Applying Statistics to Litigation Consulting

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Applying Statistics to Litigation Consulting

• Shawn Roeske Chin Yu

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WARNINGPELIGRO
When Using Statistical Analysis

• Pros
• Provides a basis or sanity check
• Widely accepted methodology
• Can develop conclusions on large amounts of data

• Cons
• Complex Equations
• Filled with hidden
• assumptions
• Many areas for opposing side to attack

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Points of Discussion

• Simple Regression Analysis
• When to use
• Interpreting regression output
• Time Series Analysis
• Problems Using Regression Analysis

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Why Use Regression?

• When we are using one variable to draw a
  conclusion about another variable
• Make predictions of one variable using another
• Test assumptions about the relation between
  variables
• Quantify the strength of the relationship between
  variables

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Linear Regression Equation
Where, i1, , n Yi Dependent Variable
b0 Intercept b1 Slope Coefficient Xi
Independent Variable ei Error term
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Linear Regression Equation (cont.)

• Linear regression assumes a linear relationship
  between the dependent and independent variables
• Linear regression computes a line that best fits
  the observations it chooses values for the
  slope, b0, and intercept, b1, that minimize the
  sum of the squared vertical distances between the
  observations and the regression line

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Sample Regression Output
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Assumptions of The Linear Regression

• A linear relation exists between the dependent
  variable and the independent variable
• The independent variable is not random
• The expected value of the error term is 0
• The variance of the error term is the same for
  all observations
• The error term is uncorrelated across
  observations
• The error term is normally distributed

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What did you just say those assumptions were?

• Assumption 1
• If the independent and dependent variables DO NOT
  have a linear relation, then estimating that
  relation with a regression model will be INVALID
• Assumptions 2 3
• Needed to ensure that the linear regression
  produces the correct estimates of b0 b1
• Assumptions 4
• Is also known as the HOMOSKEDASTICITY assumption
  or having equal variances
• Assumptions 5
• Necessary for correctly estimating the variances
  of the estimated parameters
• Assumptions 6
• Allows us to easily test a particular hypothesis
  about a linear regression model

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Sample Regression Output
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Coefficients

• Coefficients correspond to the bs in a standard
  linear regression model
• Y b0 b1X1 ei
• In our example, b1 422.09
• This means for every time period that passes, in
  this case quarters, revenues will increase by
  422.09
• The standard error measures the precision of the
  coefficients as an estimate of the model
  parameter

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Sample Regression Output
y 13950.75 422.09x
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Sample Regression Output
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R-Squared (R2) or Coefficient of Determination

• Goodness of fit for the regression model
• Measures the proportion or of the total
  variation in the dependent variable explained by
  the model

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R-Squared (R2) or Coefficient of Determination
Where, RSS Regression Sum of Squares TSS
Total Sum of Squares
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Sample Regression Output
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Caveats for Using R2

• A high R2 does not imply causality
• Just because your R2 is high does not mean your
  regression is reliable
• What does it mean? It means you must look at
  other factors along with your R2

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Standard Error Estimate

• The Standard Error Estimate (SEE) or Standard
  Error of the Regression measures the standard
  deviation of the error estimate
• SEE computes the difference between the actual
  and predicted values for each dependent variable
  observation

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Standard Error Equation
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Standard Error Equation (cont.)
Where, n observations SSR Sum of
Squared Residuals
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Sample Regression Output
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Testing t-Statistic

• A t-test is used to test the significance of
  individual estimated coefficients
• i.e. it tests whether a single regression
  coefficient is significantly different from zero

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Testing t-Statistic (cont)

• The critical t-value is obtained by using a
  t-distribution table and applying an error rate
  and the correct df (df n-2 in simple
  regression)
• The t-stat in our example is 8.81 which is
  greater than the critical value of 2.145

t - Distribution
a 1 - .95 .05
Critical Region at a/2 .025
0
Critical t -2.145
Critical t 2.145
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Sample Regression Output
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Confidence Intervals

• A confidence interval is an interval that we
  believe includes the true parameter value, b1,
  with a given degree of confidence
• To compute a confidence interval, we must
• Select the significance level for the test
• Know the standard error of the estimated
  coefficient

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Confidence Interval Equation
Where, CI Confidence Interval b1
Coefficient of the Independent Variable tc
Critical t-value Sb1 Standard Error of the
Coefficient
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Sample Regression Output
Look in Students- t table
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Types of Data Used

• Cross-sectional Data
• Observations of the independent and dependent
  variables for the same time period
• Times-series Data
• Observations of the independent and dependent
  variables over time

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Time-Series Analysis

• A Time-series is a set of values of a particular
  variable in different time periods (in this case
  time is the independent variable)
• Can be used in forecasting by estimating a linear
  trend in a time-series and using that trend to
  predict future values

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Linear Trend

• The simplest type of trend is a linear trend
  which is expressed in the following equation.

Were regressing time (independent) with the
desired variable (dependent)
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Seasonality

• Seasonality is shows a regular pattern of
  movements in a given time period
• If significant seasonality exists, we can correct
  this by adding a seasonal lag

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Potential Problems

• Heteroskedasticity
• Variance of the errors differs across
  observations
• Causes relationships to exist when they really do
  not
• Causes incorrect standard errors

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Sample Regression Output
Linear Pattern
Linear Pattern
No Pattern
Exponential Pattern
Parabolic Pattern
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Potential Problems (cont.)

• Serial Correlation (a.k.a. Autocorrelation)
• Regression errors are correlated across
  observations
• Causes incorrect standard errors
• Parameters are accurate as long as none of the
  independent variables is a lagged value of the
  dependent variable

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Detecting Problems

• Heteroskedasticity
• Breusch and Pagan Test
• Serial Correlation
• Durbin-Watson Test Statistic
• EXCEL DOES NOT PERFORM
• EITHER TEST!!!

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Caveats Using Regression Analysis

• Your estimate is only relevant for the time
  period you are looking at, but it may be the best
  estimator you have for the future
• Excel works great if you want a quick dirty
  regression, but has many limitations, for example
  Excel does not contain tools to test for
  heteroskedasticity and autocorrelation

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Questions
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Sources

• http//www.ats.ucla.edu/stat/
• http//www.surveysystem.com
• Basic Econometrics, Gujarati, Damodar N., McGraw
  Hill, 1995.
• The Basic Practice of Statistics, Moore, Dennis
  S., W.H. Freeman and Company, 1995.
• Quantitative Methods for Investment Analysis,
  Defusco, McLeavey, Pinto, Runkle, Association for
  Investment Management and Research, 2001