Questions on (linear or nonlinear) regression, the fitting of functions that best approximate empirical data.

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questions about distribution of multivariate normal

I'm looking at this past exam question, For A) Cbhat~N(CU,C(summation)C') B)I have very faint idea of what to do, I tried finding some theroems about distribution but couldn't find any that ...
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1answer
39 views

Linear regression involving angles in a triangle.

In a survey experiment, three independent measurements $29.5^{\circ}$, $30.5^{\circ}$, $120.5^{\circ}$ are obtained from the three angles $\alpha,\beta,\gamma$ of a triangle. Formulate the ...
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1answer
32 views

Question on regression

So I've been given this formula For regression $R^2=1 - \sum \frac{{(y_i - \hat{y}_i)}^2}{(y_1-\bar{y})^2}$ Now an obvious question that has come to me is why $R^2$ stays the same in certain ...
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Trying to find similarity between collection of points

This is a kind of weird problem, and I'm not sure what the best Stack Exchange to post this on is, but I assume Mathematics could help the most. I have many sets of points in 3D space (xyz ...
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1answer
21 views

how binary quantile regression divides the dependent variable into quantiles

I am not very clear with binary quantile regression. As if it was ordinary quantile regression, it would divide the dependent variable's value by its ascending value into quantiles. But I cannot ...
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1answer
33 views

Recalculate R^2 deleting 1 point

Is there a way to recalculate $R^2$ of a regression that I delete a point (for example an outlier point)? The idea is to get the $R^2$ without a point but without recalculating all the regression. ...
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2answers
62 views

Parameter optimization using a regression model.

I am working on an optimization problem. I build a regression model to understand the behavior of a system which depends on two variables which are functions of another two variables. My regression ...
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1answer
47 views

Proof of Correlation Coefficients

Good evening, I have a problem with an exercise: Let $X$ and $Y$ be two real square integrable random variables with var$X>0$, var$Y>0$. The correlation Corr$(X,Y)$ quantifices how far $X$ and ...
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1answer
15 views

Asymptotic var of IV estimator

This post is asked again due to lack of answers first time around. Suppose we have a linear model $y=Q+Rx+error$, where $E(error)=0$, and $z$ is an instrument for $x$ (endogenous) where the ...
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2answers
119 views

Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
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30 views

Using Mean vs Median for Rapidly Changing Data

I am writing a report on global GDP per capita trends. As many of you know, were large shifts in the growth rates of these figures before and after the Great Recession. There has been some analysis on ...
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1answer
126 views

Incorrect angle detected between two planes

I want to calculate the angle between 2 planes, Reference plane and Plane1. When I feed the X,Y,Z co-ordinates of pointCloud to the function plane_fit.m (by Kevin Mattheus Moerman), I get the output ...
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28 views

Algebraic expression of a regression matrix

Let's say I'm doing a multivariate regression between a set of input $n$-dimension vectors (noted by the matrix $X=\{X_1,X_2,...X_m\}$) and a transformed version (noted by the matrix ...
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44 views

Hypothesis testing - Beta coefficient

Apologies if this is trivial, just linke me somewhere. I'm currently taking statistics 101, I can't wrap my head around the hypothesis testing of coefficients. As follows, the t-test reads $$T=\frac ...
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13 views

MSE in case of log-transformed dependent variable

Let's consider the following log-linear model: $log(Y_i) = \alpha + X_i\beta + \epsilon_i$ for i = 1, ..., N The fitted value is: $\widehat{log(Y)} = \hat{\alpha} + X\hat{\beta}$ Assuming ...
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Curve fitting using a graph extracted from an article, is it possible?

I want to curve fit a graph from an article which I can only extract from the pdf file as a screenshot. Therefore, I do not have the coordinates of the data points explicitly, yet I know that the ...
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Regression question (details inside). Measuring the incremental impact on the dependent variable of one category over other categories

Formulate a regression equation you would use to test for the differences in ROE between firms that used tier 1 investment banks as their advisors and those that used tier 2 or tier 3 banks (note: ...
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1answer
46 views

Optimization Problem (Linear Algebra)

I am not trying to cheat or anything, so any reference to online literature or MOOCs, that teach this stuff, will be highly appreciated. The problem is to prove that the following optimization ...
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Statistics linear modelling in R

