Mastering Linear Regression

The purpose of linear regression in econometrics is to find the relation between two variables, one being the dependent variable and the other being the independent variable, in order to predict and analyze their values.

The regression coefficients in linear regression, denoted as β1 and β2, are unknown but fixed parameters that represent the intercept and slope of the regression line.

The stochastic disturbance term, represented as ui, is the error term in regression models that accounts for all the omitted or neglected variables that may affect the dependent variable but are not included in the model.

<p>The conditional expected values in linear regression are the mean values of the dependent variable (Y) given the values of the independent variable (X). They are denoted as E(Y | X).</p> Signup and view all the answers

<p>The population regression curve in linear regression is obtained by plotting the conditional mean values of the dependent variable (Y) against the independent variable (X). It represents the relationship between the two variables in the population.</p> Signup and view all the answers

<p>The formula for the population regression function (PRF) in econometrics is $Y = \beta_0 + \beta_1X + u$, where $Y$ represents the dependent variable, $X$ represents the independent variable, $\beta_0$ and $\beta_1$ represent the population regression coefficients, and $u$ represents the stochastic disturbance term.</p> Signup and view all the answers

<p>The formula for the sample regression function (SRF) in econometrics is $\hat{Y} = b_0 + b_1X$, where $\hat{Y}$ represents the predicted or estimated value of the dependent variable, $X$ represents the value of the independent variable, $b_0$ and $b_1$ represent the sample regression coefficients.</p> Signup and view all the answers

<p>The first order conditions for minimizing the sum of squares of errors in econometrics require the following terms to be equal to 0: $\frac{\partial}{\partial\beta_0}(\sum_{i=1}^{n}(Y_i - \hat{Y}<em>i)^2) = 0$ and $\frac{\partial}{\partial\beta_1}(\sum</em>{i=1}^{n}(Y_i - \hat{Y}_i)^2) = 0$, where $Y_i$ represents the observed values of the dependent variable, and $\hat{Y}_i$ represents the predicted or estimated values of the dependent variable.</p> Signup and view all the answers

<p>The normal equations in econometrics are derived from the first order conditions for minimizing the sum of squares of errors. They are: $\frac{\partial}{\partial\beta_0}(\sum_{i=1}^{n}(Y_i - \beta_0 - \beta_1X_i)^2) = 0$ and $\frac{\partial}{\partial\beta_1}(\sum_{i=1}^{n}(Y_i - \beta_0 - \beta_1X_i)^2) = 0$, where $Y_i$ represents the observed values of the dependent variable, $X_i$ represents the values of the independent variable, and $\beta_0$ and $\beta_1$ represent the population regression coefficients.</p> Signup and view all the answers

<p>The assumptions of the Gaussian Model (CLRM) in econometrics are: Assumption 1 - The model must be linear in parameters. Assumption 2 - The values taken by the regressor $X$ may be considered fixed in repeated samples (the case of fixed regressor) or they may be sampled along with the dependent variable $Y$ (the case of stochastic regressor). In the latter case, it is assumed that the $X$ variable(s) and the error term $u$ are independent, that is, $Cov(X_i, u_i) = 0$.</p> Signup and view all the answers