Questions tagged [regression]

This tag is for questions on (linear or nonlinear) regression, which is a way of describing how one variable, the outcome, is numerically related to predictor variables. The dependent variable is also referred to as $~Y~$, dependent or response and is plotted on the vertical axis (ordinate) of a graph.

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24 views

Finding Conditional Expected Value and Conditional Variance

Very stuck with this problem. Not meant to make any more assumptions than laid outin the question. I have a regression model: $\ln(𝑠𝑎𝑣_𝑖)$ = $𝛽_0 + 𝛽_1\cdot \ln (𝑖𝑛𝑐_i) + 𝑢_i$ (where sav ...
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7 views

struggling to understand R output for linear regression with categorical variables

If i where to fit some random linear regression model like fit<-lm(Height~factor(Location)+Daily.calcium.intake+Fathers.daily.prtintake,data=some data set ) as a random example where Location ...
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10 views

Comparison of regression estimator when data matrix is scalar. [on hold]

regression Hi guys, I am wondering about this regression. How to find the GLS estimator for this regression? And how to show the Var(Ybar/Xbar)>Var(GLS)? If Y=(1,2,-3,0)and X=(43-6-1), then how to ...
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0answers
13 views

How to show “conditionally unbiased” and “consistency” and “asymptotically distribution” in this regression?

regression Hi guys, I am wondering about this regression. How to show that the β ̃is conditionally unbiased and consistent? And how to find the asymptotically distribution of β ̃? Please tell me ...
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11 views

Multiple Linear Regression using expected values instead of observations

Normally when doing multiple linear regression we use multiple observations of the features to estimate the coefficients, in my case I want to minimize the square error. This formula normally is: $\...
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10 views

Probit vs Linear Probability Model for binary independent variable

I am new to binary regression models. From my understanding, the probit is better than the LPR because it allows for varying marginal effects and constraint Y_hat to be within 0 & 1. However, if ...
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0answers
17 views

How would a simple linear regression model look like?

for example i would have 4 groups with 6 results in each of them. if i was to do a hypothesis test to see if there was a relationship between the explanatory variable and response variable i was ...
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0answers
12 views

Combining Curve Fitting Confidence Intervals and Standard Deviations for Scaling Arguments

I have done some curve fitting in MatLab so that I can generate a trend for different terms taken from my CFD simulations. In my simulations, I have three wing lengths measured by the aspect ratio $AR ...
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0answers
9 views

Logical alternative to random forest for one variable?

I have a large dataset I am analysing with a random forest regression. Part of my analysis aims to show that combining all features is better than subgroups. To highlight improvement, I want to run a ...
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12 views

Correllation between a difference and subtrahend

I have measured a total concentration of a substance (phosphorus), called totP, and one specific form of the substance (phosphate), called P, and need to understand how the diffence between these, ...
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1answer
19 views

How to calculate fitted values and residuals from a set of data

Given a set of data with 11 observations of two variables (response and predictor), I've been asked to "calculate the fitted values $\hat y_i = \hatα + \hatβx'_i $ and residuals $e_i = y_i − \hat y_i$ ...
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19 views

Create design matrix from quadratic equation [closed]

I'm trying to create an X matrix from an equation \begin{equation} y = x^2 + 2x + 3 \end{equation} I simulated 20 Y in the following way: ...
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1answer
11 views

Dividing observed entries by their variance in univariate linear regression

Suppose I have two random variables $X,Y$ with corresponding observations$(x_1,y_1), \ldots (x_n,y_n)$. Then for linear regression $y=ax+b$ I know that $\hat{a} = \bar{y}-\hat{b}\bar{x}$ and that $\...
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2answers
18 views

linear regression on calculator

I have the equation T(t)=T_0 e^kt . and some coordinates (t,T). t: (10,20,30,40,50,60). (T: 45,40,37,33,27,24) I transform it to y = ae^bx = lna+bx. I then change the coordinates to the form (x, lny)...
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0answers
21 views

Help! How to solve this question? [closed]

$$Y_i = α + βX_i + u_i$$ $$ u_i \sim N (0, \sigma^2)$$ Question1: Cov[a^,b^] ?? (σ2=given) Question2 $$ Xi×10=X∗i ...Yi = α∗ + β∗X∗i + u∗i, $$ $$ u_i \sim N (0, \sigma^2) $$ Least squares estimation
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4 views

Endogeneity and rMSPE in Regression

Given a regression model: $$y = \beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\epsilon$$ how would root mean squared predictive error $(\frac{\sum (y_{i} - \hat{y})^{2}}{n})^{1/2}$ change (a) if $x_{1}$ ...
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0answers
19 views