Suppose I have a date set of the form: Test Subject/Sex/No. of mistakes made in the morning/ No. of mistakes made in the afternoon A / M / 2 / 5 B / F / 1 / 4 C / M / 3 / 5 D / F / 1 / 5 Suppose ...
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Cannot figure out autocavariance

The moving average model of order q has the form $$Y_t =β_0 +e_t +b_1e_{t−1} +b_2e_{t−2} +...+b_qe_{t−q}$$ where $e_t$ is a serially uncorrelated random variable with mean $0$ and variance $σ^2_e$. ...
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18 views

Time series regression

What if I have a stationary independent variable and 2 non-stationary dependent variables, and I want to run a regression on them, what model is the most appropriate?
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40 views

Is there such a thing as a combination of linear and non-linear regression in one form?

Let's say I have a dataset D with many variables. I can get a multiple linear regression from that in the form ...
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1answer
37 views

How to take the derivative of Matrices

I was browsing the derivation of the Least Squares estimates and stumbled about this problem. It said that: $$E = (Y + XB)^2$$ $$\frac{dE}{dB} = -X^TY + X^TXB$$ It is to my understanding that the ...
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1answer
40 views

Size of sample and correlation coefficient

$X$ and $Y$'s correlation coefficient is $r=0.5$. What is the size of sample when the correlation is significant at $\alpha=0.05$ with two sided test? Is there a more "formal" way to solve this ...
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1answer
19 views

what is one basic/intermediate regression analysis standard textbook that is math intense

What is one basic/intermediate regression analysis standard textbook that is math intense with proofs/derivations? Also, i need that one to be comprehensive yet the diffculty is suitable for self ...
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43 views

Matrix form for Weighted Least Squares

If we have the following weighted least-squares regression, with $\hat{\beta} = (X'WX)^{-1}X'WY$ How can we express the squared errors, MSE and the fitted values in matrix form? These are the OLS ...
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1answer
33 views

How to find the deviated form of beta 1 in OLS

How to find the deviated form of beta 1 in OLS Y=β1+β2X+u estimated β1=(ΣX^2ΣY-ΣXΣXY)/(nΣX^2-(ΣX)^2) I do not know how to turn this part (ΣX^2ΣY-ΣXΣXY)into deviated form. I found that estimated ...
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38 views

If we make the lambda very small in ridge regression, why do the beta hats still decrease?

I understand that the betas estimated from minimizing the function (y hat - y)^2 + beta' beta decrease as lambda increase. However, looking at the picture below, why cannot we choose a lambda so ...
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1answer
25 views

Help Linear Regression

It would be really nice if someone could help me out and suggest me with his expert opinion. Going through the table and looking at Part (a) of this question. I believe I should use Scatter plot to ...
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1answer
44 views

Standard Error in OLS Regression

Assuming I have the following linear regression set-up: $y_i = \alpha + x_i * \beta + \epsilon_i$ for $i = 1,2,..., n$. When I run the regession, I get a $\beta$ and $\alpha$ estimates, along ...
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1answer
40 views

Multiple Regression with Categorical Predictor Variables of More than Two Levels

I'm planning on running a hierarchical multiple regression in SPSS. In the first step, I would like to enter demographic characteristics, second step continuous predictor variables of interest, and ...
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Simplifying $\sum_i[z_i'(\delta-\hat{\delta})]^2x_ix_i'$ to apply a law of large numbers

I'm in the context of linear regressions. Let $n$ be the sample size and for $i=1,\ldots,n$, let $$ \underbrace{x_i}_{K\times 1},\quad \underbrace{z_i}_{L\times 1}\quad(n>K\geq L) $$ be column ...
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multiple linear regression analysis and case of p>>n

In the case for simple linear regression and multiple linear regression, if we have p >> n, is it undefined? And in the case for ridge regression, p>>n requires $\lambda$ to increase b/c more ...
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Outlier Contained in Prediction Interval (Tme series Forecasting Problem)

In my stats class today, the professor was showing us some output from MINITAB on a prediction interval that was calculated (from time series data using standard linear regression). For one of the ...
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152 views

Inclined Elliptic Tank Volume Calculation

Can someone help me determine an equation for calculating the volume of an elliptical cylinder on it's side and inclined 5 degrees from horizontal? The tank has flat ends. I have found several ...
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1answer
44 views

For a general linear regression , are Y and Y hat independent?

