Panel data and causal effects
ENT5587B - Research Design & Theory Testing II
Brian S. Anderson, Ph.D.
Assistant Professor
Department of Global Entrepreneurship & Innovation
andersonbri@umkc.edu
© 2017 Brian S. Anderson
- Paper revision and reviewer responses due 24 April
- Data collection discussion
- Measuring change
- Instrument variables in panel data
- Hybrid methods
- Lab 20 April – Panel data paper assessment
- Seminar 24 April – LDV models and panel data // special topics
Training time out.
What are the big questions you have about panel models?
Lets start off by getting some data. This is the same panel data that we used last week.
library(readr)
library(tidyverse)
my.panel.ds <- read_csv("http://a.web.umkc.edu/andersonbri/Panel.csv")
my.panel.df <- my.panel.dsWe often refer to panel data as “time-series cross-sectional” data.
The reason being is that you can think of each \(t\) observation as being a snapshot in time of the corresponding \(i's\) condition.
If you collected one \(t\) for each \(i\), you would have cross-sectional data. Since you collected multiple \(t's\) per \(i\), you have time-series cross-sectional data.
The material point is this…
Just because you have panel data DOES NOT mean that you can estimate causal effects. Or even that it helps identify causal relationships.
As we will talk about, with panel data you may be more likely to make an incorrect nomological conclusion, because of the noise in the data and the difficulty in isolating causal effects.
Just to review…
- \(y\) = Value of the \(t^{th}\) observation of \(y\) for the \(i^{th}\) entity
- \(\alpha\) = The value of \(y\) when \(x\) equals zero across all \(i, t's\)
- \(\beta\) = The expected change in \(y\) for an average \(i\) across time \(t\). Note that this interpretation gets more complicated, quickly.
- \(x\) = Value of the \(t^{th}\) observation of \(x\) for the \(i^{th}\) entity
- \(\mu\) = The portion of the disturbance term unique to \(i\) and that is constant over time \(t\).
- \(\epsilon\) = The variance in the \(t^{th}\) observation of \(y\) for the \(i^{th}\) entity that is not explained by the variance in the \(t^{th}\) observation of \(x\) for the \(i^{th}\) entity or \(\mu_{i}\).
Recall that we can decompose \(x_{it}\) into the following:
For each \(x_{it}\), there is going to be a between-component, \(\gamma\), that never changes over time for each \(i\) in the sample (the firm, for example). But there is also going to be a within-component, \(\tau\), that can change for each \(i\) over time (the firm’s sales, for example).
In OLS or the pooled model, we assume that \(\gamma_i\) = 0, or otherwise is constant for every \(i\) in the sample and, hence, meaningless for analysis purposes.
If \(\gamma_i\) ≠ 0, or if \(\gamma_i\) varies among the \(i's\) in the sample, then the OLS (or pooled) model will produce biased estimates of \(\beta\).
In a panel model, we are assuming that the higher order entity (\(\gamma_i\)), has some meaningful effect on the observation of \(x_{it}\).
The problem is that we don’t directly observe this effect. So any unobserved factors of \(i\) that do correlate with \(x_{it}\) will show up in the disturbance term of \(\epsilon_{it}\).
What’s this problem called again?
You guys rock!!!
So in our panel equation:
We need to take some steps to control for the \(\mu_{i}\) effect. We can do that by using the dummy variable approach, or using the within-transformation. This is the fixed effect model.
Sidebar…
How does a fixed effect model differ from a random effect model?
So full disclosure, but we’re not going to spend much time on random effect models, despite the obsession with these models in the psych/OB literature and which, very often, are improperly specified.
One exception is the hybrid method, which we’ll talk about later.
…End sidebar
The dummy variable method, while effective, is a little clunky. So most estimators—like the plm package, use the within-transformation.
Using the within-transformation, our equation looks like this…
In this equation, we’re removing the time invariant component by demeaning each term. This effectively gets rid of any information between \(i'\), because we’re holding that effect constant (just like with the dummy variables).
You will sometimes see the fixed effect model written like this…
Because \(\mu_{i}\) is constant over time, \((\mu_{i}-{\bar{\mu}}_{i})\) equal 0, so we drop it from the equation. We’ve removed the effect of \(i\) on the other variables of interest by subtracting out the constant.
What we are left with is the average within-firm effect over time for, effectively, an average firm in the sample.
Side note…
What happened to the intercept \(\alpha\) and why?
BTW…
This approach is also called “group mean centering,” which is a very common term/approach in the psychology literature using multi-level modeling.
Yes, I know that multi-level modeling is the same thing as panel modeling. It’s a classic case of two disciplines really, really, really, not talking to each other.
Lets revisit our model from yesterday…
library(plm)
my.panel.df$log.revt <- log(my.panel.df$revt)
panel.plm.df <- pdata.frame(my.panel.df, index=c("gvkey","fyear"),
drop.index=TRUE)
fixed.year.model <- plm(log.revt ~ dltt, data = panel.plm.df,
index=c("gvkey", "fyear"),
model="within", effect = c("twoways"))
summary(fixed.year.model)## Twoways effects Within Model
##
## Call:
## plm(formula = log.revt ~ dltt, data = panel.plm.df, effect = c("twoways"),
## model = "within", index = c("gvkey", "fyear"))
##
## Unbalanced Panel: n=845, T=1-21, N=9994
##
## Residuals :
## Min. 1st Qu. Median 3rd Qu. Max.
## -5.0200 -0.1700 0.0104 0.1900 1.9000
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## dltt 4.3075e-07 6.1306e-08 7.0263 2.272e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 1336.7
## Residual Sum of Squares: 1329.5
## R-Squared: 0.0053795
## Adj. R-Squared: -0.088874
## F-statistic: 49.3696 on 1 and 9128 DF, p-value: 2.272e-12We interpret \(\beta\) as the average expected change in firm revenue for a given (average) firm for each unit increase in long term debt.
Specifically, we expect a 0.00000043075 change in log transformed revenue for an average firm.
Or to better interpret a continuous IV with a log transformed DV…
x.effect <- summary(fixed.year.model)$coefficients[1,1]
y.delta <- 100*(exp(x.effect)-1)
y.delta## [1] 4.307544e-05We expect a .000043075% change in revenue for every one million dollar increase in long term debt for an average firm across time holding the effect of differences in time constant. We could also say we expect a 0.0431% change for every \(billion\) dollar increase in long term debt.
Not exactly a BIG effect, but hey, it’s statistically significant!
That we can interpret \(\beta\) as a percentage change in revenue (because revenue was log transformed) is pretty handy, because that’s often what we’re really interested in with panel models—change.
Lets say though that we’re getting deeper with the whole change thing. We’re thinking that long-term debt isn’t really related to the level of sales, but to sales growth rate.
What we want to model then is how much a change in the level of long term debt influences the rate of change in revenue.
So lets go back to our panel data frame and create a sales growth measure. While we are at it, lets also calculate a growth rate measure for long term debt.
library(tidyverse) # Love dplyr!
my.panel.df <- my.panel.df %>%
arrange(gvkey, fyear) %>% # Sort by firm and then by year
group_by(gvkey) %>% # Tell dplyr to go within each firm
mutate(sg.diff = revt - lag(revt), sgr = (100*(sg.diff/lag(revt))),
dgr.diff = revt - lag(dltt), dgr = (100*(dgr.diff/lag(dltt))))Lets take a look at something…
panel.head <- my.panel.df %>%
select(gvkey, fyear, revt, sgr, dltt, dgr)
head(panel.head, 10)## Source: local data frame [10 x 6]
## Groups: gvkey [2]
##
## gvkey fyear revt sgr dltt dgr
## <chr> <int> <dbl> <dbl> <dbl> <dbl>
## 1 001062 2007 101.160 NA 0.0 NA
## 2 001062 2008 169.921 67.972519 0.0 Inf
## 3 001062 2009 69.026 -59.377593 0.0 Inf
## 4 001177 1995 13310.500 NA 977.2 NA
## 5 001177 1996 15360.900 15.404380 2380.0 1471.9300
## 6 001177 1997 18744.900 22.029959 2346.2 687.6008
## 7 001177 1998 20754.000 10.718115 2521.2 784.5793
## 8 001177 1999 26452.700 27.458321 2677.9 949.2107
## 9 001177 2000 26949.000 1.876179 0.0 906.3483
## 10 001177 2001 25377.900 -5.829901 1591.3 InfOne issue with calculating rate of changes is that we lose one year of observations for each firm. If the firm only showed up once in the sample, it’s gone now.
