How do lawyers set fees
Download
1 / 31

How do Lawyers Set fees? - PowerPoint PPT Presentation


  • 96 Views
  • Uploaded on

How do Lawyers Set fees?. Learning Objectives. Model i.e. “Story” or question Multiple regression review Omitted variables (our first failure of GM) Dummy variables. Model. An example of how we can use the tools we have learned

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' How do Lawyers Set fees?' - leandra-winters


An Image/Link below is provided (as is) to download presentation

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.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Learning objectives
Learning Objectives

  • Model i.e. “Story” or question

  • Multiple regression review

  • Omitted variables (our first failure of GM)

  • Dummy variables


Model
Model

  • An example of how we can use the tools we have learned

  • Simple analyses that don’t have a complicated structure can often be useful

  • Question: Lawyers claim that they set fees to reflect the amount of legal work done

  • Our suspicion is that fees are set to reflect the amount of money at stake

    • Form of second degree price discrimination


Model1
Model

  • How to translate a story into econometrics and then test the story?

  • Our Idea: Fees are determined by the size of the award rather than the work done

    • Percentage fees

    • Price discrimination

  • Careful to consider alternatives: Insurance


Analysis
Analysis

  • As always summarize and describe the data

  • Graph variables of interest (see over)

  • Regression to find percentage price rule


How do lawyers set fees

reg ins_allow award

Source | SS df MS Number of obs = 91

-------------+------------------------------ F( 1, 89) = 133.05

Model | 2.7940e+09 1 2.7940e+09 Prob > F = 0.0000

Residual | 1.8689e+09 89 20999331.5 R-squared = 0.5992

-------------+------------------------------ Adj R-squared = 0.5947

Total | 4.6629e+09 90 51810441.4 Root MSE = 4582.5

------------------------------------------------------------------------------

ins_allow | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

award | .1519855 .0131763 11.53 0.000 .1258046 .1781665

_cons | 5029.183 827.947 6.07 0.000 3384.07 6674.296

------------------------------------------------------------------------------


Formulate story as hypothesis
Formulate Story as Hypothesis

  • Story is that lawyers charge a fee based on award

  • So null hypothesis is that coefficient on award is not zero

    • H0: b= 0 H1: b≠ 0

  • Test hypothesis that award is not statistically significant

    • Stata does it automatically


  • How do lawyers set fees

    • H0: b= 0 H1: b≠ 0

    • Calculate the test statistic assuming that H0 is true.

      t=(0.1519855-0)/0.0131763)=11.53

    • Either find the test statistic on the t distribution and calculate p-value

      Prob (t>11.53)=0.000

      Or compare with one of the traditional threshold (“critical”) values:

      N-k degrees of freedom

      5% significance level: 1.96

    • |t|>all the critical values and Prob (t>11.53)=0.0005

    • So we reject the null hypothesis


    Type 1 error
    Type 1 error

    • Note how we set up the hypothesis test

    • Null was that percentage charge was zero

    • Type one error is reject the null when it is true

    • The prob of type 1 error is the significance level

    • So there is a 5% chance of saying that lawyers charge a % fee when they do not


    Some comments
    Some Comments

    • You could formulate the test as one sided

      • H0: b> =0 H1: b< 0

      • H0: b<= 0 H1: b> 0

  • Exercise to do this and think about which is best

  • Could also test a particular value

    • H0: b= 0.2 H1: b≠ 0.2


  • Omitted variables
    Omitted Variables

    • Our first Failure of GM Theorem

    • Key practical issue

      • Always some variables missing (R2<1)

    • When does it matter?

      • When they are correlated with the included variables

      • OLS becomes inconsistent and biased

    • Often a way to undermine econometric results

    • Discuss in two ways

      • State the issue formally

      • Use the lawyers example


    Formally
    Formally

    • Suppose we have model with z omitted

      yi = a+ xi + gzi + ui true model

      yi = a + bxi + uiestimated

    • Then we will have:

      • E(b) 

      • b is a biased estimator of effect of x on y

      • also inconsistent: bias does not disappear as N 

    • The bias will be determined by the formula

      • E(b) =  + mg

      •  = direct effect of x on y

      • g = direct effect of z on y

      • m= effect of z on x (from regression of z on x)


    In practice
    In Practice

    • OLS erroneously attributes the effect of the missing z to x

      • Violates GM assumption that E(u|x)=0

    • From the formula, the bias will go away if

      • g=0 : the variable should be omitted as it doesn’t matter

      • m=0: the missing variable is unrelated to the included variable(s)

    • In any project ask:

      • are there missing variables that ought to be included (g≠0)?

      • could they be correlated with any included variables (m≠0) ?

      • What is the direction of bias?


    Lawyers example
    Lawyers Example

    • Suppose we had the simple model of lawyers fees as before.

    • A criticism of this model is that it doesn’t take account of the work done by lawyers

      • i.e. measure of quantity and quality of work are omitted variables

      • This invalidates the est of b

      • This is how you could undermine the study


    How do lawyers set fees

    • Is the criticism valid?

      • these variables ought to be included as they plausibly affect the fee i.e. g≠0

      • They could be correlated with the included award variable (m≠0)

        • it is plausible that more work may lead to higher award

        • or higher wards cases may require more work

    • Turns out not to matter in our case because award and trial are uncorrelated

    • Not always the case: use IV


    Dummy variables
    Dummy Variables

    • Record classifications

      • Dichotomous: “yes/no” e.g. gender, trial, etc

      • Ordinal e.g. level of education

    • OLS doesn’t treat them differently

    • Need to be careful about how coefficients are interpreted

    • Illustrate with “trial” in the fee regression

      • Trial =1 iff case went to court

      • Trial =0 iff case settled before court


    How do lawyers set fees

    • Our basic model is

      feei = 1 + 2awardi + ui

    • This can be interpreted a predicting fees based on awards i.e.

