$\hat{Y}=X\hat{\beta}$ and

$E[e]=E(Y-\hat{Y})=E[X{\beta}_{0}-X\hat{\beta}]=X({\beta}_{0}-E(\hat{\beta}))=0$

since $\hat{\beta}$ is an unbiased estimator of ${\beta}_{0}$.

Am I making a mistake?

Pierre Holmes
2022-07-21
Answered

Assume $Y=X{\beta}_{0}+\u03f5$ where $\u03f5$ is zero mean and $X$ is fixed. I know that under certain conditions on the design matrix $X$ in OLS, the sample mean of the residuals $\overline{e}$ is $0.$ Can we say the same for the true population mean of residuals as well?

$\hat{Y}=X\hat{\beta}$ and

$E[e]=E(Y-\hat{Y})=E[X{\beta}_{0}-X\hat{\beta}]=X({\beta}_{0}-E(\hat{\beta}))=0$

since $\hat{\beta}$ is an unbiased estimator of ${\beta}_{0}$.

Am I making a mistake?

$\hat{Y}=X\hat{\beta}$ and

$E[e]=E(Y-\hat{Y})=E[X{\beta}_{0}-X\hat{\beta}]=X({\beta}_{0}-E(\hat{\beta}))=0$

since $\hat{\beta}$ is an unbiased estimator of ${\beta}_{0}$.

Am I making a mistake?

You can still ask an expert for help

slapadabassyc

Answered 2022-07-22
Author has **21** answers

If we are treating $X$ as fixed, which I take to mean non-random, then by assumption

$E[\u03f5]=0$

You are sort of going in a circle because you say $E[Y]=X{\beta}_{0}$, but this is because of our prior assumption. It is a result not an assumption. We have

$\begin{array}{rl}Y& =X{\beta}_{0}+\u03f5\\ E[Y]& =E[X{\beta}_{0}+\u03f5]\\ E[Y]& =E[X{\beta}_{0}]+E[\u03f5]\\ E[Y]& =X{\beta}_{0}+0\\ E[Y]& =X{\beta}_{0}\end{array}$

where the fourth line is a result of our assumption and $X$ being non-random (as well as ${\beta}_{0}$ being a constant).

$E[\u03f5]=0$

You are sort of going in a circle because you say $E[Y]=X{\beta}_{0}$, but this is because of our prior assumption. It is a result not an assumption. We have

$\begin{array}{rl}Y& =X{\beta}_{0}+\u03f5\\ E[Y]& =E[X{\beta}_{0}+\u03f5]\\ E[Y]& =E[X{\beta}_{0}]+E[\u03f5]\\ E[Y]& =X{\beta}_{0}+0\\ E[Y]& =X{\beta}_{0}\end{array}$

where the fourth line is a result of our assumption and $X$ being non-random (as well as ${\beta}_{0}$ being a constant).

asked 2020-12-22

Regarding a scatterplot matrix:

a. Identify two of its uses.

b. What property should the scatterplots of the response versus each predictor variable have in order to proceed to obtain the regression plane for the data?

a. Identify two of its uses.

b. What property should the scatterplots of the response versus each predictor variable have in order to proceed to obtain the regression plane for the data?

asked 2021-05-04

Make a scatterplot for each set of data.

Hits: 7 8 4 11 8 2 5 9 1 4

Runs: 3 2 2 7 4 2 1 3 0 1

Hits: 7 8 4 11 8 2 5 9 1 4

Runs: 3 2 2 7 4 2 1 3 0 1

asked 2021-12-03

From the data, answer the following With Solving solution:

1. What is the probability that it is a male or in managerial position?

2. What is the probability that it is neither male nor in managerial position

3. What is the probability that it is female or clerical portion?

$$\begin{array}{|cccc|}\hline Position& Male& Female& Total\\ Managerial& 8& 4& 12\\ Engineer& 18& 7& 25\\ Accountant& 3& 5& 8\\ Clerical& 15& 28& 40\\ Total& 44& 44& 88\\ \hline\end{array}$$

1. What is the probability that it is a male or in managerial position?

2. What is the probability that it is neither male nor in managerial position

3. What is the probability that it is female or clerical portion?

asked 2022-06-25

How do I calculate the quartiles for this problem?

I have the following list of numbers, and I'm trying to calculate the quartiles:

2, 4, 4, 5, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9

I'm running into a bit of confusion because the median turns out to be 7. Now I don't know how to demarcate the lower group from the upper group since there are a whole bunch of 7's. Consequently, I don't know how to calculate ${Q}_{1}$ and ${Q}_{3}$

I have the following list of numbers, and I'm trying to calculate the quartiles:

2, 4, 4, 5, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9

I'm running into a bit of confusion because the median turns out to be 7. Now I don't know how to demarcate the lower group from the upper group since there are a whole bunch of 7's. Consequently, I don't know how to calculate ${Q}_{1}$ and ${Q}_{3}$

asked 2021-02-19

Make a scatterplot for each set of data. Tell whether the data show a linear association or a nonlinear association.

asked 2021-11-30

A learning management system allows users to take up to 4 different practice exams before taking the actual exam. Data is collected from 40 users of the product regarding the number of practice exams each used.

$$\begin{array}{|cccccc|}\hline \text{Number of Practice Exams Taken}& 0& 1& 2& 3& 4\\ Frequency& 5& 10& 13& 9& 3\\ \hline\end{array}$$

Based on the data, find the estimated probability (relative frequency) that a user will take more than one practice exam.

Given your answer as a decimal.

The probability is ?

Based on the data, find the estimated probability (relative frequency) that a user will take more than one practice exam.

Given your answer as a decimal.

The probability is ?

asked 2021-12-06

Suppose that, based on historical data, we believe that the annual percentage salary increases for the chief executive officers of all midsize corporations are normally distributed with a mean of 12.2% and a standard deviation of 3.6%. A random sample of nine observations is obtained from this population, and the sample mean is computed. What is the probability that the sample mean will be greater than 14.4%?