semibruin
  • Member for 8 years, 6 months
  • Last seen more than a month ago
  • Los Angeles, CA
Properties of bivariate standard normal and implied conditional probability in the Roy model
Accepted answer
9 votes

First, in the Roy model, $\sigma_{\varepsilon}^{2}$ is normalized to be $1$ for identification reason (c.f. Cameron and Trivedi: Microeconometrics: methods and applications). I will maintain this ...

View answer
Introductory reading on Copulas
8 votes

Chris Genest has another introductory paper "Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask".

View answer
Estimating logistic regression coefficients in a case-control design when the outcome variable is not case/control status
Accepted answer
8 votes

This is a variation of the selection model in econometrics. The validity of the estimates using only the selected sample here depends on the condition that $\Pr\left(Y_{i}=1\mid X_{i},D_{i}=1\right)=\...

View answer
Interpretation of Saturated Model vs. Model with Interaction and One Main Effect
Accepted answer
5 votes

Throughout my answer, the usual conditional mean independence $\mathbb{E}(\varepsilon_{i}\vert X_{i},Z_{i})=0$ is maintained. It is instructive to consider a concrete example. Let $X_{i}$ be a dummy ...

View answer
Why can't I use OLS for switching regression?
Accepted answer
4 votes

Simply put: the simple OLS estimator will be inconsistent when $U_{1}$ or $U_{0}$ is not mean indepent with $\varepsilon$, i..e $\mathrm{E}\left(U_{1}\mid\varepsilon\right)\neq0$ or $\mathrm{E}\left(...

View answer
Bounded in probability and finite expectation
4 votes

Here is a counter example. Let $X_{n}$, $n=1,2,\ldots$, be a sequence of random variables, whose distributions are defined as follows, $$ X_{n}=\begin{cases} 0, & \mbox{with probability }\frac{n-1}...

View answer
comparing groups in repeated measures FE models, with a nested error component, estimated using plm
Accepted answer
3 votes

The following code implemented the practice of putting interaction between Female dummy and year. The F test at the bottom test your null $\beta_{Female} = \beta_{Male}$. The t-statistic from plm ...

View answer
Given $P(X\perp Y\;|\;Z)$ and $P(X\perp Y\;|\;W)$, prove or disprove that $P(X\perp Y\;|\;Z, W)$
Accepted answer
3 votes

The intuition is that it is possible that the pair $(Y,Z)$ or $(Y,W)$ cannot determine $X$; but $(Y,Z,W)$ together can determine $X$. The following counter example is constructed from this intuition. ...

View answer
Regression with interaction term that is a function of other regressors
Accepted answer
3 votes

Let me rephrase your question a little bit. The original regression $(1)$ is $$ Y=\beta_1 X +\beta_2M+\beta_3XM+\varepsilon, $$ where $\varepsilon$ is the error term. To study the partial effect of $X$...

View answer
Convergence in distribution of sum implies marginal convergence?
2 votes

I may prove it under the assumption that $\mathrm{E}\left|Y\right|<\infty$. In order to prove $X_{n}\rightarrow_{d}X$, we wish to show that $\mathrm{E}f\left(X_{n}\right)\rightarrow\mathrm{E}f\left(...

View answer
Statistics of sample correlation for uniformly distributed samples
2 votes

Your question is indeed asking for the finite sample distribution of $r_{N}$. To address your question, let me rephrase it in terms of linear regressions. So a linkage between $r_{N}$ and the ordinary ...

View answer
Is it true that if one coefficient in a linear model is endogenous, then any individual coefficient will be inconsistent?
Accepted answer
1 votes

Yes and no. If $x$ and $w$ are correlated, the endogeneity of $w$ will bias the estimation of $\beta$, partial effect of $x$. Let $Y$, $X$, $\hat{X}$ and $W$ be the matrices by stacking the ...

View answer
Distribution of the a random variable defined on the index of a set of independent random variables
Accepted answer
1 votes

For $n = 2$, $p(\gamma(x) = 1) = p(x_{1} - x_{2} > 0)$. Let $z_{1,2} = x_{1} - x_{2}$. Obviously, $z_{1,2} \sim N(\delta_{1} - \delta_{2}, 2)$. Hence, $p(\gamma(x) = 1) = 1 - \Phi((\delta_{1} - \...

View answer
Understanding Big/Little $O_p$/$o_p$ Notation for Estimators
1 votes

We have $\sqrt{h_n} = o_p(1)$. Note that $\sqrt{h_n}O_p(1) = O_p(\sqrt{h_n})$. Using the result that $o_p(1)O_p(1)=o_p(1)$. We have the conclusion that $O_p(\sqrt{h_n}) = \sqrt{h_n}O_p(1) = o_p(1)$.

View answer
identification of simultaneous equation model
1 votes

For notational simplicity, I drop the subindex $i$ in the sequel. Observe that $\mathbb{E}\left(vx\right)=0$ implies that $$ \mathbb{E}\left\{ \left(z-x\delta\right)x\right\} =0. $$ Hence, we have $\...

View answer
Is the fixed effect in a fixed effect model a random variable or not?
1 votes

I think the main difference between fixed and random effects is that the unobserved individual effect $\alpha_i$ is purely random in the random effect paradigm in the sense that its distribution does ...

View answer
BIC and AIC(c) and group data
Accepted answer
1 votes

I did not see the rationale of using your "group BIC/AIC". Your proposed group BIC/AIC is comparing a different pair of models. Let $f(\mathbf{x}_i \mid \alpha)$ and $g(\mathbf{y}_i \mid \beta)$ be ...

View answer
Recovering fixed effects after Arrellano-Bond estimation in panel data
Accepted answer
0 votes

For simplicity, consider $y_{i,t} = \rho y_{i,t-1} + a_{i} + e_{i,t}$. Let $\hat{\rho}$ be the Arellano-Bond or any consistent estimator of $\rho$. We have $$ \hat{e}_{i,t} \equiv y_{i,t} - \hat{\rho}...

View answer
Drawing smooth 2D curves with 2D spline in R
0 votes

Your question can be treated as a polynomial interpolation problem. The following example might help you. You don't need extra package. x <- c(1 : 4) y <- rnorm(4) plot(x, y) # plot the four ...

View answer
The homogeneity of error variance in MMR with continuous moderator?
0 votes

You could use heteroskedasticity robust standard error estimators, e.g. White's estimator, when you suspect the homoskedasticity assumption. White's test can also be used to test the homoskedasticity ...

View answer