Regression Analysis Concepts

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Questions and Answers

What is the problem with including too many variables in a regression model?

  • It can lead to overfitting and an inaccurate prediction of the dependent variable.
  • It can lead to misspecification, where the model's form does not accurately represent the relationship between the variables.
  • It can lead to underspecification, where important variables are omitted, causing omitted variable bias.
  • It can lead to overspecification, where irrelevant variables are included, but it does not bias the coefficients. (correct)

What is the potential problem with including too few variables in a regression model?

  • It can lead to misspecification, where the model's form does not accurately represent the relationship between the variables.
  • It can lead to overspecification, where irrelevant variables are included, but it does not bias the coefficients.
  • It can lead to overfitting, where the model is too closely fit to the data and may not generalize well to new data.
  • It can lead to underspecification, where important variables are omitted, causing omitted variable bias. (correct)

What is omitted variable bias?

  • The bias that occurs when the dependent variable is not measured accurately.
  • The bias that occurs when the model is overspecified, including irrelevant variables.
  • The bias that occurs when important variables are omitted from the regression model. (correct)
  • The bias that occurs when the independent variables are not independent of each other.

When does omitted variable bias occur?

<p>When the omitted variable correlates with both the dependent variable and at least one independent variable in the model. (D)</p> Signup and view all the answers

Which of the following is NOT a problem associated with omitted variable bias?

<p>Reducing the R-squared value of the model. (C)</p> Signup and view all the answers

What does the adjusted R-squared statistic measure?

<p>The proportion of variance in the dependent variable that is explained by the independent variables, adjusted for the number of variables in the model. (A)</p> Signup and view all the answers

Which of the following is a naive approach to variable selection in regression analysis?

<p>Stepwise regression. (B)</p> Signup and view all the answers

What is a 'kitchen sink' regression?

<p>A regression model that includes all variables, regardless of their significance. (D)</p> Signup and view all the answers

What is the consequence of multicollinearity in regression analysis?

<p>Biased standard errors (B)</p> Signup and view all the answers

Which of the following issues can lead to biased coefficients in regression analysis?

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

What is the first step in the recipe for conducting a regression analysis?

<p>Start with assumptions about relationships between variables (C)</p> Signup and view all the answers

How does heteroscedasticity specifically affect regression analysis?

<p>It leads to biased standard errors. (D)</p> Signup and view all the answers

In regression analysis, what is an essential consideration regarding the data sample used?

<p>The sample must be a random sample of the population. (D)</p> Signup and view all the answers

What is the Gauss-Markov Theorem primarily concerned with?

<p>The use of random samples in regression analysis (C)</p> Signup and view all the answers

Which rule of thumb is commonly used regarding observations in regression analysis?

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What aspect of sample selection can lead to non-problems if based on the independent variable?

<p>Exogenous sample selection (B)</p> Signup and view all the answers

What is a consequence of perfect collinearity in a regression model?

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

Which of the following is a method to test for multicollinearity in regression analysis?

<p>Variance Inflation Factor (VIF) (A)</p> Signup and view all the answers

Which statement about independent variables in a regression model is correct?

<p>They should ideally be independent of each other (D)</p> Signup and view all the answers

What is meant by the 'dummy trap' in regression analysis?

<p>Including a dummy variable for each category (C)</p> Signup and view all the answers

If a researcher is only surveying future students among high school graduates, what is the result of this sampling method?

<p>Highly selective sample (B)</p> Signup and view all the answers

In regression analysis, when is multicollinearity considered a problem?

<p>When one independent variable perfectly predicts another (A)</p> Signup and view all the answers

What should be done if a variable causes perfect collinearity in a regression model?

<p>Exclude the perfectly collinear variable (A)</p> Signup and view all the answers

What is a primary advantage of using Principal Component Analysis (PCA) in regression?

<p>It reduces the complexity of the regression model. (A), It eliminates multicollinearity among variables. (B)</p> Signup and view all the answers

Which of the following is a disadvantage of Principal Component Analysis?

<p>It does not provide intuitive interpretation of results. (C)</p> Signup and view all the answers

In what context is Principal Component Analysis commonly applied?

<p>Generating socio-economic status indices. (C)</p> Signup and view all the answers

What does PCA primarily address when multiple variables are included in a regression model?

<p>Reducing variable correlations. (A)</p> Signup and view all the answers

Why might researchers prefer to use PCA before regression analysis?

<p>To enhance the power of statistical tests. (B)</p> Signup and view all the answers

Which of the following is NOT a reason to use Principal Component Analysis?

<p>Statistical significance of regression variables. (D)</p> Signup and view all the answers

What is often a result of applying PCA in data analysis?

<p>Reduction in data dimensionality. (B)</p> Signup and view all the answers

What mathematical characteristic is significant when performing PCA?

<p>It is driven by mathematical and statistical principles. (D)</p> Signup and view all the answers

What is autocorrelation?

<p>A condition where residuals are dependent on each other (C)</p> Signup and view all the answers

Which method can be used for detecting autocorrelation?

<p>Residuals vs. observation order visual inspection (C)</p> Signup and view all the answers

In which situation may autocorrelation occur?