For unbalanced panels, the year lost will also be a function of when in the sample the firm first shows up. Further, it could be the case that you missed a year in between observations of the firm because it may have skipped a year of rate of change
We see this happening with the dgr variable. There are a few years that have a 0 value. Well, the firm might have paid off its debt that year (or it could be a data entry error), but that creates a division by zero problem. So, the next year’s growth rate may be HUGE by comparison, skewing the distribution.
There’s not a whole lot to do about it, other than really getting to know your data. Average smoothing or rolling averages may also help.
One thing to keep in mind with ALL data but especially when working with secondary financial data is outliers. Take a look at this…
Lets take a look at observations with a greater than 500% sales growth rate…
extreme.df <- my.panel.df %>%
select(gvkey, conm, fyear, sgr) %>%
filter(sgr > 500) %>%
arrange(desc(sgr))
extreme.df## Source: local data frame [7 x 4]
## Groups: gvkey [7]
##
## gvkey conm fyear sgr
## <chr> <chr> <int> <dbl>
## 1 031521 INTL FCSTONE INC 2007 4240.4735
## 2 025895 RADIAN GROUP INC 2008 1370.5761
## 3 160312 PRIMUS GUARANTY LTD 2009 894.0807
## 4 160293 NORTHSTAR REALTY FINANCE CP 2013 718.9489
## 5 001487 AMERICAN INTERNATIONAL GROUP 2009 700.9000
## 6 127377 BGC PARTNERS INC 2008 671.8695
## 7 023485 COVANTA HOLDING CORP 2002 557.8243Now, you would need to get in to the data to really understand what is going on here. For now, lets just drop these observations on the assumption that they are so far outside the mainstream to bias our results
Yes, that is a big assumption—don’t just do this without thinking!
panel.no.df <- my.panel.df %>%
filter(sgr < 500) # Note that this will also get rid of 0 values for SGR!Take a look now though at the distribution of sales growth rate…
Ok, so lets create a new plm dataframe, and lets estimate a fixed effect model of the level of long term debt (not dgr) predicting sales growth rate without our outliers.
no.plm.df <- pdata.frame(panel.no.df, index=c("gvkey","fyear"),
drop.index=TRUE)
sgr.model <- plm(sgr ~ dltt, data = no.plm.df, index=c("gvkey", "fyear"),
model="within", effect = c("twoways"))
summary(sgr.model)## Twoways effects Within Model
##
## Call:
## plm(formula = sgr ~ dltt, data = no.plm.df, effect = c("twoways"),
## model = "within", index = c("gvkey", "fyear"))
##
## Unbalanced Panel: n=822, T=1-20, N=9149
##
## Residuals :
## Min. 1st Qu. Median 3rd Qu. Max.
## -157.00 -9.93 -1.75 6.70 440.00
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## dltt 2.5155e-05 4.6848e-06 5.3696 8.104e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 6815900
## Residual Sum of Squares: 6792400
## R-Squared: 0.0034589
## Adj. R-Squared: -0.097431
## F-statistic: 28.8327 on 1 and 8307 DF, p-value: 8.104e-08Ok, your turn. How do we interpret \(\beta\)?
Modeling change is actually pretty difficult. It is, however, theoretically interesting and practically important.
The best way to learn about modeling change is to grab a longitudinal dataset and start p-hacking. Just to be clear though, no, you can’t publish what you p-hacked!
Lets move on to instrument variable models.
No causation without manipulation.
In panel data like we have been working with, it’s entirely observational. That means that we’re dealing with an omitted variable problem from any number of sources.
One nice thing though about fixed effect models is that we’ve already controlled for a substantial omitted variable problem—the unobserved heterogeneity existing at the \(i\) level that correlates with the \(t\) level observation.
Basically, all of the unobserved ‘stuff’ existing at the firm (or manager, or team, etc.) that you didn’t measure but that can influence the \(x_{it}\)—> \(y_{it}\) relationship.
So with a fixed effect model, we’re already a long way to where we need to be to isolate the causal effect of \(x_{it}\) on \(y_{it}\).
But…
There may also be other factors at the \(t\) level that are unobserved, and that represent an omitted variable problem. So, we need some help.
If you remember, the great thing about instrument variable models is that once you understand the basics, you can extend the framework into more complicated models.
Instruments must be…
- Individually and jointly valid predictors of the endogenous variable
- Properly excluded from the second stage equation (exogenous)
Note the | operator to delineate the instruments in the model.
inst.model <- plm(sgr ~ dltt | lt + ch,
data = no.plm.df, index=c("gvkey", "fyear"),
model="within", effect = c("twoways"))
summary(inst.model)## Twoways effects Within Model
## Instrumental variable estimation
## (Balestra-Varadharajan-Krishnakumar's transformation)
##
## Call:
## plm(formula = sgr ~ dltt | lt + ch, data = no.plm.df, effect = c("twoways"),
## model = "within", index = c("gvkey", "fyear"))
##
## Unbalanced Panel: n=822, T=1-20, N=9149
##
## Residuals :
## Min. 1st Qu. Median 3rd Qu. Max.
## -157.00 -9.92 -1.76 6.68 440.00
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## dltt 1.7049e-05 7.2804e-06 2.3418 0.01921 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 6815900
## Residual Sum of Squares: 6794800
## R-Squared: 0.0034589
## Adj. R-Squared: -0.097431
## F-statistic: 25.8294 on 1 and 8307 DF, p-value: 3.811e-07Unfortunately, 2SLS with panel data gets really complicated, really fast. The big thing to remember at this stage of your training is that yes, you can—and likely need to—integrate instruments into your fixed effect panel models.
If your research takes you down that path, we’ll get you additional training :)
A criticism—completely accurate—of fixed effect models is that they are a blunt instrument.
By eliminating all of the between-effect variance, any variable that exists at the \(i\) (firm) level, or that is otherwise invariant across time for each \(i\), can’t be used in the model.
Why?
Well, lets do an experiment.
In our dataframe is the firm’s SIC code, which represents the firm’s Standard Industrial Classification–basically, the firm’s primary industry.
Now lets say that we want to control for the firm’s industry. Makes sense, right? Firms are nested within industries, so it follows that there might be factors at the industry level that might impact how all, or some, or none, of the firms within that industry on whatever research question we are interested in.
The problem though is that there are not any cases of firm’s within the sample changing industries. Lets take a look at how many industries there are in our dataset…
# We're first going to create a new factor variable based on SIC code
panel.no.df$industry <- as.factor(panel.no.df$sic)
nlevels(panel.no.df$industry) # Count the instries## [1] 43This will be easier to see using the dummy variable method…
sgr.ind.model <- lm(sgr ~ dltt + factor(gvkey)-1 +
factor(industry)-1, data = panel.no.df)