      E[feei]= 1 + 2E[awardi]

    • Suspect that fee is systematically different if case goes to trial

      feei = 1 + 2awardi + 3Triali + ui


    How do lawyers set fees

    • Now theprediction becomes:

      E[feei]= 1 + 2 E[awardi]+ 3 iff trial

      E[feei]= 1 + 2 E[awardi] iff not

    • Note that “trial” disappears when it is zero

    • This translates into separate intercepts on the graph

    • The extra € for bringing a case to trial

    • Testing if 3 is significant is test of significant difference in fees between the two groups

    • For price discrimination story: award still significant


    How do lawyers set fees

    regress ins_allow award trial

    Source | SS df MS Number of obs = 91

    -------------+------------------------------ F( 2, 88) = 78.43

    Model | 2.9871e+09 2 1.4936e+09 Prob > F = 0.0000

    Residual | 1.6758e+09 88 19043267.3 R-squared = 0.6406

    -------------+------------------------------ Adj R-squared = 0.6324

    Total | 4.6629e+09 90 51810441.4 Root MSE = 4363.9

    ------------------------------------------------------------------------------

    ins_allow | Coef. Std. Err. t P>|t| [95% Conf. Interval]

    -------------+----------------------------------------------------------------

    award | .1489103 .0125847 11.83 0.000 .1239009 .1739197

    trial | 5887.706 1848.795 3.18 0.002 2213.616 9561.797

    _cons | 4798.368 791.7677 6.06 0.000 3224.896 6371.84

    ------------------------------------------------------------------------------


    Interaction
    Interaction

    • While the intercept could be different the slope could be also i.e. the degree of price discrimination could be different between the two groups

    • Model this by an “interaction term”

      feei = 1 + 2awardi + 3Triali +

      4 awardi*Triali + ui


    How do lawyers set fees

    • Now theprediction becomes:

      E[feei]= 1 + (2 + 4 )*E[awardi]+ 3 iff trial

      E[feei]= 1 + 2 E[awardi] iff not

    • Note that “trial” disappears when it is zero

    • This translates into separate intercepts and slopes on the graph

    • The extra € for bringing a case to trial and an extra %

    • Testing if 4 is significant is test of significant difference in % fee between the two groups


    How do lawyers set fees

    gen interact=trial*award

    regress ins_allow award trial interact

    Source | SS df MS Number of obs = 91

    -------------+------------------------------ F( 3, 87) = 52.34

    Model | 3.0004e+09 3 1.0001e+09 Prob > F = 0.0000

    Residual | 1.6625e+09 87 19109443.6 R-squared = 0.6435

    -------------+------------------------------ Adj R-squared = 0.6312

    Total | 4.6629e+09 90 51810441.4 Root MSE = 4371.4

    ------------------------------------------------------------------------------

    ins_allow | Coef. Std. Err. t P>|t| [95% Conf. Interval]

    -------------+----------------------------------------------------------------

    award | .1468693 .012842 11.44 0.000 .1213445 .1723941

    trial | 2444.119 4526.143 0.54 0.591 -6552.081 11440.32

    interact | .0561776 .0673738 0.83 0.407 -.0777352 .1900904

    _cons | 4901.306 802.6927 6.11 0.000 3305.868 6496.745

    ------------------------------------------------------------------------------


    Multiple hypotheses
    Multiple Hypotheses

    • A little weird that the interact and trial variables are insignificant

    • Possible that they are jointly significant

    • Formally: H0: 4=0 and 3=0

      H1: 4≠0 and 3≠0

    • This is not the same as two t-tests in sequence

    • Use F-test of “Linear Restriction”

    • Turns out t-test is a special case


    Procedure
    Procedure

    • Estimate the model assuming the null is true i.e. impose the restriction

      • Record R2 for the restricted model

      • R2r=0.5992

  • Estimate the unrestricted model i.e. assuming the null is false

    • Record the R2 for the unrestricted model

    • R2u= 0.64350.5992


  • How do lawyers set fees

    • Form the Test statistic

      r = number of restrictions (count equals signs)

      N = number of observations

      Ku = number of variables (and constant) in the unrestricted model

    • Compare with the critical value from F tables: F (r, N- Ku)

      • If test statistic is greater than critical value: reject H0

      • F(2,87)= 3.15 at 5% significance level


    Comments intuition
    Comments/Intuition

    • Imposing a restriction must make the model explain less of the dep variable

    • If it is “a lot” less then we reject the restriction as being unrealistic

    • How much is “a lot”?

      • Compare the two R2 (not “adjusted R2”)

      • Scale the difference

      • Compare to a threshold value

    • Critical value is fn of 3 parameters: df1, df2, significance level

    • Note doesn’t say anything about the component hypotheses

    • Could do t-tests this way: stata does

    • Sata automatically does H0: 2=0 …k=0


    Conclusions
    Conclusions

    • We had four learning objectives

      • Model i.e. “Story” or question

      • Multiple regression review

      • Dummy variables

      • Omitted variables (the first failure of GM)

    • What’s Next?

      • More examples

      • More problems for OLS