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

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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
- 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

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

- As always summarize and describe the data
- Graph variables of interest (see over)
- Regression to find percentage price rule

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

- 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

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

- 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

- 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

- 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

- 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

- 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

- 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

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

- 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

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

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

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

- 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

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

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

- 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

- 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

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

- 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

- 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

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