summary(sgr.ind.model)##
## Call:
## lm(formula = sgr ~ dltt + factor(gvkey) - 1 + factor(industry) -
## 1, data = panel.no.df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -152.27 -11.16 -1.99 7.81 430.10
##
## Coefficients: (42 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## dltt 1.610e-05 4.813e-06 3.346 0.000823 ***
## factor(gvkey)001062 4.297e+00 2.093e+01 0.205 0.837285
## factor(gvkey)001177 8.756e+00 6.789e+00 1.290 0.197169
## factor(gvkey)001257 1.051e+01 7.641e+00 1.376 0.168843
## factor(gvkey)001274 3.045e+01 7.909e+00 3.851 0.000119 ***
## factor(gvkey)001414 -6.133e+00 1.208e+01 -0.508 0.611734
## factor(gvkey)001447 3.845e+00 6.790e+00 0.566 0.571204
## factor(gvkey)001449 6.471e+00 6.789e+00 0.953 0.340528
## factor(gvkey)001487 1.355e+00 6.981e+00 0.194 0.846126
## factor(gvkey)001526 4.032e+00 6.789e+00 0.594 0.552569
## factor(gvkey)001545 -9.211e+00 1.118e+01 -0.823 0.410256
## factor(gvkey)001618 -1.112e+01 1.118e+01 -0.994 0.320218
## factor(gvkey)001949 3.631e+00 2.959e+01 0.123 0.902352
## factor(gvkey)001982 9.767e+00 7.641e+00 1.278 0.201196
## factor(gvkey)002002 3.002e+00 6.789e+00 0.442 0.658352
## factor(gvkey)002005 -2.189e+00 6.789e+00 -0.322 0.747093
## factor(gvkey)002019 8.846e+00 6.789e+00 1.303 0.192610
## factor(gvkey)002176 2.448e+01 6.790e+00 3.606 0.000313 ***
## factor(gvkey)002547 3.554e+00 6.789e+00 0.524 0.600602
## factor(gvkey)002558 -1.839e+00 6.789e+00 -0.271 0.786464
## factor(gvkey)002620 6.318e+00 6.789e+00 0.931 0.352051
## factor(gvkey)002849 9.727e+00 7.641e+00 1.273 0.203051
## factor(gvkey)002968 7.364e+00 6.815e+00 1.081 0.279904
## factor(gvkey)003024 4.792e+00 6.789e+00 0.706 0.480299
## factor(gvkey)003082 9.813e+00 6.789e+00 1.445 0.148380
## factor(gvkey)003221 7.559e+00 6.789e+00 1.113 0.265535
## factor(gvkey)003231 -4.315e-01 6.789e+00 -0.064 0.949327
## factor(gvkey)003238 2.125e+00 6.789e+00 0.313 0.754316
## factor(gvkey)003243 9.952e+00 6.844e+00 1.454 0.145915
## factor(gvkey)003278 4.276e+01 1.323e+01 3.231 0.001238 **
## factor(gvkey)003410 -8.911e+00 1.118e+01 -0.797 0.425631
## factor(gvkey)003562 1.144e+01 7.909e+00 1.447 0.147915
## factor(gvkey)003643 6.410e+00 6.789e+00 0.944 0.345141
## factor(gvkey)004193 5.740e+00 7.641e+00 0.751 0.452508
## factor(gvkey)004201 1.273e+01 6.789e+00 1.876 0.060725 .
## factor(gvkey)004601 6.510e+00 8.607e+00 0.756 0.449416
## factor(gvkey)004605 6.489e+00 7.909e+00 0.820 0.411981
## factor(gvkey)004640 9.010e+00 6.789e+00 1.327 0.184493
## factor(gvkey)004666 2.217e+01 7.909e+00 2.803 0.005074 **
## factor(gvkey)004674 1.025e+01 6.789e+00 1.509 0.131249
## factor(gvkey)004678 5.468e+00 6.789e+00 0.805 0.420611
## factor(gvkey)004685 2.367e+00 6.789e+00 0.349 0.727387
## factor(gvkey)004690 4.111e+00 6.789e+00 0.606 0.544793
## factor(gvkey)004699 8.580e+00 6.789e+00 1.264 0.206342
## factor(gvkey)004723 2.712e+01 6.790e+00 3.994 6.54e-05 ***
## factor(gvkey)004737 7.305e-01 6.789e+00 0.108 0.914312
## factor(gvkey)004740 1.421e+01 8.207e+00 1.731 0.083501 .
## factor(gvkey)004842 3.846e+00 7.909e+00 0.486 0.626818
## factor(gvkey)004885 1.506e+01 6.789e+00 2.218 0.026566 *
## factor(gvkey)005048 -3.569e+00 7.715e+00 -0.463 0.643629
## factor(gvkey)005072 -4.347e+00 2.093e+01 -0.208 0.835450
## factor(gvkey)005149 3.698e+00 7.641e+00 0.484 0.628391
## factor(gvkey)005342 4.821e+01 9.358e+00 5.152 2.63e-07 ***
## factor(gvkey)005543 2.794e+01 7.909e+00 3.533 0.000413 ***
## factor(gvkey)005735 4.608e+00 6.794e+00 0.678 0.497655
## factor(gvkey)005763 7.355e+00 8.922e+00 0.824 0.409783
## factor(gvkey)005786 3.381e+00 6.789e+00 0.498 0.618461
## factor(gvkey)005849 7.685e+00 7.909e+00 0.972 0.331220
## factor(gvkey)005862 -1.691e+00 8.207e+00 -0.206 0.836806
## factor(gvkey)006239 1.452e+01 6.789e+00 2.139 0.032503 *
## factor(gvkey)006333 6.181e-02 7.641e+00 0.008 0.993546
## factor(gvkey)006653 1.887e+01 6.975e+00 2.706 0.006822 **
## factor(gvkey)006682 3.354e+01 9.864e+00 3.401 0.000676 ***
## factor(gvkey)006742 5.478e+00 6.789e+00 0.807 0.419766
## factor(gvkey)006781 -9.033e-01 6.789e+00 -0.133 0.894156
## factor(gvkey)006791 7.336e+00 9.864e+00 0.744 0.457052
## factor(gvkey)007063 1.753e+01 6.975e+00 2.513 0.011986 *
## factor(gvkey)007525 9.362e+00 1.480e+01 0.633 0.526948
## factor(gvkey)007647 1.110e+01 6.823e+00 1.627 0.103678
## factor(gvkey)007982 5.772e+00 6.789e+00 0.850 0.395248
## factor(gvkey)008007 1.553e+01 6.799e+00 2.284 0.022387 *
## factor(gvkey)008148 1.218e+01 2.093e+01 0.582 0.560686
## factor(gvkey)008240 -3.683e+00 8.207e+00 -0.449 0.653621
## factor(gvkey)008245 6.611e+00 6.790e+00 0.974 0.330263
## factor(gvkey)008363 1.362e+01 7.641e+00 1.782 0.074760 .
## factor(gvkey)008431 2.733e+00 6.789e+00 0.403 0.687244
## factor(gvkey)008457 1.270e+01 7.909e+00 1.606 0.108221
## factor(gvkey)008605 5.530e-02 8.207e+00 0.007 0.994625
## factor(gvkey)008898 1.306e+01 6.789e+00 1.924 0.054359 .
## factor(gvkey)009061 6.904e+00 6.789e+00 1.017 0.309258
## factor(gvkey)009083 1.819e+01 8.543e+00 2.130 0.033220 *
## factor(gvkey)009256 7.954e+00 1.046e+01 0.760 0.447127
## factor(gvkey)009317 1.088e+01 6.789e+00 1.602 0.109134
## factor(gvkey)009783 -1.396e+00 6.789e+00 -0.206 0.837103
## factor(gvkey)010035 9.380e+00 6.789e+00 1.382 0.167115
## factor(gvkey)010086 1.286e+01 6.975e+00 1.844 0.065261 .
## factor(gvkey)010096 8.727e+00 7.909e+00 1.103 0.269879
## factor(gvkey)010121 2.266e+00 6.799e+00 0.333 0.738902
## factor(gvkey)010137 9.100e+00 8.927e+00 1.019 0.308053
## factor(gvkey)010187 5.840e+00 6.789e+00 0.860 0.389707
## factor(gvkey)010390 1.260e+00 1.323e+01 0.095 0.924136
## factor(gvkey)010614 3.593e+00 6.789e+00 0.529 0.596695
## factor(gvkey)010713 1.465e+00 1.709e+01 0.086 0.931647
## factor(gvkey)010894 6.272e+00 7.177e+00 0.874 0.382201
## factor(gvkey)010903 1.402e+01 7.641e+00 1.835 0.066555 .
## factor(gvkey)010916 3.230e+00 6.789e+00 0.476 0.634255
## factor(gvkey)010917 1.852e+01 2.093e+01 0.885 0.376244
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## factor(gvkey)011220 1.012e+01 7.641e+00 1.324 0.185523
