Post on 23-Feb-2018
transcript
7/24/2019 Lecture14 Panel Data
1/40
ECON4150 - Introductory Econometrics
Lecture 14: Panel data
Monique de Haan
(moniqued@econ.uio.no)
Stock and Watson Chapter 10
7/24/2019 Lecture14 Panel Data
2/40
2
OLS: The Least Squares Assumptions
Yi=0+ 1Xi+ui
Assumption 1: conditional mean zero assumption: E[ui|Xi] =0
Assumption 2: (Xi,Yi)are i.i.d. draws from joint distribution
Assumption 3: Large outliers are unlikely
Under these three assumption the OLS estimators are unbiased,consistent and normally distributed in large samples.
Last week we discussed threats to internal validity
In this lecture we discuss a method we can use in case of omittedvariables
Omitted variable is a determinant of the outcomeYi Omitted variable is correlated with regressor of interestXi
7/24/2019 Lecture14 Panel Data
3/40
3
Omitted variables
Multiple regression model was introduced to mitigate omitted variablesproblem of simple regression
Yi=0+ 1X1i+2X2i+3X3i+...+kXki+ui
Even with multiple regression there is threat of omitted variables:
some factors are difficult to measure
sometimes we are simply ignorant about relevant factors
Multiple regression based on panel data may mitigate detrimental effectof omitted variableswithout actually observing them.
7/24/2019 Lecture14 Panel Data
4/40
4
Panel data
Cross-sectional data:
A sample of individuals observed in 1 time period
2010
Panel data: same sample of individuals observed in multiple time periods
2010
2011
2012
7/24/2019 Lecture14 Panel Data
5/40
5
Panel data; notation
Panel data consist of observations onnentities (cross-sectional units)andTtime periods
Particular observation denoted with two subscripts (iandt)
Yit =0+ 1Xit+uit
Yitoutcome variable for individuali in yeart
For balanced panel this results innT observations
6
7/24/2019 Lecture14 Panel Data
6/40
6
Advantages of panel data
More control over omitted variables.
More observations.
Many research questions typically involve a time component.
7
7/24/2019 Lecture14 Panel Data
7/40
7
The effect of alcohol taxes on traffic deaths
About 40,000 traffic fatalities each year in the U.S.
Approximately 25% of fatal crashes involve driver who drunk alcohol.
Government wants to reduce traffic fatalities.
One potential policy: increase the tax on alcoholic beverages.
We have data on traffic fatality rate and tax on beer for 48 U.S. states in1982 and 1988.
What is the effect of increasing the tax on beer on the traffic fatality rate?
8
7/24/2019 Lecture14 Panel Data
8/40
8
Data from 1982
0
1
2
3
4
Trafficfa
tality
rate
0 .5 1 1.5 2 2.5 3
Tax on beer (in real dollars)
Traffic deaths and alcohol taxes in 1982
FatalityRatei,1982 = 2.01 + 0.15 BeerTaxi,1982
(0.
14) (0.
18)
9
7/24/2019 Lecture14 Panel Data
9/40
9
Data from 1988
0
1
2
3
4
Trafficfa
tality
rate
0 .5 1 1.5 2 2.5
Tax on beer (in real dollars)
Traffic deaths and alcohol taxes in 1988
FatalityRatei,1988 = 1.86 + 0.44 BeerTaxi,1988
(0.
11) (0.
16)
10
7/24/2019 Lecture14 Panel Data
10/40
10
Panel data: before-after analysis
Both regression using data from 1982 & 1988 likely suffer from omitted
variable bias
We can use data from 1982 and 1988 together as panel data
Panel data withT =2
Observed areYi1,Yi2 andXi1,Xi2
Suppose model is
Yit=0+ 1Xit+2Zi+uit
and we assumeE(uit|Xi1,Xi2,Zi) =0
Zi are (unobserved) variables that vary between states but not over time
(such as local cultural attitude towards drinking and driving)
Parameter of interest is 1
11
7/24/2019 Lecture14 Panel Data
11/40
Panel data
12
7/24/2019 Lecture14 Panel Data
12/40
Panel data: before
Consider cross-sectional regression for first period (t=1):
Yi1 =0+ 1Xi1+ 2Zi+ui1 E[ui|Xi1,Zi] =0
Ziobserved: multiple regression ofYi1 on constant,Xi1 andZileads tounbiased and consistent estimator of1
Zinot observed: regression ofYi1 on constant andXi1 only results inunbiased estimator of1 whenCov(Xi1,Zi) =0
What can we do if we dont observe Zi?
