The basics of panel data
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
- Reviews due!
- Paper revision and reviewer responses due 24 April
- Data collection discussion
- Panel data basics
- Lab this afternoon – Panel data assessment
- Seminar 17 April – Panel data and causal effects
This is a tough topic. I’m not kidding.
For the next three weeks we will, maybe, scratch the surface. We’re going to cover the basics, which should help you to make sense of papers that use panel data.
If you think—and I think must of you are in this boat—that you will investigate multilevel research questions, let me point you to two very good resources.
The Wooldridge book is a classic text, and you should buy it. You’ll see references to this book across several different literatures. The Heiss book is new—and free to read!—and leverages the Wooldridge text within an R framework. Think of Wooldridge as the theory and the proofs, and Heiss as the ‘how-to’ in R.
First up, some definitions…
Panel, multilevel, longitudinal, and hierarchical data are all the same thing. You have observations of some lower level object, \(t\), nested within a higher order entity, \(i\).
So \(x_{it}\) is the observation of \(x\) for entity \(i\) at time \(t\).
REALLY REALLY REALLY IMPORTANT NOTE
In the psychology literature (and I sometimes do this myself), you will see the lower level represented as \(i\) and the higher as \(j\), as in the \(i^{th}\) observation of the \(j^{th}\) entity, which is sometimes written as \(x_{ij}\) (the inverse order of the preceding slide). I’m going to use the \(i,t\) notation because it’s closer to how economists (and most strategy people) present panel data, and the way I learned it.
The difference generally has to do with whether the lower order entity is a temporal observation (econ/strategy), or is an entity within another entity (OB/psych).
THE MATH IS THE SAME, but watch out on confusing your \(i's\), \(j's\), and \(t's\), because order does matter!
Lets also bust a common myth in our field right now. You do NOT need special software to analyze panel data. Period dot.
You do, however, need to use different kinds of estimation techniques that account for the nested structure of the data.
Side note…
Time series data is not quite the same thing. It could be, in the sense that you might have multiple \(i's\) in a data set. But you could also just have one \(i\) and you are trying to model the \(t\) observations for that one \(i\). For example, think about analyzing the S&P 500 Index over time—there is one \(i\), the index, but tens of thousands of \(t's\), the daily value of the index.
In the hard sciences, they work a lot with time series data. In the social sciences, we deal mostly with panel data. The analytic techniques are generally comparable, but the language used can be a little different. We’re going to focus on panel data in this class, but we’ll do a quick time series example.
We will be working with this equation a lot.
- \(y\) = 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 \(\mu_{i}\).
WARNING
Note that failure to understand the assumptions underlying this equation is, as Antonakis et al (2010) rightly noted, the reason why the vast majority—I’d say greater than 80%—of panel studies in our literature are misspecified.
END WARNING
You, however, will know better right from day one.
Thank me later :)
Lets start with a time series example.
In a time series model, we’re generally interested in analyzing the pattern of \(t's\) for a given \(i\). It is one dimensional data, similar to cross-sectional data, although in this case it’s one \(i\) and many \(t's\), as opposed to one \(t\) for many \(i's\).
Generally we assume that observations close together will correlate more than observations farther apart. Just as with any sample, the greater the observations of \(t\), the more accurate our estimation of this temporal dependency.
While it’s interesting to fit times series parameters, ultimately the value of time series modeling is prediction/forecasting. For example, trying to forecast the magnitude of future earthquakes based on previous earthquakes.
Or trying to forecast the weather. Or trying to forecast tomorrow’s value of a stock price :)
Lets get some data. I’ve got the daily closing stock price for Illinois Tool Works from 3 Jan 2000 through 31 Dec 2009.
Where did I get it do you ask? Right here.
library(readr)
library(tidyverse)
my.ds <- read_csv("http://a.web.umkc.edu/andersonbri/ITWDailyReturns.csv")
my.df <- my.ds
head(my.df, 3)## # A tibble: 3 × 2
## Date ClosingPrice
## <chr> <dbl>
## 1 03jan2000 31.9375
## 2 04jan2000 30.5000
## 3 05jan2000 30.8750Take a look at the Date variable. It’s actually being stored as a string right now. That’s not going to work for us. Frequently when dealing with secondary data sources we need to manipulate the data to make it usable for analyses.
class(my.df$Date)## [1] "character"Fortunately, the lubridate package makes it easy to work with dates. We don’t have enough time to go through all of it, but I’d encourage you to take a look at this tutorial.