For a general linear regression , are $Y$ and $\hat{Y}$ independent? $Y$=XB+e $\hat{Y}$=X$\hat{B}$ I think they are dependent, because if they rely on the same data, they should have some sort of ...
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1answer
40 views

Inference about the true intercept of the model and the OLS being BLUE

Consider the following population regression model: $$y_{i} = \beta _{1} + \beta_{2}x_{i} + \epsilon _{i},$$ where $i=1,...,n$. Assume $\epsilon \sim iid$, with the pdf in equation: $f(\epsilon ) = ...
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What is a common name for the resulting function?

Consider the following population regression model: $$y_{i} = \beta _{1} + \beta_{2}x_{i} + \epsilon _{i},$$ where $i=1,...,n$. Assume $\epsilon \sim iid$, with the pdf in equation: $f(\epsilon ) = ...
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derive the distribution(a multiple regression problem)

(Multiple regression model with p's predictor variables.) Derive the distribution of $$\frac{(b-\beta)X'X(b-\beta)}{MSE\cdot p}$$ As far as I know, $b\sim N(\beta,\sigma^2 (X'X)^{-1})$ $b-\beta ...
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1answer
40 views

Proving an implication in a linear regression

Suppose we have a linear regression: $$ y_i=X_i'\beta+u_i,\quad i=1,\ldots,T. $$ Here $y_i$ and $u_i$ are scalars, $X_i$ and $\beta$ $k\times 1$. $\beta$ is a (non-stochastic) vector of ...
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1answer
16 views

Conditional expectation; regression

Assume that E[$y_{it}|c_i, x_{it}$] = $c_i$ + $x'_{it}\beta$ Eliminate $c_i$ by taking the expectation with respect to $c_i$, leading to E[$y_{it}|x_{it}$] = E[$c_i|x_{it}$] + $x'_{it}\beta$ ...
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23 views

Question about regression model

Suppose you fit (estimate the parameters of) a regression model, obtaining $\hat{Y}$, $\hat{B}$, and $\hat{E}$. And you fit a second regression model , using $\hat{Y}$ x matrix from previous model ...
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31 views

how to show for a simple regression with an intercept and one independent variable$ R^2 = r ^2$ , where $r$ is the ordinary correlation coefficient.

how to show for a simple regression with an intercept and one independent variable $R^2 = r ^2$, where $r$ is the ordinary correlation coefficient. Here is where I'm at. $R^2= ...
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Regression on even function?

Is there any test for whether or not the regression function is even? Suppose we have a model: $Y=g(X, \epsilon)$, where $Y, X$ are both one dimensional. My questions is how do we test for $g$ is an ...
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The Hessian Matrix I calculate is twice as much as it should be. Why?

I have a function "fkt." In this example, let it be as simple as $y=a \cdot x+b$. I have a real dataset with values obeying to the model. After regression of the points to the model, I find the ...
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1answer
91 views

When is Block-Partitioned Matrix Invertible?

Suppose I have a block partitioned matrix \begin{equation} \begin{bmatrix} \mathbf{X}_1^{\top}\mathbf{X}_1 & \mathbf{X}_1^{\top}\mathbf{X}_2 \\ \mathbf{X}_2^{\top}\mathbf{X}_1 & ...
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Multivariate regression analyisis with grouped data. How do I 'un-group'

I am trying to determine if either of two non-numeric variables effect the percentage of people who take a specific action. There is another variable that needs to be controlled for. In real world ...
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1answer
9 views

Do multiclass logistic regressions obey Kolmogorov's second axiom?

Logistic regressions were taught to me using the intuition that they approximate $\mathbb{P}(Y=y|x;\theta)$. Multiclass regressions use one-vs-all classification, selecting one $y$ and classifying ...
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2answers
46 views

Show $E\left(\mathbf{X}_i \otimes \mathbf{u}_i\right)=\mathbf{0}$ implies $E\left(\mathbf{X}_i^{\top}\mathbf{G}\mathbf{u}_i\right)=\mathbf{0}$

Let $\mathbf{X}_i$ be a $G \times K$ random matrix, and let $\mathbf{u}_i$ be a $G \times 1$ random vector, and suppose we have a sample of $i=1,\ldots,N$ of each. Suppose the following condition ...
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How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix Expression

Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that \begin{align} \sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i &= \begin{bmatrix} ...