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Econometrics

Advanced Panel Data Techniques

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Advanced Panel Data Topics

Fixed Effects estimationSTATA stuff: xtregAutocorrelation/Cluster correction But first! Review of heteroskedasticity

Probably won’t get to details: Random Effects estimation Hausman test

Other kinds of panel data

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Panel Data with two periodsNotation:yit = 0 + 0d2t + 1xit1 +…+ kxitk + ai + uit

ai = “person effect” (etc) has no “t” subscript All unobserved influences which are fixed for a

person over time (e.g., “ability”) uit = “idiosyncratic error”

Person (firm, etc) i… …in period t

Dummy for t= 2 (intercept shift)

ai = time-constant component of the composite error,

third subscript: variable #

vit

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Fixed Effects EstimationTwo periods of data. The population model is yit = 0 + 0d2t + 1xit1 +…+ kxitk + ai + uit

ai is unknown, but we could estimate it… Estimate âi by including a dummy variable for

each individual, i! For example, in a dataset with 46 cities each

observed in two years (1982, 1987) we would have 45 dummies (equal to one for only two observations each)

d2t is a dummy for the later year (e.g., 1987)

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crmrte unem d87 dcity1 dcity2 … dcity45

73.3 14.9 0 1 0 … 0

63.7 7.7 1 1 0 … 0

169.3 9.1 0 0 1 … 0

164.5 2.4 1 0 1 … 0

96.1 11.3 0 0 0 … 0

120.0 3.9 1 0 0 … 0

116.3 5.3 0 0 0 … 0

169.5 4.6 1 0 0 … 0

… … … … … … …

70.8 6.9 0 0 0 … 1

72.5 6.2 1 0 0 … 1

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Fixed Effects EstimationWe are essentially estimating: yit = 0 + 0d2t + 1xit1 +…+ kxitk + a1d(i=1) + a2d(i=2) +

… + a45d(i=45) + uit

But, for short, we just write,

yit = 0 + 0d2t + 1xit1 +…+ kxitk + ai + uit

Estimated âi are the slope coefficients on these dummy variables

These are called “fixed effects” The dummies control for anything – including

unobservable characteristics – about an individual which is fixed over time

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More on fixed effects…In two-period data (only), including fixed effects equivalent to differencing That is, either way should get you exactly the same

slope estimates Can see this in difference eq’s for predicted values:

Period 2: ŷi2 = b0 + d0∙1 +b1xi21 +…+ bkxi2k + âi

Period 1: ŷi1 = b0 + d0∙0 +b1xi11 +…+ bkxi1k + âi

Diff: ŷi = d0 +b1xi1 +…+ bkxik

Intercept in differences same as coefficient on year dummy

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Fixed Effects In STATA: three ways

1. In STATA, can estimate fixed effects by creating dummies for every individual and including them in your regression E.g.: tab city, gen(citydummy)

2. The “areg” command does the same, w/o showing the dummy coefficients (the “fixed effects”) [we don’t usually care anyway!] a = “absorb” the fixed effects Syntax: areg y x yrdummies, absorb(city)

3. xtreg …, fe (below)Variable identifying cross-sectional units

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Fixed effects regression. areg crmrte unem d87, absorb(area) robust

Linear regression, absorbing indicators Number of obs = 92 F( 2, 44) = 4.50 Prob > F = 0.0166 R-squared = 0.8909 Adj R-squared = 0.7743 Root MSE = 14.178

------------------------------------------------------------------------------ | Robust crmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- unem | 2.218 .8155056 2.72 0.009 .5744559 3.861543 d87 | 15.4022 5.178907 2.97 0.005 4.964803 25.8396 _cons | 75.40837 8.916109 8.46 0.000 57.43913 93.3776-------------+---------------------------------------------------------------- area | absorbed (46 categories)

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First difference regressionc = “change” =

. reg ccrmrte cunem, robust

Linear regression Number of obs = 46 F( 1, 44) = 7.40 Prob > F = 0.0093 R-squared = 0.1267 Root MSE = 20.051

------------------------------------------------------------------------------ | Robust ccrmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- cunem | 2.217999 .8155056 2.72 0.009 .5744559 3.861543 _cons | 15.4022 5.178907 2.97 0.005 4.964803 25.8396------------------------------------------------------------------------------

Same as coefficienst on unemp, d87 in fixed effects estimates!