<p>Time series data with repeated measurements (A)</p> Signup and view all the answers

What is a consequence of failing to meet the OLS assumptions?

<p>You may utilize transformations or different estimators (C)</p> Signup and view all the answers

Which of the following is NOT an assumption outlined in the Gauss-Markov Theorem?

<p>Normal distribution of independent variables (C)</p> Signup and view all the answers

What is the purpose of the normality assumption in regression analysis?

<p>To allow for significance testing via p values (B)</p> Signup and view all the answers

How can one approximate normality in residuals if the sample size is large?

<p>Collecting a sufficiently large sample (D)</p> Signup and view all the answers

What is indicated when OLS is described as BLUE?

<p>It consistently provides the best linear unbiased estimation (A)</p> Signup and view all the answers

What is the primary concern regarding the selection of the sample in regression analysis?

<p>The sample selection can influence the accuracy of the results. (A)</p> Signup and view all the answers

Why is having a larger sample size beneficial in regression analysis?

<p>It increases the reliability of statistical inferences. (C)</p> Signup and view all the answers

What does the formula for degrees of freedom in regression output represent?

<p>The total sample size minus one and one for the intercept. (B)</p> Signup and view all the answers

What is the suggested minimum number of observations per variable when constructing a regression model?

<p>10 observations to ensure stability. (B)</p> Signup and view all the answers

What is a significant drawback of selecting a sample based on the dependent variable?

<p>It may introduce selection bias that complicates inference. (C)</p> Signup and view all the answers

How does an increase in degrees of freedom affect regression predictions?

<p>It leads to lower critical values of t and improves prediction accuracy. (B)</p> Signup and view all the answers

What is the relationship between sample size and the inclusion of independent variables in regression analysis?

<p>More independent variables necessitate a larger sample size. (D)</p> Signup and view all the answers

What is a common rule of thumb regarding the total number of observations in regression analysis?

<p>The more variables included, the more observations required. (B)</p> Signup and view all the answers

Flashcards

Principal Component Regression

A regression technique using PCA to handle correlated variables.

Principal Component Analysis (PCA)

A mathematical method to transform correlated variables into uncorrelated variables.

Multicollinearity

The presence of high correlations among independent variables in regression.

Dimensionality Reduction

The process of decreasing the number of variables under consideration.

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Advantages of PCA

Reduces complexity and handles multicollinearity in regression analysis.

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Disadvantages of PCA

Lacks intuitive interpretation; driven by mathematics, not theory.

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Applications of PCA

Commonly used to create new variables for regression analysis, like wealth indicators.

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PCA-generated variables

Variables created through PCA that can be included in regressions for analysis.

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Sampling Selection

Choosing participants based on specific criteria such as their education level (e.g., Abitur).

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Independent Variable

A variable that is manipulated to observe its effect on a dependent variable.

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Exogenous Sample Selection

Choosing a sample based on an independent variable, affecting the outcome indirectly.

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Perfect Collinearity

When one independent variable perfectly predicts another, creating redundancy.

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Variance Inflation Factor (VIF)

A measure used to detect multicollinearity in regression models.

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Stratified Sampling

A method of sampling that divides the population into subgroups before selection.

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Dummy Variable Trap

Occurs when dummy variables for all categories are included in a model, causing perfect multicollinearity.

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Random Sample

A sample where each member of the population has an equal chance of being selected.

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Sample Size

The number of observations in a statistical sample.

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Degrees of Freedom

The number of independent values in a statistical calculation, calculated as n - 1 - k.

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Gauss-Markov Theorem

States that the best linear unbiased estimator is obtained under certain conditions, including random sampling.

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Rule of Thumb for Observations

A guideline of at least 10 observations per variable in regression analysis.

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Critical Value of t

The value that the test statistic must exceed to reject the null hypothesis, which decreases with more degrees of freedom.

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Non-linearity of parameters

Leads to biased coefficients and biased standard errors in regression models.

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Biased sample

A non-random sample that leads to biased selection and non-representative results.

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Endogeneity

A situation where an explanatory variable is correlated with the error term, resulting in biased coefficients.

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Heteroscedasticity

Unequal variances in the error terms across observations leading to biased standard errors.

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Autocorrelation

A condition where residuals are not independent, indicating a potential violation of regression assumptions.

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Panel Analysis

A statistical method treating data in repeated measurements to analyze effects over time across multiple subjects.

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Durbin-Watson Statistic

A test statistic used to detect the presence of autocorrelation in the residuals from a regression analysis.

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BLUE

Best Linear Unbiased Estimator - the best estimation method according to OLS under the Gauss-Markov assumptions.

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Normality Assumption

An additional assumption that the unobserved error in a regression is normally distributed, important for significance testing.

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Significance Testing

A statistical method to determine if results are meaningful, typically using p-values derived from t and F statistics.

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Countermeasure for Normality

Collecting a sufficiently large sample (>200) to ensure that the distribution of residuals approximates normality.

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Overspecification

The inclusion of irrelevant variables in a regression model, leading to unnecessary complexity without biasing coefficients.

