Two-way fixed-effect models
Difference in difference
Two-way fixed effects
Balanced panels
i=1,2,3.N groups
t=1,2,3.T observations/group
Easiest to think of data as varying across
states/time
Write model as single observation
Yit= + Xit + ui + vt +it
Xit is (1 x k) vector
2
Three-part error structure
ui group fixed-effects. Control for
permanent differences between groups
vt time fixed effects. Impacts common to
all groups but vary by year
it -- idiosyncratic error
Current excise tax rates
Low: SC($0.07), MO ($0.17), VA($0.30)
High: RI ($3.46), NY ($2.75); NJ($2.70)
Average of $1.32 across states
Average in tobacco producing states:
$0.40
Average in non-tobacco states, $1.44
Average price per pack is $5.12
4
Do taxes reduce consumption?
Law of demand
Fundamental result of micro economic theory
Consumption should fall as prices rise
Generated from a theoretical model of
consumer choice
Thought by economists to be fairly
universal in application
Medical/psychological view certain
goods not subject to these laws
7
Starting in 1970s, several authors began
to examine link between cigarette prices
and consumption
Simple research design
Prices typically changed due to state/federal
tax hikes
States with changes are treatment
States without changes are control
8
Near universal agreement in results
10% increase in price reduces demand by 4%
Change in smoking evenly split between
Reductions in number of smokers
Reductions in cigs/day among remaining smokers
Results have been replicated
in other countries/time periods, variety of
statistical models, subgroups
For other addictive goods: alcohol, cocaine,
marijuana, heroin, gambling
9
Taxes now an integral part of antismoking
campaigns
Key component of Master Settlement
Surgeon Generals report
raising tobacco excise taxes is widely
regarded as one of the most effective tobacco
prevention and control strategies.
Tax hikes are now designed to reduce
smoking
10
11
12
13
14
Caution
In balanced panel, two-way fixed-effects
equivalent to subtracting
Within group means
Within time means
Adding sample mean
Only true in balanced panels
If unbalanced, need to do the following
15
Can subtract off means on one dimension
(i or t)
But need to add the dummies for the other
dimension
16
* generate real taxes
gen s_f_rtax=(state_tax+federal_tax)/cpi
label var s_f_rtax "state+federal real tax on cigs,
cents/pack"
* real per capita income
gen ln_pcir=ln(pci/cpi)
label var ln_pcir "ln of real real per capita income"
* generate ln packs_pc
gen ln_packs_pc=ln(packs_pc)
* construct state and year effects
xi i.state i.year
17
* run two way fixed effect model by brute force
* covariates are real tax and ln per capita income
reg ln_packs_pc _I* ln_pcir s_f_rtax
* now be more elegant take out the state effects by areg
areg ln_packs_pc _Iyear* ln_pcir s_f_rtax, absorb(state)
* for simplicity, redefine variables as y x1 (ln_pcir)
* x2 (s-f_rtax)
gen y=ln_packs_pc
gen x1=ln_pcir
gen x2=s_f_rtax
18
* sort data by state, then get means of within state
variables
sort state
by state: egen y_state=mean(y)
by state: egen x1_state=mean(x1)
by state: egen x2_state=mean(x2)
* sort data by state, then get means of within state
variables
sort year
by year: egen y_year=mean(y)
by year: egen x1_year=mean(x1)
by year: egen x2_year=mean(x2)
19
* get sample means
egen y_sample=mean(y)
egen x1_sample=mean(x1)
egen x2_sample=mean(x2)
* generate the devaitions from means
gen y_tilda=y-y_state-y_year+y_sample
gen x1_tilda=x1-x1_state-x1_year+x1_sample
gen x2_tilda=x2-x2_state-x2_year+x2_sample
* the means should be maching zero
sum y_tilda x1_tilda x2_tilda
20
* run the regression on differenced values
*since means are zero, you should have no constant
* notice that the standard errors are incorrect
* because the model is not counting the 51 state dummies
* and 19 year dummies. The recorded DOF are
* 1020 - 2 = 1018 but it should be 1020-2-51-19=948
* multiply the standard errors by
sqrt(1018/948)=1.036262
reg y_tilda x1_tilda x2_tilda, noconstant
21
. * run two way fixed effect model by brute force
. * covariates are real tax and ln per capita income
