Statistical Analysis of Economic Relations Outline

1
Statistical Analysis of Economic RelationsOutline

• Data Summary and Description
• Measures of Central Tendency
• Measures of Dispersion
• Hypothesis Testing
• Regression Analysis
• Regression Statistics
• Additional Econometric Issues

2
Data Summary and Description

• Population Parameters Summary and descriptive
  measures for the population.
• Sample Statistics Summary and descriptive
  measures for a sample.
• NOTE We rarely have data for the population.
  Hence we need to be able to draw inferences from
  a sample.

3
Measures of Central Tendency

• Mean The average
• Issue You must note the distribution of the
  sample. If it is unbalanced the mean may be
  misleading.
• Median Middle observation
• Mode Most common value

4
Symmetrical vs. Skewness

• Symmetrical A balanced distribution.
• Median Mean
• Skewness A lack of balance.
• Skewed to the left Median gt Mean
• Skewed to the right Median lt Mean
• If skewness is observed one may with to examine a
  sub-sample of the data or consider a different
  distribution in estimating the model.

5
Measures of Dispersion

• Range Difference between the largest and
  smallest sample observations.
• Only considers the extremes of the sample.
• Used to identify what is possible.
• Variance and Standard Deviation
• Sample Variance Average squared deviation from
  the sample mean.
• Sample Standard Deviation Squared root of the
  sample variance.

6
Coefficient of Variation

• Coefficient of variation Standard deviation
  divided by the mean.
• A measure that does not rely on the size of the
  observations or the unit of measurement.
• This is used to compare relative dispersion
  across a variety of data.

7
Hypothesis Testing

• Hypothesis Testing Statistical experiment used
  to measure the reasonableness of a given theory
  or premise
• NOTE WE DO NOT PROVE A THEORY
• Type I Error Incorrect rejection of a true
  hypothesis.
• Type II Error Failure to reject a false
  hypothesis.
• You cannot eliminate both Type I and Type II
  errors.

8
Testing an Hypothesis

• Steps in testing a hypothesis
• Formally state the basic premise or the null
  hypothesis H0
• State the alternative hypothesis HA
• Collect data
• Analyze data with respect to H0 and HA

9
Regression AnalysisDefinitions

• Regression analysis statistical method for
  describing the relationship between a dependent
  variable Y and independent variable(s) X.
• Deterministic Relation An identity
• A relationship that is known with certainty.
• Statistical Relation An inexact relation

10
Regression AnalysisTypes of Data

• Time series A daily, weekly, monthly, or annual
  sequence of data. i.e. GDP data for the United
  States from 1950 to 2001
• Cross-section Data from a common point in time.
  i.e. GDP data for OECD nations in 1986.
• Panel data Data that combines both
  cross-section and time-series data. i.e. GDP data
  for OECD nations from 1960 to 1992.

11
Steps in Regression Analysis

• Specify the dependent and independent variable(s)
  to be analyzed
• Obtain reliable data.
• Estimate the model.
• Interpret the regression results.

12
Specifying the Regression AnalysisThe choice of
independent variables

• Univariate analysis Simple regression model - A
  regression model with only one independent
  variable.
• Issue Cannot impose ceteris paribus
• Multivariate analysis Multiple regression model
  - A regression model with multiple independent
  variables.

13
The Least Squares Model

• Ordinary Least Squares a statistical method that
  chooses the regression line by minimizing the
  squared distance between the data points and the
  regression line.
• Why not sum the errors? Generally equals zero.
• Why not take the absolute value of the errors? We
  wish to emphasize large errors.

14
Estimating a Univariate ModelDefinitions

• Y a0 a1X e
• Y Dependent Variable
• a0 the constant term
• a1 slope coefficient
• e error term

15
How does OLS work? The Slope Coefficient

• OLS selects estimates of a0 and a1 so that the
  sum of squared residuals is minimized.
• a1 S(Xi - mean of X) (Yi - mean of Y) /
  S (Xi - mean of X)2
• Intuition ß1 equals the joint variation of X and
  Y (around their means) divided by the variation
  of X around its mean. Thus it measures the
  portion of the variation in Y that is associated
  with variations in X.

16
How does OLS work?The constant term

• a0 mean of Y a1 mean of X
• a0 is defined to ensure that the regression
  equation does indeed pass through the means of Y
  and X.
• The mean value of the error term is zero, which
  will only be true if the constant term is
  included.
• Note The value of the constant term is often
  outside the realm of what is possible. Hence the
  interpretation of the constant term is often
  avoided.

17
The Error Term

• Error term (e) random, included because we do
  not expect a perfect relationship.
• Sources of error
• Omitted variables
• Measurement error
• Incorrect functional form

18
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.

19
Regression Statistics

• Standard Error of the Estimate
• Coefficient of Determination
• Adjusted Coefficient of Determination
• The F-Statistic
• The t-statistic

20
Coefficient of Determination

• Coefficient of Determination Percentage of
  Y-variation explained by the regression model.
• Also referred to as R2
• R2 Variation Explained by Regression
• R2 ranges from 0 to 1.

21
R-squared

• 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
• R2 ESS/TSS R2 1 - RSS/TSS

22
Adjusted R2

• 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

23
The F-Statistic

• F-Statistic Offers evidence if explained
  variation in Y is significant.
• F-statistics are both provided and evaluated in
  Excel.

24
Judging the significance of a variable

• The t-statistic estimated coefficient / standard
  deviation of the coefficient.
• 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.
• Rule of thumb if tgt2 we believe the coefficient
  is statistically different from zero. WHY?
• Understand the difference between statistical
  significance and economic significance.

25
Multicollinearity

• Multicollinearity - more than two independent
  variables exhibit a linear correlation.
• Consequences
• Standard errors will rise, t-stats will fall
• Estimates will be sensitive to changes in
  specification
• Overall fit of regression will be unaffected

26
Other Econometric Issues

• Omitted Variable Bias You cannot impose ceteris
  paribus if relevant independent variables are not
  included in the model.
• Small Sample Bias You cannot adequately assess a
  relationship with an inadequate sample.
  Remember, we are trying to learn about the
  underlying population.

27
More Econometric Issues

• Serial Correlation violation of the assumption
  that the observations of the error terms are
  uncorrelated.
• Consequence Standard errors are underestimated.
• Primarily occurs in time series data.
• Heteroskedasticity violation of the assumption
  that the variance of the error term is constant.
• Consequence Standard errors are underestimated.
• Primarily occurs in cross-sectional data