13
7/24/2019 Lecture14 Panel Data
13/40
Panel data: after
We also observeYi2 andXi2, hence model for second period is:
Yi2 =0+ 1Xi2+ 2Zi+ui2
Similar to argument before cross-sectional analysis for period 2 mightfail
Problem is again the unobserved heterogeneity embodied inZi
14
7/24/2019 Lecture14 Panel Data
14/40
Before-after analysis (first differences)
We have
Yi1 =0+ 1Xi1+ 2Zi+ui1
and
Yi2 =0+ 1Xi2+ 2Zi+ui2
Subtracting period 1 from period 2 gives
Yi2 Yi1 = (0+ 1Xi2+ 2Zi+ui2) (0+ 1Xi1+ 2Zi+ui1)
Applying OLS to:
Yi2 Yi1 =1(Xi2 Xi1) + (ui2 ui1)
will produce an unbiased and consistent estimator of1
Advantage of this regression is that we do not need data onZ
By analyzing changes in dependent variable we automatically control fortime-invariant unobserved factors
15
7/24/2019 Lecture14 Panel Data
15/40
Data from 1982 and 1988
1
.4
1.2
1
.8
.6
.4
.2
0
.2
.4
.6
.8
F
atality
rate
1988
Fatatlity
rate
1982
.6 .4 .2 0 .2 .4
Beer tax 1988 Beer tax 1982
Traffic deaths and alcohol taxes: beforeafter
Fatalityi,1988 Fatalityi,1982 = 0.07 1.04 (BeerTaxi,1988 BeerTaxi,1982)(0.06) (0.42)
16
7/24/2019 Lecture14 Panel Data
16/40
Panel data with more than 2 time periods
17
7/24/2019 Lecture14 Panel Data
17/40
Panel data with more than 2 time periods
Panel data withT2
Yit =0+ 1Xit+2Zi+uit, i=1, ...,n; t=1, ...,T
Yit is dependent variable;Xit is explanatory variable;Ziare state
specific, time invariant variables
Equation can be interpreted as model with n specific intercepts (one foreach state)
Yit =1Xit+i+uit, with i=0+ 2Zi
i,i=1, ...,nare called entity fixed effects
imodels impact of omitted time-invariant variables onYit
18
7/24/2019 Lecture14 Panel Data
18/40
State specific intercepts
0
.5
1
1.5
2
2.5
3
3.5
4
Predicted
fatalityrate
0 .5 1 1.5 2 2.5 3
beer tax
Alabama Arizona
Arkansa California
19
7/24/2019 Lecture14 Panel Data
19/40
Fixed effects regression modelLeast squares with dummy variables
Having data on Yit and Xithow to determine1?
Population regression model:Yit=1Xit+i+uit
In order to estimate the model we have to quantifyi
Solution: createndummy variablesD1i, ...,Dni
withD1i=1 ifi=1 and 0 otherwise, withD2i=1 ifi=2 and 0 otherwise,....
Population regression model can be written as:
Yit=1Xit+1D1i+2D2i+...+nDni+uit
20
Fi d ff i d l
7/24/2019 Lecture14 Panel Data
20/40
Fixed effects regression modelLeast squares with dummy variables
Alternatively, population regression model can be written as:
Yit =0+ 1Xit+2D2i+...+nDni+uit
with0 =1 andi= i 0 fori>1
Interpretation of 1 identical for both representations
Ordinary Least Squares (OLS): choose 0, 1, 2..., nto minimizesquared prediction mistakes (SSR):
n
i=1
T
t=1
Yit 0 1Xit2D2i ...nDni
2
SSRis function of 0,1, 2..., n
21
Fi d ff t i d l
7/24/2019 Lecture14 Panel Data
21/40
Fixed effect regression modelLeast squares with dummy variables
ni=1
Tt=1
Yit 0 1Xit2D2i ...nDni
2
OLS procedure:
Take partial derivatives ofSSRw.r.t. 0,1, 2...,n
Equal partial derivatives to zero resulting inn+1 equations withn+1unknown coefficients
Solutions are the OLS estimators 0,1, 2...,n
22
Fi d ff t i d l
7/24/2019 Lecture14 Panel Data
22/40
Fixed effect regression modelLeast squares with dummy variables
Analytical formulas require matrix algebra
Algebraic properties OLS estimators (normal equations, linearity) sameas for simple regression model
Extension to multipleXs straightforward: n+knormal equations
OLS procedure is also labeled Least Squares Dummy Variables (LSDV)method
Dummy variable trap: Never include allndummy variables and theconstant term!