. reg ln_packs_pc _I* ln_pcir s_f_rtax
Source |
SS
df
MS
-------------+-----------------------------Model | 73.7119499
71 1.03819648
Residual | 4.35024662
948 .004588868
-------------+-----------------------------Total | 78.0621965 1019
.07660667
Number of obs
F( 71,
948)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
1020
226.24
0.0000
0.9443
0.9401
.06774
-----------------------------------------------------------------------------ln_packs_pc |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------_Istate_2 |
.0926469
.0321122
2.89
0.004
.0296277
.155666
_Istate_3 |
.245017
.0342414
7.16
0.000
.1778192
.3122147
Delete results
_Iyear_1998 | -.3249588
.0226916
-14.32
0.000
-.3694904
-.2804272
_Iyear_1999 | -.3664177
.0232861
-15.74
0.000
-.412116
-.3207194
_Iyear_2000 |
-.373204
.0255011
-14.63
0.000
-.4232492
-.3231589
ln_pcir |
.2818674
.0585799
4.81
0.000
.1669061
.3968287
s_f_rtax | -.0062409
.0002227
-28.03
0.000
-.0066779
-.0058039
_cons |
2.294338
.5966798
3.85
0.000
1.123372
3.465304
------------------------------------------------------------------------------
22
Source |
SS
df
MS
-------------+-----------------------------Model | 3.99070575
2 1.99535287
Residual | 4.35024662 1018 .004273327
-------------+-----------------------------Total | 8.34095237 1020 .008177404
Number of obs =
F( 2, 1018)
Prob > F
R-squared
Adj R-squared
Root MSE
1020
= 466.93
= 0.0000
= 0.4784
= 0.4774
= .06537
-----------------------------------------------------------------------------y_tilda |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------x1_tilda |
.2818674
.05653
4.99
0.000
.1709387
.3927961
x2_tilda | -.0062409
.0002149
-29.04
0.000
-.0066626
-.0058193
------------------------------------------------------------------------------
SE on X1 0.05653*1.036262 = 0.05858
SE on X2 0.0002149*1.036262 = 0.0002227
23
Difference in difference models
Maybe the most popular identification
strategy in applied work today
Attempts to mimic random assignment
with treatment and comparison sample
Application of two-way fixed effects model
24
Problem set up
Cross-sectional and time series data
One group is treated with intervention
Have pre-post data for group receiving
intervention
Can examine time-series changes but,
unsure how much of the change is due to
secular changes
25
Y
True effect = Yt2-Yt1
Estimated effect = Yb-Ya
Yt1
Ya
Yb
Yt2
t1
ti
t2
time
26
Intervention occurs at time period t 1
True effect of law
Ya Yb
Only have data at t1 and t2
If using time series, estimate Yt1 Yt2
Solution?
27
Difference in difference models
Basic two-way fixed effects model
Cross section and time fixed effects
Use time series of untreated group to
establish what would have occurred in the
absence of the intervention
Key concept: can control for the fact that
the intervention is more likely in some
types of states
28
Three different presentations
Tabular
Graphical
Regression equation
29
Difference in Difference
Before
Change
After
Change
Difference
Group 1
(Treat)
Yt1
Yt2
Yt
Group 2
(Control)
Yc1
Difference
= Yt2-Yt1
Yc2
Yc
=Yc2-Yc1
Y
Yt Yc
30
Y
Treatment effect=
(Yt2-Yt1) (Yc2-Yc1)
Yc1
Yt1
Yc2
Yt2
control
treatment
t1
t2
time
31
Key Assumption
Control group identifies the time path of
outcomes that would have happened in
the absence of the treatment
In this example, Y falls by Yc2-Yc1 even
without the intervention
Note that underlying levels of outcomes
are not important (return to this in the
regression equation)
32
Y
Yc1
Treatment effect=
(Yt2-Yt1) (Yc2-Yc1)
Yc2
Yt1
control
Treatment
Effect
Yt2
treatment
t1
t2
time
33
In contrast, what is key is that the time
trends in the absence of the intervention
are the same in both groups
If the intervention occurs in an area with a
different trend, will under/over state the
treatment effect
In this example, suppose intervention
occurs in area with faster falling Y
34
Y
Estimated treatment
Yc1
Yc2
Yt1
True treatment effect
Yt2
treatment
t1
t2
control
True
Treatment
Effect
time
35
Basic Econometric Model
Data varies by
state (i)
time (t)
Outcome is Yit
Only two periods
Intervention will occur in a group of
observations (e.g. states, firms, etc.)
