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Questions tagged [conditional-variance]

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GLMs and their conditional expectation and variance

Let the density of the distribution of response $y_i | x_i$ in GLMs denote as: $$f(y; \theta, \phi) = \exp\left(\frac{y\theta - b(\theta)}{\phi} + c(y; \phi)\right)$$ Then conditional expectation and ...
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36 views

Conditional Variance of $Z_i|\sum_i\beta_iZ_i$

Let's assume I have $K$ i.i.d. standard normal random variables $Z_1,...,Z_K$. Hence, I know that $V[Z_i] = 1$ and $E[Z_i] = 0$ for all $i\in K$. I am faced with computing the following conditional ...
BMBE's user avatar
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3 votes
1 answer
67 views

Interpretation of $\sigma$ in Gaussian mixture

I have a distribution of a variable that was normalized with plt.hist and then fitted with a sum of gaussian curves $g_M = \displaystyle\sum_i\frac{w_i}{\sigma_i \...
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BIvariate Normal and Conditional Expectation

I am working on a problem where I must show that the conditional distribution of Y given X follows the distribution with mean and variance shown below. In the previous question, we were given that X ...
Harry Lofi's user avatar
1 vote
0 answers
101 views

The expected value and variance for the sum of 4 dice rolls only if a coin gives Head

Edited: toss a fair coin 4 times and then roll a fair 6-side dice whenever the coin gives a head H. Let X be the sum of the dice rolls. How to calculate E[X] and <...
Sandra Sukarieh's user avatar
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0 answers
24 views

Conditional volatility and notation to express it

I'm taking a risk modelling class, and we're discussing conditional vs unconditional volatility. We have standard volatility $\sigma = \sqrt{\mathbb{E}[r_t - \mathbb{E}(r_t)]^2}$ But what is ...
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37 views

finding conditional variance in kyle model

While desribing properties of the single auction equilibrium defined by theorem 1 in "Continuous Auctions and Insider Trading" A. Kyle conjectured that one-half of the insider's private ...
V013's user avatar
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Constant conditional variance

Let $X$ and $U$ be two independent random variables, and let $Z= X+U$. Under which conditions is $Var(X|Z)$ constant? I know this holds for instance when $X$ and $Z$ are jointly normal. I'm ...
Nidjsi's user avatar
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1 answer
350 views

Conditional variance notation

Why does the formula $\operatorname{Var} (Y\mid X)= E \left (\left(Y- E (Y\mid X)\right )^2\mid X\right )$ condition $EY$, but not $Y$, on $X$?
Roman's user avatar
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2 votes
1 answer
126 views

Loss function for estimating the conditional variance by fitting $y_i^2$

I'm trying to detect anomolies in a dataset $i \in \{1,2,...,N\}$ where a random variable $y_i$ is expected to be drawn from a normal distribution with mean $\mu_i=0$ and variance $\sigma_i^2 (X_i)$ ...
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1 vote
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Having difficulty finding regression function and conditional variance [duplicate]

The random vector distribution X = (Y, X, Z) is Gaussian with mean µ = (1, 2, 4)T and a covariance matrix Σ is equal to: \begin{pmatrix} 2 & 3 & 1\\ 3 & 5 & 2\\ 1 & 2 & 6 \end{...
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62 views

How to calculate power for a 2-level model with two L1-predictors?

I am new to power analysis in multi-level models. I am looking for a possibility to do a power analysis for the following 2-level model: Y = y00 + y10D1 + y20D2+y01Z +y11D1Z+y21*D2Z. In this model, I ...
Janna's user avatar
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0 answers
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Actual definition of $\sigma$ algebra generated by information available up to time t-1 in ARCH, GARCH models

I'm trying to figure out the proper mathematical definition in sigma-algebra generated by infinite past in ARCH, GARCH type models. Let $\epsilon_t = \sqrt{h_t} z_t $, where $z_t$ $i.i.d$ random ...
mathstat's user avatar
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0 answers
49 views

Conditional variance $\hat{\beta}_1|x_1, x_2,\ldots, x_n$

Why do we need to estimate the variance of $\hat{\beta}_1|x_1, x_2,\ldots, x_n$ when we want to test a hypothesis about $\beta_1$?
Fimi's user avatar
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4 votes
2 answers
320 views

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 ...
Dave's user avatar
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2 votes
2 answers
196 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}...
johnsmith's user avatar
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2 votes
1 answer
207 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 ...
shenflow's user avatar
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1 vote
0 answers
332 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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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 + ...
Albert Zevelev's user avatar
1 vote
1 answer
433 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 ...
quevivasbien's user avatar
0 votes
1 answer
237 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 ...
canthandlehtml's user avatar
1 vote
0 answers
230 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 ...
shenflow's user avatar
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5 votes
1 answer
558 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 $Y$, ...
goopy's user avatar
  • 98
2 votes
2 answers
740 views

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}) - \...
Adrian's user avatar
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0 votes
0 answers
56 views

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 ...
Stasya7's user avatar
  • 31
-1 votes
1 answer
923 views

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^...
Stasya7's user avatar
  • 31
2 votes
1 answer
165 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^...
tneulinger's user avatar
0 votes
1 answer
277 views

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 ...
Star's user avatar
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1 vote
1 answer
82 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. ...
Guinness's user avatar
3 votes
1 answer
206 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 ...
Caio C.'s user avatar
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0 answers
63 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 ${\...
eisendon's user avatar
  • 109
1 vote
1 answer
33 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 ...
eisendon's user avatar
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3 votes
1 answer
2k 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 ...
Ben's user avatar
  • 1,884
0 votes
0 answers
355 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 ...
Julien's user avatar
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1 vote
1 answer
2k 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 ...
Tim's user avatar
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0 answers
111 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 ...
MTP's user avatar
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0 answers
285 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 ...
Alemu Assefa's user avatar
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0 answers
67 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 ...
Dadoo's user avatar
  • 107
0 votes
1 answer
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 ...
welp's user avatar
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1 vote
0 answers
179 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 ...
mpmsa2013's user avatar
3 votes
0 answers
596 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 ...
Michael L.'s user avatar
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