Questions tagged [conditional-variance]

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Does endogeneity imply heteroskedasticity?

Consider two random variables $X,Y$, with supports $\mathcal{X}$ and $\mathcal{Y}$, respectively, finite for simplicity. Assume that the map $$ x\in \mathcal{X} \mapsto E(Y|X=x)\in \mathbb{R} $$ is ...
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1answer
16 views

Var(e|X), when e=Xu (from Hansen's Econometrics book)

I was working through Bruce Hansen's Econometrics book/notes and got tripped up over something that should be very simple. See the snapshot below, which comes from page 25 of his book Econometrics. ...
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1answer
39 views

How to show that an m.d.s is not independent?

I have to prove that this Martingale Difference: $x_t = u_t u_{t-1}$ where $u_t \sim^{iid} (0, \sigma^2)$ is not serially independent, but am failing to do such thing. I also have to prove that it's ...
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23 views

Analytical Approximation for Conditional Moments

Say I have a function of a latent variable: $F(X_{t+1})$. $F(X_{t+1})=-log(\sum\limits_{\substack{k \neq j}}\alpha^{k}_{j}\frac{S^{k}_{t+1}}{S^{j}_{t+1}})$ The term in brackets is $X_{t+1}$. I know ...
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14 views

Consistent estimator of E(Var(Y|X)) for continuous random variables

Consider a set of $n$ i.i.d. observations $\{(x_i,y_i)\}_{i=1}^n$ of a pair of random variables $(X,Y)$ with finite mean and variance. I am interested in estimating the unexplained variance $\mathbb{E}...
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42 views

Nonlinearity of ${\mathbb E}[\epsilon_i|{\mathbf X}]$ and also ${\mathbb E}[\epsilon_i^2|{\mathbf X}]$ under linear regression

With two of the crucial assumptions of the Classical Linear Regression Model (CLRM) being the zero conditional mean of the error term (${\epsilon_i}$) and the constant conditional variance of ${\...
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1answer
30 views

Identity of ${{\mathit f}({\mathbf z} {\mid} {\mathbf x)}}$ and ${\mathit f}$($\mathbf {z}$) under normality - a peculiar case

I am a newbie to econometrics, so kindly excuse me if I sound too naive. This is what Fumio Hayashi says on page 34 of "Econometrics": Recall from probability theory that the normal distribution ...
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1answer
122 views

Can I predict the variance of a random variable using a machine learning regression model that predicts expected outcomes?

For example, suppose I'm using some machine learning model like gradient boosting that, given some input $x_i$ predicts the expected output $f(x_i) = y_i$. However, I'm also interested in estimating ...
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42 views

The concepts of conditional mean and variance in time series: semantic issues

In time series, the concepts of a "conditional mean" $E_{t}(X_{t+1})$ and "conditional variance" $V_{t}(X_{t+1})$ is semantically unclear to me. Would anyone be able to clearly explain (references ...
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1answer
396 views

Between-cluster variance in k-means - derivation using total variance

Follow-up to this older post (have to make it a question since I can't post comments yet). Specifically, could anyone kindly show how $$\operatorname{Var}[\operatorname E[X\mid K]]$$ (in total ...
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31 views

Finite second moments inhertitable to conditional variables?

Assume a random vector $\mathbf{x}=(x_1,\ldots,x_n)^\top$ that has finite second moments, i.e., $$\int\mathbf{x}\mathbf{x}^\top\rho(\mathbf{x})\,\text{d}\mathbf{x} < \infty.$$ Does it follow that ...
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150 views

variance of multinomial distribution

Assume $A_{kj} \sim$Multinomial$(1, \;\underbrace{(1/m, 1/m, ..., 1/m)}_{\textrm{m times}})$, where $k=1,2, ... m$ and $j=1,2, ... n$. It is clear to see that $\sum_{k=1}^mA_{kj}=1$. If we impose a ...
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Is Cov(X,Y|Z).x always positive? (with X,Y,Z, normal random vectors and x>0)

Let x be a vector of positive values, we know that for multivariate normal distributions of X, Y and Z, $Cov(X,Y|Z)x=(\Sigma_{XZ}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YZ})x$ does not depend on the given ...
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1answer
465 views

Independent variables in the conditional variance GARCH(1,1)

I am using a GARCH(1,1) model, and I would like to add some variables to my conditional variance. I have the data for these variables, but I was wondering if I have to change these variables to ...
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96 views

GARCH model with large conditional variance

I have an extremely volatile series (capital flows). Due to heteroskedasticity I tried to estimate GARCH type models. However, none of the variants (I tried altering process equation, as well as ...
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304 views

Truncated mulitvariate normal: first two moments

Let $X\in \mathbb{R}$ be a univariate random varible for which it holds that $$ X \sim N(\mu,\sigma^2).$$ where $\mu\in \mathbb{R}$ gives the expected value and $\sigma^2>0$ is the variance. If ...