QM 2

Question 1

What is conditional heteroskedasticity?

Answer 1

It occurs when the error variance in a regression is correlated with or conditional on the values of the independent variables.

Question 2

What are the main consequences of conditional heteroskedasticity for statistical inference?

Answer 2

Standard errors of regression coefficients are underestimated t-statistics are inflated and Type I errors rejecting a true null hypothesis are more likely. The F-test for overall significance also becomes unreliable.

Question 3

What is the purpose of the Breusch-Pagan (BP) test?

Answer 3

To test for conditional heteroskedasticity in a regression model.

Question 4

What is the null hypothesis of the Breusch-Pagan (BP) test?

Answer 4

There is no conditional heteroskedasticity meaning the regression’s squared residuals are uncorrelated with the independent variables.

Question 5

How is the Breusch-Pagan (BP) test statistic calculated and what is its distribution?

Answer 5

The BP test statistic is n multiplied by R-squared from a regression of the squared residuals on the original independent variables. It is approximately chi-square distributed with k degrees of freedom where k is the number of independent variables.

Question 6

How do you correct for conditional heteroskedasticity?

Answer 6

By computing robust standard errors also known as heteroskedasticity-consistent or White-corrected standard errors.

Question 7

What is serial correlation or autocorrelation in a regression model?

Answer 7

It occurs when regression errors are correlated across observations typically in time-series regressions.

Question 8

What are the consequences of serial correlation if one of the independent variables is a lagged value of the dependent variable?

Answer 8

All parameter estimates (coefficients) become inconsistent and invalid. Standard errors are also invalid.

Question 9

What are the consequences of positive serial correlation if no independent variable is a lagged dependent variable?

Answer 9

Coefficient estimates are consistent but standard errors are typically underestimated leading to inflated t-statistics and more Type I errors. The F-statistic may also be inflated.

Question 10

What is the Breusch-Godfrey (BG) test used for?

Answer 10

To detect serial correlation of a pre-designated order p in regression residuals.

Question 11

What is the null hypothesis of the Breusch-Godfrey (BG) test?

Answer 11

There is no serial correlation in the model’s residuals up to the specified lag p.

Question 12

How are robust standard errors (like Newey-West) useful in the context of serial correlation?

Answer 12

They adjust the coefficient standard errors to account for serial correlation and also correct for conditional heteroskedasticity allowing for more reliable statistical inference.

Question 13

What is multicollinearity in a multiple regression model?

Answer 13

It occurs when two or more independent variables are highly correlated or when there is an approximate linear relationship among independent variables.

Question 14

What are the primary consequences of multicollinearity?

Answer 14

It makes coefficient estimates imprecise and unreliable inflates standard errors and diminishes t-statistics making it difficult to distinguish the individual impacts of independent variables.

Question 15

What is a classic symptom of multicollinearity?

Answer 15

A high R-squared and a significant overall F-statistic but t-statistics for individual slope coefficients that are not significant.

Question 16

What does the Variance Inflation Factor (VIF) measure?

Answer 16

It quantifies the extent to which an independent variable’s variance is inflated due to its correlation with other independent variables in the model.

Question 17

What is a common rule of thumb for interpreting VIF values?

Answer 17

A VIF greater than 5 warrants further investigation and a VIF greater than 10 indicates serious multicollinearity requiring correction.

Question 18

What are possible solutions for correcting multicollinearity?

Answer 18

Excluding one or more correlated variables using a different proxy for one of the variables or increasing the sample size.

Question 19

What is omitted variable bias?

Answer 19

It’s a bias in coefficient estimates that arises when an important independent variable is omitted from a regression model.

Question 20

What happens if an omitted variable is correlated with an included independent variable?

Answer 20

The estimated coefficient of the included variable will be biased and inconsistent and its standard error will also be inconsistent.

Question 21

What is a high-leverage point in regression analysis?

Answer 21

A data point having an extreme value for an independent variable.

Question 22

What is an outlier in regression analysis?

Answer 22

A data point having an extreme value for the dependent variable relative to its predicted value resulting in a large residual.

Question 23

What does the leverage measure (hii) indicate and what is a rule of thumb for its use?

Answer 23

Leverage measures the distance of an observation’s independent variable value from the mean. If hii > 3(k+1)/n it is a potentially influential high-leverage point.

Question 24

What are studentized residuals used for and what is a rule of thumb?

Answer 24

To identify influential outlying Y observations. If the absolute value of the studentized residual exceeds the critical t-value for n-k-2 degrees of freedom the observation is potentially influential.

Question 25

If you want to distinguish among n categories using dummy variables how many dummy variables should you include in the regression?

Answer 25

You should include n-1 dummy variables. The nth category becomes the base or control group.

Question 26

What does an intercept dummy variable do in a regression model?

Answer 26

It adds to or reduces the original intercept if a specific condition (represented by the dummy being 1) is met shifting the regression line up or down.

Question 27

What does a slope dummy variable do in a regression model?

Answer 27

It allows for a change in the slope of the relationship between an independent variable and the dependent variable if a specific condition (represented by the dummy being 1) is met.

Question 28

When is logistic regression typically used?

Answer 28

When the dependent variable is qualitative or categorical often binary (e.g. event happens or does not happen).

Question 29

What transformation is applied to the dependent variable in logistic regression?

Answer 29

The logistic transformation which converts the probability P of an event into the log-odds ln(P/(1-P)).

Question 30

How are slope coefficients in a logistic regression model interpreted?

Answer 30

As the change in the log-odds that the event happens per unit change in the independent variable holding all other independent variables constant.

Question 31

What is the Likelihood Ratio (LR) test used for in logistic regression?

Answer 31

To assess the fit of logistic regression models by comparing the log-likelihood of a restricted model to an unrestricted model similar to an F-test in OLS regression.