## factor(gvkey)011301 5.973e+00 7.641e+00 0.782 0.434418
## factor(gvkey)011340 7.321e+00 7.641e+00 0.958 0.338002
## factor(gvkey)011687 9.817e+00 6.789e+00 1.446 0.148208
## factor(gvkey)011729 2.174e+01 6.789e+00 3.203 0.001367 **
## factor(gvkey)011770 2.139e+01 7.641e+00 2.800 0.005129 **
## factor(gvkey)011819 2.245e+01 1.323e+01 1.697 0.089827 .
## factor(gvkey)011842 9.463e+00 6.789e+00 1.394 0.163413
## factor(gvkey)011856 1.003e+01 6.789e+00 1.477 0.139611
## factor(gvkey)011861 4.148e+00 6.789e+00 0.611 0.541252
## factor(gvkey)011896 3.091e+00 6.789e+00 0.455 0.648897
## factor(gvkey)012124 1.042e+01 6.632e+00 1.572 0.116072
## factor(gvkey)012138 1.346e+01 6.789e+00 1.982 0.047520 *
## factor(gvkey)012140 1.289e+01 7.398e+00 1.742 0.081512 .
## factor(gvkey)012407 -4.331e+00 2.959e+01 -0.146 0.883641
## factor(gvkey)012544 2.098e+01 7.641e+00 2.746 0.006040 **
## factor(gvkey)012603 4.293e+00 6.789e+00 0.632 0.527172
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## factor(gvkey)013041 2.518e+00 6.789e+00 0.371 0.710762
## factor(gvkey)013125 1.598e+01 7.909e+00 2.021 0.043357 *
## factor(gvkey)013142 -5.792e-01 7.909e+00 -0.073 0.941626
## factor(gvkey)013189 1.426e+01 9.358e+00 1.524 0.127542
## factor(gvkey)013294 -5.626e+00 1.480e+01 -0.380 0.703801
## factor(gvkey)013341 1.075e+01 6.789e+00 1.583 0.113479
## factor(gvkey)013342 8.513e+00 6.789e+00 1.254 0.209884
## factor(gvkey)013453 1.208e+01 6.789e+00 1.779 0.075240 .
## factor(gvkey)013510 4.635e+00 7.641e+00 0.607 0.544147
## factor(gvkey)013561 1.718e+01 7.398e+00 2.322 0.020256 *
## factor(gvkey)013562 6.594e+00 7.641e+00 0.863 0.388133
## factor(gvkey)013579 5.411e+00 7.909e+00 0.684 0.493875
## factor(gvkey)013580 2.991e+01 1.709e+01 1.750 0.080075 .
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## factor(gvkey)014140 8.355e+00 6.844e+00 1.221 0.222237
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## factor(gvkey)014219 5.958e+00 6.789e+00 0.878 0.380210
## factor(gvkey)014253 1.489e+00 6.789e+00 0.219 0.826364
## factor(gvkey)014275 1.500e+01 6.789e+00 2.210 0.027138 *
## factor(gvkey)014401 1.532e+01 1.480e+01 1.036 0.300378
## factor(gvkey)014403 1.438e+01 1.480e+01 0.972 0.331274
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## factor(gvkey)014828 7.731e+00 7.909e+00 0.978 0.328328
## factor(gvkey)015101 2.326e+01 8.207e+00 2.834 0.004603 **
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## factor(gvkey)015208 1.103e+01 7.599e+00 1.452 0.146535
## factor(gvkey)015261 4.551e+00 7.909e+00 0.575 0.565032
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## factor(gvkey)015363 3.410e+00 6.789e+00 0.502 0.615510
## factor(gvkey)015364 1.748e+01 7.641e+00 2.288 0.022187 *
## factor(gvkey)015505 5.246e+00 7.178e+00 0.731 0.464927
## factor(gvkey)015509 3.636e+00 7.423e+00 0.490 0.624298
## factor(gvkey)015532 3.753e+00 8.293e+00 0.453 0.650835
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## factor(gvkey)015576 1.896e+00 6.823e+00 0.278 0.781126
## factor(gvkey)015634 9.320e+00 7.668e+00 1.215 0.224217
## factor(gvkey)015679 1.110e+01 8.208e+00 1.352 0.176362
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## factor(gvkey)016305 1.091e+01 8.208e+00 1.329 0.183766
## factor(gvkey)016348 -1.394e+01 2.095e+01 -0.666 0.505722
## factor(gvkey)016549 1.464e+01 2.093e+01 0.700 0.484189
## factor(gvkey)016668 1.533e+00 6.789e+00 0.226 0.821329
## factor(gvkey)016681 5.039e+00 6.975e+00 0.722 0.470080
## factor(gvkey)016698 7.712e+00 8.207e+00 0.940 0.347433
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## factor(gvkey)016714 9.308e+00 7.641e+00 1.218 0.223187
## factor(gvkey)016716 6.483e+00 7.641e+00 0.848 0.396205
## factor(gvkey)016720 4.930e+02 2.959e+01 16.659 < 2e-16 ***
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## factor(gvkey)016821 6.540e+00 6.789e+00 0.963 0.335391
## factor(gvkey)016832 1.392e+01 7.177e+00 1.940 0.052438 .
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## factor(gvkey)016878 5.677e+00 8.922e+00 0.636 0.524647
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## factor(gvkey)016967 5.015e+00 1.709e+01 0.294 0.769133
## factor(gvkey)016981 1.208e+01 6.789e+00 1.780 0.075153 .
## factor(gvkey)016989 6.473e-01 9.864e+00 0.066 0.947684
## factor(gvkey)017035 6.740e+00 7.641e+00 0.882 0.377720
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## factor(gvkey)017073 5.952e+00 7.641e+00 0.779 0.436044
## factor(gvkey)017074 1.711e+01 7.641e+00 2.239 0.025184 *
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## factor(gvkey)017120 1.740e+01 6.789e+00 2.563 0.010381 *
## factor(gvkey)017130 2.303e+01 6.789e+00 3.392 0.000696 ***
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## factor(gvkey)017132 5.211e+00 6.789e+00 0.768 0.442776
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## factor(gvkey)017145 2.990e+00 6.789e+00 0.440 0.659627
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## factor(gvkey)017195 4.671e+00 6.789e+00 0.688 0.491438
## factor(gvkey)017197 2.937e+01 6.975e+00 4.211 2.57e-05 ***
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## factor(gvkey)017245 1.239e-01 6.789e+00 0.018 0.985439
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## factor(gvkey)017252 6.992e+00 7.641e+00 0.915 0.360154
## factor(gvkey)017266 2.733e+01 7.641e+00 3.576 0.000350 ***
## factor(gvkey)017269 -4.337e+00 6.789e+00 -0.639 0.522970
## factor(gvkey)017367 2.216e+00 1.208e+01 0.183 0.854438
## factor(gvkey)017388 -1.602e+00 8.922e+00 -0.179 0.857553
## factor(gvkey)017438 1.248e+01 1.480e+01 0.844 0.398934
## factor(gvkey)017451 3.269e+01 1.709e+01 1.913 0.055765 .
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## factor(gvkey)017556 2.503e+01 2.093e+01 1.196 0.231607
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## factor(gvkey)017724 8.577e+00 9.864e+00 0.869 0.384616
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## factor(gvkey)018037 7.079e-01 7.909e+00 0.090 0.928677
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## factor(gvkey)018110 1.014e+00 6.789e+00 0.149 0.881326
## factor(gvkey)018184 5.744e+01 2.093e+01 2.745 0.006064 **
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## factor(gvkey)019057 -4.801e+00 7.909e+00 -0.607 0.543846
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## factor(gvkey)019124 3.183e+00 6.789e+00 0.469 0.639194
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## factor(gvkey)019159 8.309e+00 2.959e+01 0.281 0.778877
## factor(gvkey)019262 1.872e+00 6.789e+00 0.276 0.782699
## factor(gvkey)019318 6.401e+00 6.789e+00 0.943 0.345753
## factor(gvkey)019355 6.459e+00 6.789e+00 0.951 0.341450
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## factor(gvkey)019570 3.515e+00 6.789e+00 0.518 0.604694
## factor(gvkey)019713 1.302e+01 7.398e+00 1.759 0.078562 .
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## factor(gvkey)019860 -1.063e+00 2.959e+01 -0.036 0.971357
## factor(gvkey)019873 7.056e+00 6.975e+00 1.012 0.311740
## factor(gvkey)019927 2.546e+01 2.959e+01 0.860 0.389694
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## factor(gvkey)020277 1.350e+01 2.093e+01 0.645 0.518860
## factor(gvkey)020280 1.677e+02 2.959e+01 5.668 1.49e-08 ***
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## factor(gvkey)020344 4.291e+00 2.093e+01 0.205 0.837527
## factor(gvkey)020422 8.723e+01 2.959e+01 2.948 0.003210 **
## factor(gvkey)020677 2.227e+01 2.959e+01 0.753 0.451689
## factor(gvkey)020761 6.533e+01 2.959e+01 2.208 0.027300 *
## factor(gvkey)020791 2.625e+01 8.543e+00 3.072 0.002131 **
## factor(gvkey)021073 2.064e+01 2.959e+01 0.697 0.485588
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## factor(gvkey)021326 1.523e+01 2.093e+01 0.728 0.466629
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## factor(gvkey)021616 -2.030e+00 2.093e+01 -0.097 0.922726
## factor(gvkey)021825 -3.623e+00 2.093e+01 -0.173 0.862561
## factor(gvkey)022025 1.995e+02 2.959e+01 6.741 1.68e-11 ***
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## factor(gvkey)022983 5.146e+00 2.959e+01 0.174 0.861960
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## factor(gvkey)023768 4.471e+01 1.208e+01 3.701 0.000216 ***
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## factor(gvkey)024233 8.031e+00 7.641e+00 1.051 0.293251
## factor(gvkey)024287 2.254e+01 8.208e+00 2.747 0.006033 **
## factor(gvkey)024318 1.448e+01 6.975e+00 2.076 0.037971 *
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## factor(gvkey)024466 1.621e+01 6.975e+00 2.323 0.020184 *
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## factor(gvkey)024725 1.361e+01 7.641e+00 1.781 0.074889 .
