.

Question 7

If the three least squares assumptions hold, then the large sample normal distribution of $\hat { \beta }$ ₁ is

Accepted Answer

The answer of If the three least squares assumptions hold,...

Question 8

(Requires Appendix)The sample average of the OLS residuals is

Accepted Answer

The sample average of the OLS residuals is simply the average of the deviations between the observed values and the predicted values. Since OLS aims to minimize the sum of squared deviations, this means that the average of the residuals will always be as close to zero as possible. Therefore, the correct answer is B, which states that the sample average of the OLS residuals is zero.

Question 9

The standard error of the regression (SER)is defined as follows

Accepted Answer

The standard error of the regression (SER) is correctly defined by option A, which represents the square root of the average squared deviations of the observed values from the values predicted by the model, adjusted for degrees of freedom (n - 2). Option D is incorrect because it divides by $n - 1$, which is typically used for the sample variance calculation, not for SER. Options B and C are incorrect as they do not represent the formula for SER; SSR refers to the sum of squared residuals, and option C is nonsensical.

Question 10

Interpreting the intercept in a sample regression function is

Accepted Answer

The intercept in a sample regression function can be interpreted if your sample contains values of the explanatory variables around the origin, which is represented by X_i. This is because the intercept represents the value of the dependent variable when all the explanatory variables are at zero.

Question 11

The OLS residuals, $\hat u$ _i, are defined as follows:

Accepted Answer

OLS residuals, denoted as $\hat{u}$, are the differences between the observed values ($Y$) and the predicted values ($\hat{Y}$). Option C accurately represents this definition.

Question 12

The OLS residuals

Accepted Answer

OLS residuals are the difference between the actual values and the predicted values (fitted values) of the dependent variable. Therefore, option B is correct. Option A is incorrect because errors are not the same as residuals - errors are the difference between the actual values of the dependent variable and the values predicted by the population regression equation, which is typically unknown in practice. Option C is incorrect because while we do not know the population regression function, we can estimate the sample regression function and use it to calculate residuals. Option D is incorrect because residuals are a necessary part of regression analysis and are used to assess the goodness of fit of the regression model.

Question 13

(Requires Appendix material)Which of the following statements is correct?

Accepted Answer

TSS (total sum of squares) is the total variability in the data. It can be decomposed into two parts- ESS (explained sum of squares) and SSR (sum of squares of residuals or errors). Hence, the correct relationship is TSS = ESS + SSR.

Question 14

The following are all least squares assumptions with the exception of:

Accepted Answer

The least squares assumptions are:
1. Linearity and additivity of the relationship between dependent and independent variables.
2. Independence of the errors or residuals.
3. Homoscedasticity (constant variance) of the errors.
4. Normality of the error distribution.
Option B is not a least squares assumption, as the explanatory variable can be non-normal, but the error distribution needs to be normal.

Question 15

The regression R² is defined as follows:

Accepted Answer

The answer of The regression R² is defined as follows: ...

Question 16

Binary variables

Accepted Answer

Binary variables can take on only two values, typically labeled as 0 or 1, true or false, yes or no, etc. They are commonly used in statistical analysis to represent categorical data or to indicate the presence or absence of a particular characteristic or event.

Question 17

In the simple linear regression model, the regression slope

Accepted Answer

The regression slope indicates by how many units Y increases, given a one unit increase in X.

Question 18

In the simple linear regression model Y_i = β₀ + β₁X_i+ u_i,

Accepted Answer

βS1U1B10S1U1B0 + βS1U1B11S1U1B0XS1U1B1i S1U1B0 represents the population regression function, which describes the expected relationship between the independent variable (X) and dependent variable (Y) in the entire population. This is the true relationship that we are trying to estimate through our sample data. The intercept (βS1U1B10S1U1B0) and slope (βS1U1B11S1U1B0) represent the parameters of this function. The intercept may or may not be small and unimportant, and the absolute value of the slope can vary widely depending on the context. The sample regression function is represented by the equation YS1U1B1iS1U1B0 = bS1U1B0 + bS1U1B1iS1U1B0X, where bS1U1B0 and bS1U1B1iS1U1B0 are the estimated intercept and slope based on the sample data.

Question 19

The OLS estimator is derived by

Accepted Answer

The OLS estimator is derived by minimizing the sum of squared residuals, which is the vertical distance between the observed and predicted values of the dependent variable. This method is known as the method of least squares and results in the estimates of the slope and intercept coefficients that provide the best linear fit to the data.

Question 20

The reason why estimators have a sampling distribution is that

Accepted Answer

The values of the explanatory variable and the error term differ across samples, which leads to different estimates each time a new sample is drawn, hence the existence of a sampling distribution for estimators.

Quiz 4: Linear Regression With One Regressor

The regression R² is a measure of

(Requires Appendix) The sample regression line estimated by OLS

The variance of Y_i is given by

When the estimated slope coefficient in the simple regression model, $\hat { \beta }$ ₁, is zero, then

The normal approximation to the sampling distribution of $\hat { \beta }$ ₁ is powerful because

The slope estimator, β₁, has a smaller standard error, other things equal, if

If the three least squares assumptions hold, then the large sample normal distribution of $\hat { \beta }$ ₁ is

(Requires Appendix) The sample average of the OLS residuals is

The standard error of the regression (SER) is defined as follows

Interpreting the intercept in a sample regression function is

The OLS residuals, $\hat u$ _i, are defined as follows:

The OLS residuals

(Requires Appendix material) Which of the following statements is correct?

The following are all least squares assumptions with the exception of:

The regression R² is defined as follows:

Binary variables

In the simple linear regression model, the regression slope

In the simple linear regression model Y_i = β₀ + β₁X_i+ u_i,

The OLS estimator is derived by

The reason why estimators have a sampling distribution is that

Quiz 4: Linear Regression With One Regressor

The regression R2 is a measure of

(Requires Appendix) The sample regression line estimated by OLS

The variance of Yi is given by

When the estimated slope coefficient in the simple regression model, β^\hat { \beta }β^​ 1, is zero, then

The normal approximation to the sampling distribution of β^\hat { \beta }β^​ 1 is powerful because

The slope estimator, β1, has a smaller standard error, other things equal, if

If the three least squares assumptions hold, then the large sample normal distribution of β^\hat { \beta }β^​ 1 is

(Requires Appendix) The sample average of the OLS residuals is

The standard error of the regression (SER) is defined as follows

Interpreting the intercept in a sample regression function is

The OLS residuals, u^\hat uu^ i, are defined as follows:

The OLS residuals

(Requires Appendix material) Which of the following statements is correct?

The following are all least squares assumptions with the exception of:

The regression R2 is defined as follows:

Binary variables

In the simple linear regression model, the regression slope

In the simple linear regression model Yi = β0 + β1Xi + ui,

The OLS estimator is derived by

The reason why estimators have a sampling distribution is that

The regression R² is a measure of

The variance of Y_i is given by

When the estimated slope coefficient in the simple regression model, $\hat { \beta }$ ₁, is zero, then

The normal approximation to the sampling distribution of $\hat { \beta }$ ₁ is powerful because

The slope estimator, β₁, has a smaller standard error, other things equal, if

If the three least squares assumptions hold, then the large sample normal distribution of $\hat { \beta }$ ₁ is

The OLS residuals, $\hat u$ _i, are defined as follows:

The regression R² is defined as follows:

In the simple linear regression model Y_i = β₀ + β₁X_i+ u_i,