36
Three key variables
Tit =1 if obs i belongs in the state that will
eventually be treated
Ait =1 in the periods when treatment occurs
TitAit -- interaction term, treatment states after
the intervention
Yit = 0 + 1Tit + 2Ait + 3TitAit + it
37
Yit = 0 + 1Tit + 2Ait + 3TitAit + it
Before
Change
After
Change
Group 1
(Treat)
0+ 1
0+ 1+ 2+ 3 Yt
Group 2
(Control)
Difference
Difference
= 2+ 3
0+ 2
Yc
= 2
Y = 3
38
More general model
Data varies by
state (i)
time (t)
Outcome is Yit
Many periods
Intervention will occur in a group of states
but at a variety of times
39
ui is a state effect
vt is a complete set of year (time) effects
Analysis of covariance model
Yit = 0 + 3 TitAit + ui + vt + it
40
What is nice about the model
Suppose interventions are not random but
systematic
Occur in states with higher or lower average Y
Occur in time periods with different Ys
This is captured by the inclusion of the
state/time effects allows covariance
between
ui and TitAit
vt and TitAit
41
Group effects
Capture differences across groups that are
constant over time
Year effects
Capture differences over time that are
common to all groups
42
Meyer et al.
Workers compensation
State run insurance program
Compensate workers for medical expenses
and lost work due to on the job accident
Premiums
Paid by firms
Function of previous claims and wages paid
Benefits -- % of income w/ cap
43
Typical benefits schedule
Min( pY,C)
P=percent replacement
Y = earnings
C = cap
e.g., 65% of earnings up to $400/week
44
Concern:
Moral hazard. Benefits will discourage return to work
Empirical question: duration/benefits gradient
Previous estimates
Regress duration (y) on replaced wages (x)
Problem:
given progressive nature of benefits, replaced wages
reveal a lot about the workers
Replacement rates higher in higher wage states
45
Yi = Xi + Ri + i
Y (duration)
R (replacement rate)
Expect > 0
Expect Cov(Ri, i)
Higher wage workers have lower R and higher
duration (understate)
Higher wage states have longer duration and
longer R (overstate)
46
Solution
Quasi experiment in KY and MI
Increased the earnings cap
Increased benefit for high-wage workers
(Treatment)
Did nothing to those already below original
cap (comparison)
Compare change in duration of spell
before and after change for these two
groups
47
48
49
Model
Yit = duration of spell on WC
Ait = period after benefits hike
Hit = high earnings group (Income>E 3)
Yit = 0 + 1Hit + 2Ait + 3AitHit + 4Xit + it
Diff-in-diff estimate is 3
50
51
Questions to ask?
What parameter is identified by the quasiexperiment? Is this an economically
meaningful parameter?
What assumptions must be true in order
for the model to provide and unbiased
estimate of 3?
Do the authors provide any evidence
supporting these assumptions?
52
Tyler et al.