## factor(gvkey)024731 7.234e+00 7.909e+00 0.915 0.360379
## factor(gvkey)024750 4.312e+01 1.709e+01 2.524 0.011636 *
## factor(gvkey)024825 1.427e+01 7.909e+00 1.805 0.071131 .
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## factor(gvkey)025339 1.329e+01 7.641e+00 1.739 0.082119 .
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## factor(gvkey)025425 -2.190e+00 2.959e+01 -0.074 0.941000
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## factor(gvkey)025570 1.143e+00 6.617e+00 0.173 0.862831
## factor(gvkey)025632 5.043e+01 1.323e+01 3.810 0.000140 ***
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## factor(gvkey)025665 2.629e+00 7.909e+00 0.332 0.739577
## factor(gvkey)025714 2.076e+01 8.209e+00 2.529 0.011461 *
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## factor(gvkey)025877 1.566e+01 7.641e+00 2.049 0.040449 *
## factor(gvkey)025895 2.433e+01 6.975e+00 3.488 0.000489 ***
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## factor(gvkey)027932 2.131e+01 9.864e+00 2.160 0.030795 *
## factor(gvkey)028022 7.584e+00 6.789e+00 1.117 0.263992
## factor(gvkey)028034 2.178e+01 6.975e+00 3.123 0.001798 **
## factor(gvkey)028119 1.339e+01 7.398e+00 1.810 0.070354 .
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## factor(gvkey)028155 6.272e+00 6.789e+00 0.924 0.355599
## factor(gvkey)028216 1.396e+01 7.641e+00 1.827 0.067725 .
## factor(gvkey)028278 2.893e+00 7.641e+00 0.379 0.704926
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## factor(gvkey)028629 7.584e+00 7.909e+00 0.959 0.337629
## factor(gvkey)028733 1.200e+01 6.975e+00 1.720 0.085518 .
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## factor(gvkey)028866 1.120e+01 1.046e+01 1.071 0.284326
## factor(gvkey)028967 7.628e+00 9.864e+00 0.773 0.439385
## factor(gvkey)029052 1.735e+01 7.398e+00 2.345 0.019068 *
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## factor(gvkey)029061 -7.836e-01 1.323e+01 -0.059 0.952785
## factor(gvkey)029082 9.030e+00 7.641e+00 1.182 0.237303
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## factor(gvkey)029101 6.524e+00 6.789e+00 0.961 0.336570
## factor(gvkey)029211 1.565e+01 7.641e+00 2.048 0.040620 *
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## factor(gvkey)029282 2.129e+01 6.789e+00 3.136 0.001719 **
## factor(gvkey)029286 1.382e+01 6.789e+00 2.036 0.041766 *
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## factor(gvkey)029868 1.614e+01 7.641e+00 2.112 0.034699 *
## factor(gvkey)029875 1.225e+01 7.398e+00 1.656 0.097795 .
## factor(gvkey)029984 2.503e+01 7.398e+00 3.384 0.000718 ***
## factor(gvkey)029994 2.847e+00 1.323e+01 0.215 0.829650
## factor(gvkey)030146 1.331e+01 6.789e+00 1.960 0.050019 .
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## factor(gvkey)030452 1.391e+01 7.910e+00 1.759 0.078685 .
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## factor(gvkey)030580 1.868e+01 7.912e+00 2.361 0.018231 *
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## factor(gvkey)030647 2.444e+00 2.959e+01 0.083 0.934179
## factor(gvkey)030822 1.654e+01 7.909e+00 2.091 0.036577 *
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## factor(gvkey)030932 6.233e+00 6.789e+00 0.918 0.358583
## factor(gvkey)030990 1.928e+01 6.789e+00 2.839 0.004535 **
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## factor(gvkey)031368 9.653e+00 7.398e+00 1.305 0.192010
## factor(gvkey)031521 6.432e+01 1.118e+01 5.751 9.19e-09 ***
## factor(gvkey)031692 1.829e+01 7.177e+00 2.548 0.010837 *
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## factor(gvkey)031718 6.915e+00 2.959e+01 0.234 0.815232
## factor(gvkey)031764 5.253e+00 9.864e+00 0.533 0.594340
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## factor(gvkey)061067 1.893e+01 6.789e+00 2.789 0.005302 **
## factor(gvkey)061129 2.958e+01 8.208e+00 3.604 0.000315 ***
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## factor(gvkey)061188 1.983e+01 6.975e+00 2.844 0.004471 **
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## factor(gvkey)061380 6.123e+01 1.208e+01 5.068 4.10e-07 ***
## factor(gvkey)061388 1.139e+01 6.789e+00 1.677 0.093499 .
## factor(gvkey)061406 -9.232e-01 9.864e+00 -0.094 0.925433
## factor(gvkey)061408 2.760e+00 6.975e+00 0.396 0.692354
## factor(gvkey)061452 1.217e+01 6.789e+00 1.793 0.073058 .
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## factor(gvkey)061544 1.468e+01 1.480e+01 0.992 0.321284
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## factor(gvkey)061586 1.676e+01 7.641e+00 2.193 0.028307 *
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## factor(gvkey)062654 6.932e+00 6.789e+00 1.021 0.307284
## factor(gvkey)062689 1.117e+01 6.789e+00 1.646 0.099797 .
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## factor(gvkey)062976 2.650e+01 1.323e+01 2.002 0.045290 *
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## factor(gvkey)063288 7.708e+00 7.909e+00 0.975 0.329809
## factor(gvkey)063501 3.664e+01 6.975e+00 5.253 1.53e-07 ***
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## factor(gvkey)063639 1.828e+01 6.975e+00 2.621 0.008789 **
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## factor(gvkey)063781 1.577e+01 7.398e+00 2.131 0.033087 *
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## factor(gvkey)064306 1.490e+01 7.177e+00 2.076 0.037909 *
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## factor(gvkey)064547 1.495e+01 7.398e+00 2.020 0.043369 *
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## factor(gvkey)064699 1.379e+01 6.975e+00 1.977 0.048101 *
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## factor(gvkey)064821 1.588e+01 7.641e+00 2.079 0.037658 *
## factor(gvkey)064925 1.272e+01 7.398e+00 1.720 0.085517 .
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## factor(gvkey)065290 1.644e+01 7.909e+00 2.078 0.037705 *
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## factor(gvkey)065540 2.117e+01 8.922e+00 2.372 0.017708 *
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## factor(gvkey)065556 3.508e+01 7.641e+00 4.591 4.48e-06 ***
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## factor(gvkey)065710 1.576e+01 7.909e+00 1.993 0.046279 *
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## factor(gvkey)122515 1.950e+01 1.118e+01 1.743 0.081327 .
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## factor(gvkey)149337 1.546e+01 8.207e+00 1.884 0.059604 .
