Notes2, revised

Notes2, revised - Introduction to Regression We have...

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Unformatted text preview: Introduction to Regression We have considered the relationship between two variables X and Y with the covariance and correlation. Economics frequently tries to explain the relationship between two variables, such as consumption and GDP, or age and wealth. Econometrics attempts to explain the relationship between two or more explanatory variables and a dependent variable. However, we do not claim that our models “explain” the dependent variable in terms of an independent variable are any more than when we say that the weatherman explains what will happen to the weather. That is, correlation does not imply causation. A useful introduction to regression is to consider the following. Suppose you want to predict how much money a student here at UCB earns in a year. Furthermore, suppose you are given the following data. Annual Earnings Years Work Experience $20,000 1 $15,000 1 $25,000 2 $35,000 5 $12,500 1 $14,750 1.5 $17,500 3 $19,000 3 $21,000 4 $45,000 12 $36,500 8 $32,000 8 $18,000 1 $47,852 15 $22,000 2 $25,124 2 $31,000 6 $21,250 1.5 $14,200 1 $39,000 8 If you are given no additional information and you want to estimate how much a UCB student earns, what would be your estimate? Given the above descriptive statistics discussion, you may look at measures of location. Mean=$25,584 Median=$21,625 Mode=N/A You may even construct a model to estimate earnings, such as: This simple model says your best earnings estimate, E-hat, is mean earnings plus a random error term, ε . However, we assume the E( ε )=0, hence, your best estimate for any student’s earnings is just the mean earnings of all students. Using mean earnings may be a good estimate of student earnings, but you may be given more information than just earnings. You may also be given years of work experience. This additional information may improve your estimated earnings. For example, if we have two students and one is just out of high school with one year of work experience and the second is returning to finish their degree at night and has 12 years of work experience, who do you think earns more? You may very well predict that the student with 12 years work experience earns more because they have accumulated skills that are rewarded with higher earnings. They’ve been in the labor force long enough to establish labor force connections or have more training. Whatever the reason, we’ll predict the association between work experience and earning is positive. ε + = E E ˆ Recall y=mx+b from your algebra courses. Statistics measures the relationship as y=mx+b+ ε . (Notice, statistics uses α to represent the y-intercept and β to represent the slope coefficient.) Your math courses suggest such a relationship between y and x is deterministic. That is, if I know x, I know y with certainty. Statistics is less certain of this relationship. We may believe the correct model specification is y=mx+b, but we know we live in a probabilistic world, not a deterministic world. In other words, y=mx+b...
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This note was uploaded on 08/06/2009 for the course ECON 140 taught by Professor Duncan during the Spring '08 term at Berkeley.

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Notes2, revised - Introduction to Regression We have...

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