library(lubridate)
my.df$Date <- dmy(my.df$Date)
class(my.df$Date)## [1] "Date"head(my.df, 3)## # A tibble: 3 × 2
## Date ClosingPrice
## <date> <dbl>
## 1 2000-01-03 31.9375
## 2 2000-01-04 30.5000
## 3 2000-01-05 30.8750Ok, the first thing we need to do is to tell R that we’ve got time series data. For this, we’re going to use the xts package.
library(xts)
my.df.ts <- xts(my.df$ClosingPrice, order.by = my.df$Date)
colnames(my.df.ts) <- "ClosingPrice"When working with time series data, we generally focus on visualizations. So lets take a look at Illinois Tool Work’s stock over the ten year period.
plot.xts(my.df.ts, main = "Illinois Tool Works Daily Closing Stock Price")
Ok, now lets do some useful things with this. We’re going to need the forecast package for this.
This isn’t even time series 101 BTW, and is a very rudimentary analysis. Generally, you want to use ARIMA models for time series which explicitly model the auto-regressive component in the data.
This is for a different (semester length) class, but lets just do an example.
library(forecast)
my.ts.2009 <- my.df.ts['2009'] # Subset our time series to just 2009
arima.model <- auto.arima(my.ts.2009) # Use an auto-fitting ARIMA model
plot(forecast(arima.model),
main = "Illinois Tool Works 2009 Stock Price ARIMA Forecast") # Make the plot
The dark grey is an 80% confidence level around the stock’s predicted value. The light grey is a 95% band. Notice that the bands get larger as time progresses.
A big part of what we want to understand with time series models is auto-correlation—the extent to which an observation at time \(t\) correlates with an observation at time \(t-1\). Auto-correlation is at the heart of time series modeling, our ability to make reasonable forecasts, and why we need to use special estimators (auto-correlation violates the i.i.d. assumption).
acf(my.ts.2009, lag.max=100,
main = "Illinois Tool Works 2009 Stock Price Autocorrelation") 
As is common with stock prices, particularly for large, established firms with low volatility, daily stock prices are highly correlated. The dotted line is for the \(\alpha\) = .05 level of significance. We have to go back more than 70 days to see no statistically significant auto-correlation.
Again, we didn’t even come close to scratching the surface here, and time series modeling can be a real beast to model well.
If you think you are interested in this kind of approach, lets talk offline about getting you additional training.
Ok, so lets move on to full panel data.
We’re going to use a different dataset for this one. It’s a 20-year panel of publicly traded companies in the U.S. (1995-2014) with sales greater than $50MM annually (non-inflation adjusted).
Where did I get it do you ask? Right here.
First up, lets get the data.
my.panel.ds <- read_csv("http://a.web.umkc.edu/andersonbri/Panel.csv")
my.panel.df <- my.panel.dsOpen up the dataset and take a look…
View(my.panel.df)
Take a look at a couple of things:
- The panel is unbalanced—firms (\(i's\)) don’t necessarily have the same number of observations (\(t's\))
- Often you will see missing values (NA) or 0.00 values, which may or may not indicate that the firm had a zero value on that variable for that year
Now take a look at this…
header.df <- subset(my.panel.df, select = c(gvkey, fyear, conm))
head(header.df, 5)## # A tibble: 5 × 3
## gvkey fyear conm
## <chr> <int> <chr>
## 1 001062 2007 ASA GOLD AND PRECIOUS METALS
## 2 001062 2008 ASA GOLD AND PRECIOUS METALS
## 3 001062 2009 ASA GOLD AND PRECIOUS METALS
## 4 001177 1995 AETNA INC
## 5 001177 1996 AETNA INCSomething unique to Compustat data is the necessity to convert the gvkey variable from a string to numeric. In R, we do that with the as.numeric function.
my.panel.df$gvkey <- as.numeric(my.panel.df$gvkey)You’re going to want to remember that if you work with Compustat data. Trust me :)
Critical to panel data analysis is that there must be a unique combination of \(i\) and \(t\) to identify each observation…
| Valid | Invalid |
|---|---|
| 1062(\(i\)); 2007(\(t\)) | 1062(\(i\)); 2007(\(t\)) |
| 1062(\(i\)); 2008(\(t\)) | 1062(\(i\)); 2007(\(t\)) |
The reason this is important is because with panel data, we are looking at two different effects of \(x\) on \(y\).