## factor(gvkey)149618 1.094e+02 2.093e+01 5.229 1.75e-07 ***
## factor(gvkey)149738 6.076e+00 7.642e+00 0.795 0.426615
## factor(gvkey)150279 3.802e+00 7.909e+00 0.481 0.630708
## factor(gvkey)150306 1.865e+01 7.909e+00 2.358 0.018391 *
## factor(gvkey)152149 2.605e+01 8.207e+00 3.173 0.001512 **
## factor(gvkey)152249 7.881e-01 9.864e+00 0.080 0.936319
## factor(gvkey)153130 2.190e+01 8.543e+00 2.564 0.010360 *
## factor(gvkey)154595 1.613e+01 1.208e+01 1.336 0.181747
## factor(gvkey)154739 2.094e+01 2.093e+01 1.001 0.317093
## factor(gvkey)154759 1.563e+01 2.093e+01 0.747 0.455201
## factor(gvkey)155174 1.177e+01 9.864e+00 1.193 0.233002
## factor(gvkey)155738 1.036e+01 8.543e+00 1.212 0.225481
## factor(gvkey)155754 3.413e+01 9.358e+00 3.647 0.000267 ***
## factor(gvkey)156156 4.076e+01 1.208e+01 3.374 0.000745 ***
## factor(gvkey)156157 6.052e+00 8.922e+00 0.678 0.497637
## factor(gvkey)156176 4.969e+00 1.046e+01 0.475 0.634813
## factor(gvkey)156383 2.576e+01 8.207e+00 3.138 0.001704 **
## factor(gvkey)156384 -4.465e+00 9.864e+00 -0.453 0.650813
## factor(gvkey)156653 -5.095e+00 8.922e+00 -0.571 0.567974
## factor(gvkey)156953 3.783e+01 8.543e+00 4.429 9.60e-06 ***
## factor(gvkey)157057 4.031e+00 8.543e+00 0.472 0.637000
## factor(gvkey)157307 -1.478e+00 1.323e+01 -0.112 0.911063
## factor(gvkey)157353 5.635e+00 9.864e+00 0.571 0.567825
## factor(gvkey)157452 2.277e+01 2.093e+01 1.088 0.276554
## factor(gvkey)157679 5.325e+01 1.323e+01 4.024 5.78e-05 ***
## factor(gvkey)157955 2.688e+01 8.922e+00 3.013 0.002598 **
## factor(gvkey)158053 8.535e+01 2.093e+01 4.079 4.56e-05 ***
## factor(gvkey)158354 -1.209e+00 8.543e+00 -0.142 0.887427
## factor(gvkey)158587 1.600e+01 9.864e+00 1.622 0.104818
## factor(gvkey)158742 1.497e+01 9.864e+00 1.517 0.129260
## factor(gvkey)160173 1.824e+01 9.358e+00 1.949 0.051359 .
## factor(gvkey)160181 1.397e+01 8.543e+00 1.636 0.101921
## factor(gvkey)160225 1.531e+01 1.323e+01 1.157 0.247443
## factor(gvkey)160233 6.455e+01 1.208e+01 5.343 9.36e-08 ***
## factor(gvkey)160293 4.997e+01 1.046e+01 4.776 1.82e-06 ***
## factor(gvkey)160312 -3.628e+01 1.709e+01 -2.124 0.033731 *
## factor(gvkey)160378 3.236e+01 1.046e+01 3.093 0.001989 **
## factor(gvkey)160417 7.869e+00 8.543e+00 0.921 0.356975
## factor(gvkey)160479 2.848e+01 9.358e+00 3.043 0.002351 **
## factor(gvkey)160541 2.224e+01 1.323e+01 1.680 0.092944 .
## factor(gvkey)160621 -3.915e-01 1.208e+01 -0.032 0.974151
## factor(gvkey)160667 3.358e+00 1.709e+01 0.197 0.844189
## factor(gvkey)160706 2.138e+01 9.864e+00 2.167 0.030235 *
## factor(gvkey)160719 1.344e+01 8.922e+00 1.506 0.132018
## factor(gvkey)160776 2.499e+01 9.358e+00 2.671 0.007581 **
## factor(gvkey)160891 1.964e+00 9.864e+00 0.199 0.842161
## factor(gvkey)160989 2.385e+01 8.922e+00 2.673 0.007531 **
## factor(gvkey)160990 1.637e+01 8.922e+00 1.834 0.066667 .
## factor(gvkey)160991 3.665e+01 8.922e+00 4.108 4.03e-05 ***
## factor(gvkey)161000 8.307e+00 1.046e+01 0.794 0.427252
## factor(gvkey)161013 2.343e+01 9.864e+00 2.376 0.017541 *
## factor(gvkey)161040 1.492e+01 9.358e+00 1.595 0.110809
## factor(gvkey)161048 2.522e+01 8.543e+00 2.952 0.003163 **
## factor(gvkey)161065 4.832e+01 1.709e+01 2.828 0.004694 **
## factor(gvkey)161853 8.252e-01 8.922e+00 0.092 0.926315
## factor(gvkey)161942 1.416e-01 8.922e+00 0.016 0.987337
## factor(gvkey)161952 1.465e+01 8.922e+00 1.642 0.100635
## factor(gvkey)161953 1.477e+01 9.864e+00 1.497 0.134429
## factor(gvkey)161966 5.507e+01 1.046e+01 5.263 1.45e-07 ***
## factor(gvkey)162160 2.557e+01 1.046e+01 2.444 0.014545 *
## factor(gvkey)162385 1.808e+01 1.046e+01 1.728 0.083984 .
## factor(gvkey)162489 1.579e+01 9.358e+00 1.687 0.091659 .
## factor(gvkey)162557 1.705e+01 9.864e+00 1.728 0.084009 .
## factor(gvkey)162559 6.839e+00 8.922e+00 0.767 0.443380
## factor(gvkey)162560 2.060e+01 1.208e+01 1.705 0.088237 .
## factor(gvkey)162925 6.598e+00 1.046e+01 0.631 0.528326
## factor(gvkey)163049 5.669e+01 2.959e+01 1.916 0.055431 .
## factor(gvkey)163610 4.622e+01 8.922e+00 5.181 2.26e-07 ***
## factor(gvkey)163678 5.139e+00 8.922e+00 0.576 0.564627
## factor(gvkey)163680 3.755e+01 1.323e+01 2.838 0.004555 **
## factor(gvkey)163863 1.775e+01 2.093e+01 0.848 0.396295
## factor(gvkey)163872 2.488e+01 2.093e+01 1.189 0.234447
## factor(gvkey)163884 1.086e+01 8.922e+00 1.218 0.223412
## factor(gvkey)163920 2.524e+01 8.922e+00 2.829 0.004679 **
## factor(gvkey)163924 7.303e+01 2.093e+01 3.490 0.000485 ***
## factor(gvkey)163963 1.094e+00 2.093e+01 0.052 0.958300
## factor(gvkey)164059 9.626e+00 2.959e+01 0.325 0.744971
## factor(gvkey)164132 3.863e-01 1.480e+01 0.026 0.979174
## factor(gvkey)164364 1.380e+01 8.922e+00 1.546 0.122089
## factor(gvkey)164365 7.756e+00 8.922e+00 0.869 0.384733
## factor(gvkey)164368 -2.341e+00 1.208e+01 -0.194 0.846361
## factor(gvkey)164404 2.077e+01 1.208e+01 1.720 0.085550 .
## factor(gvkey)164555 6.028e+00 2.093e+01 0.288 0.773306
## factor(gvkey)164572 4.959e+00 9.864e+00 0.503 0.615155
## factor(gvkey)164633 2.533e+01 9.358e+00 2.707 0.006812 **
## factor(gvkey)164708 6.684e+00 8.922e+00 0.749 0.453816
## factor(gvkey)165264 6.825e+00 8.922e+00 0.765 0.444361
## factor(gvkey)165284 3.856e+00 1.046e+01 0.369 0.712457
## factor(gvkey)166005 3.962e+01 9.864e+00 4.016 5.96e-05 ***
## factor(gvkey)166368 2.243e+01 9.358e+00 2.397 0.016574 *
## factor(gvkey)166582 1.335e+01 1.480e+01 0.902 0.366912
## factor(gvkey)166705 7.203e+00 1.208e+01 0.596 0.551058
## factor(gvkey)170375 2.447e+01 1.480e+01 1.654 0.098161 .
## factor(gvkey)170396 4.331e+01 1.709e+01 2.535 0.011258 *
## factor(gvkey)170419 -3.043e+00 1.480e+01 -0.206 0.837062
## factor(gvkey)171023 -9.115e-02 2.093e+01 -0.004 0.996524
## factor(gvkey)174022 -1.437e+00 9.358e+00 -0.154 0.878003
## factor(gvkey)174053 7.874e+00 9.864e+00 0.798 0.424775
## factor(gvkey)174159 1.011e+02 2.093e+01 4.833 1.37e-06 ***
## factor(gvkey)174301 3.903e+00 1.046e+01 0.373 0.709117
## factor(gvkey)174313 7.492e+00 9.358e+00 0.801 0.423361
## factor(gvkey)174647 5.145e+00 1.046e+01 0.492 0.622873
## factor(gvkey)174729 4.950e+00 9.864e+00 0.502 0.615830
## factor(gvkey)174744 2.341e+01 1.046e+01 2.238 0.025259 *
## factor(gvkey)175131 3.854e-01 1.046e+01 0.037 0.970619
## factor(gvkey)175263 4.851e+00 9.358e+00 0.518 0.604199
## factor(gvkey)175307 6.143e+00 9.358e+00 0.656 0.511565
## factor(gvkey)175575 2.690e+01 1.709e+01 1.574 0.115466
## factor(gvkey)175646 -3.611e+00 2.959e+01 -0.122 0.902870
## factor(gvkey)175674 -7.671e+00 1.118e+01 -0.686 0.492832
## factor(gvkey)175688 -1.835e+00 9.864e+00 -0.186 0.852392
## factor(gvkey)176239 5.263e+00 1.046e+01 0.503 0.614931
## factor(gvkey)176268 1.583e+01 9.864e+00 1.605 0.108524
## factor(gvkey)176351 1.204e+01 1.046e+01 1.151 0.249804
## factor(gvkey)176375 2.023e+01 2.093e+01 0.967 0.333616
## factor(gvkey)176591 6.530e+00 9.358e+00 0.698 0.485348
## factor(gvkey)176592 1.926e+01 9.358e+00 2.058 0.039624 *
## factor(gvkey)176595 6.946e+00 1.046e+01 0.664 0.506807
## factor(gvkey)176637 1.351e+02 1.709e+01 7.908 2.95e-15 ***
## factor(gvkey)176701 4.736e+00 9.864e+00 0.480 0.631120
## factor(gvkey)176703 2.074e+00 1.118e+01 0.185 0.852905
## factor(gvkey)176725 1.854e+00 1.709e+01 0.109 0.913597
## factor(gvkey)176766 -1.057e+00 9.864e+00 -0.107 0.914648
## factor(gvkey)176828 4.675e-01 1.323e+01 0.035 0.971822
## factor(gvkey)176973 4.024e+01 1.118e+01 3.598 0.000323 ***
## factor(gvkey)177088 1.456e+01 1.118e+01 1.302 0.192978
## factor(gvkey)177216 5.203e+01 1.709e+01 3.045 0.002332 **
## factor(gvkey)177255 1.890e+01 1.118e+01 1.690 0.091060 .
