Class Outline The Meaning of Regression Data The Population Regression Function (PRF) Stochastic Specification of the PRF The Sample Regression Function (SRF) The Nature of the Stochastic Error Term Reading: Chapter 1 and2 Textbook The Meaning of Regression
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Reading: Chapter 1 and2 Textbook
Regression analysis is concerned with the study of the relationship between one variable called explained, or dependent, variable and one or more other variables called independent, or explanatory, variables.
Statistical versus Deterministic Relationships
We are concerned with what is known as the statistical, not functional or deterministic, dependence among variables. We deal with random or stochastic variables
Regression versus Causation
Regression does not imply causation. We need a theory to explain causation.
Regression versus Correlation
Correlation measure the strength or degree of linear association between two variables
Regression estimates or predict the average value of one variable on the basis of the fixed values of other variables
The dependent variable is assumed to be statistical, random, or stochastic. The explanatory variables are assumed to have fixed values
Example: assume that we want to estimate the average consumption of 60 families in a community
Population = 60 families
We want to analyze the relationship between the Consumption expenditure of each family (Y) depending on the level of income (X)
The PRL gives the average, or mean, value of the dependent variable corresponding to each value of the independent variable, in the population as a whole
Since the PRL is approximately linear we can express it mathematically
The PRL is a line that passes through the conditional means of Y. The mathematical equation is called Population Regression Function (PRF)
Where 1 and 2 are the parameters of the model.
By linear, we mean linearity on the parameters.
We can express the deviation of an specific Yi around its expected value as
Where the deviation ui is an unobservable random variable taking positive or negative values known as the stochastic disturbance (error term)
This specification has two main parts:
If we take the expected value of the PRF, we obtain the following
The error term contains all the factors explained by other variables. Why not to include other variables?
where = estimator of E(Y/Xi) the estimator of the population conditional mean
=estimator of 1
=estimator of 2
Not all the sample data lie exactly on the respective sample regression line. Then, we need to develop the stochastic model, which we write as
where = estimator of ui
represents the difference between the actual Y values and their estimated values from the sample regression, that is
In solving this estimation problem we do not observe 1, 2 and u. What we observe are their proxies