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Underspecification

Leaving out important variables from a regression model, potentially leading to omitted variable bias.

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Omitted Variable Bias

The bias resulting from excluding relevant variables from a model, altering estimates of the effect.

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Adjusted R-squared

A statistical measure that indicates the proportion of variance explained by the independent variables, adjusted for the number of predictors in the model.

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Stepwise Regression

A method of variable selection for regression models that involves adding or removing predictors based on statistical significance.

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Forward Selection

A stepwise regression method that starts with no variables and adds significant ones sequentially.

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Backward Selection

A stepwise regression technique that begins with all available variables and removes the least significant ones iteratively.

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Theory-driven Variable Selection

The ideal method of choosing variables based on theoretical understanding rather than arbitrary methods.

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Study Notes

Quantitative Methods in Empirical Economic Geography

  • This is a lecture on linear regression models, part III
  • Lecturer: Christian Hundt
  • Slides presented by Christian Hundt and Kerstin Nolte
  • Location: Institute of Economic and Cultural Geography, Leibniz University Hannover

OLS Assumptions: The Gauss-Markov Theorem

  • OLS stands for Ordinary Least Squares
  • OLS yields consistent estimators for parameters β₀, β₁, ..., βₙ
  • This is only true under certain assumptions.
  • A consistent estimator converges to the true value of the parameter as sample size increases.

Assumptions in a Linear Regression Model

  • Linear in parameters: A linear relationship exists between the dependent and independent variables.
  • Random Sample: The sample must be representative of the population from which it was drawn.
  • No perfect collinearity: No strong correlations between independent variables; one variable cannot perfectly predict another.
  • Exogeneity of the predictors: Predictor variables are not correlated with the error term.
  • Homoscedasticity: Error terms have constant variance.
  • No autocorrelation: Error terms are not correlated with each other.

Checking for Linearity

  • Use a residuals vs. fits plot to check for linearity.
  • The residuals should be randomly scattered around a horizontal line at y = 0.

Logarithmic Transformation

  • If a model is not linear in parameters, transform the variables.
  • Logarithmic transformation is a common method to transform non-linear models into linear ones.
  • Example: Cobb-Douglas production function can be transformed into linear form by using logarithms.

Random Sample of Size n

  • If the sample is not random, it may be difficult to make inferences about the population.
  • Collect data yourself.
  • Use probability sampling methods.
  • Probability sampling implies every member of the population has a nonzero chance of being picked in the sample.

No Perfect Collinearity

  • Avoid perfect correlation between independent variables.
  • Use variance inflation factor (VIF) to measure the strength of the correlation between variables in the model.
  • A high VIF value indicates that an independent variable is correlated with other independent variables.
  • High VIF values suggest that your model may be problematic; consider removing variables or using PCA.
  • Variables that are perfectly correlated should be removed from the model.

Exogeneity of Predictors

  • If a predictor variable(s) is correlated with the error term, it means that the predictor variable(s) is(are) endogenous.
  • This includes, but is not limited to: misspecification, omitted variable, simultaneity.

Misspecification

  • Could mean that the underlying data generating process is not correctly characterized or that some variables are not included in the regression.
  • A missing variable in the regression will introduce bias to the other parameters, because it is correlated with at least one of the independent variables that were included in the model.

Omitted Variable

  • If an important variable is missing in the model—this variable is correlated with one or more of the independent variables included in the model—the predictor variable coefficient estimates will also be biased.

Simultaneity

  • An endogenous variable can be caused by other variables or it is a cause for other variables in a model.
  • To fix this, instrument variables that are strongly correlated with the predictor variables but are not correlated with the dependent variable(s) can be used.

Homoscedasticity

  • Examine the plot of the model's residuals vs. predicted values.
  • If the residuals aren't randomly dispersed around 0, the variance isn't homoscedastic.
  • Transforming the data—such as using logarithms—may fix heteroscedasticity. Calculating robust standard errors can also help.

No Autocorrelation

  • Examine correlation plots among observations to determine if there is a trend.
  • A violation of this assumption indicates that autocorrelation is present.
  • This is true in the case of panel analysis and in the case of observations that are clustered (such as repeated measures from the same person or group).

Violations of the Assumptions

  • A violation causes biased coefficients and biased standard errors.
  • Address each issue individually.

Model Diagnostics and Strategy

  • Examine the model assumptions after creating the model using diagnostics.
  • Appropriately address violations. If assumptions are violated, re-evaluate the model's parameters and variables.
  • If necessary, transform the data or use an alternative estimator (beyond OLS).

Creating a Regression Equation

  • Start with a theory-driven variable selection: focus on variables with a clear theoretical relationship with the dependent variable.
  • Use alternative methods (such as stepwise regression) only if theory isn't clear or if you have a lot of variables.
  • A trade-off exists between including many variables and keeping the model simple and reliable.

Principal Component Analysis (PCA)

  • Use PCA to combine multiple highly correlated variables to streamline data analysis.
  • Creates a small set of uncorrelated variables from highly correlated ones.

Further Reading

  • Wooldridge (2013), Introductory Econometrics. A Modern Approach (5th ed).
  • Online resources for further details

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