Linear Regression Coefficients and R-squared Quiz

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Questions and Answers

What is the effect of an increase of living room surface of 5 sqm on the price, according to the estimated linear regression model?

The price increases by $3.00

The price increases by $0.60

The price increases by $0.15

The price increases by $1.50 (correct)

Which of the following is the correct interpretation of the house price elasticity of 0.3%?

A 1% increase in demand leads to a 3% increase in house price

A 1% increase in house price leads to a 0.3% increase in demand (correct)

A 1% increase in house price leads to a 3% increase in demand

A 1% increase in demand leads to a 0.3% increase in house price

If the OLS assumption E(u|X)=0 is violated, which of the following is true?

The OLS estimators are no longer Gaussian random vectors

The OLS estimators are biased (correct)

The OLS estimators are unbiased, but inefficient

The OLS estimators can't be computed

Which of the following is not a solution to the problem of multicollinearity?

Robust estimation of the standard errors Signup and view all the answers

What is the correct test of significance for the linear regression model Y=b0 + b1X1 + b2X2?

A test F on the null hypothesis b0=b1=b2=0 Signup and view all the answers

Which of the following is the correct interpretation of the statement that the logarithm of house price increases by 5%?

The house price increases by 50% Signup and view all the answers

What does the beta1 coefficient represent in the context of a simple linear model of Y=scores in math vs the binary variable D=gender?

The average difference in scores between male and females Signup and view all the answers

Which statement about R-squared is incorrect?

Always between -1 and 1 Signup and view all the answers

How can the variance of the regression errors be estimated?

The square of the SER, equal to the sum of the squares of the residuals divided by (n-k1) Signup and view all the answers

What is the first assumption of OLS called?

Homoscedasticity of errors Signup and view all the answers

Under assumptions A1 to A4, what can be said about the OLS estimator of the beta coefficient?

Best linear unbiased (BLUE) Signup and view all the answers

In a logit model, what does the linear regression function $x^T\beta$ represent?

The log(ODDS) of the event $Y=1$ Signup and view all the answers

What does the beta coefficient associated with an independent variable $X$ in a logit model represent?

The effect of a unit increment of $X$ on the log-odds of the dependent variable Signup and view all the answers

Which of the following correctly describes the fraction of correctly predicted cases in a logit model?

The sum of the diagonal elements in the cross-tabulation of the predicted and the observed $Y$ Signup and view all the answers

In a logit model, what transformation is applied to the linear regression function to obtain the ODDS of the event $Y=1$?

Exponential transformation Signup and view all the answers

What does the probability that $Y=1$, conditional on the regressors, represent in a logit model?

A Gaussian transformation of the data Signup and view all the answers

In a logit model, what transformation is applied to the ODDS of the event $Y=1$ to obtain the linear regression function?

Logarithmic transformation Signup and view all the answers

Which assumption is NOT a standard assumption of the fixed effect model (within estimation)?

Errors at time t must be correlated with both past and future values of the predictors Signup and view all the answers

When is a panel considered balanced?

No units leave or are added to the cross-sectional dimension for the whole length of the panel Signup and view all the answers

What does the term 'attrition' refer to in panel data analysis?

A portion of units exiting the panel before the end of the time period considered Signup and view all the answers

What does the expression Cov(u(i,t), u(i,s)| X(i,t), X(i,s))=0 for t≠s imply?

The errors are independent across units but not necessarily over time Signup and view all the answers

Which of the following is NOT a requirement for the fixed effect model (within estimation)?

No time autocorrelation of the error component is allowed Signup and view all the answers

What is a potential issue associated with the term 'attrition' in panel data analysis?

The potential for biased estimates due to non-random sample attrition Signup and view all the answers

What is the main advantage of using panel data over cross-sectional data?

It allows you to control for some types of omitted variables without actually observing them. Signup and view all the answers

In the fixed effects regression model, what does the coefficient of the binary variable for an entity indicate?

