Simple Linear Regression Model
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Questions and Answers

What is the dependent variable in a simple linear regression model?

  • y (correct)
  • x
  • β
  • ε

    • The residuals are exactly equal to the true disturbances.

    False

    What is the predicted value of y given an estimate of the parameter vector β?

    y^ = β^0 + β^1x

    An ordinary least squares (OLS) estimate minimizes the sum of squared __________.

    <p>residuals</p> Signup and view all the answers

    What is the formula for the OLS estimator β^1 in simple linear regression?

    <p>cor(x, y) / sd(x)</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Dependent variable = y Explanatory variable = x Disturbance = ε Estimated value of the disturbance = ε^</p> Signup and view all the answers

    The matrix notation for a linear regression model is required to understand the lecture.

    <p>False</p> Signup and view all the answers

    What is the purpose of an ordinary least squares (OLS) estimate?

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

    What is the rewritten form of the OLS estimator?

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

    The OLS estimator is a constant value.

    <p>False</p> Signup and view all the answers

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

    <p>The OLS estimator is a linear transformation of the true parameters.</p> Signup and view all the answers

    The OLS estimate is a realization of the OLS estimator, i.e. the value for particular draws of ____________________ and ____________________.

    <p>y and ε</p> Signup and view all the answers

    What is the purpose of conducting a simulation study in R?

    <p>To compare the distribution of the OLS estimates with the true value of β</p> Signup and view all the answers

    The distribution of the OLS estimator β^1 depends on the sample size.

    <p>True</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Estimator = A random variable Estimate = A realization of the estimator Standard Error = The estimated standard deviation of the estimator OLS = Ordinary Least Squares</p> Signup and view all the answers

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

    <p>We can estimate the OLS estimator more precisely.</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 describing unobserved influences.
    • The model assumes some distribution for εt, and uses the letters ε, u, and η to denote disturbances.
    • The model is represented by a vector of true coefficients β = (β0, β1), and data y = (y1,..., yT ) and x = (x1,..., xT ).

    Estimate, Predicted Value, and Residuum

    • An estimate of the true parameter vector β is denoted by β^.
    • The predicted values (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 are close to the true disturbances if the estimate β^ is close to the true parameters.

    Ordinary Least Squares (OLS) Estimation

    • The OLS estimate minimizes the sum of squared residuals.
    • The OLS estimator for simple linear regression has the formula β^ = argmin Σ t=1 T ε^2t.
    • The formula can be rewritten in terms of empirical correlation and standard deviation: β^1 = cor(x, y) * sd(y) / sd(x).

    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 is then given by β^ = (X'X)^-1X'y.
    • The matrix notation is not required for understanding this lecture.

    Estimators and Estimates

    • The OLS estimator can be rewritten as β^ = β + (X'X)^-1X'ε, showing that it is a linear transformation of the true parameters and disturbance.
    • The OLS estimator is a random variable, and the OLS estimate is a realization of the OLS estimator.
    • An estimator is a random variable, and an estimate is a realization of that estimator.

    Monte-Carlo Simulation in R

    • A Monte-Carlo simulation can be performed in R to analyze the OLS estimator.
    • The simulation involves estimating the demand function using a simple linear regression model.
    • The estimated coefficients are stored and plotted to show their distribution, which changes depending on the sample size.

    Distribution of the OLS Estimator

    • The OLS estimator has a distribution that depends on the sample size.
    • The distribution of the estimator is shown for different sample sizes and true values of β1.

    Standard Error of the OLS Estimator

    • The standard deviation of the OLS estimator can be estimated by (Σ t=1 T ε^2t / (T - 2))^(1/2).
    • This estimate is called the standard error of the OLS estimator.
    • The standard error decreases with a larger sample size and more variation in the explanatory variable.

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    Description

    Learn about the simple linear regression model, its assumptions, and representation. Understand the relationship between dependent and explanatory variables.

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