Podcast Beta
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 __________.
Signup and view all the answers
What is the formula for the OLS estimator β^1 in simple linear regression?
Signup and view all the answers
Match the following terms with their definitions:
Signup and view all the answers
The matrix notation for a linear regression model is required to understand the lecture.
Signup and view all the answers
What is the purpose of an ordinary least squares (OLS) estimate?
Signup and view all the answers
What is the rewritten form of the OLS estimator?
Signup and view all the answers
The OLS estimator is a constant value.
Signup and view all the answers
What is the relationship between the OLS estimator and the true parameters?
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 ____________________.
Signup and view all the answers
What is the purpose of conducting a simulation study in R?
Signup and view all the answers
The distribution of the OLS estimator β^1 depends on the sample size.
Signup and view all the answers
Match the following terms with their definitions:
Signup and view all the answers
What is the effect of increasing the sample size on the estimation of the OLS estimator?
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Learn about the simple linear regression model, its assumptions, and representation. Understand the relationship between dependent and explanatory variables.