## factor(gvkey)177300 3.565e+00 9.864e+00 0.361 0.717828
## factor(gvkey)177376 7.853e+00 9.864e+00 0.796 0.425986
## factor(gvkey)177640 -5.633e+00 2.959e+01 -0.190 0.849036
## factor(gvkey)177782 2.408e+01 1.118e+01 2.153 0.031353 *
## factor(gvkey)177996 3.198e+01 1.046e+01 3.056 0.002247 **
## factor(gvkey)178371 3.819e+01 1.208e+01 3.161 0.001577 **
## factor(gvkey)178529 3.433e+01 1.480e+01 2.320 0.020362 *
## factor(gvkey)178539 3.701e+01 1.709e+01 2.166 0.030342 *
## factor(gvkey)178545 9.752e-02 1.323e+01 0.007 0.994121
## factor(gvkey)178610 3.963e+01 1.118e+01 3.543 0.000398 ***
## factor(gvkey)178703 1.255e+00 1.046e+01 0.120 0.904489
## factor(gvkey)178811 2.869e+01 1.208e+01 2.375 0.017587 *
## factor(gvkey)178823 4.634e+01 2.093e+01 2.215 0.026806 *
## factor(gvkey)178834 4.093e+00 9.864e+00 0.415 0.678210
## factor(gvkey)178862 2.743e+01 1.323e+01 2.073 0.038235 *
## factor(gvkey)179077 1.389e+01 9.864e+00 1.408 0.159152
## factor(gvkey)179298 1.400e+01 1.118e+01 1.251 0.210816
## factor(gvkey)179361 -4.372e+00 2.093e+01 -0.209 0.834510
## factor(gvkey)179534 2.110e+01 1.046e+01 2.017 0.043755 *
## factor(gvkey)179889 6.084e+01 1.323e+01 4.597 4.34e-06 ***
## factor(gvkey)179974 4.223e+01 1.480e+01 2.854 0.004322 **
## factor(gvkey)180183 8.769e+00 2.093e+01 0.419 0.675194
## factor(gvkey)180193 4.571e+01 1.709e+01 2.675 0.007484 **
## factor(gvkey)180228 4.027e+01 1.480e+01 2.722 0.006506 **
## factor(gvkey)180272 5.907e+00 1.208e+01 0.489 0.624882
## factor(gvkey)180423 2.997e+00 1.118e+01 0.268 0.788732
## factor(gvkey)182701 2.445e+01 1.208e+01 2.024 0.043022 *
## factor(gvkey)182788 5.508e+01 1.208e+01 4.559 5.21e-06 ***
## factor(gvkey)183247 4.259e-02 1.118e+01 0.004 0.996962
## factor(gvkey)183324 7.328e+01 1.709e+01 4.289 1.81e-05 ***
## factor(gvkey)183388 8.498e+01 1.480e+01 5.744 9.59e-09 ***
## factor(gvkey)183603 6.706e+01 1.709e+01 3.925 8.75e-05 ***
## factor(gvkey)183606 3.966e+01 1.709e+01 2.321 0.020290 *
## factor(gvkey)183780 6.080e+01 1.709e+01 3.559 0.000375 ***
## factor(gvkey)183797 3.459e+01 1.323e+01 2.614 0.008964 **
## factor(gvkey)183826 3.417e+01 1.323e+01 2.582 0.009852 **
## factor(gvkey)183830 1.159e+02 1.480e+01 7.832 5.40e-15 ***
## factor(gvkey)183963 8.630e+01 1.480e+01 5.832 5.67e-09 ***
## factor(gvkey)184009 2.578e+01 1.208e+01 2.134 0.032897 *
## factor(gvkey)184167 -3.109e+00 1.208e+01 -0.257 0.796905
## factor(gvkey)184287 4.383e+00 1.323e+01 0.331 0.740504
## factor(gvkey)184498 7.168e+01 1.480e+01 4.845 1.29e-06 ***
## factor(gvkey)184500 7.686e+00 1.323e+01 0.581 0.561417
## factor(gvkey)184571 2.819e+01 1.323e+01 2.130 0.033208 *
## factor(gvkey)184688 3.948e+01 1.709e+01 2.311 0.020870 *
## factor(gvkey)184689 3.846e+01 2.093e+01 1.838 0.066109 .
## factor(gvkey)184735 2.831e+01 2.093e+01 1.353 0.176178
## factor(gvkey)184899 1.557e+00 1.208e+01 0.129 0.897444
## factor(gvkey)185177 1.683e+01 1.709e+01 0.985 0.324528
## factor(gvkey)185339 4.698e+01 1.709e+01 2.750 0.005981 **
## factor(gvkey)185370 1.308e+01 2.959e+01 0.442 0.658555
## factor(gvkey)185396 4.632e+01 2.093e+01 2.214 0.026874 *
## factor(gvkey)185453 1.622e+01 2.959e+01 0.548 0.583641
## factor(gvkey)185518 4.977e+00 2.093e+01 0.238 0.812016
## factor(gvkey)185549 2.015e+01 1.323e+01 1.523 0.127809
## factor(gvkey)185585 3.539e+00 1.480e+01 0.239 0.810950
## factor(gvkey)185618 3.068e+01 1.709e+01 1.796 0.072565 .
## factor(gvkey)185824 6.411e-01 1.480e+01 0.043 0.965439
## factor(gvkey)186230 3.696e+01 2.959e+01 1.249 0.211765
## factor(gvkey)186344 6.052e+00 1.323e+01 0.457 0.647462
## factor(gvkey)186363 1.850e+01 1.323e+01 1.398 0.162207
## factor(gvkey)186428 5.403e+00 1.480e+01 0.365 0.715021
## factor(gvkey)187164 1.596e+01 1.323e+01 1.206 0.227752
## factor(gvkey)187252 1.702e+01 2.093e+01 0.813 0.416080
## factor(gvkey)187253 2.396e+01 1.480e+01 1.620 0.105368
## factor(gvkey)187549 6.722e+00 1.709e+01 0.393 0.694010
## factor(gvkey)189517 1.080e+01 2.093e+01 0.516 0.605832
## factor(gvkey)190963 -5.252e+00 1.709e+01 -0.307 0.758546
## factor(gvkey)192458 1.669e+01 1.709e+01 0.977 0.328650
## factor(gvkey)200664 5.573e+00 1.480e+01 0.377 0.706459
## factor(gvkey)211732 2.674e+01 9.358e+00 2.858 0.004275 **
## factor(gvkey)223148 3.602e+01 7.909e+00 4.554 5.33e-06 ***
## factor(gvkey)241366 1.377e+01 7.641e+00 1.803 0.071476 .
## factor(gvkey)241388 -9.684e+00 2.093e+01 -0.463 0.643523
## factor(gvkey)243588 6.681e+00 1.709e+01 0.391 0.695770
## factor(gvkey)248136 -3.050e+00 9.918e+00 -0.307 0.758502
## factor(gvkey)252819 3.636e+00 8.208e+00 0.443 0.657809
## factor(gvkey)252940 6.421e+00 8.557e+00 0.750 0.453058
## factor(gvkey)258664 2.341e+01 9.358e+00 2.502 0.012375 *
## factor(gvkey)260774 2.069e+01 8.543e+00 2.422 0.015477 *
## factor(gvkey)260778 2.739e+01 8.543e+00 3.207 0.001347 **
## factor(gvkey)260779 7.348e+00 8.543e+00 0.860 0.389745
## factor(gvkey)264395 1.059e+01 8.543e+00 1.240 0.215099
## factor(gvkey)264510 7.588e+01 2.959e+01 2.564 0.010365 *
## factor(gvkey)266214 2.096e+01 8.543e+00 2.454 0.014156 *
## factor(gvkey)266216 -1.273e-01 1.709e+01 -0.007 0.994055
## factor(gvkey)266257 7.936e+01 1.046e+01 7.585 3.67e-14 ***
## factor(gvkey)266315 1.450e+01 9.864e+00 1.470 0.141679
## factor(gvkey)275661 2.670e+00 2.959e+01 0.090 0.928119
## factor(gvkey)285313 2.067e+01 1.119e+01 1.848 0.064708 .
