Basic Econometrics

1
Basic Econometrics

• Literally defined as economic measurement
• Econometrics may be defined as the social science
  in which the tools of economic theory,
  mathematics, and statistical inference are
  applied to the analysis of economic phenomena.
  Arthur S. Goldberger, 1964

2
Ordinary Least Squares

• Ordinary Least Squares a statistical method that
  chooses the regression line by minimizing the
  squared distance between the points and the line.
• i.e. minimize the sum of S, where S ?e2
• where e the distance between the line and the
  point, or the error term.
• Error term (e) random, included because we do
  not expect a perfect relationship.
• Why not sum the errors? Generally equals zero
• Why not take the absolute value of the errors? We
  wish to emphasize large errors.

3
Source of Error

• Omitted variables
• Measurement error
• Incorrect functional form

4
Univariate Analysis

• Y a bX Where
• Y The Dependent Variable, or what you are
  trying to explain (or predict).
• X The Independent Variable, or what you believe
  explains Y.
• a the y-intercept or constant term.
• b the slope or coefficient

5
The t-statistic

• To evaluate the quality of our slope coefficient
  we refer to the
• t-statistic.
• T-statistic slope coefficient / standard error
• Standard error - estimated standard deviation of
  the coefficient
• Rule of thumb T-stat gt 2
• Why? We want to know whether or not the
  coefficient is statistically different from zero.
• The t-statistic is used to test the null
  hypothesis (H0) that the coefficient is equal to
  zero. The alternative hypothesis (HA) is that the
  coefficient is different than zero.

6
The slope coefficient

• How do we interpret the slope coefficient?
• Example
• Wins -0.1508 0.0063PTS per game
• Each additional one point per game results in a
  0.0063 increase in winning percentage.
• How many wins is this? 0.513 over an 82 game
  season.
• Is this the truth? We never know the truth, we
  are simply attempting to derive estimates.
• Is this a good estimate? Clearly points alone
  do not explain wins.

7
The constant term

• How do we interpret the constant term?
• The constant term must be included in the
  regression, or else we are forcing the regression
  line through zero.
• The constant term is used to impose a zero mean
  for the error term, hence it acts as a garbage
  collector. In other words, it captures all the
  factors not explicitly utilized in the equation.
• The constant term is theoretically the value of Y
  when X is zero. Frequently this is outside the
  range of possibility, and therefore the constant
  term should not be interpreted.

8
Multivariate Analysis

• Introducing the idea of ceteris paribus.
• One cannot impose ceteris paribus unless all
  relevant variables are included in the model.
• Wins a b(ORB)
• As illustrated earlier, the estimated impact of
  offensive rebounds (ORB) on wins is negative.
• In other words, b lt 0
• Wins c d(Missed Shots)
• The value of d lt 0
• ORB e f(Missed Shots)
• the value of f gt 0
• In other words, missed shots and offensive
  rebounds are positively related. So when we
  estimate wins as a function of offensive
  rebounds, we are simply picking up the
  relationship between wins and missed shots.

9
R-squared

• How do we know how accurate our equation is?
• R-squared Explained Sum of Squares / Total Sum
  of Squares
• Total sum of squares Sum of the squared
  difference between the actual Y and the mean of
  Y, or,
• TSS ?(Yi - mean of Y)2
• Explained sum of squares Sum of the squared
  differences between the predicted Y and the mean
  of Y, or,
• ESS ?(Y - mean of Y)2
• Residual sum of squares Sum of the squared
  differences between the actual Y and the
  predicted Y, or,
• RSS ? e2
• The coefficient of determination Ratio of the
  explained sum of squares to the total sum of
  squares. (Provide example)
• R2 ESS/TSS R2 1 - RSS/TSS
• 1 - ? e2 / ? (Yi - mean of Y)2

10
Adjusted R-Squared

• Adding any independent variable will increase R2.
  To combat this problem,we often report the
  adjusted R2.
• Adjusted R2 1 - RSS/(n-K-1) / TSS/(n-1)
• where n observations
• K number of coefficients