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I want to show that $\hat{\beta}$ is identical,no matter if I do a multivariate regression or the partialling out approach. Therefore consider the following partialling out approach (without constant! …

We say that in "standard" OLS regression the residuals $û$ are uncorrelated with $k$-th explanatory variable $x_k$. I know the argument can be intuitively derived from geometry of OLS. There are a lot …

For various reasons we often assume that there is independence across observations in linear regression models. When does that assumption hold? Only when the data is collected by random sampling? …

Given the following likelihood: $$\prod_{i=1}^{N} \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left\{-\frac{(y_i - x_i' \beta)^2}{2\sigma^2}\right\}$$ Thanks to the information matrix equality we have two cho …

Given the estimator for $\hat{\beta}_1$ in the Frisch-Waugh-Lovell Theorem: $$\hat{\beta}_1 = \left(X_1^\prime{} M_2 X_1\right)^{-1} X_1' M_2 y$$ I can arbitrarily use either the $y$ in the auxiliary …

In the book I am currently reading, they propose the following proof: $\hat{Y} = X_1 \hat{\beta}_1 + X_2 \hat{\beta}_2$ Now because $M_1 = I - P_1$, we can always derive a formula for $X_2$: $X_2 = P_ …

What exactly is the difference between those two models: model 1: $Income_i = \beta_0 + \beta_1 \text{female}_i + \beta_2 \text{experience}_i + \beta_3 \text{female}_i \cdot \text{experience}_i + u_i$ …

I got two questions which are somehow closely related: Let's take the famous study of Acemoglu et al. (The Colonial Origins of Comparative Development: An Empirical Investigation) as an example. When …

Let's denote the asymptotic variance under heteroscedasticity as: $$\hat{\text{Avar}}(\hat{\beta}) = 1/N * \left(\frac{1}{N} \sum_{i}{x_i x_i'}\right)^{-1} \left(\frac{1}{N} \sum_{i} \hat{u}^2_i x_i x …

If I know about an omitted variable in the regression and the data for it is available. But I also know that this omitted variable is endogenous, because the dependent variable and the omitted variabl …

Suppose we want to do an instrumental variable regression. We have an endogenous variable X, exogenous instrument Z, dependent variable Y and an error term U. In order for the approach to be valid, we …

If in general one wants to apply the Frisch-Waugh-Lovell "Partialling Out"-approach, should we include constants and in which of the following regressions? (1) In the first stage where we regress X1 …

Suppose we create a dummy variable male (1=male, 0=female) and dummy variable female (1=female, 0=male). Does the dummy variable trap, also occur, if we include them into interaction terms: $Y_i = β_0 …

One of the OLS assumptions concerning the $X$-matrix (with a constant) is that the columns $(1, x_{i1}, \ldots , x_{iK})$ are not linearly dependent. This looks intuitive to me, because of the dummy-v …

I am referring to the question here: Determining the Number of Restrictions in an F-test. But I want to derive a more general rule than counting the equality signs. If I denote the null hypothesis in …