10 Quantitative Methods - OLS Regression Flashcards
What is the formula for OLS regression?
y=a+bx
The equation y = a + bx is a linear regression model that describes the relationship between two variables: y and x. In this equation, b represents the slope of the regression line, which tells us how much y changes for each unit change in x. The intercept a is the value of y when x equals zero.
Questions on: Mataic, Dane R. (2018): “Countries Mimicking Neighbors: The Spatial Diffusion of Governmental Restrictions on Religion”. Journal for the Scientific Study of Religion 57(2): 221-237.
(1) Read the abstract and introduction. What is the research question?
What are the impacts a country takes on matters of religion (e.g. restrictions) on its direct neighbours?
(3) Now go to the section “Discussion and conclusion” (p.234-235). What are the main findings of the study?
In general, clear connections were made between the religious economies model and governance dimensions and the level of government restrictions on religion. Favoritism of particular religions was associated with higher levels of restrictions on minority religions. Countries with generally lower levels of democracy were also associated with higher levels of restrictions. The results provide robust support for Hypothesis 1; countries adopt policies that impose restrictions on minority religions following the adoption of policies in neighboring countries. Governmental restrictions on minority religions are significantly clustered, where countries with high levels of restrictions neighbor countries with similar levels.
(4) Go back to “Data and Methods” (p. 227-232) and read this section 2-3 times if necessary.
- What is the dependent variable? * What are the main independent variables?
- What is the dependent variable? “Restrictions of minority religions” consisting of 27 items of restrictions towards minority religions, with potential scores ranging between 0 and 65. Higher values on this modified index represent countries with more restrictions on minority religions
- What are the main independent variables? 1. the religious economies measure (government favoritism through the level of funding religions receive from the national government), 2. governance dimensions, 3. controls that were significant in prior research predicting restrictions on religion.
Questions on: Wagschal, Uwe (1999): Statistik für Politikwissenschaftler. Oldenbourg, München u.a. (pages to read: chapter 12).
What is the purpose of regression analysis?
Truly meaningful correlations require causality between variables, meaning a cause-effect relationship. Regression analysis explores the functional relationship between variables, with bivariate regression focusing on the influence of one or more independent variables X on a dependent variable Y. The bivariate relationship is expressed as Y = a + b1 * X1, with a denoting the constant and b1 the slope of the line.
(2.1) Make sure you understand the equation of the straight line.
Straight line: a in the equation of Y = a + b1 * X1
Before the actual regression analysis is presented, a simple mathematical equation for a straight line and its corresponding notation should be discussed. The expression Y = a + b1 * X1 is the mathematical function of a straight line. Here, Y denotes the dependent variable and X1 denotes the independent variable. A change in X causes a change in Y. Conventionally, in scatterplots, the dependent variable is plotted on the y-axis and the independent variable is plotted on the x-axis. The y-intercept on the y-axis is denoted by a. The y-intercept obtains the value a when the variable X is zero, or set to zero. The y-intercept is also referred to as the constant. The slope of the line is denoted by b1. It indicates by how many units Y changes when X changes by one unit. In mathematical notation, one writes: b1 = ΔY/ΔX
Depending on the sign (positive or negative), it indicates the functional direction of the relationship, that is, whether there is a positive or negative influence. The absolute value of the slope is also important because it allows us to investigate whether the influence of X on Y is actually significant and has a substantial impact.
What does the “ordinary least squares” refer to?
“Ordinary least squares” (OLS) is a statistical method used in linear regression analysis to estimate the unknown parameters in a linear regression model. The method calculates the best-fitting line by minimizing the sum of the squared differences between the observed values and the predicted values. The “ordinary” in “ordinary least squares” distinguishes this method from other types of regression analysis, such as weighted least squares or nonlinear least squares.
(2.3) What are the similarities and differences of the regression equation and the equation of the straight line?
The regression equation and the equation of the straight line are both mathematical formulas that represent the relationship between two variables. The primary similarity between the two is that they both represent a linear relationship between variables, meaning that the relationship between the variables can be described using a straight line.
However, there are also some differences between the two equations. The regression equation is typically used to model the relationship between a dependent variable and one or more independent variables. The equation of the straight line, on the other hand, is typically used to describe the relationship between two continuous variables, where one variable is the predictor or independent variable, and the other variable is the response or dependent variable.
The regression equation is expressed as Y = a + bX, where Y is the dependent variable, X is the independent variable, a is the intercept, and b is the slope of the line. In contrast, the equation of the straight line is expressed as y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the intercept.
Another difference is that the regression equation can involve multiple independent variables and can be used to determine the contribution of each variable to the variation in the dependent variable. The equation of the straight line, on the other hand, only involves two variables and does not account for the effects of other variables. The regression equation can include an error term leading to the equation y = a+bX + e.
What do “explained variation” and “unexplained variation” mean?
In the context of regression analysis, “explained variation” and “unexplained variation” refer to the total variability of the dependent variable (Y) that can be attributed to the independent variable(s) (X) and the variability that cannot be explained by the independent variable(s), respectively.
“Explained variation” is also known as “explained variance” or “regression sum of squares.” It is the variation in the dependent variable that is accounted for by the independent variable(s). In other words, it is the amount of variation in Y that can be explained by the regression equation.
“Unexplained variation” is also known as “unexplained variance” or “residual sum of squares.” It is the variation in the dependent variable that is not accounted for by the independent variable(s). In other words, it is the amount of variation in Y that is left over after accounting for the regression equation. This variation is typically attributed to other factors that are not included in the regression equation or to random error.
