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,ECONOMETRICS TOPICS
Chapter 1: An Overview of Regression Analysis
What is econometrics?
Econometrics: economic measurement
Uses of econometrics:
1. Describing economic reality
2. Testing hypothesis about economic theory
3. Forecasting future economic activity
Alternative economic approaches
Steps necessary for any kind of quantitative research:
1. Specifying the models or relationships to be studied
2. Collecting the data needed to quantify the models
3. Quantifying the models with the data
Single-equation linear regression analysis is one particular economic
approach that is the focus of this book.
What is regression analysis?
Dependent variables, independent variables, and causality
Regression analysis: a statistical technique that attempts to explain
movements in one variable, the dependent variable, as a function of
movements in a set of other variables, called the independent (or
explanatory) variables, through the quantification of a single equation.
A regression result, no matter how statistically significant, cannot prove
causality. All regression analysis can do is test whether a significant
quantitative relationship exists.
Single-equation linear models
Betas: the coefficients that determine the coordinates of the straight line at
any point.
Beta-null: the constant or intercept term; it indicates the value of Y when
X equals zero.
Beta-one: the slope coefficient; it indicates the amount that Y will change
when X increases by one unit.
An equation is linear in the variables if plotting the function in terms of X
and Y generates a straight line.
An equation is linear in the coefficients only if the coefficients appear in
the simplest form – they are not raised to any powers are not multiplied or
divided by other coefficients, and do not themselves include some sort of
function.
The stochastic error term
Stochastic error term: a term that is added to a regression equation to
introduce all of the variation in Y that cannot be explained by the included
X’s.
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, The deterministic component: B 0 + B1X; can be thought of as the expected
value of Y given X, the mean value of the Ys associated with a particular
value of X.
The stochastic error term must be present in a regression equation because
there are at least four sources of variation in Y other than the variation in
the included Xs:
1. Many minor influences on Y are omitted from the equation (for
example, because data are unavailable).
2. It is virtually impossible to avoid some sort of measurement
error in at least one of the equation’s variables.
3. The underlying theoretical equation might have a different
functional form than the one chosen for the regression. For
example, the underlying equation might be nonlinear in the
variables for a linear regression.
4. All attempts to generalize human behavior must contain at least
some amount of unpredictable or purely random variation.
Extending the notation
The meaning of the regression coefficient beta-one: the impact of a one
unit increase in X-one on the dependent variable Y, holding constant the
other included independent variables.
Multivariate regression coefficients: serve to isolate the impact on Y of a
change in one variable from the impact on y of the changes in the other
variables.
The estimated regression equation
Estimated regression equation: a quantified version of the theoretical regression
equation
Estimated regression coefficients: empirical best guesses of the true regression
coefficients and are obtained from a sample of the Xs and Ys; denoted by beta-
hats
A simple example of regression analysis
Using regression to explain housing prices
Chapter 2: Ordinary Least Squares
Estimating single-independent-variable models with OLS
Ordinary least squares (OLS): a regression estimation technique that calculates
the beta-hats so as to minimize the sum of the squared residuals.
Why use ordinary least squares?
1. OLS is relatively easy to use.
2. The goal of minimizing the sum of the squared residuals is quite
appropriate fro a theoretical point of view.
3. OLS estimates have a number of useful characteristics:
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, a. The estimated regression line goes through the means of Y and
X. That is, if you substitute Y-bar and X-bar into the equation
it holds exactly.
b. The sum of the residuals is exactly zero.
c. OLS can be shown to be the best estimator possible under a set
of fairly restrictive assumptions.
Estimator: a mathematical technique that is applied to a sample of data to
produce real-world numerical estimates of the true population regression
coefficients (or other parameters). OLS is an estimator.
How does OLS work?
Regression model equation
Estimate equations of beta-one and beta-null
Total, unexplained, and residual sum of squares
Total sum of squares (TSS): the squared variations of Y around its mean
as a measure of the amount of variation to be explained by the regression.
Explained sum of squares (ESS): measures the amount of the squared
deviation of Yi from its mean that is explained by the regression line.
Residual sum of squares (RSS): the unexplained portion of the total sum
of squares.
OLS minimizes the RSS and therefore maximizes the ESS.
An illustration of OLS estimation
Estimating multivariate regression models with OLS
The meaning of multivariate regression coefficients
Multivariate regression coefficient: indicates the change in the dependent
variable associated with a one-unit increase in the independent variable in
question holding constant the other independent variables in the equation.
OLS estimate of multivariate regression models
Estimate equations for multivariate regression coefficients
An example of a multivariate regression model
Evaluating the quality of a regression equation
1. Is the equation supported by sound theory?
2. How well does the estimated regression as a whole fit the data?
3. Is the data set reasonably large and accurate?
4. Is OLS the best estimator to be used for this equation?
5. How well do the estimated coefficients correspond to the expectations
developed by the researcher before the data were collected?
6. Are all the obviously important variables included in the equation?
7. Has the most theoretically logical functional form been used?
8. Does the regression appear to be free of major econometric problems?
Describing the overall fit of the estimated model
R-squared, the coefficient of determination
Coefficient of determination: the ratio of the ESS to the TSS
Equation for R-squared
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