Residual sum of squares for ridge regression

I define RSS as $$RSS = (Y-X\hat{\beta})'(Y - X\hat{\beta})$$ where $\hat{\beta}$ is the solution to the ridge regression problem and is given by the closed form formula $$\hat{\beta} = (X'X + \...
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1answer
12 views

$\operatorname E(Y_i^2\mid X_i=x) $in regression

I have a regression equation of the form \begin{equation} Y_i = f(X_i) + \varepsilon_i \end{equation} I'm trying to figure out what $\operatorname E(Y_i^2\mid X_i=x)$ in this case. I know that \...
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0answers
19 views

Standard errors of panel regression constant dependent variable across panels

I'm running a two stage MLE model in Stata, where my first stage is instrumenting a variable, $r$, and the second stage does a spatial panel regression with fixed effects. $r$ is an interest rate, and ...
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1answer
23 views

Polynomial regression for 2D data (paraboloid fitting)

I want to create cool 2D parabolic fits like these figures However, I do not know how to mathematically formulate the problem. I know how to formulate the 1D polynomial fitting problem, which is ...
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0answers
12 views

Estimation of Variance, Covariance of estimators of linear regression

I am examining this exercise: https://i.imgur.com/eDMtCaa.png From this image https://i.imgur.com/1WBO6iA.png I think the desired values are given directly by the provided matrix. However, I am a ...
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0answers
21 views

Data fitting (REGRESSIONS)

For linear and non-linear regressions, what are the differences between the following fitting targets? Which one should be used to fit the points more precisely? Lowest sum of squared absolute error ...
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2answers
41 views

The form of the surface $z(x,y)$

In $3$D space, we have For any fixed $x$, $z$ is of the form: $z=ay+b$ For any fixed $y$, $z$ is of the form: $z=c \ln(x) +d$ In other words, we have $z(x=\text{constant},y)=A(x)y+B(x)$, and $z(x,...
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1answer
32 views

How to to choose theta value when calculating hypothesis on linear regression?

I am new to data science and my math skills are really rusty. I am try to understand linear regression, but unfortunately there is one thing that is not clear to me. Assuming I have these data (or ...
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0answers
39 views

x,y,z regression [linear and logarithmic] relaions

Say we have the following set of data: We know that for a fixed value of $x$, $z=Ay+B$ [linear]. We know that for a fixed value of $y$, $z=C \ln(x)+D$ [logarithmic]. What could be the form of $...
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6 views

Regression Analysis in Expenditure based problems [migrated]

I have a doubt regarding prediction of expenditure of a single person's data. For example, if we have data of expenditure of 5 days say 'food', can we predict the the expense of 6th day using ...
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2answers
29 views

least square estimates to coefficient of analytic regression curve

This is a bit of an embarrassing question because I thought I knew much (and have been using linear regression since forever) about regression until I read something in a text that challenged this. ...
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0answers
12 views

In the simple linear regression model, ˆb is the sum of independent normally distributed random variables.

$y = \hat{a} + \hat{b}x$ is the regression model equation. is it true that .''In the simple linear regression model, $\hat{b}$ is the sum of independent normally distributed random variables? and why?
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19 views

Form a linear model based on this setting

Let $Y_1,.., Y_4$ be independent random variables with normal distributions and $Y_i \sim N(i\beta, \sigma^2) \quad (i=1,,..,4).$ 1) Form a linear model based on this setting. 2) Form an $F$ test ...
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17 views

The Distribution of Residual Sum of Squares (RSS)

I understand the R.S.S to be a measure of the discrepancy between observed data and an estimation model. My current Statistics course has a section based upon the distribution of the Residual Sum of ...
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1answer
23 views

Finding the constants in sigmoid equations

We have a sigmoid with the form $$y=\frac{e^{x+a}}{b+ce^{x+d}}$$ where $a,b,c,d$ are real constants. We have known points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ all satisfying the equation of ...
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0answers
16 views

Show that the components of the vector Y are mutually independent

Suppose that Y is a d-dimensional multivariate normal random variable with mean μ and covariance matrix D. If D is a diagonal matrix, show that the components of Y are mutually independent. My ...
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1answer
19 views

Ordinary Least Squares with right hand size all zeros

I have a system of equations $X\beta = 0$. The standard Ordinary Least Squares solution to $y = X\beta$: $$ \hat\beta = (X^TX)^{-1}X^Ty $$ then only provides the trivial $\hat\beta = 0$ solution. ...
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0answers
11 views

Hard time understanding the parameters of a prediction interval

I stumpled upon the definition of a prediction interval from https://www.statisticshowto.datasciencecentral.com/prediction-interval/ In my setup I have a type of tree-based regression model with non-...
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1answer
42 views

Least Square Error and how to find it

I am working on this problem and having trouble starting. Consider the regression model $$y_t = \beta_1 y_{t-1}+e_t$$ where $e_t$ is the white noise with zero-mean and variance $\...
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0answers
18 views

Convergence rate of linear regression without chain rule

To minimize an MSE, a common method is to perform a gradient descent on the objective. For example, the derivative is: $\frac{d}{dw} \sum_{i=1}^n (t_i - w x_i)^2 = \sum_{i=1}^n 2 (t_i - w x_i) x_i$. ...
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1answer
18 views

Is this a typo in this econometric exercise?