What is the R-squared and how is it calculated (you do not need to know the math, but rather understand the principle)?
R-squared, also known as the coefficient of determination, is a statistical measure that represents the proportion of variation in the dependent variable that can be explained by the independent variable(s) included in the regression model. It is a value between 0 and 1, where 0 indicates that none of the variation is explained by the model and 1 indicates that all of the variation is explained.
R-squared is calculated by dividing the explained variation (SSR) by the total variation (SST), and subtracting the result from 1.
R-squared = 1 - (SSR/SST)
where SSR is the sum of squared residuals (the difference between the predicted and actual values of the dependent variable), and SST is the total sum of squares (the difference between the actual values of the dependent variable and the mean of the dependent variable).
R-squared is often used as a measure of the goodness of fit of a regression model, where a higher R-squared indicates a better fit of the model to the data. However, it is important to note that a high R-squared does not necessarily mean that the model is the best possible model or that the independent variables are causally related to the dependent variable.
What are the assumptions of OLS-regression? Please explain each of the assumptions in detail.
- No specification error.
a. The relationship between X and Y is linear.
b. No relevant independent variable has been excluded.
c. No irrelevant independent variable has been included. - No measurement errors. All variables have been accurately measured.
- The following assumptions concern the error term e (i.e. the residuals):
a. The expected value for each residual is zero: E(i) = 0.
b. The variance of the residuals for all values of xi is constant (homoscedasticity) E(²) = ².
c. No autocorrelation. The residuals are uncorrelated.
d. The independent variables are uncorrelated with the residuals.
e. The error term (i.e. the residuals) is normally distributed. - The independent variables are not correlated with each other (= no multicollinearity).
- The regression coefficients must be the same for repeated samples (reliability).
What happens if these assumptions of OLS-regression are not met?
Jede dieser Annahmen kann verletzt werden und gravierende Konsequenzen nach sich ziehen (siehe unten). Gelten die Annahmen 1 bis 3d, dann nennt man den Schätzer „BLUE” = Best Linear Unbiased Estimator. Dies ist derjenige Schätzer, der am besten den postulierten Zusammenhang in der Grundgesamtheit vorhersagt.
Oft wird man nichts gegen Verletzungen der Annahmen machen können. Im Forschungsbericht sind sie jedoch auf jeden Fall zu erwähnen, weshalb weiter unten Verfahren zu ihrer Identifikation vorgestellt werden. Nachdem in den vorangegangenen Abschnitten die Ermittlung der Regressionsfunktion dargestellt wurde, sollen nun die Ergebnisse dahingehend geprüft werden, ob sie auch statistisch signifikant sind. Die methodische Vorgehensweise besteht darin, mit Hilfe statistischer Testverfahren die Qualität der Regressionsgleichung zu prüfen. Für solche Testverfahren sind bei der Regression - strenggenommen - Stichproben notwendig, da nur von einer Stichprobe auf eine Grundgesamtheit zurückgeschlossen werden kann.
What is multiple regression, and how is it different from bivariate regression?
Multiple regression is a statistical technique used to examine the relationship between a dependent variable and multiple independent variables. In other words, it is a method of predicting a dependent variable based on several independent variables.
Bivariate regression, on the other hand, is a statistical technique used to examine the relationship between two variables, where one variable is considered the independent variable and the other is the dependent variable.
The key difference between multiple regression and bivariate regression is the number of independent variables involved in the analysis. Bivariate regression involves only one independent variable, whereas multiple regression involves two or more independent variables. Multiple regression allows for the examination of the unique effects of each independent variable on the dependent variable, while controlling for the effects of other independent variables.
In multiple regression, the relationship between the dependent variable and each independent variable is tested, while holding the other independent variables constant. The result is a set of regression coefficients that represent the effect of each independent variable on the dependent variable, holding all other variables constant.
What is multicollinearity and why is it a problem?
Multicollinearity is a statistical phenomenon where two or more independent variables in a multiple regression model are highly correlated with each other. In other words, there is a linear relationship between two or more independent variables. This high degree of correlation can lead to problems in the multiple regression model because it becomes difficult to distinguish the individual effects of each independent variable on the dependent variable.
Multicollinearity can cause the following problems:
It makes the estimation of regression coefficients unstable, which means small changes in the data can lead to large changes in the regression coefficients.
It decreases the precision of the estimates of the regression coefficients, which means that the coefficients are less reliable.
It can make it difficult to interpret the results of the regression analysis, as it becomes difficult to determine which independent variable is actually having an effect on the dependent variable.
Therefore, multicollinearity is generally considered to be a problem in multiple regression analysis, and researchers often try to minimize its effects by removing or combining highly correlated variables or by using other techniques such as principal component analysis.
What is heteroskedasticity and why is it a problem?
Heteroskedasticity is a problem that occurs in regression analysis when the variance of the errors (residuals) is not constant across all levels of the independent variables. In other words, the errors have different variances for different values of the independent variables. This violates one of the key assumptions of regression analysis, which is homoscedasticity or the assumption that the variance of the errors is constant.
Heteroskedasticity can be problematic because it can lead to biased and inefficient estimates of the regression coefficients. Specifically, when heteroskedasticity is present, the estimated standard errors of the coefficients can be biased, leading to incorrect statistical inference. Heteroskedasticity can also make it difficult to identify the true functional form of the relationship between the dependent variable and the independent variables.
