All Questions
Tagged with bias least-squares
45
questions
1
vote
0
answers
45
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Question on nonlinear least squares
Consider the following equation for $Y>0$:
$$
(1) \quad \log(Y)=\log(\gamma)+\log(\alpha+\beta X)+\epsilon.
$$
Assume that $E(\epsilon| X)=c\neq 0$. What are the consequences of this assumption on ...
- 889
2
votes
1
answer
79
views
Granular difference-in-differences with non-repeating unit of observation
I want to analyze changes in characteristics of job postings around an (exogenous) event. However, rather than conducting the analysis at the job poster level (e.g., a company or geographic area), my ...
- 77
6
votes
3
answers
470
views
Do autocorrelated residuals cause OLS coefficients to be biased?
I see different answers everywhere. Intuitively, I would think if residuals are autocorrelated then there is some information that you are not incorporating into your model and is a sign of a biased ...
- 372
3
votes
2
answers
119
views
Instrumental variable as a control variable
I understand that instrumental variable is used to address endogeneity bias since there could be correlation between the variable of interest and the error term.
Suppose now we want to see the ...
- 33
1
vote
0
answers
44
views
Regress y on residuals of x and z [closed]
I have the following set up:
$y_i = \beta_0 + \beta_1 x_i + \beta_2 z_i + e_i$, where $e_i$ is extracted from a Normal (0,1) distribution independently of $x$ and $z$. The true values are $\beta_0 = \...
- 11
0
votes
0
answers
37
views
Determine direction of bias with measurement error
We want to estimate the following population model:
$$y_i=\beta x_i+\epsilon_i$$
with $E[y_i]=E[x_i]=$ and $E[x_i\epsilon_i]=0$. We cannot observe $x_i$ directly, but we observe two variables $x_i^a$ ...
- 151
1
vote
0
answers
96
views
OLS with $iid$ Cauchy errors still unbiased?
A comment to this question suggests that the OLS estimate of linear model parameters is unbiased, even when the error term is Cauchy. Given that Cauchy distributions lack an expected value, I am ...
- 65k
2
votes
1
answer
177
views
Does an endogenous variable bias the coefficient of the exogenous one?
We have the following model:
$$ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon. $$
We know that:
\begin{align*}
\operatorname{Cov}(x_1, \epsilon) &\neq 0 \\
\operatorname{Cov}(x_2, \epsilon) &...
- 2,301
1
vote
2
answers
777
views
Proof OLS is Biased if a Regressor is Correlated with the error
Consider the basic linear model:
$$y_i =\beta_0 +\beta_1 x_i +\varepsilon_i$$
I am aware that if $E[\varepsilon_i|x_i ]=0$ then $E[\hat{\beta_1}]=\beta_1$ (unbiasedness) and also that if $Cov(x_i,\...
- 140
1
vote
1
answer
603
views
OLS Estimation, Bias and Causality
I wish to ask about the bias of an OLS estimator. In what follows I assume that the regression that we are dealing with is an approximation to a linear conditional expectations function. That is we ...
- 303
2
votes
1
answer
135
views
Bias-variance trade-off in case of biased estimators: is the bias zero?
Consider a data generating process (DGP) that is AR(1): $y_t=\varphi_1 y_{t-1}+\varepsilon_t$ with $\varepsilon_t\sim i.i.D(0,\sigma^2)$ for some distribution $D$ with mean zero and variance $\sigma^2$...
- 68.6k
2
votes
2
answers
119
views
If a fitted OLS regression model is mis-specified, is it possible to produce a second model that is unbiased?
Let’s say I want to build a linear regression model to conduct some sort of statistical inference. I plan on using the least squares method to fit the model. My understanding is that you need to ...
- 43
1
vote
0
answers
20
views
In a linear model, if regressor is positively correlated with error, will OLS estimated coefficient be upward biased? why or why not?
Suppose the dependent variable $Y_i$ is generated by $Y_i=\alpha_0+X_i\beta_0+e_i$, where $e_i$ is the unobserved error and $cov(X_i,e_i)>0$. Using data $\{Y_i,X_i\}_{i=1}^{n}$, I get the OLS ...
- 2,966
4
votes
1
answer
4k
views
Is the MSE of a vector a scalar or a matrix? [duplicate]
Suppose $Y = X\beta + \epsilon,$ where $Y$ is $n \times 1$, $X$ is $n \times p$, and $\beta$ is $p \times 1$, and $\epsilon$ is $n \times 1$ with mean 0 and variance $\sigma^2$. The OLS estimator of $\...
- 2,909
0
votes
1
answer
919
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Unbiased estimator and biased error
I'm having some trouble relating unbiased estimators and bias error. By bias error, I mean the bias error we talk about when analyzing "bias-variance tradeoffs." Is this bias error and an unbiased ...
- 363