# Questions tagged [least-squares]

Refers to a general estimation technique that selects the parameter value to minimize the squared difference between two quantities, such as the observed value of a variable, and the expected value of that observation conditioned on the parameter value. Gaussian linear models are fit by least squares and least squares is the idea underlying the use of mean-squared-error (MSE) as a way of evaluating an estimator.

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### Least squares estimators with a restriction [closed]

Assume the model $y_i = \beta_1 log(x_{1i}) + \beta_2 /x_{2i} + \epsilon_i$ with the restriction that $\beta_2 = 2 \beta_1$. Find the least-squares estimators of the regression coefficients. I am ...
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### Why am I getting incorrect predictions from ols model applied on standard scaled data set? [closed]

I have applied standardization to my dataframe with the following code: ...
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### Ordered Logistic Regression or OLS

I have five employee-level variables. Z1 = employee wage Z2 = employee tenure Z3 = employee age Y = employee perception of fairness (ordinal survey item with 5 = very fair... 1 = very unfair) X = ...
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### Why would bootstrap OLS standard errors differ from ML estimate?

Let's say I have a regression dataset (paired x and y) such that the response variable (y) has an unknown distribution (but definitely not Gaussian) and is large enough such that the central limit ...
1 vote
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### OLS assumptions for weighted errors

As far as I know, under satisfied assumptions for OLS, the estimates acquire qualities like BLUE, MVUE, MLE. But in the case where there is a priori knowledge of the influence of each data point, it ...
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### Proof of a statement about the variance of an OLS estimator

I am currently taking an econometrics course and in the lecture notes there is a statement about the variance of an OLS estimator that I am unable to prove. The statement is as follows: Suppose the ...
1 vote
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### Omitted variable problem

I'm studying the cases in which the endogeneity problem arises in OLS regression. Suppose we have the following population equation: $y=\beta_0 +\beta_1 x_1 + ... + \beta_k x_k + \gamma q + \epsilon$ ...
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### When the endogenous variable in a regression is population changes by age group, do I run a separate regression for each age group?

I am running a regression on how trade shocks affect population by age group at the county level. We bin ages into three age groups. I am confused about whether I should run one big regression, like ...
1 vote
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### If we have a binary variable in our linear regression, the VIF for its coefficient estimate uses the $R^2$ of a linear probability model. What gives?

The variance inflation factor (VIF) in an ordinary least squares linear regression coefficient is calculated using the $R^2$ of a linear model that uses the other features to predict the feature to ...
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### Different time lags for different explanatory variables

I have built a Dynamic Model with an AR(1) term and some other explanatory variables for example inflation, GDP and interest rate. I have used different time lags for the different explanatory ...
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### Testing implications of Conditional Mean Independence

In some empirical studies as a validation exercise some people regress some variables on the variable of interest controlling for key control variables. The reason for doing this I think comes from ...
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### What does it mean for $\hat\beta_1$ and $\hat\beta_0$ to have a variance?

With regarding to OLS estimators why $\hat\beta_1$ and $\hat\beta_0$ have a variance?
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### Show that $E [u] = 0$ and $cov(u,x_j)=0$ does not imply $E[u|x]=0$

In the regression model $$y= x'\beta + u, \quad x = (1, x_2,...,x_K)$$ with $$E[u |x]=0,$$ we know that it implies: $E [u] = 0$ and $cov(u,x_j)=0$, for $j=1,...,K$. I think that the reciprocal is not ...
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### Why is i.i.d. an OLS assumption?

Assume the following linear relationship: $Y_i = \beta_0 + \beta_1 X_i + u_i$, where $Y_i$ is the dependent variable, $X_i$ a single independent variable and $u_i$ the error term. According to Stock &...
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### R squared for a regression plane without observations

Assuming we have three random variables $X$, $Y$, and $Z$, and we want to estimate a least squares regression plane of the form $Z = a + bX + cY$. We do not know the individual observations, but we ...
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### Performing 3 multivariate linear regressions at once

I have 3 variables X, Y and Z. I want to perform 3 OLS regressions: X dependent on Y and Z, Y dependent on X and Z, and Z dependent on X and Y. Instead of doing the 3 of them sepparately, I want to ...
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### OLS regression with fixed effects when dependent variables are measured differently by group

I have data where each observation is a region within a country, and each observation falls into one of two large groups (each group makes up half the dataset). Each group corresponds to a seven-year ...
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### Cannot understand why total SS＝explained SS+unexplained SS

I cannot understand why total SS＝explained SS+unexplained SS because geometrically the sum of two small squares is not equal to a big square. I wish someone could explain that to me. Thank you.
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### Estimating differences slopes between groups with y as the computed slope

I'm estimating a basic OLS model, y ~ x, and I want to know the difference in the slope between two groups. Here is a simulated example in R: ...
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### How to adjust team win probability based on in in game win probability

I have a formula to calculate each team's win probability based on their win rates and home court advantage. So lets say I use it to determine that team A has 45% chance to win and team B has 55% ...
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### Least squares estimator for parameter p in binomial distribution

I am trying to find the least square estimator for the parameter $p$ in $Bin(n,p)$ but is it even possible? Isn't it the same as finding it using MLE?
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### What is the difference between standard errors and residuals in OLS?

I'm trying to get a deeper understanding of how OLS works. One thing that I thought I understood is the difference between standard errors and residuals. Here are two definitions Standard errors: The ...
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### Price elasticity of demand estimation with single store level data

I have weekly data per single SKU and for more than 500 point of sales. Data embeds base price, quanity sold, holidays, temperature, market activities (cut-price, display, leaflets) and so on. I want ...
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### Least squares problem penalising the number of non-zero coefficients

Consider a constrained least squares problem of minimizing $(y-X\beta)^T(y-X\beta)$ subject to a constraint or penalty on the number of non-zero coefficients $\beta_i$. This seems related to LASSO or ...
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### How to structure combinations of dataframes for regression, without corruption/loss?

I have a data set, redacted sample below. My goal is linear regression. My question is: Have I created unintended results, due to how I structured the df, using concat and/or div? For example, ...
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### Have I implemented WLS correctly?

I have to predict the amount of charitable donations with my dataset. The OLS model I initially used is this: ...
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### for linear regression, when is MLE the same as least squares? [duplicate]

Under what conditions is MLE the same as the least squares estimate for ordinary linear regression? I have seen statements saying that these two methods are not entirely the same. But so far, using ...
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### Interpretation of rescaled coefficients in OLS

Context To make the coefficients of regression coefficients comparable, one usually rescales by the standard deviation (see case 3 below). This means that the different regression coefficients can be ...
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### OLS - Why coefficient Beta has Normal Distribution but not t-Distribution

I have a little trouble understanding the solution to the following problem. To my understanding, coefficients in OLS should follow t-distribution. However, the solution says it follows Normal. Please ...
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### How many control variables can I use in experiment / observational study without overfitting?

Suppose I run a regression looking at the correlation of a training program on wages and I include dummies for different levels of for age, education and industry:  wage_i = \beta_0 + \beta_1 ...
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I need to answer to the following problem: In an heteroschedasticity setting, let $n$ be the index of the n-th statistical unit with $n=1, \dots, N$. Suppose a multiple linear regression setting with ...