- The within effect, which is the average effect of \(x\) on \(y\) for a given (average) \(i\) across time \(t\)
- The between effect, which is the average effect of \(x\) on \(y\) across \(i's\) over time \(t\)
If we couldn’t uniquely identify each observation, there would be a perfectly co-linear combination and the estimator wouldn’t know which one was the “right” one to use to estimate each effect.
Think about it this way…
We can decompose any \(x_{it}\) 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 \(\gamma\) over time (the firm’s sales, for example).
We’re going to get in to within and between effects a lot more next week, although understandin the within and the between are central to panel data estimators. For right now, lets just focus on setting up our model.
Panel data isn’t always secondary financial data. Panels could be…
- Repeated measures in an experiment
- Students nested within classrooms
- Employees nested within managers
- Top management teams nested within firms
- Firms nested within industries
- Counties nested within states
- …
One thing though with secondary panel data is that, most often, the data violates the normalcy assumption, which can bias our parameter estimates.
Lets do a QQ Plot of the ‘revt’ variable, which is the firm’s annual reported revenue in millions of USD.
qqnorm(my.panel.df$revt, main = "QQ Plot of Firm Revenue")
For a reminder, this is what a QQ plot looks like for a normally distributed variable…
Often you will employ variable transformations with secondary financial data. The most common is taking the natural log of the variable…
my.panel.df$log.revt <- log(my.panel.df$revt)
qqnorm(my.panel.df$log.revt, main = "QQ Plot of Natural Log Firm Revenue")
That’s better. Not great, but better. There are related concepts like skewness and kurtosis when it comes to evaluating normalcy, and we’ll touch on these issues as we go along.
Ok, so lets start with a simple, naive model of the relationship between the level of the firm’s long term debt—the dltt variable in our dataset—and log transformed revenue.
Keep in mind that this model has absolutely no relevance in the real world—big firms tend to have bigger debt loads.
ols.model <- lm(log.revt ~ dltt, data = my.panel.df)
summary(ols.model)##
## Call:
## lm(formula = log.revt ~ dltt, data = my.panel.df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.0498 -1.4602 -0.3204 1.2037 5.4340
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.320e+00 1.811e-02 348.93 <2e-16 ***
## dltt 5.845e-06 2.046e-07 28.57 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.797 on 9992 degrees of freedom
## (15 observations deleted due to missingness)
## Multiple R-squared: 0.0755, Adjusted R-squared: 0.07541
## F-statistic: 816 on 1 and 9992 DF, p-value: < 2.2e-16OLS makes the assumption that each observation is random to estimate a consistent parameter—\(\beta\)—for the change in \(y\) expected for each one unit change in \(x\).
With panel data, we violate that assumption. Given that each \(t\) occurs as a function of the \(i\) it nests under, we assume that there is some type of clustering of each \(t\) around its respective \(i\). If this is true, then our OLS estimator will be inconsistent.
Lets see how this works in practice.
You’re going to need a new package: plm. We’ll be using this package a lot.
We also need to create a new, plm formatted dataframe, which tells R that we’re using panel data. This new dataframe creates an index based on the unique identifier of each observation. In our case, that is the gvkey variable that identifies each firm, and fyear which identifies the fiscal year of the observation. Oh, go ahead and View the dataframe when you are done.
library(plm)
panel.plm.df <- pdata.frame(my.panel.df, index=c("gvkey","fyear"), drop.index=TRUE)We’re going to estimate a pooled model in plm. A pooled model is the same thing as regular OLS. It assumes that there is no connection between each \(t\) and the \(i\) it nests under.
pooled.model <- plm(log.revt ~ dltt, data = panel.plm.df,
index=c("gvkey","fyear"), model="pooling")
summary(pooled.model)## Pooling Model
##
## Call:
## plm(formula = log.revt ~ dltt, data = panel.plm.df, model = "pooling",
## index = c("gvkey", "fyear"))
##
## Unbalanced Panel: n=845, T=1-21, N=9994
##
## Residuals :
## Min. 1st Qu. Median 3rd Qu. Max.
## -13.00 -1.46 -0.32 1.20 5.43
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## (Intercept) 6.3198e+00 1.8112e-02 348.928 < 2.2e-16 ***
## dltt 5.8450e-06 2.0462e-07 28.566 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 34906
## Residual Sum of Squares: 32271
## R-Squared: 0.075499
## Adj. R-Squared: 0.075406
## F-statistic: 815.987 on 1 and 9992 DF, p-value: < 2.22e-16Look familiar?
In OLS, or the pooled model, we make the assumption that \(\beta{_{it}}\) = \(\beta\) for all \(i, t\). Basically, this means that there is no meaningful difference in the effect of \(x\) on \(y\) as a function of \(i\) or of \(t\).