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Fixed effects vs. first differences

First difference and fixed effects (f.e.) equivalent only when there are exactly two periods of data

When there are more than two periods of data, f.e. equivalent to “demeaning” data

Fixed effects model: Individuals’ means over t: Difference…

itiitit uaxy ...110

iitiitiit uuxxyy ...111

iiii uaxy ...110

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Fixed Effects vs. First Differences

Textbook writes as i.e. where etc.

Also known as the “within” estimator Idea: only using variation “within” individuals

(or other cross-sectional units) over time, and not the variation “between” individuals “Between” estimator, in contrast, uses just the

means, and none of the variation over time:

iitiitiit uuxxyy ...111

ititit uxy ...11

iii uxy ...110

iitit yyy

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With T>2…First Differences vs Fixed Effects

F.E. estimation more common than differences Probably because it’s easier to do (no “differencing”

required) not necessarily because it’s better Advantages:

Fixed effects easily implemented for “unbalanced” panels (not all individuals are observed in all periods)

Also pooled cross-section: don’t need to see the same ‘individuals’ over time, just, e.g., the same cities

Fixed effects estimation is more efficient than differencing if no autocorrelation of in the uit’s Intuition: first differences (estimating in changes) removes

more of the individual variation over time than fixed effects

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Aside: fixed effects using “xtreg” command

In STATA, there are a powerful set of commands, beginning with “XT,” which allow you do carry out many panel techniques (fixed effect, random effects – below)

Step 0 in using these commands: tell STATA the name of the variables that contain… The cross-sectional unit of observation (“X”) The time-series (period) unit of observation (“T”)

Command is: xtset xsecvar timevar

e.g., city, person, firm

e.g., year

xtset:

. xtset area year, delta(5) panel variable: area (strongly balanced) time variable: year, 82 to 87 delta: 5 units

• “area” is the cross section unit variable• “year” is the year variable• delta(5) is an option tells STATA that a one-unit

change in time is 5 years (in this case – 82 to 87)• After that, can run xtreg…

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. xtreg crmrte unem d87, fe

Fixed-effects (within) regression Number of obs = 92Group variable: area Number of groups = 46

R-sq: within = 0.1961 Obs per group: min = 2 between = 0.0036 avg = 2.0 overall = 0.0067 max = 2

F(2,44) = 5.37corr(u_i, Xb) = -0.1477 Prob > F = 0.0082

------------------------------------------------------------------------------ crmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- unem | 2.218 .8778658 2.53 0.015 .4487771 3.987222 d87 | 15.4022 4.702117 3.28 0.002 5.92571 24.8787 _cons | 75.40837 9.070542 8.31 0.000 57.12789 93.68884-------------+---------------------------------------------------------------- sigma_u | 28.529804 sigma_e | 14.178068 rho | .80194672 (fraction of variance due to u_i)------------------------------------------------------------------------------F test that all u_i=0: F(45, 44) = 7.97 Prob > F = 0.0000

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First DifferencesAfter you run the Xtset command, you can also do first differences with a “D.” in front of any variable:

. reg D.crmrte D.unem

Source | SS df MS Number of obs = 46-------------+------------------------------ F( 1, 44) = 6.38 Model | 2566.43744 1 2566.43744 Prob > F = 0.0152 Residual | 17689.5497 44 402.035219 R-squared = 0.1267-------------+------------------------------ Adj R-squared = 0.1069 Total | 20255.9871 45 450.133047 Root MSE = 20.051

------------------------------------------------------------------------------ D.crmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- unem | D1. | 2.217999 .8778658 2.53 0.015 .4487771 3.987222 _cons | 15.4022 4.702117 3.28 0.002 5.92571 24.8787------------------------------------------------------------------------------

