Consider the standard simple regression model y = Bo + Bix + u under standard assumptions The usual OLS estimators Bo and ß, are unbiased for their respective population parameters. Let B, be the estimator of ß obtained by assuming the intercept is zero. (i) Find E(B,) in terms of the x;, Bo, and B1. Verify that ß, is unbiased for B, when the population intercept (Bo) is zero. Are there other cases where B, is unbiased? (ii) Find the variance of B. (Hint: The variance does not depend on Bo-) (iii) Show that Var(Š)) < Var(ß,). [Hint: For any sample of data, 2 > E,(x; - x)?, with strict inequality unless i = 0.] (iv) Comment on the tradeoff between bias and variance when choosing between B, and ßỊ.
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.
![Consider the standard simple regression model y = Bo + Bx + u under standard assumptions
The usual OLS estimators B, and B, are unbiased for their respective population parameters. Let
3, be the estimator of B obtained by assuming the intercept is zero.
(i)
Find E(B) in terms of the x;, Bo, and B1. Verify that B, is unbiased for B, when the population
intercept (Bo) is zero. Are there other cases where B, is unbiased?
(ii) Find the variance of B1. (Hint: The variance does not depend on Bo-)
(iii) Show that Var(ß)) < Var(B,). [Hint: For any sample of data, E x > E(x; - x)?, with
strict inequality unless = 0.]
(iv) Comment on the tradeoff between bias and variance when choosing between B, and BỊ.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83c107df-6c4a-453f-b7f6-0fe8a1db354a%2F5efe0731-cc42-467c-b04f-b97d7b1cb636%2Fo5m84t_processed.png&w=3840&q=75)
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