I'm trying to solve an exercise in OLS estimator: I'm not sure if $\mathrm{E}(u | \boldsymbol{x}, q)=0$ is a typo. Moreover, $\mathrm{E}(u | \boldsymbol{x})=0$ is used in the solution. Should it ...
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0answers
8 views

comparing effect of dummies to continuous explanatory variables in linear regression

I have a 30 explanatory variables, among which are some dummy variables. The dependent variable is in log format and so are almost all the explanatory variables. So my model looks like: ...
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0answers
20 views

Computing the variance of $Y$ given $Y_i=a+bX_i+u_i$, where the distribution of $u_i$ is known

I'm working on the following problem, which I quote verbatim from the assignment. Suppose that in the regression equation $$Y_i=a+bX_i+u_i$$ the coefficients are $a=1$ and $b=2$. Suppose also ...
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0answers
11 views

Definition of the spectral expansion

I'm from Germany and neither heard or read about spectral expansion. I saw it in Hall and Horowitz, Functional linear regression. Does the term just summarize all expansion types, like Fourier-series-...
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17 views

Formulate loss function to perform regression with a catmull-rom curve

I'm trying to use robust regression (https://scipy-cookbook.readthedocs.io/items/robust_regression.html), to fit a parametric curve like the catmull-rom curve to a set of unordered noisy data points ...
1
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1answer
28 views

How does it follow that $\operatorname{Var} (g({X}) {\epsilon} | {X}) = (g({X}) )\operatorname{Var} ( {\epsilon} | {X}) (g({X}) )^{\prime}$?

Good morning, I'm reading lecture slides bout the BLUE properties of OLS estimator. Conditional unbiasedness Conditional variance My question: I have two equalities from the two slides: $$...
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0answers
22 views

Question regarding linear regression weighting matrix

Consider the linear regression model $$b = Xy + e, \quad E[e] = 0, \quad E[ee'] = V$$ Assume that the matrix $X$ has linearly independent columns. It is well known that the minimum variance affine ...
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0answers
8 views

How does Locally Weighted Regression work for test sets far outside the train set bound

I was following CS229 machine learning course where I came across the Locally Weighted Regression algorithm. We have to minimize $\sum\limits_{i}w^i(y^i - \theta^T x^i)$ and output $\theta^Tx$ ...
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1answer
30 views

Prove this formula about residuals in case there is intercept in the OLS estimator

I'm learning OLS estimator with difficulty with computing the $R^2$. First are the notations used in my lecture note: $X_{i} \equiv\left(\begin{array}{c}{X_{i 1}} \\ {X_{i 2}} \\ {\vdots} \\ {X_{i K}}...
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0answers
13 views

Prove this formula for $R^2$ in the case there is one explanatory variable in the OLS estimator

I'm learning OLS estimator with difficulty with computing the $R^2$. First are the notations used in my lecture note: $X_{i} \equiv\left(\begin{array}{c}{X_{i 1}} \\ {X_{i 2}} \\ {\vdots} \\ {X_{i K}}...
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0answers
15 views

multiple regression

Given a regression model $y_i=\beta_0 + \beta_1 x_i +\beta_2 (x^2 -2)+ \epsilon_i$ for the sample values of $x_1=-1, x_2=0, x_3=1, x_4=1, x_5=0, x_6=-1.$ Prove that the estimators $\beta_0$ and $\...
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1answer
14 views

Why use residual plots in linear regression for assessing normality?

Let's take the case of condition simple linear regression for example where we are assuming: $$Y|X=x = \beta_0 + \beta_1 x + \epsilon,$$ where $\epsilon$ represent the random noise. In order to ...
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0answers
48 views

Unbiased estimators for linear regression

Consider the following linear regression model $Y = Xw + \epsilon$, where $w \in \mathbb{R}^d$, $X \in \mathbb{R}^{(n \times d)}$, and $\epsilon \sim N(0,\sigma^2I)$. We want to show that if for $...
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0answers
32 views

Show that W is a Linear Estimator.

Hello I need help doing a proof. I need to show that W is a linear estimator of the following equation don't now where to start. ...