Questions tagged [conditional-variance]

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Loss function for conditional variance?

Minimizing square loss results in predicting conditional means. Minimizing absolute loss results in predicting conditional medians. What loss function results in predicting conditional variances? I ...
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2 votes
2 answers
47 views

Conditional expectation of the square of a random variable

Let the joint PDF of X and Y, $f(x, y) = \frac{1}{2}e^{-x}$ if $x \geq 0$, $|y| < x$, $f(x, y) = 0$ everywhere else. Calculate $\mathbb{V}(Y|X = x)$. By definition, $\mathbb{V}(Y|X = x) = \mathbb{E}...
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Difference in difference for a conditional outcome variable

Suppose I want to study the impact of a policy on two outcome variables using the Difference in Difference (DID) method. The first outcome is the decision of whether to produce content in an online ...
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2 votes
1 answer
55 views

Regression of squared residuals

I have read in several papers, that one can regress the squared residuals of some conditional mean regression of a variable $X$ on a set of predictor variables and interpret the fitted values as the ...
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Variance of a normal random variable when conditioning on a correlated normal random variable being above a threshold

Suppose $X$ and $Y$ are correlated with correlation coefficient $\rho$. They are jointly normal with means $\mu_X$ and $\mu_Y$ respectively. Then what is $Var[X | Y \geq T]$? Feel free to add ...
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1 vote
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159 views

When is it acceptable to compute (conditional) subset-averaged coefficients?

I'm running an ecological study and I have 4 dependent variables (DVs) that I would like to explain (my interest thus lies in inference and not in prediction). For each one of these variables, I built ...
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56 views

Law of Total Variance Issue

The Law of Total Variance says: if the variance of X is finite then $V(X) = E(V(X|Z)) + V(E(X|Z))$ Suppose $X\sim N(0,1)$, $Y\sim \text{Cauchy}(0,1)$, $X$ and $Y$ are independent. Define $Z \equiv X + ...
1 vote
1 answer
224 views

Conditional distributions of correlated normal random variables

Suppose that $X$ and $Y$ are normally distributed with mean zero and nonzero covariance. I want to know the distributions of $X | X - Y > c$ and $Y | X - Y > c$, which I believe should be ...
0 votes
1 answer
122 views

What is the intuition of a GARCH model without fitting ARMA for the conditional mean?

I wanted to ask, as I've seen this used a couple of times before, about the logic of fitting a GARCH model in absence of estimating ARMA for a series that is clearly an ARMA process (Fitting a GARCH ...
1 vote
0 answers
111 views

Modelling the Conditional Variance in a Panel Setting

I am familiar with ARCH-type models to estimate the conditional volatility of some variable of interest in a univariate setting. I know that there also exists the concept of multivariate ARCH-type ...
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5 votes
1 answer
212 views

How to model conditional variance?

Sorry if this question has been asked before; I'd love to read any discussion around this. There's got to be a better way to summarize this question as well. I've got covariates $X$ and response $Y$, ...
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Conditional bias-variance decomposition of MSE

The MSE can be decomposed as follows: \begin{align*} \mathbb{E}\left[(\hat{\theta} - \theta)^2\right] &= \mathbb{E}\left[\left(\hat{\theta} - \mathbb{E}(\hat{\theta}) + \mathbb{E}(\hat{\theta}) - \...
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What is conditional normal variance of $X$ given that dependent $Y$ is bigger than a real number $M$? [duplicate]

There are two dependent normal variables with the same distribution and the correlation coefficient $\rho$: $X,Y \sim N(\mu, \sigma^2)$. I would like to get $P(X>M|Y>M)$. For that I need to know ...
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-1 votes
1 answer
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The conditional normal distribution [duplicate]

I would like to find the conditional bivariate normal distribution. There are two dependent normal variables with the same distribution and the correlation coefficient $\rho$: $X,Y \sim N(\mu, \sigma^...
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2 votes
1 answer
95 views

Confusion about parameter covariance using least squares method

I am using the method of least squares to estimate parameter values for a nonlinear model with three parameters: $a$, $b$, and $c$. Call the sum of the squares of the residuals $\chi^2$. I plot $\chi^...
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1 answer
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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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1 vote
1 answer
60 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. ...
2 votes
1 answer
137 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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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 ${\...
1 vote
1 answer
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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 ...
2 votes
1 answer
1k 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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0 votes
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221 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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1 answer
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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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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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246 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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50 views

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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1k 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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1 vote
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141 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 ...
3 votes
0 answers
533 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 ...
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