23
Fi ed effect regression model
7/24/2019 Lecture14 Panel Data
23/40
Fixed effect regression modelWithin estimation
Typicallynis large in panel data applications
With largencomputer will face numerical problem when solving system
ofn+1 equations
OLS estimator can be calculated in two steps
First step: demeanYit andXit
Second step: use OLS on demeaned variables
24
Fixed effect regression model
7/24/2019 Lecture14 Panel Data
24/40
Fixed effect regression modelWithin estimation
We have
Yit = 1Xit+i+uit
Yi = 1Xi+i+ui
Yi = 1T
Tt=1Yit, etc. is entity mean
Subtracting both expressions leads to
Yit Yi = (1Xit+i+uit) (1Xi+i+ui)
Yit =1
Xit+uit
Yit =Yit Yi,etc. is entity demeaned variable
ihas disappeared; OLS on demeaned variables involves solving onenormal equation only!
25
Fixed effect regression model
7/24/2019 Lecture14 Panel Data
25/40
Fixed effect regression modelWithin estimation
26
Fixed effect regression model
7/24/2019 Lecture14 Panel Data
26/40
Fixed effect regression modelWithin estimation
Entity demeaning is often called the Within transformation
Within transformation is generalization of "before-after" analysis to morethanT =2 periods
Before-after: Yi2 Yi1 =1(Xi2 Xi1) + (ui2 ui1)
Within: Yit Yi=1(Xit Xi) + (uitui)
LSDV and Within estimators are identical:
FatalityRateit = 0.66 BeerTaxit + State dummies(0.19)
( FatalityRateit FatalityRate) = 0.66 (BeerTaxit BeerTax)(0.19)
27
Fixed effects regression model
7/24/2019 Lecture14 Panel Data
27/40
Fixed effects regression modeltime fixed effects
In addition to entity effects we can also include time effects in the model
Time effects control for omitted variables that are common to all entitiesbut vary over time
Typical example of time effects: macroeconomic conditions or federalpolicy measures are common to all entities (e.g. states) but vary over
time
Panel data model with entity and time effects:
Yit=1Xit+i+t+uit
28
Fixed effects regression model
7/24/2019 Lecture14 Panel Data
28/40
Fixed effects regression modeltime fixed effects
OLS estimation straightforward extension of LSDV/Within estimators ofmodel with only entity fixed effects
LSDV: createTdummy variablesB1t....BTt
Yit= 0+ 1Xit+2D2i+...+nDni
+2B2t+3B3t+...+TBTt+uit
Within estimation: DeviatingYit andXitfrom their entityand time-periodmeans
The effect of the tax on beer on the traffic fatality rate:
FatalityRateit = 0.64 BeerTaxit + State dummies+Time dummies(0.20)
29
Fixed effects regression model
7/24/2019 Lecture14 Panel Data
29/40
Fixed effects regression modelstatistical properties OLS
Yit =1Xit+i+t+uit
statistical assumptions are:
ASS #1: E(uit|Xi1, ...,XiT, i, t) =0
ASS #2: (Xi1, ...,XiT,Yi1, ...,YiT)are i.i.d. over the cross-section
ASS #3: large outliers are unlikely
ASS #4: no perfect multicollinearity
ASS #5: cov(uit,uis|Xi1, ...,XiT, i, t) =0 fort=s
30
Fixed effects regression model
7/24/2019 Lecture14 Panel Data
30/40
Fixed effects regression modelstatistical properties OLS
ASS #1 to ASS #5 imply that:
OLS estimator 1 isunbiasedandconsistentestimator of1
OLS estimators approximately have a normal distribution
remarks:
ASS #1 is most important
extension to multipleXs straightforward
Yit=1X1it+2X2it+...+kXkit+i+t+uit
additional assumption ASS #5 implies that error terms are uncorrelatedover time (no autocorrelation)
31
Fixed effects regression model
7/24/2019 Lecture14 Panel Data
31/40
Fixed effects regression modelClustered standard errors
Violation of assumption #5: error terms are correlated over time:(Cov(uit, uis)=0)
uitcontains time-varying factors that affect the traffic fatality rate (but
that are uncorrelated with the beer tax)
These omitted factors might for a given entity be correlated over time
Examples: downturn in local economy, road improvement project
Not correcting for autocorrelation leads to standard errors which areoften too low
32
Fixed effects regression model
7/24/2019 Lecture14 Panel Data
32/40
Fixed effects regression modelClustered standard errors
Solution: compute HAC-standard errors (clustered ses)
robust to arbitrary correlation within clusters (entities) robust to heteroskedasticity assume no correlation across entities
Clustered standard errors valid whether or not there isheteroskedasticity and/or autocorrelation
Use of clustered standard errors problematic when number of entities isbelow 50 (or 42)
In stata: command, cluster(entity)
33
The effect of a tax on beer on traffic fatalities
7/24/2019 Lecture14 Panel Data
33/40
Dependent variable: traffic fatality rate (number of deaths per 10 000)
Beer tax 0.36*** -0.66*** -0.64*** -0.59*** -0.59*(0.06) (0.19) (0.20) (0.18) (0.33)
State fixed effects - yes yes yes yesTime fixed effects - - yes yes yes
Additional control variables - - - yes yes
Clustered standard errors - - - - yes
N 336 336 336 336 336
Note:* significant at 10% level, ** significant at 5% level, *** significant at 1% level. Control variables: Unemployment rate, per capitaincome, minimum legal drinking age.
