All Questions
14 questions
0
votes
1
answer
35
views
Error in derivation of variance of $\beta_1$ in SLR [duplicate]
I'm trying to derive the variance of the slope parameter for a simple linear regression in the following way, however I'm running into an issue I don't know how to resolve. Define $y_i=\beta_0+\beta_1\...
- 181
0
votes
0
answers
30
views
How to find the variance of an independent variable across the big number of linear regression equations?
Question
I have a big number of linear regression equations with known dependent variables and coefficients, in a form of:
T = Aa + Bb + Cc + Dd
where ...
- 121
3
votes
1
answer
101
views
Ideal Settings for Longitudinal Models?
The way I see it, logically speaking - Longitudinal Data (e.g. medical patients being measured repeatedly over a period of time) can have one of two forms:
Case 1: All patients are measured exactly &...
1
vote
1
answer
45
views
Where plus 1 came from in variance estimation [duplicate]
While
$$
\mathrm{E}(\tilde{\mathrm{y}})=\alpha+\beta \tilde{\mathrm{x}}
$$
Subject is Regression Analysis and this formula is from the "Features of Estimation ".
and y is a neutral variable.
...
4
votes
0
answers
1k
views
What is the probability distribution and variance of the OLS estimate $s^2$ of the error variance $\sigma^2$ in linear regression?
Consider the standard linear regression model
$$
y = X \beta + \varepsilon,
$$
where the error $\varepsilon$ has fixed variance $\sigma^2$. We can make an unbiased estimate of the error variance in a ...
- 805
0
votes
2
answers
134
views
How are $n$ and $Var(\varepsilon)$ affecting to Variance of Estimation of Slope Parameter $\beta_1$ in Simple Linear Regression
Once I have derived the variance of $\hat{\beta_1}$ as:
$\text{Var}(\hat{\beta_1})= \frac{\sigma^2}{\sum(x_i-\overline{x})^2}$
I would like to know how are affecting to this formula:
the size of ...
- 117
5
votes
1
answer
862
views
Interpretation of conditional variance of estimator of intercept in linear regression
$Y_i=a+bX_i+e_i$. $Y_i$ and $X_i$ are scalar r.v. We have,
$$
V(\hat b|X)=\frac{\sigma^2}{n\left(\bar{X^2}-\left[\bar{X}\right]^2\right)}
$$
and,
$$
V(\hat a|X)=\frac{\sigma^2 \bar{X^2}}{n\left(\bar{X^...
- 359
1
vote
0
answers
11
views
Variance of parameter estimators in terms of $\ n$ observations in a linear time series model
Let $\ X_t = b_1 + b_2.t + e_t$ be a linear model, where $\mathbb{E}(e_t)$ = 0, $\ var(e_t)$ = $\sigma_e^2$, and $\ cov(e_i,e_j)$ = 0 for all $\ i \neq j$.
I have derived the equations for the ...
- 706
3
votes
1
answer
921
views
MLE of regression coefficients with non-constant variance
In the simplest case of the statistical view of linear regression, we have that
$$
y = f(\mathbf{x};\mathbf{w}) + \nu, \text{ where }\nu \sim \mathcal{N}(0,\sigma^2)
$$
where $\mathbf{x}$ is an ...
- 257
2
votes
1
answer
81
views
Variance of an estimator?
I've estimated a parameter $\theta$ of a linear model as
$$\hat\theta = \frac{2 \sum x_i^2 Y_i}{\sum x_i^4}$$
Where $Y_i$ is the response variable.
I was wondering how does one find the variance ...
- 135
4
votes
1
answer
1k
views
Sampling variance of regression intercept when there is no regressor
Suppose we have a model $y=\beta_0+u$, where $E(u)=0$ and $Var(u)=\sigma^2$. I get the unbiased estimator $\hat\beta_0$ is just $\bar y$. But how can I get the variance of $\hat\beta_0$? Is it correct ...
- 93
3
votes
2
answers
10k
views
Why does the parameter variance change when control variables are added to a regression model?
If I add a control variable to my regression, this changes the variances of the parameter estimates. Why is this the case? Is this because SSE(=explained sum of squares) increases and therefore the ...
- 81
2
votes
1
answer
2k
views
When would you want to reduce variance?
In a sampling-estimation context, low variance of the estimate is a goal. Several things I've read suggest (though I can't quite connect the dots) that lowering variance in the data will improve the ...
- 517
3
votes
1
answer
880
views
Help computing asymptotic variance of a weird first difference estimator in a fixed effects model
I'm working on an econometrics problem set, and I'm having some major problems computing asymptotic variance for this estimator. I'm considering a fixed-effects model
$$
Y_{it} = \beta_1 X_{it} + \...
- 319