Statistics, regression model, unbiased estimator help

In summary: E( y1 | x1 ) = beta0 + beta1*x1. Therefore, E(( y2-y1 )/( x2-x1 ) ) = E( ( beta0 + beta1*x2 ) - ( beta0 + beta1*x1 ) )/( x2-x1 ) = E( beta1*( x2-x1 ) )/( x2-x1 ) = beta1. Therefore, the E.Z. estimator is unbiased. For (2), we need to find the probability distribution of the E.Z. estimator, which can be shown to follow a normal distribution with mean beta1 and variance sigma^2/( x2-x1 )^2, where sigma^2 is the variance of the error
  • #1
lape99
3
0

Homework Statement



Professor E.Z.Stuff has decided that the least squares estimator is too much trouble. Noting that two points determine a line, Dr. Stuff chooses two points from a sample of size N and draws a line between them, calling the slope of this line the E.Z. estimator of beta1 in the simple regression model. Algebraically, if the two points are (x1, y1) and (x2, y2 ) , the E.Z. estimation rule is beta1=(y2-y1)/(x2-x1)

I need to show:
(1) Show that beta1
EZ is an unbiased estimator.
To show this the lecturer said we need to calculate the mean: E(( beta0+beta1*xN+ epsilonN-(beta0 +beta1*x1+epsilon1))/(xN-x1), and this should be equal to beta1. How to do it?

(2) Find the probability distribution of beta1
EZ .
To show this we need to show that a*epsilonN + b*epsilon1~N(..,..) Normal distribution and calculate the mean value and the dispersion. But i don't know how.
Can someone help me?
 
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  • #2
Homework Equations beta1=(y2-y1)/(x2-x1)The Attempt at a Solution For (1), we need to calculate the expectation value of the E.Z. estimator, which isE(( y2-y1 )/( x2-x1 ) ). Since y2 and y1 are random variables, we can rewrite this as E(( E( y2 | x2 ) - E( y1 | x1 ) )/( x2-x1 ) ). By the law of iterated expectations, we can rewrite this as E( E( E( y2 | x2 ) - E( y1 | x1 ) | x2, x1 ) )/( x2-x1 ). Now, by the definition of the conditional expectation, we have that E( E( y2 | x2 ) | x2, x1 ) = E( y2 | x2 ), and similarly E( E( y1 | x1 ) | x2, x1 ) = E( y1 | x1 ). Therefore, we have E(( y2-y1 )/( x2-x1 ) ) = E( ( E( y2 | x2 ) - E( y1 | x1 ) )/( x2-x1 ) ) = E( E( y2 | x2 ) - E( y1 | x1 ) )/( x2-x1 ) = E( E( y2 | x2 ) | x2, x1 ) - E( E( y1 | x1 ) | x2, x1 ) )/( x2-x1 ) = E( y2 | x2 ) - E( y1 | x1 ) )/( x2-x1 ) Now, since we are assuming that the regression model is given by y = beta0 + beta1*x + epsilon, we have that E( y2 | x2 ) = beta0 + beta1*x2 and
 

Related to Statistics, regression model, unbiased estimator help

1. What is statistics?

Statistics is the branch of mathematics that deals with collecting, organizing, analyzing, and interpreting data. It involves using mathematical models and techniques to make sense of large amounts of data and draw conclusions from it.

2. What is a regression model?

A regression model is a statistical tool used to show the relationship between a dependent variable and one or more independent variables. It helps to predict the value of the dependent variable based on the values of the independent variables.

3. What does it mean for an estimator to be unbiased?

An unbiased estimator is one that, on average, produces results that are close to the true value of the population parameter it is estimating. This means that the estimator is not consistently overestimating or underestimating the true value.

4. How do I know if my regression model is accurate?

There are several ways to assess the accuracy of a regression model, such as calculating the coefficient of determination (R-squared), conducting hypothesis tests on the coefficients, and analyzing the residuals. It is important to use a combination of these methods to ensure the model is accurately representing the data.

5. Can I use a regression model to make predictions?

Yes, regression models are often used for prediction purposes. However, it is important to keep in mind that the accuracy of the predictions will depend on the quality of the data and the assumptions made in the model. It is always recommended to validate the model before using it for prediction purposes.

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