Question 32

What does a pseudo-R-squared in logistic regression measure?

Answer 32

It attempts to capture the explained variation similar to R-squared in OLS but must be interpreted with care and is mainly for comparing different specifications of the same model.

Question 33

What is overfitting in a machine learning model?

Answer 33

When a model fits the training data too well capturing noise and random fluctuations such that it does not generalize well to new out-of-sample data.

Question 34

What is the difference between bias error and variance error in ML models?

Answer 34

Bias error is the degree to which a model fits training data (high bias means underfitting). Variance error is how much model results change with new data (high variance means overfitting).

Question 35

What is k-fold cross-validation used for?

Answer 35

To estimate a model's out-of-sample error and mitigate overfitting by repeatedly splitting the data into training and validation sets.

Question 36

What is the primary goal of penalized regression techniques like LASSO?

Answer 36

To prevent overfitting by adding a penalty term to the sum of squared residuals that increases with the number of included features or the magnitude of coefficients.

Question 37

What does LASSO (Least Absolute Shrinkage and Selection Operator) do?

Answer 37

It minimizes the sum of squared residuals plus a penalty term based on the sum of the absolute values of coefficients effectively shrinking some coefficients to zero and performing feature selection.

Question 38

What is a Support Vector Machine (SVM) primarily used for in supervised learning?

Answer 38

For classification tasks aiming to find the optimal hyperplane that separates data points into distinct classes with the maximum margin.

Question 39

How does the K-Nearest Neighbor (KNN) algorithm classify a new observation?

Answer 39

By finding the k most similar (nearest) observations in the existing labeled dataset and assigning the new observation to the majority class among those k neighbors.

Question 40

What are Classification and Regression Trees (CART) used for?

Answer 40

For both classification (predicting a categorical target) and regression (predicting a continuous target) by partitioning data into subgroups based on feature values.

Question 41

What is ensemble learning in machine learning?

Answer 41

A technique of combining the predictions from a collection of models to typically produce more accurate and stable predictions than any single model.

Question 42

What is a random forest classifier?

Answer 42

An ensemble learning method that constructs a multitude of decision trees during training and outputs the class that is the mode of the classes (classification) or mean prediction (regression) of the individual trees.

Question 43

What is Principal Components Analysis (PCA) used for in unsupervised learning?

Answer 43

For dimension reduction by transforming a set of correlated features into a smaller set of uncorrelated composite variables (principal components) that retain most of the original variance.

Question 44

What is the goal of clustering algorithms like K-Means?

Answer 44

To sort observations into groups (clusters) such that observations within the same cluster are more similar to each other than to observations in other clusters.

Question 45

How does K-Means clustering work?

Answer 45

It partitions observations into a pre-specified number (k) of clusters by iteratively assigning observations to the nearest cluster centroid and then recalculating the centroids.

Question 46

What is hierarchical clustering?

Answer 46

An unsupervised algorithm that builds a hierarchy of clusters either agglomeratively (bottom-up starting with individual observations) or divisively (top-down starting with one large cluster).

Question 47

What is a neural network (NN) in machine learning?

Answer 47

A flexible algorithm inspired by the human brain with layers of interconnected nodes (neurons) that transform inputs non-linearly to learn complex patterns.

Question 48

What distinguishes a Deep Neural Network (DNN) from a shallow neural network?

Answer 48

A DNN has multiple hidden layers (typically at least 2) allowing it to learn more complex hierarchical features from data.

Question 49

What is reinforcement learning (RL)?

Answer 49

A type of machine learning where an agent learns to make decisions by performing actions in an environment to maximize a cumulative reward often through trial and error.

Question 50

What is the primary challenge of working with time-series data in regression?

Answer 50

The assumptions of linear regression such as uncorrelated errors and constant variance (homoskedasticity) are often violated.

Question 51

What is a covariance stationary time series?

Answer 51

A time series whose mean variance and autocovariances are constant and finite over time.

Question 52

What happens if you estimate an autoregressive (AR) model for a time series that is not covariance stationary?

Answer 52

The estimation results (coefficients standard errors) will likely be invalid and have no economic meaning.

Question 53

How can you test if the residuals of an AR model are serially correlated?

Answer 53

By examining the t-statistics of the residual autocorrelations. The Durbin-Watson test is generally invalid for AR models.

Question 54

What is mean reversion in a time series?

Answer 54

A tendency for the series to fall when its level is above its long-run mean and rise when its level is below its mean. Covariance stationary series are mean reverting.

Question 55

What is the root mean squared error (RMSE) used for in time-series forecasting?

Answer 55

To compare the out-of-sample forecasting accuracy of different models. A smaller RMSE indicates greater accuracy.

Question 56

What is a random walk process?

Answer 56

A time series where the value in one period is the value in the previous period plus an unpredictable random error. It is not covariance stationary.

Question 57

What is a unit root in a time series?

Answer 57

A characteristic of a non-stationary time series (like a random walk) where the coefficient on the first lag in an AR(1) model is equal to 1.

Question 58

How can a time series with a unit root often be transformed to achieve stationarity?

Answer 58

By first-differencing the time series (i.e. taking the difference between consecutive observations).

Question 59

How can seasonality in a time-series model often be addressed?

Answer 59

By including seasonal lags in an autoregressive model (e.g. adding a 4th lag for quarterly data or a 12th lag for monthly data).

Question 60

What does Autoregressive Conditional Heteroskedasticity (ARCH) model?

Answer 60

It models the variance of the error term in a time series as dependent on the variance of previous errors meaning volatility can cluster.

Question 61

If two time series both have unit roots when can linear regression between them be valid?

Answer 61

If the time series are cointegrated meaning there is a long-term stable economic relationship between them and their linear combination is stationary.