Suppose that you are interested in estimating the causal relationship between y and x1. For this purpose, you can collect data on two control variables, x2 and x3. For concreteness, you might think of y as final exam score, x1 as class attendance, x2 as GPA up through the previous semester, and x3 as SAT or ACT score. Let β˜1 be the simple regression estimate from y on x1 and let βˆ1 be the multiple regression estimate from y on x1, x2, x3. 1) If x1 is highly correlated with x2 and x3 in the sample, and x2 and x3 have large partial effects on y, would you expect β˜1 and βˆ1 to be similar or very different? Explain. 2) If x1 is almost uncorrelated with x2 and x3, but x2 and x3 are highly correlated, will β˜1 and βˆ1 tend to be similar or very different? Explain. 3) If x1 is highly correlated with x2 and x3, and x2 and x3 have small partial effects on y, would you expect se(β˜1) or se(βˆ1) to be smaller? Explain. 4) If x1 is almost uncorrelated with x2 and x3, and x2 and x3 have large partial effects on y, and x2 and x3 are highly correlated, would you expect se(β˜1) or se(βˆ1) to be smaller? Explain.
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
Suppose that you are interested in estimating the causal relationship between y and x1. For this purpose, you can collect data on two control variables, x2 and x3. For concreteness, you might think of y as final exam score, x1 as class attendance, x2 as GPA up through the previous semester, and x3 as SAT or ACT score. Let β˜1 be the simple regression estimate from y on x1 and let βˆ1 be the multiple regression estimate from y on x1, x2, x3.
1) If x1 is highly
2) If x1 is almost uncorrelated with x2 and x3, but x2 and x3 are highly correlated, will β˜1 and βˆ1 tend to be similar or very different? Explain.
3) If x1 is highly correlated with x2 and x3, and x2 and x3 have small partial effects on y, would you expect se(β˜1) or se(βˆ1) to be smaller? Explain.
4) If x1 is almost uncorrelated with x2 and x3, and x2 and x3 have large partial effects on y, and x2 and x3 are highly correlated, would you expect se(β˜1) or se(βˆ1) to be smaller? Explain.
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