Impact of GED on wages
General education development degree
Earn a HS degree by passing an exam
Exam pass rates vary by state
Introduced in 1942 as a way for veterans
to earn a HS degree
Has expanded to the general public
53
In 1996, 760K dropouts attempted the
exam
Little human capital generated by studying
for the exam
Really measures stock of knowledge
However, passing may signal something
about ability
54
Identification strategy
Use variation across states in pass rates
to identify benefit of a GED
High scoring people would have passed
the exam regardless of what state they
lived in
Low scoring people are similar across
states, but on is granted a GED and the
other is not
55
NY
CT
Increasing scores
Passing score NY
Passing Scores CT
56
Groups A and B pass in either state
Group D passes in CT but not in NY
Group C looks similar to D except it does
not pass
57
What is impact of passing the GED
Yis=earnings of person i in state s
Lis = earned a low score
CTis = 1 if live in a state with a generous
passing score
Yis = 0 + Lis1 + CT2 + LisCTis 3 + is
58
Difference in Difference
CT
NY
Difference
Test score D
is low
(D-C)
Test score B
is high
(B-A)
Difference
(D-C)
(B-A)
59
How do you get the data
From ETS (testing agency) get social
security numbers (SSN) of test takes,
some demographic data, state, and test
score
Give Social Security Admin. a list of SSNs
by group (low score in CT, high score in
NY)
SSN gives you back mean, std.dev. # obs
per cell
60
61
62
More general model
Many within group estimators that do not have
the nice discrete treatments outlined above are
also called difference in difference models
Cook and Tauchen. Examine impact of alcohol
taxes on heavy drinking
States tax alcohol vary over time
Examine impact on consumption and results of
heavy consumption death due to liver cirrhosis
63
Yit = 0 + 1 INCit + 2 INCit-1
+ 1 TAXit + 2 TAXit-1 + ui + vt + it
i is state, t is year
Yit is per capita alcohol consumption
INC is per capita income
TAX is tax paid per gallon of alcohol
64
65
66
Some Keys
Model requires that untreated groups
provide estimate of baseline trend would
have been in the absence of intervention
Key find adequate comparisons
If trends are not aligned, cov(TitAit,it) 0
Omitted variables bias
How do you know you have adequate
comparison sample?
67
Do the pre-treatment samples look similar
Tricky. D-in-D model does not require means
match only trends.
If means match, no guarantee trends will
However, if means differ, arent you
suspicious that trends will as well?
68
Develop tests that can falsify model
Yit = 0 + 3 TitAit + ui + vt + it
Will provide unbiased estimate so long as
cov(TitAit, it)=0
Concern: suppose that the intervention is
more likely in a state with a different trend
If true, coefficient may show up prior to
the intervention
69
Add leads to the model for the treatment
Intervention should not change outcomes
before it appears
If it does, then suspicious that covariance
between trends and intervention
70
Yit = 0 + 3 TitAit + 1TitAit+1 + 2 TitAit+2 +
3TitAit+3 + ui + vt + it
Three leads
Test null: Ho: 1=2=3=0
71
Grinols and Mustard
Impact of a casino opening on crime rates
Concern: casinos are not random
opened in struggling areas
Data at county/year level simple dummy
that equals 1 in year of intervention, =0
otherwise
72
Cit Ait Lit ui vt it
Cit crime rate, county i, year t
ui , vt county and year effects
Lit vector of 0 /1 dummies for laws
Ait county characteristics
73
74
75
76
Pick control groups that have
similar pre-treatment trends
Most studies pick all untreated data as
controls
Example: Some states raise cigarette taxes.
Use states that do not change taxes as
controls
Example: Some states adopt welfare reform
prior to TANF. Use all non-reform states as
controls
Intuitive but not likely correct
77
Can use econometric procedure to pick
controls
Appealing if interventions are discrete and
few in number
Easy to identify pre-post
78
Card and Sullivan
Examine the impact of job training
Some men are treated with job skills,
others are not
Most are low skill men, high
unemployment, frequent movement in and
out of work
Eight quarters of pre-treatment data for
treatment and controls
79
Let Yit =1 if i worked in time t
There is then an eight digit sequence of
outcomes
11110000 or 10100111
Men with same 8 digit pre-treatment
sequence will form control for the treated
People with same pre-treatment time
series are matched
80
Intuitively appealing and simple procedure
Does not guarantee that post treatment
trends would be the same but, this is the
best you have.
81
More systematic model
Data varies by individual (i), state (s), time
Intervention is in a particular state
Yist = 0 + Xist 2+ 3 TstAst + us + vt + ist
Many states available to be controls
How do you pick them?