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Not xtreg: reg D.crmrte D.unem

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Autocorrelation

Yit = β0 + β1Xit1 + … ai + uit Model implies autocorrelation = errors correlated

across periods ai is perfectly correlated over time, for example (uit’s may also have autocorrelation) We continue assume no error correlation between different

individuals

Consequences similar to heteroskedasticity…: OLS calculates biased standard errors OLS is inefficient (not “BLUE”)

it -- composite error term

Aside: se formula derived…(bivariate case); N = #people; T = # of time periods

OLS standard errors calculated assuming:

Homoskedasticity No autocorrelation Cov(different v’s) = 0

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Other time periods (if T>2)

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1. “cluster” correction

To correct OLS se’s for the possibility of correlation of errors over time, “cluster” Clustering on “person” variable (i) adds error

interaction terms from above to se calculation: w/ = person i, period 2

residual This provides consistent se’s if you have a

large # of people to average over (N big) True se formula has cov(vi1,vi2); averaged over

large #of people, distinction not important

...ˆˆ

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“Cluster”In STATA:reg crmrte lpolpc, cluster(area) Usually makes standard errors larger

because cov(vi1,vi2)>0 Other intuition:

OLS treats every observation as independent information about the relationship

With autocorrelation, that’s not true – observations are not independent

Estimates are effectively based on less information, they should be less precise

Cross-sectional unit with autocorrelation across periods

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. reg crmrte lpolpc d87, robust noheader------------------------------------------------------------------------------ | Robust crmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- lpolpc | 41.09728 9.527411 4.31 0.000 22.16652 60.02805 d87 | 5.066153 5.78541 0.88 0.384 -6.429332 16.56164 _cons | 66.44041 7.324693 9.07 0.000 51.8864 80.99442------------------------------------------------------------------------------

. reg crmrte lpolpc d87, cluster(area) noheader (Std. Err. adjusted for 46 clusters in area)------------------------------------------------------------------------------ | Robust crmrte | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- lpolpc | 41.09728 12.5638 3.27 0.002 15.79249 66.40208 d87 | 5.066153 3.027779 1.67 0.101 -1.032107 11.16441 _cons | 66.44041 8.99638 7.39 0.000 48.32077 84.56005------------------------------------------------------------------------------

Coefficient estimates exactly the same – both OLSStandard errors larger

With and without “cluster” correction

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2. Efficiency Correction = “Random Effects”

“Random effects” is a data transformation to get rid of autocorrelation in errors Like WLS or feasible GLS, random effects

transforms data to produce error w/o autocorrelation

Transformed data meets Gauss-Markov assumptions, and so estimates are efficient

If other Gauss-Markov assumptions hold, random effects will be unbiased and “BLUE” Important: random effects assumes ** ai is

uncorrelated with the x’s ** If not, random effects estimates are biased (o.v. bias!)

and inconsistent

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What is Random Effects?Define Interpretation: fraction of error variance due

to factors fixed across periods If = 0, then = 1.

“Quasi-demeaning” by this factor gets rid of error correlation across periods

212221 auu T

it = ai + uit

2u

iitikitkk

iitiit

xx

xxyy

...

1 1110

(can show): new composite error has no correlation across periods

Slopes theoretically same, (though OLS estimates may not be if errors correlated w/X’s)

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What is Random Effects?

Also can be interpreted as a weighted average of OLS and fixed effects If = 1 (error = fixed), then random effects = regular

demeaning = fixed effects If = 0 (error = entirely unfixed – no autocorrelation)

– pure OLS, ignoring panel structure

iitikitkk

iitiit

xx

xxyy

...

