Linked Questions

0 votes
0 answers
4k views

Covariance between estimates of slope and intercept [duplicate]

I wonder why the covariance between estimates of slope ($\hat{\alpha}$) and intercept ($\hat{\beta}$) is $-\bar{X}\times Var(\hat{\beta})$. How can I derive this solution by not using matrix? I add a ...
Dayeon's user avatar
  • 29
1 vote
0 answers
1k views

Covariance Between $\hat{\beta_0}$ and $\hat{\beta_1}$ [duplicate]

Our model is $Y=\beta_0+\beta_1X+U$. We know that $$\hat{\beta_0} = \beta_0 + \sum\limits_{n=1}^N c_nu_n \quad \text{ and }\quad \hat{\beta_1} = \beta_1 + \sum\limits_{n=1}^N k_nu_n \,,$$ where $$k_n ...
Jon's user avatar
  • 11
16 votes
1 answer
16k views

Why does the standard error of the intercept increase the further $\bar x$ is from 0?

The standard error of the intercept term ($\hat{\beta}_0$) in $y=\beta_1x+\beta_0+\varepsilon$ is given by $$SE(\hat{\beta}_0)^2 = \sigma^2\left[\frac{1}{n}+\frac{\bar{x}^2}{\sum_{i=1}^n(x_i-\bar{x})^...
elexhobby's user avatar
  • 855
3 votes
1 answer
8k views

Interpreting the output from a gls model [closed]

I am quite new to R and coding so please forgive the lack of in depth information I may provide. I am also new at using linear models, particularly with large data sets. I have used the gls function ...
Lizaan de Necker's user avatar
3 votes
1 answer
2k views

Fixed effects of independent variables highly correlated with the intercept. Is it an issue?

In a linear mixed model run with lmer() I got this output for the correlation between fixed effects. My independent variable is almost perfectly correlated (...
Sergio Nolazco's user avatar
4 votes
1 answer
477 views

Zero Covariance vs Independence of Slope and Intercept Estimators in Linear Models with Least Squares

$\newcommand{\Cov}{\operatorname{Cov}}$Problem Statement: Under the assumptions of Exercise 11.16, find $\Cov\big(\hat\beta_0,\hat\beta_1\big).$ Use this answer to show that $\hat\beta_0$ and $\hat\...
Adrian Keister's user avatar
0 votes
1 answer
1k views

Independent variable collinear with intercept

I have high collinearity between one of the covariates (credit score variable with values ranging from 600 to 800) and the intercept term, when regressing a continuous dependent variable on some ...
Nagesh Rathi's user avatar
6 votes
2 answers
204 views

Identifying the correlation between a slope and a level

Throughout this post, I assume at least second moments exist. Consider a heterogeneous linear treatment effect model of the form: $$Y_i = \alpha_i + \beta_i X_i$$ where $\alpha_i, \beta_i$ are ...
stats_model's user avatar
  • 2,435
1 vote
0 answers
155 views

How to derive correlation using regression without empirical proof?

I just finished learning MLE, Regression, Covariance and now in to Correlation.I want to transform logically from Regression to Correlation using Covariance. Regression: A simple regression model ...
Parthiban Rajendran's user avatar
2 votes
0 answers
102 views

What is the meaning of correlations between regression coefficients? [duplicate]

I understand that coefficient correlations can arise in the presence of correlated covariates, essentially indicating that our inference for those parameters is coming from information that cannot be ...
overdisperse's user avatar
1 vote
1 answer
60 views

Slope-intercept correlation for Gaussian GLM with log link function

I am fitting an exponential model using GLM regression (assuming Gaussian error and a log link function) to 1000 trials, giving me 1000 slope-intercept pairs that are moderately correlated. I want to ...
Applesauce26's user avatar
0 votes
0 answers
85 views

How to explain correlation between populations of slope and intercept coefficients

According to this post, the expected correlation between the sampling distributions for the slope and intercept in OLS regression is given by E(Corr) = -E(X) / sqrt(E(X^2)). Now, let's consider an ...
Applesauce26's user avatar
0 votes
0 answers
80 views

Independent variable is correlated with intercept, creating singularities

I am building a logistic model with about 20 variables. I have used the following code: `fullmod = glm(cancer ~ B_SEX+BLINE_AGE_AT_BASELINE+B_BMI+B_chro+B_fdrc+B_hrt+B_LMET+ B_MET+B_FV+B_EDU+B_INC+...
Fahmida Yeasmin's user avatar
0 votes
0 answers
55 views

Running model without intercept term? [duplicate]

I have the true model set as $y = b_0+b_1 x+u$. Now supposing that I'm running the model without the intercept term, Under what circumstances would the coefficient term in the model (without the ...
jalinda a.'s user avatar