82
Restrict sample to pre-treatment period
State 1 is the treated state
State k is a potential control
Run data with only these two states
Estimate separate year effects for the
treatment state
If you cannot reject null that the year
effects are the same, use as control
83
Unrestricted model
Pretreatment years so TstAst not in model
M pre-treatment years
Let Wt=1 if obs from year t
Yist = 0 + Xist 2+ t=2tWt + t=2 t TiWt + us
+ ist
Ho: 2= 3= m=0
84
85
Acemoglu and Angrist
ada_jpe.do
ada_jpe.log
86
Americans with Disability Act
Requires that employers accommodate
disabled workers
Outlaws discrimination based on
disabilities
Passes in July 1990, effective July 1992
May discourage employment of disabled
Costs of accommodations
Maybe more difficult to fire disabled
87
Econometric model
Difference in difference
Have data before/after law goes into effect
Treated group disabled
Control non-disabled
Treatment variable is interaction
Diabled * 1992 and after
88
Yit = Xit + Di + Yeartt + Yeart Ditt + it
Yit = labor market outcome, person i year t
Xit vector of individual characteristics
Dit =1 if disableld
Yeart = year effect
Yeart Dit = complete set of year x disability
interactions
89
Coef on is should be zero before the law
May be non zero for years>=1992
90
91
92
Data
March CPS
Asks all participants employment/income
data for the previous year
Earnings, weeks worked, usual hours/week
Data from 1988-1997 March CPS
Data for calendar years 1987-1996
Men and women, aged 21-58
Generate results for various subsamples
93
Constructs sets of dummies
For year, region and age
. * get work year and region fixed effects;
. xi i.yearw i.region i.age;
i.yearw
_Iyearw_1987-1996
(naturally coded; _Iyearw_1987 omitted)
i.region
_Iregion_11-42
(naturally coded; _Iregion_11 omitted)
i.age
_Iage_21-39
(naturally coded; _Iage_21 omitted)
.
.
.
.
.
* the authors interact the disabled dummy with all year effect;
* and include all interactions in the model. if the d-in-d;
* assumptions are correct, the interactions prior to 1992 should;
* all be zero;
gen d_y88=_Iyearw_1988*disabled;
.
(repeated text deleted)
.
. gen d_y94=_Iyearw_1994*disabled;
. gen d_y95=_Iyearw_1995*disabled;
. gen d_y96=_Iyearw_1996*disabled;
Generate year x Disability
interactions
94
Table 2
ADA not in effect
Effective years of ADA
95
Model with few
controls
After adding extensive
list
Of controls, results change
little
96
reg wkswork1 _Iy* disabled d_y*;
Include all variables
that begin with
_ly
Include all variables
that begin with d_y
97
# obs close to what is
Reported in paper
Disability main effect
. * run basic diff-in-diff model, column 2, table 3;
. * do not control for covariates;
. reg wkswork1 _Iy* disabled d_y*;
Source |
SS
df
MS
-------------+-----------------------------Model | 6036918.11
19 317732.532
Residual | 46189618.9195694 236.029816
-------------+-----------------------------Total |
52226537195713 266.852672
Number of obs
F( 19,195694)
Prob > F
R-squared
Adj R-squared
Root MSE
= 195714
= 1346.15
= 0.0000
= 0.1156
= 0.1155
= 15.363
-----------------------------------------------------------------------------wkswork1 |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------Year effects 1988 1996 delete
disabled |
-23.7888
.500911
-47.49
0.000
-24.77057
-22.80703
d_y88 | -.7371178
.7328373
-1.01
0.314
-2.173461
.6992259
d_y89 | -.7552183
.7189482
-1.05
0.294
-2.16434
.653903
d_y90 | -2.612262
.7073555
-3.69
0.000
-3.998661
-1.225862
d_y91 | -2.176184
.7040983
-3.09
0.002
-3.556199
-.7961677
d_y92 | -1.567489
.700199
-2.24
0.025
-2.939862
-.1951153
d_y93 | -3.113591
.707372
-4.40
0.000
-4.500023
-1.727159
d_y94 | -4.044365
.7328552
-5.52
0.000
-5.480743
-2.607986
d_y95 | -3.563268
.7626032
-4.67
0.000
-5.057952
-2.068584
d_y96 | -4.472773
.7514502
-5.95
0.000
-5.945597
-2.999948
_cons |
43.86073
.1085045
404.23
0.000
43.64807
44.0734
------------------------------------------------------------------------------
Need to delete one year effect
Since constant is in model
Disability law
interactions
98
Run different model
One treatment variable: Disabled x after
1991
. gen ada=yearw>=1992;
. gen treatment=ada*disabled;
Add year effects to model, disabled, them
ADA x disabled interaction
99
Regression statement
.
.
.
.