We can actually test this assumption using something called the Breusch-Pagan Lagrange Multiplier Test, under the null hypothesis that \(\beta{_{it}}\) = \(\beta\) for all \(i, t\).
plmtest(pooled.model, type=c("bp"))##
## Lagrange Multiplier Test - (Breusch-Pagan) for unbalanced panels
##
## data: log.revt ~ dltt
## chisq = 59000, df = 1, p-value < 2.2e-16
## alternative hypothesis: significant effectsRejecting the null, as we did in this case, indicates that there is a meaningful difference in the slope of the effect of \(x\) on \(y\) across \(i's\), but we really don’t know much about the nature of that difference (more on that later).
So, we can safely assume that our OLS (or pooled) model is inconsistent.
There’s another wrinkle here though, and it has to do with the standard errors. We know that the slopes are different across \(i's\), so it stands to reason that the standard error for each observation will also cluster around the \(i\).
So with panel data, you have to be concerned with consistency of parameters, and consistency of inference.
We will fix the consistency of inference issue using cluster-robust standard errors in a little bit.
So it is not the case that \(\beta{_{it}}\) = \(\beta\) for all \(i, t\) in our data. What do we do?
Well, we need to add a term to our equation:
The \(\mu_{i}\) term is another predictor of \(y_{it}\). In this case, it is an unobserved portion of the disturbance term (\(\epsilon\)) that represents the between \(i\) differences for each \(t\). The kicker is that the model assumes that \(\mu_{i}\) has a mean of zero and constant variance.
Translated, this means that each firm’s slope may vary, but it does so randomly around a mean of zero—the differences between \(i's\) effectively wash each other out.
You can also think about it as yes, each \(i\) has an effect on it’s \(t's\), so we need to account for it in our estimate of \(\beta_{x}\), but we really don’t care what the specific \(i\) effect is (just the average effect) because the differences occur at random.
Importantly, in the random effects model, you are explicitly making the assumption that \(\mu_{i}\) is NOT correlated with the focal predictor(s) in the model—any \(i\) effect does not correlate with \(x_{it}\).
Does this sound familiar at all?
Failing to acknowledge this critical assumption—that the \(i\) effect is exogenous—is the driving reason why so many of our panel/multilevel models are improperly specified.
So there is an empirical consideration when it comes to assumptions about \(i\), but the bigger reason is theoretical.
With a random effect model, you are explicitly interested in the differences across \(i's\) in your sample over time. You want, effectively, to find out the differences among firms.
Firms that take on larger levels of long term debt will have higher revenue than firms with lower levels of long term debt.
We can fit a random effect model using plm…
random.model <- plm(log.revt ~ dltt, data = panel.plm.df,
index=c("gvkey","fyear"), model="random")
summary(random.model)## Oneway (individual) effect Random Effect Model
## (Swamy-Arora's transformation)
##
## Call:
## plm(formula = log.revt ~ dltt, data = panel.plm.df, model = "random",
## index = c("gvkey", "fyear"))
##
## Unbalanced Panel: n=845, T=1-21, N=9994
##
## Effects:
## var std.dev share
## idiosyncratic 0.2398 0.4897 0.079
## individual 2.8046 1.6747 0.921
## theta :
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.7194 0.9159 0.9271 0.9192 0.9348 0.9363
##
## Residuals :
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -4.7600 -0.2530 0.0295 0.0184 0.3140 2.1500
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## (Intercept) 5.9800e+00 5.8147e-02 102.842 < 2.2e-16 ***
## dltt 1.0335e-06 7.8117e-08 13.231 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 2485.4
## Residual Sum of Squares: 2409.3
## R-Squared: 0.044305
## Adj. R-Squared: 0.04421
## F-statistic: 315.501 on 1 and 9992 DF, p-value: < 2.22e-16Ok, lets walk through the output.
The first thing that should pop out is the very different coefficient estimate for the effect of long term debt on revenue (OLS = 5.84x10-6; Random effect = 1.03x10-7).
The idiosyncratic effect in the output is the variance of \(\epsilon_{it}\). This is analogous to the disturbance term in OLS regression, and it doesn’t account for much of the variance in the combined (\(\mu\) and \(\epsilon\)) disturbance term (8%).