The difference in fixed effects between that entity and the excluded entity. Signup and view all the answers

What is true about the fixed effects regression model?

In a log-log model, it may include logs of binary variables to control for fixed effects. Signup and view all the answers

Which statement is true about the division of errors by regressors in different time periods?

It is always zero. Signup and view all the answers

What is the assumption about the errors in panel data models?

The errors are uncorrelated over time, conditional on the regressors. Signup and view all the answers

Which statement about the residuals in panel data models is true?

There is no correlation over time in the residuals. Signup and view all the answers

Study Notes

Logit Model

The probability of Y=1, conditional on the regressors, is a Gaussian transformation of the data.
The log(ODDS) of the event Y=1 is modelled by a linear regression function xTb.
The ODDS of the event Y=1 are modelled by a linear regression function.
The exp(ODDS) of the event Y=1 are modelled by a linear regression function.

Logit Coefficients

The beta coefficient associated with an independent variable X in a logit model represents the effect of a unit increment of X on the log-odds of the dependent variable.
It is not the effect of a percent increment of X on the dependent variable.
It is not equal to the log-odds ratio of Y and X.

Predictive Accuracy

The fraction of correctly predicted cases is not the total number of cases where both the predicted and observed value of Y is equal to 1.
It is not the sum of the first column in the cross tabulation of the predicted and the observed Y.
It is not the sum of the first row in the cross tabulation of the predicted and the observed Y.
It is the sum of the diagonal elements in the cross tabulation of the predicted and the observed Y.

House Price Elasticity

If house price per Sqm increases by 5%, the logarithm of house price increases by 5%.
If house price increases by 50$ per sqm, house price elasticity is not 0.01%.
If house price increases by 50$ per sqm, the effect on the logarithm of house price is not 50%.

Linear Regression Model

The OLS estimators are biased if the OLS assumption A1 (E(u|X)=0) fails.
The OLS estimators are not unbiased, but inefficient if the OLS assumption A1 (E(u|X)=0) fails.
A test F on the null hypothesis b0=b1=b2=0 is used to test the significance of the linear regression model.
A chi-squared test is not used to test the significance of the linear regression model.

Multicollinearity

Using the Ridge estimator as an alternative to the OLS is a solution to the problem of multicollinearity.
Robust estimation of the standard errors is not a solution to the problem of multicollinearity.
Identification of the variables that cause multicollinearity and estimation of a model without those variables is a solution to the problem of multicollinearity.
Principal component analysis is a solution to the problem of multicollinearity.

Panel Data

The main advantage of using a panel data over cross-sectional data is that it allows controlling for some types of omitted variables without actually observing them.
Panel data allows analyzing behavior across time, but not across entities.
Division of errors by regressors in different time periods is not always zero.
Conditional on the regressors, the error are not always uncorrelated over time.

Fixed Effects Regression Model

The coefficient of the binary variable in the fixed effects regression model indicates the level of the fixed effect of the i-th entity.
The fixed effects regression model is such that the slope coefficients are allowed to differ across entities, but the intercept is “fixed” (remains unchanged).
In a log-log model, the fixed effects regression model may include logs of binary variables, which control for the fixed effects.
The fixed effects regression model has n different intercepts.

R-Squared

The R-squared is always between 0 and 1.
The R-squared is a measure of the goodness of fit of the linear model.
The R-squared is not always non-decreasing.

Variance of Regression Errors

The variance of the regression errors can be estimated by the sum of the squares of the residuals divided by (n-k-1).
It can also be estimated by the HC estimator.
It cannot be estimated by the R-squared.

Assumptions of OLS

The first assumption of the OLS is called Homoscedasticity of errors.
Under the assumptions from A1 to A4, the OLS estimator of the beta coefficient is unbiased and normally distributed for large n.
It is not normally distributed for every sample of size n.

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Test your knowledge on interpreting beta coefficients in linear regression models and understanding the properties of R-squared. Questions cover topics such as average score differences between genders and the range and interpretation of R-squared values.