## factor(industry)2011 NA NA NA NA
## factor(industry)2741 NA NA NA NA
## factor(industry)3714 NA NA NA NA
## factor(industry)4950 NA NA NA NA
## factor(industry)5063 NA NA NA NA
## factor(industry)6020 NA NA NA NA
## factor(industry)6035 NA NA NA NA
## factor(industry)6036 NA NA NA NA
## factor(industry)6099 NA NA NA NA
## factor(industry)6111 NA NA NA NA
## factor(industry)6141 NA NA NA NA
## factor(industry)6153 NA NA NA NA
## factor(industry)6159 NA NA NA NA
## factor(industry)6162 NA NA NA NA
## factor(industry)6163 NA NA NA NA
## factor(industry)6172 NA NA NA NA
## factor(industry)6199 NA NA NA NA
## factor(industry)6200 NA NA NA NA
## factor(industry)6211 NA NA NA NA
## factor(industry)6282 NA NA NA NA
## factor(industry)6311 NA NA NA NA
## factor(industry)6321 NA NA NA NA
## factor(industry)6324 NA NA NA NA
## factor(industry)6331 NA NA NA NA
## factor(industry)6351 NA NA NA NA
## factor(industry)6361 NA NA NA NA
## factor(industry)6411 NA NA NA NA
## factor(industry)6500 NA NA NA NA
## factor(industry)6512 NA NA NA NA
## factor(industry)6513 NA NA NA NA
## factor(industry)6531 NA NA NA NA
## factor(industry)6552 NA NA NA NA
## factor(industry)6726 NA NA NA NA
## factor(industry)6797 NA NA NA NA
## factor(industry)6798 NA NA NA NA
## factor(industry)6799 NA NA NA NA
## factor(industry)7370 NA NA NA NA
## factor(industry)7374 NA NA NA NA
## factor(industry)7389 NA NA NA NA
## factor(industry)7510 NA NA NA NA
## factor(industry)9995 NA NA NA NA
## factor(industry)9997 NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 29.59 on 8326 degrees of freedom
## (8 observations deleted due to missingness)
## Multiple R-squared: 0.2586, Adjusted R-squared: 0.1854
## F-statistic: 3.529 on 823 and 8326 DF, p-value: < 2.2e-16The key part is this line of the output…
Coefficients: (42 not defined because of singularities)
Remember, we had 43 industries, and used 42 (k-1) industry dummies. The lm function got mad and said we’re not going to estimate coefficients for these.
Why?
With fixed effect models, you can’t estimate what’s called cross-level interactions. These are very popular in OB/psychology, but you know where I stand on those models.
Bottom line? If you fail the Hausman test and need to retain a fixed effect model, you’re not going to be able to include any variable (or interaction effect) that is time invariant within the panel (the \(i\)).
Unless you use the hybrid method.
This is pretty common in other literatures, and ironically it’s often done in OB/pysch (although for a different reason).
It combines, in a sense, the best element of the fixed effect model while still being able to estimate a between (and hence \(i\) level) effect.
We don’t have time to go through all of the details, but the approach is straightforward, and so is the interpretation. What you do with this model is to split the predictor into its between and within elements.
We haven’t seen it used a lot to date, but I have a hunch we will in the near future.
What about lagged values?
Temporal sequencing can be a great thing! But…
The same rules about establishing causality still apply…damn omitted variable problem!
I actually have a bias against using one year lags (which is very common in strategy) in our models. The reason being is that the choice of the lag structure is—almost—always a function of the lag structure imposed by the data provider.
There’s nothing theoretical about it—it’s just a one year lag because that’s the way Compustat, etc., spits out the data.
My challenge with the one year lag is that there is usually very little reason to justify that the relationship between \(x\) and \(y\) manifests over a one-year period.
In fact, there may be little reason to expect that this is the ‘correct’ timeframe. Furthermore, you may be more likely to induce omitted variable bias by including the lag, why?
Still, lets estimate a model with a one year lag of long term debt on sales growth rate to show how it’s done.
There are a couple of ways to create a lagged variable, but I like using dplyr (part of the tidyverse).
panel.no.df <- panel.no.df %>%
arrange(gvkey, fyear) %>% # Sort by firm and then by year
group_by(gvkey) %>% # Tell dplyr to go within each firm
mutate(lag.dltt = lag(dltt)) # Generate a new lagged dltt value
lag.plm.df <- pdata.frame(panel.no.df, index=c("gvkey","fyear"),
drop.index=TRUE)lag.model <- plm(sgr ~ lag.dltt, data = lag.plm.df, index=c("gvkey", "fyear"),
model="within", effect = c("twoways"))
summary(lag.model)## Twoways effects Within Model
##
## Call:
## plm(formula = sgr ~ lag.dltt, data = lag.plm.df, effect = c("twoways"),
## model = "within", index = c("gvkey", "fyear"))
##
## Unbalanced Panel: n=779, T=1-19, N=8328
##
## Residuals :
## Min. 1st Qu. Median 3rd Qu. Max.
## -146.00 -9.31 -1.60 6.51 436.00
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## lag.dltt -1.6327e-05 4.8252e-06 -3.3837 0.0007188 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 5586100
## Residual Sum of Squares: 5577700
## R-Squared: 0.0015182
## Adj. R-Squared: -0.10416
## F-statistic: 11.4495 on 1 and 7530 DF, p-value: 0.00071877FWIW, the sign flip isn’t all that surprising with panel data, and particularly with small effect sizes. Remember, these are really, really small effects with a large N dataset.
BTW, the plm package can estimate a lag model just using inline code…
lag.il.model <- plm(sgr ~ lag(dltt,1), data = panel.no.df, index=c("gvkey", "fyear"),
model="within", effect = c("twoways"))
summary(lag.il.model)## Twoways effects Within Model
##
## Call:
## plm(formula = sgr ~ lag(dltt, 1), data = panel.no.df, effect = c("twoways"),
## model = "within", index = c("gvkey", "fyear"))
##
## Unbalanced Panel: n=777, T=1-19, N=8250
##
## Residuals :
## Min. 1st Qu. Median 3rd Qu. Max.
## -146.00 -9.24 -1.55 6.44 436.00
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## lag(dltt, 1) -1.6294e-05 4.7052e-06 -3.4629 0.0005374 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 5257000
## Residual Sum of Squares: 5248500
## R-Squared: 0.0016062
## Adj. R-Squared: -0.10488
## F-statistic: 11.9916 on 1 and 7454 DF, p-value: 0.00053741NOTE
Keep in mind that the manual lag and the auto lag work a bit differently. The coefficients are close, but the number of firms is different. The reason being is how the inline model handles missing data.
Purely FWIW, I like the manual way to preserve power.
PARTING NOTE
DO NOT DO NOT DO NOT lag the DV and include it as a predictor in the panel model.
Yes, this happens a lot, under the logic that the value of the DV at time \(t\) is often a function of the value of the DV at \(t-1\). Makes sense, but with panel estimators (or other OLS estimators) including a lagged DV term biases all of the other \(\beta\)’s in the model. That and the lagged DV will be endogenous by definition, so you can’t interpret it anyway.
Properly specified lagged DV models are, however, very cool. We call them dynamic panel estimators, and the plm package can estimate these kind of models.
This is, though, pretty advanced stuff, so if you really want to ask these kinds of research questions, lets get you some additional training.
Wrap-up.
Lab 20 April – Panel data paper critique
Seminar 24 April – LDV models and panel data // special topics