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Questions and Answers
What is the dependent variable in a simple linear regression model?
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 __________.
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What is the formula for the OLS estimator β^1 in simple linear regression?
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Match the following terms with their definitions:
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The matrix notation for a linear regression model is required to understand the lecture.
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What is the purpose of an ordinary least squares (OLS) estimate?
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What is the rewritten form of the OLS estimator?
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The OLS estimator is a constant value.
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What is the relationship between the OLS estimator and the true parameters?
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The OLS estimate is a realization of the OLS estimator, i.e. the value for particular draws of ____________________ and ____________________.
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What is the purpose of conducting a simulation study in R?
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The distribution of the OLS estimator β^1 depends on the sample size.
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Match the following terms with their definitions:
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What is the effect of increasing the sample size on the estimation of the OLS estimator?
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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.