The individual effect is the variance of \(\mu_{i}\). Remember, this is the unobserved estimate in the \(i\) effect variance. It accounts for 92% of the variance in the combined (\(\mu\) and \(\epsilon\)) disturbance term. We’ll come back to this in a minute.
One of the challenges though with interpreting the effect of \(x\) on \(y\) in a random effect model is that the estimate of \(\beta_{x}\) contains variance between \(i's\) and within \(i's\).
Think about it this way. Is the effect of long term debt on revenue because of differences among firms, or because of what a given firm does to itself? In a random effect model, you are saying both differences matter, but all of the differences are included in the estimate of \(\beta\).
We can also look at the problem through the lens of the Interclass Correlation Coeffecient, which is the extent to which measurements of some trait within a given \(i\) resemble each other.
library(ICC)
ICCbare(gvkey, log.revt, data = my.panel.df)## Warning in ICCbare(gvkey, log.revt, data = my.panel.df): 'x' has been
## coerced to a factor## [1] 0.9301151Not surprisingly, our observations of a firm’s revenue are highly correlated with each other.
Sometimes though, the presence of a meaningful \(i\) effect precludes asking the between-effect research question (although there are options). The reason being if there is systematic variance in \(\mu_{i}\) and \(\mu_{i}\) correlates with \(x_{it}\) from unobserved heterogeneity/omitted variables then the estimate of \(\beta{x}\) will be inconsistent.
Other times, we specifically want to isolate, or control for, the \(i\) effect. Our theory is interested only in the within-firm variance of \(x\) on \(y\). We believe, effectively, that not only is the \(i\) effect present, but that’s it is theoretically meaningful and likely correlates with \(x\).
We don’t want any between differences then, we want to go deep within the firm…
For a given (average) firm, increasing long term debt will increase revenue.
If we just want the within-firm effect of \(x\) on \(y\), then we need to control, effectively, for all other sources of between-firm variance that influences \(\beta{x_{it}}\).
We need to stop \(\mu_{i}\) from being random, and hold its effect as fixed.
A straightforward way of doing this is to return to our OLS model and include a \(k-1\) dummy variable (\(\gamma\)) for each firm in the sample.
ols.dummy.model <- lm(log.revt ~ dltt + factor(gvkey)-1, data = my.panel.df)
summary(ols.dummy.model)##
## Call:
## lm(formula = log.revt ~ dltt + factor(gvkey) - 1, data = my.panel.df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.1222 -0.2087 0.0287 0.2604 2.0158
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## dltt 9.788e-07 7.807e-08 12.538 < 2e-16 ***
## factor(gvkey)1062 4.662e+00 2.827e-01 16.491 < 2e-16 ***
## factor(gvkey)1177 1.016e+01 1.095e-01 92.832 < 2e-16 ***
## factor(gvkey)1257 5.132e+00 1.224e-01 41.925 < 2e-16 ***
## factor(gvkey)1274 7.194e+00 1.264e-01 56.898 < 2e-16 ***
## factor(gvkey)1414 7.298e+00 1.851e-01 39.431 < 2e-16 ***
## factor(gvkey)1447 1.015e+01 1.095e-01 92.667 < 2e-16 ***
## factor(gvkey)1449 9.477e+00 1.095e-01 86.552 < 2e-16 ***
## factor(gvkey)1487 1.086e+01 1.096e-01 99.058 < 2e-16 ***
## factor(gvkey)1526 7.827e+00 1.095e-01 71.486 < 2e-16 ***
## factor(gvkey)1545 5.073e+00 1.731e-01 29.306 < 2e-16 ***
## factor(gvkey)1618 