1 1110

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Example: panel of firms from 1987-89

storage display valuevariable name type format label variable label------------------------------------------------------------------------------

-year int %9.0g 1987, 1988, or 1989fcode float %9.0g firm code numberemploy int %9.0g # employees at plantsales float %9.0g annual sales, $avgsal float %9.0g average employee salaryscrap float %9.0g scrap rate (per 100 items)rework float %9.0g rework rate (per 100 items)tothrs int %9.0g total hours trainingunion byte %9.0g =1 if unionizedgrant byte %9.0g = 1 if received grantd89 byte %9.0g = 1 if year = 1989d88 byte %9.0g = 1 if year = 1988hrsemp float %9.0g tothrs/totrain

. xtset fcode year

Time variable

Cross-section variable

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More on XTreg in STATA

After doing “xtset,” “xtreg” same as “reg” but can do panel techniques Random effects (default): xtreg y x1…, re Fixed effects: xtreg y x1 x2…, fe

Xtreg can handle all the other stuff that we have used in “reg”: Robust, cluster, weights, etc.

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. xtset fcode year

. xtreg lscrap hrsemp lsales tothrs union d89 d88, re

Random-effects GLS regression Number of obs = 135Group variable: fcode Number of groups = 47

R-sq: within = 0.2010 Obs per group: min = 1 between = 0.0873 avg = 2.9 overall = 0.0977 max = 3

Random effects u_i ~ Gaussian Wald chi2(6) = 25.50corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0003------------------------------------------------------------------------------ lscrap | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- hrsemp | .0029383 .0042795 0.69 0.492 -.0054494 .011326 lsales | -.1935359 .1553334 -1.25 0.213 -.4979837 .1109119 tothrs | -.0060263 .0044724 -1.35 0.178 -.0147921 .0027395 union | .9214526 .4627995 1.99 0.046 .0143823 1.828523 d89 | -.3653513 .1191926 -3.07 0.002 -.5989645 -.1317381 d88 | -.0831037 .1130728 -0.73 0.462 -.3047223 .1385149 _cons | 3.287955 2.326674 1.41 0.158 -1.272241 7.848152-------------+----------------------------------------------------------------

Fixed vs. Random Effects

Always remember: random effects estimated assuming ai uncorrelated with X’s Unlike fixed effects, r.e. does not remove any

omitted variables bias; just more “efficient” assuming no omitted variables bias

Put another way: random effects can only reduce standard errors, not bias

The ai assumption is testable! If it holds, fixed effects estimates should be

statistically indistiguishable from r.e. estimates

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Hausman testHausman test intuition: H0: cov(ai,xit)=0; estimate with random effects since

it’s the most efficient under this assumption Then estimate with fixed effects, and if the

coefficient estimates are significantly different reject then null

IMPORTANT: as always, failure to reject the null ≠ there is no bias in random effects We never “accept the null” (we just lack sufficient

evidence to reject it) For example, both random effects and fixed effects could

be biased by a similar amount; or standard errors just big

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Hausman testMore broadly, Hausman tests are specification tests comparing two estimators where… One estimator is efficient (in this case, random

effects) if the null hypothesis is true (cov[ai,xi]=0) One estimator is consistent (in this case, fixed

effects) if the null hypothesis is falseRelated to latter, important caveat on this test and all Hausman tests: We must assume (without being able to test!) that

there is no omitted variables bias in the alternative (fixed effects) estimator Reasoning: need an unbiased estimate of the “true” slopes If not true, Hausman test tells us nothing

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Hausman test in STATAHow do you compare your fixed effects and random effects estimates? Steps:1. Save your fixed effects and random effects

estimates using the “estimates store” command (example below)

2. Feed to the “Hausman” command Hausman command calculates standard errors

on difference in whole list of coefficient estimates and tests whether they are jointly significantly different

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Hausman test example: Does job training reduce “scrap” (error) rates: re or fe?. xtset fcode year. xtreg lscrap hrsemp lsales tothrs union d89 d88, re

Random-effects GLS regression Number of obs = 135Group variable: fcode Number of groups = 47

R-sq: within = 0.2010 Obs per group: min = 1 between = 0.0873 avg = 2.9 overall = 0.0977 max = 3

