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# Econometrics: Simple Linear Regression Model

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@StatelyHelium

### What is the primary purpose of the disturbance term in the simple linear regression model?

To model the unobserved influences on the dependent variable

(β0, β1)

y^ = β^0 + β^1x

### What is the formula for the ordinary least squares (OLS) estimator in the simple linear regression model?

<p>β^1 = cor(x, y) / (sd(x) * sd(y))</p> Signup and view all the answers

### What is the purpose of the residuals in the simple linear regression model?

<p>To measure the difference between the observed and predicted values of the dependent variable</p> Signup and view all the answers

### What is the notation for the matrix of explanatory variables in the linear regression model in matrix notation?

<p>X = (1, ..., 1, x1, ..., xT)^T</p> Signup and view all the answers

### What is the assumption made about the distribution of the disturbance term in the simple linear regression model?

<p>The disturbance term is assumed to be normally distributed</p> Signup and view all the answers

### What is the purpose of the OLS estimator in the simple linear regression model?

<p>To minimize the sum of squared residuals</p> Signup and view all the answers

### What is the OLS estimator in the context of linear regression?

<p>A linear transformation of the true parameters and the disturbance</p> Signup and view all the answers

### What is the purpose of conducting a simulation study in econometrics?

<p>To understand the distribution of the OLS estimator</p> Signup and view all the answers

### What is the difference between the OLS estimator and the OLS estimate?

<p>The OLS estimator is a random variable, while the OLS estimate is a fixed value</p> Signup and view all the answers

### What is the formula for the OLS estimator in simple linear regression?

<p>β^ = (X′X)−1X′y</p> Signup and view all the answers

### What is the effect of increasing the sample size on the distribution of the OLS estimator?

<p>The distribution of the OLS estimator becomes less dispersed</p> Signup and view all the answers

### What is the standard error of the OLS estimator in simple linear regression?

<p>The square root of the variance of the disturbance</p> Signup and view all the answers

### What is the purpose of the simulation study in R?

<p>To conduct a Monte-Carlo simulation of the ice cream model</p> Signup and view all the answers

### What is the relationship between the OLS estimator and the true parameters?

<p>The OLS estimator is both unbiased and consistent</p> Signup and view all the answers

## Study Notes

### Simple Linear Regression Model

• A simple linear regression model satisfies the relationship yt = β0 + β1xt + εt for all observations, where yt is the dependent variable, xt is the explanatory variable, and εt is a random variable that describes unobserved influences.
• β0 and β1 are the true coefficients, and εt is the disturbance (or error term).
• We will typically make some assumptions on the distribution of εt.

### Estimate, Predicted Value, and Residuum

• β^ is an estimate of the true parameter vector β.
• The predicted values (or fitted values) of y are given by y^ = β^0 + β^1x.
• The residuals (estimated values of the disturbance) are given by ε^ = y - y^ = y - β^0 - β^1x.
• The residuals ε^ are close to the true disturbances ε if our estimate β^ is close to the true parameters β.

### Ordinary Least Squares (OLS) Estimation

• An OLS estimate minimizes the sum of squared residuals.
• The OLS estimator β^ is given by the formula β^ = argmin Σ ε^2.
• In simple linear regression, the OLS estimator has the formula β^1 = Cov(xt, yt) / Var(xt).

### Linear Regression Model in Matrix Notation

• The linear regression model can be written in matrix notation as y = Xβ + ε, where X is a matrix of explanatory variables.
• The OLS estimator β^ is given by β^ = (X'X)^-1X'y, where X' is the transpose of X.

### Estimators and Estimates

• The OLS estimator β^ is a linear transformation of the true parameters β and the disturbance ε.
• The OLS estimator β^ is a random variable, and the OLS estimate β^ is a realization of the OLS estimator for particular draws of ε.
• To understand econometrics and statistics, it's essential to distinguish between an estimator (a random variable) and an estimate (a realization of the estimator).

### Distribution of OLS Estimator

• A Monte-Carlo simulation study shows that the OLS estimator β^ has a distribution that depends on the sample size T.
• The distribution of β^ changes if we change the sample size T.

### Standard Error of OLS Estimator

• In a simple linear regression, the standard deviation of the OLS estimator β^ can be estimated by σ / sqrt(Σ (xi - x̄)^2), where σ is the standard deviation of εt.

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## Description

Learn about the simple linear regression model, its components, and the endogeneity problem in market analysis with econometrics and machine learning.

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