34
Panel data: an example
7/24/2019 Lecture14 Panel Data
34/40
preturns to schooling
Yit =1Xit+i+uit
Yitis logarithm of individual earnings;Xitis years of completededucation
iunobserved ability
Likely to be cross-sectional correlation between Xit andi, hencestandard cross-sectional analysis with OLS fails
However, in this case panel data does not solve the problem becauseXit
typically lacks time series variation (Xit=Xi)
We have to resort to cross-sectional methods (instrumental variables) toidentify returns to schooling
35
Panel data: Cigarette taxes and smoking
7/24/2019 Lecture14 Panel Data
35/40
g g
Is there an effect of cigarette taxes on smoking behavior?
Yit =1Xit+i+uit
Yitnumber of packages per capita in state i in yeart,Xitis real tax oncigarettes in statei in yeart
i is a state specific effect which includes state characteristics which areconstant over time
Data for 48 U.S. states in 2 time periods: 1985 and 1995
36
Panel data: Cigarette taxes and smoking
7/24/2019 Lecture14 Panel Data
36/40
g g
Lpackpc = log number of packages per capita in state iin yeart
rtax = real avr cigarette specific tax during fiscal year in statei
Lperinc = log per capita real income
. r egr ess l packpc r t ax l per i nc
Source | SS df MS Number of obs = 96
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 2, 93) = 21. 25Model | 1. 76908655 2 . 884543277 Pr ob > F = 0. 0000
Resi dual | 3. 87049389 93 . 041618214 R- squared = 0. 3137
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squared = 0. 2989
Tot al | 5. 63958045 95 . 059364005 Root MSE = . 20401
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
l packpc | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
r t ax | - . 0156393 . 0027975 - 5. 59 0. 000 - . 0211946 - . 0100839
l per i nc | - . 0139092 . 158696 - 0. 09 0. 930 - . 3290481 . 3012296
_cons | 5. 206614 . 3781071 13. 77 0. 000 4. 455769 5. 95746
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
37
Panel data: Cigarette taxes and smoking
7/24/2019 Lecture14 Panel Data
37/40
Before-After estimation
. gen di f f _r t ax= r t ax1995- r t ax1985
. gen di f f _l packpc= l packpc1995- l packpc1985
. gen di f f _l per i nc= l per i nc1995- l per i nc1985
. r egr ess di f f _ l packpc di f f _r t ax di f f _ l per i nc, nocons
Source | SS df MS Number of obs = 48
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 2, 46) = 145. 66
Model | 3. 33475011 2 1. 66737506 Pr ob > F = 0. 0000
Resi dual | . 526571782 46 . 011447213 R- squar ed = 0. 8636
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squared = 0. 8577
Tot al | 3. 86132189 48 . 080444206 Root MSE = . 10699
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
di f f _ l packpc | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
di f f _r t ax | - . 0169369 . 0020119 - 8. 42 0. 000 - . 0209865 - . 0128872
di f f _l per i nc | - 1. 011625 . 1325691 - 7. 63 0. 000 - 1. 278473 - . 7447771
38
Panel data: Cigarette taxes and smoking
7/24/2019 Lecture14 Panel Data
38/40
Least squares with dummy variables (no constant term)
. r egr ess l packpc r t ax l per i nc st at eB*, nocons
Source | SS df MS Number of obs = 96
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 50, 46) = 7317. 61
Model | 2094. 15728 50 41. 8831457 Pr ob > F = 0. 0000
Resi dual | . 263285891 46 . 005723606 R- squared = 0. 9999
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squar ed = 0. 9997
Tot al | 2094. 42057 96 21. 8168809 Root MSE = . 07565
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
l packpc | Coef . St d. Err . t P>| t | [ 95% Conf . I nt er val ]
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
r t ax | - . 0169369 . 0020119 - 8. 42 0. 000 - . 0209865 - . 0128872
l per i nc | - 1. 011625 . 1325691 - 7. 63 0. 000 - 1. 278473 - . 7447771
st at eB1 | 7. 663688 . 3037711 25. 23 0. 000 7. 052229 8. 275148
st at eB2 | 7. 834448 . 2926539 26. 77 0. 000 7. 245367 8. 42353
st at eB3 | 7. 678433 . 3121525 24. 60 0. 000 7. 050103 8. 306763
st at eB4 | 7. 66627 . 3392221 22. 60 0. 000 6. 983451 8. 349088
......