Random effects u_i ~ Gaussian Wald chi2(6) = 25.50corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0003------------------------------------------------------------------------------ lscrap | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- hrsemp | .0029383 .0042795 0.69 0.492 -.0054494 .011326 lsales | -.1935359 .1553334 -1.25 0.213 -.4979837 .1109119 tothrs | -.0060263 .0044724 -1.35 0.178 -.0147921 .0027395 union | .9214526 .4627995 1.99 0.046 .0143823 1.828523 d89 | -.3653513 .1191926 -3.07 0.002 -.5989645 -.1317381 d88 | -.0831037 .1130728 -0.73 0.462 -.3047223 .1385149 _cons | 3.287955 2.326674 1.41 0.158 -1.272241 7.848152-------------+----------------------------------------------------------------

Estimates storeAfter any regression command, can save the estimates for later. Purposes: Look at the results again later (some regressions

take a long time to estimate) - “estimates replay” Feed to another command (like Hausman).

Here, after estimating by random effects, type:. estimates store reff -- stores ests in “reff”

(More generally, estimates store anyname)

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Fixed effects:. xtreg lscrap hrsemp lsales tothrs union d89 d88, fe

Fixed-effects (within) regression Number of obs = 135Group variable: fcode Number of groups = 47

R-sq: within = 0.2021 Obs per group: min = 1 between = 0.0016 avg = 2.9 overall = 0.0224 max = 3

F(5,83) = 4.20corr(u_i, Xb) = -0.0285 Prob > F = 0.0019------------------------------------------------------------------------------ lscrap | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- hrsemp | .0025011 .0044336 0.56 0.574 -.0063171 .0113194 lsales | -.1294226 .2084326 -0.62 0.536 -.5439866 .2851414 tothrs | -.0060857 .0046726 -1.30 0.196 -.0153793 .0032079 union | (dropped) d89 | -.3692155 .1247804 -2.96 0.004 -.6173986 -.1210324 d88 | -.0841784 .1162988 -0.72 0.471 -.3154921 .1471353 _cons | 2.627201 3.18534 0.82 0.412 -3.708313 8.962715-------------+----------------------------------------------------------------. estimates store feff

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Hausman test STATA commandSyntax: hausman consistent_est efficient_est …:

. hausman feff reff

---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | feff reff Difference S.E.-------------+---------------------------------------------------------------- hrsemp | .0025011 .0029383 -.0004372 .0011587 lsales | -.1294226 -.1935359 .0641133 .1389808 tothrs | -.0060857 -.0060263 -.0000594 .001353 d89 | -.3692155 -.3653513 -.0038642 .0369224 d88 | -.0841784 -.0831037 -.0010747 .0272022------------------------------------------------------------------------------ b = consistent under Ho and Ha; obtained from xtreg B = inconsistent under Ha, efficient under Ho; obtained from xtreg

Test: Ho: difference in coefficients not systematic

chi2(5) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 0.66 Prob>chi2 = 0.9851

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Large p-value: fail to reject H0

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Fixed Effects or Random?My view: Don’t use random effects. Random effects is just an efficiency correction

Key assumption that fixed unobservables are uncorrelated with x’s is almost always implausible

My mantra: if you are worried about efficiency, you just don’t have enough data

Bottom line: just correct the standard errors using “cluster” and forget about efficiency

Analog of my view on heteroskedasticity

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Autocorrelation/heteroskedasticity:Problems and solutions

Problem: Heteroskedasticity

Autocorrelation

OLS SE’s biased “robust” produces consistent SE’s

“cluster” produces consistent SE’s

OLS inefficient- could get smaller SE’s from same data

Weighted least squares (or “feasible GLS”)

Random effects

Not a good idea except in cases when you know the form of heteroskedasticity (prone to manipulation)

Not a good idea: requires implausible assumption that there is no ov bias from fixed unobservables

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Other Uses of Panel Methods

It’s possible to think of models where there is an unobserved fixed effect, even if we do not have true panel data A common example: observe different members of

the same family (but not necessarily over time) Or individual plants of larger firms, etc.

We think there is an unobserved family effect

Examples: difference siblings, twins, etc. Can estimate “family fixed effect” model