* run basic diff-in-diff model, column 2, table 3;
* do not control for covariates;
* have only one treatment variable;
reg wkswork1 _Iy* disabled treatment;
Source |
SS
df
MS
-------------+-----------------------------Model | 6027911.45
11
547991.95
Residual | 46198625.5195702
236.06619
-------------+-----------------------------Total |
52226537195713 266.852672
Number of obs
F( 11,195702)
Prob > F
R-squared
Adj R-squared
Root MSE
= 195714
= 2321.35
= 0.0000
= 0.1154
= 0.1154
= 15.364
-----------------------------------------------------------------------------wkswork1 |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------_Iyearw_1988 |
.3950318
.1526629
2.59
0.010
.0958162
.6942475
Do not show all year effects
_Iyearw_1996 |
1.293786
.1604476
8.06
0.000
.9793123
1.608259
disabled | -25.07033
.2274473 -110.22
0.000
-25.51612
-24.62454
treatment | -1.970964
.3280719
-6.01
0.000
-2.613977
-1.327951
_cons |
43.92087
.1064727
412.51
0.000
43.71218
44.12955
------------------------------------------------------------------------------
ADA reduced work by almost 2 weeks/year
100
Should you cluster?
Intervention varies by year/disability
Should be within-year correlation in errors
People are in the sample two years in a
row so there should be some correlation
over time
Cannot cluster on years since # groups
too small
101
Need larger set that makes sense
Two options (many more)
Cluster on state
Cluster on state/disability
102
. gen disabled_state=100*disabled+statefip;
reg wkswork1 _Ia* _Iy* _Ir* white black hispanic lths
hsgrad somecol disabled treatment, cluster(statefip);
.reg wkswork1 _Ia* _Iy* _Ir* white black hispanic lths
hsgrad somecol disabled treatment,
cluster(disabled_state);
103
Summary of results for cluster
Coefficient on treatment (standard error)
Regular OLS
-1.998 (0.315)
Cluster by state:
-1.998 (0.487)
Cluster by state/disab. -1.998 (0.532)
104
Dranove et al.
105
Introduction
Increased use of report cards, especially
in health care and education
Two best examples:
NCLB legislation for education
NYs publication of coronary artery bypass
graft (CABG) mortality rates for surgeons and
hospitals
106
Disagreement about usefulness
For: Better informed consumers make better
decisions, makes markets more efficient
Choose best doctors
Provides incentives for schools and docs to
improve care
Against
May give incomplete evidence. Can risk adjust
but not on all characteristics
Docs can manipulate rankings by selecting
patients with the highest expected success rate,
decreasing access to care for the sickest patients
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This paper
Uses data on al heart attack patients in Medicare
in from 1987-94
Impact of reports cards in NY and PA
Examines three sets of outcomes associated with
report cards
Matching of patients to providers: is there a match of
the sickest patients to best providers?
Incidence and quantity of CABG
Do total surgeries go up or down?
Shift to healthier patients?
Is there a substitution into other forms of treatment
NOT measured by the report card?
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Report Cards
NY
Hospital specific, risk adjusted CABG
mortality rates based on 1990
Physician specific rates in 1992
PA hospital specific data in 1992
Effective dates impact patient decision
making in 1991 (NY) and 1993 (PA)
concerning hospitals, 1993 in both states
for physicians
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Data
Population potentially impacted are those
with acute myocardial infarctions (AMI) in
Medicare
Easily obtained from Medicare claims data
Large fraction treated with CABG
Selection into the sample unlikely impacted
by report cards
Physicians treating AMI likely to have multiple
treatment options (e.g., heart cath., medical
treatment, etc.)
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Hospital Model
ln(hlst ) As Bt Z lst g Lst p N st q elst
l hospital , s state, t time
hlst mean hospital level severity AMI patients
As , Bt are state, time effects
Z lst are hospital characteristics
N st is a polynomial in no. of hospitals
Lst law dummy, 1 in 91 in NY , 93 in PA
111
Individual model
Ckst ) As Bt Z kst g Lst p ekst
k person, s state, t time
Ckst 1 if patient had CABG w / in year of AMI
As , Bt are state, time effects
Z kst are person characteristics
Lst law dummy, 1 93 in both NY / PA
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