...st at eB45 | 7. 844359 . 3193189 24. 57 0. 000 7. 201603 8. 487114
st at eB46 | 7. 92666 . 3154175 25. 13 0. 000 7. 291758 8. 561563
st at eB47 | 7. 644741 . 2936826 26. 03 0. 000 7. 053589 8. 235894
st at eB48 | 7. 825943 . 3275694 23. 89 0. 000 7. 16658 8. 485306
39
Panel data: Cigarette taxes and smoking
7/24/2019 Lecture14 Panel Data
39/40
Least squares with dummy variables with constant term
. r egr ess l packpc r t ax l per i nc st at eB*
Source | SS df MS Number of obs = 96
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - F( 49, 46) = 19. 17
Model | 5. 37629455 49 . 109720297 Pr ob > F = 0. 0000
Resi dual | . 263285891 46 . 005723606 R- squared = 0. 9533
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Adj R- squar ed = 0. 9036
Tot al | 5. 63958045 95 . 059364005 Root MSE = . 07565
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -l packpc | Coef . St d. Er r. t P>| t | [ 95% Conf . I nt erval ]
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
r t ax | - . 0169369 . 0020119 - 8. 42 0. 000 - . 0209865 - . 0128872
l per i nc | - 1. 011625 . 1325691 - 7. 63 0. 000 - 1. 278473 - . 7447771
st ateB1 | - . 1530275 . 0900694 - 1. 70 0. 096 - . 3343279 . 0282728
st at eB2 | . 0177322 . 1005272 0. 18 0. 861 - . 1846185 . 220083
..
.
..
.
..
.st ateB42 | - . 771239 . 0918679 - 8. 40 0. 000 - . 9561594 - . 5863186st at eB43 | ( dr opped)
st at eB44 | . 1757536 . 0854144 2. 06 0. 045 . 0038233 . 347684
st at eB45 | . 0276429 . 0948094 0. 29 0. 772 - . 1631985 . 2184843
st at eB46 | . 1099444 . 0918156 1. 20 0. 237 - . 0748708 . 2947597
st ateB47 | - . 1719747 . 0959042 - 1. 79 0. 080 - . 3650198 . 0210705
st at eB48 | . 0092272 . 0787188 0. 12 0. 907 - . 1492255 . 16768
_cons | 7. 816716 . 3458507 22. 60 0. 000 7. 120554 8. 512877
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
40
Panel data: Cigarette taxes and smoking
7/24/2019 Lecture14 Panel Data
40/40
Within estimation
. xt r eg l packpc rt ax l per i nc, f e i ( STATE)
Fi xed- ef f ect s ( wi t hi n) r egr essi on Number of obs = 96
Gr oup var i abl e: STATE Number of groups = 48
R- sq: wi t hi n = 0. 8636 Obs per group: mi n = 2
bet ween = 0. 0896 avg = 2. 0
overal l = 0. 2354 max = 2
F( 2, 46) = 145. 66cor r ( u_i , Xb) = - 0. 5687 Prob > F = 0. 0000
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
l packpc | Coef . St d. Er r . t P>| t | [ 95% Conf . I nt er val ]
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
r t ax | - . 0169369 . 0020119 - 8. 42 0. 000 - . 0209865 - . 0128872
l per i nc | - 1. 011625 . 1325691 - 7. 63 0. 000 - 1. 278473 - . 7447771
_cons | 7. 856714 . 3150362 24. 94 0. 000 7. 222579 8. 490849
- - - - - - - - - - - - - +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
si gma_u | . 25232518
si gma_e | . 07565452
r ho | . 91751731 ( f r act i on of var i ance due t o u_i )
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
F t est t hat al l u_i =0: F( 47, 46) = 13. 41 Prob > F = 0. 0000