4.771e+00 1.731e-01 27.559 < 2e-16 ***
## factor(gvkey)1949 4.197e+00 3.462e-01 12.122 < 2e-16 ***
## factor(gvkey)1982 5.252e+00 1.224e-01 42.907 < 2e-16 ***
## factor(gvkey)2002 7.790e+00 1.095e-01 71.145 < 2e-16 ***
## factor(gvkey)2005 6.732e+00 1.095e-01 61.489 < 2e-16 ***
## factor(gvkey)2019 9.039e+00 1.095e-01 82.550 < 2e-16 ***
## factor(gvkey)2176 1.089e+01 1.095e-01 99.444 < 2e-16 ***
## factor(gvkey)2547 9.937e+00 1.095e-01 90.752 < 2e-16 ***
## factor(gvkey)2558 9.359e+00 1.095e-01 85.473 < 2e-16 ***
## factor(gvkey)2620 8.021e+00 1.095e-01 73.259 < 2e-16 ***
## factor(gvkey)2849 7.543e+00 1.224e-01 61.622 < 2e-16 ***
## factor(gvkey)2968 1.094e+01 1.099e-01 99.586 < 2e-16 ***
## factor(gvkey)3024 9.237e+00 1.095e-01 84.359 < 2e-16 ***
## factor(gvkey)3082 6.666e+00 1.095e-01 60.882 < 2e-16 ***
## factor(gvkey)3221 8.983e+00 1.095e-01 82.039 < 2e-16 ***
## factor(gvkey)3231 8.088e+00 1.095e-01 73.865 < 2e-16 ***
## factor(gvkey)3238 6.931e+00 1.095e-01 63.301 < 2e-16 ***
## factor(gvkey)3243 1.117e+01 1.103e-01 101.240 < 2e-16 ***
## factor(gvkey)3278 9.396e+00 1.999e-01 47.000 < 2e-16 ***
## factor(gvkey)3410 4.905e+00 1.731e-01 28.335 < 2e-16 ***
## factor(gvkey)3562 5.439e+00 1.264e-01 43.020 < 2e-16 ***
## factor(gvkey)3643 6.554e+00 1.095e-01 59.855 < 2e-16 ***
## factor(gvkey)4193 4.990e+00 1.224e-01 40.761 < 2e-16 ***
## factor(gvkey)4201 6.424e+00 1.095e-01 58.672 < 2e-16 ***
## factor(gvkey)4601 9.796e+00 1.369e-01 71.576 < 2e-16 ***
## factor(gvkey)4605 6.155e+00 1.264e-01 48.684 < 2e-16 ***
## factor(gvkey)4640 8.537e+00 1.095e-01 77.973 < 2e-16 ***
## factor(gvkey)4666 5.655e+00 1.264e-01 44.725 < 2e-16 ***
## factor(gvkey)4674 8.422e+00 1.095e-01 76.915 < 2e-16 ***
## factor(gvkey)4678 6.612e+00 1.095e-01 60.387 < 2e-16 ***
## factor(gvkey)4685 6.313e+00 1.095e-01 57.657 < 2e-16 ***
## factor(gvkey)4690 6.825e+00 1.095e-01 62.335 < 2e-16 ***
## factor(gvkey)4699 7.944e+00 1.095e-01 72.555 < 2e-16 ***
## factor(gvkey)4723 9.173e+00 1.095e-01 83.774 < 2e-16 ***
## factor(gvkey)4737 7.643e+00 1.095e-01 69.808 < 2e-16 ***
## factor(gvkey)4740 4.613e+00 1.309e-01 35.248 < 2e-16 ***
## factor(gvkey)4842 7.133e+00 1.264e-01 56.422 < 2e-16 ***
## factor(gvkey)4885 8.218e+00 1.095e-01 75.059 < 2e-16 ***
## factor(gvkey)5048 1.065e+01 1.235e-01 86.193 < 2e-16 ***
## factor(gvkey)5072 9.161e+00 2.827e-01 32.402 < 2e-16 ***
## factor(gvkey)5149 4.376e+00 1.224e-01 35.744 < 2e-16 ***
## factor(gvkey)5342 6.874e+00 1.476e-01 46.560 < 2e-16 ***
## factor(gvkey)5543 6.238e+00 1.264e-01 49.343 < 2e-16 ***
## factor(gvkey)5735 9.023e+00 1.096e-01 82.355 < 2e-16 ***
## factor(gvkey)5763 4.410e+00 1.414e-01 31.198 < 2e-16 ***
## factor(gvkey)5786 7.858e+00 1.095e-01 71.765 < 2e-16 ***
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## factor(gvkey)5862 7.101e+00 1.309e-01 54.260 < 2e-16 ***
## factor(gvkey)6239 7.239e+00 1.095e-01 66.117 < 2e-16 ***
## factor(gvkey)6333 6.130e+00 1.224e-01 50.073 < 2e-16 ***
## factor(gvkey)6653 7.492e+00 1.123e-01 66.690 < 2e-16 ***
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## factor(gvkey)6742 8.954e+00 1.095e-01 81.776 < 2e-16 ***
## factor(gvkey)6781 9.724e+00 1.095e-01 88.811 < 2e-16 ***
## factor(gvkey)6791 5.514e+00 1.548e-01 35.608 < 2e-16 ***
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## factor(gvkey)8007 1.044e+01 1.096e-01 95.239 < 2e-16 ***
## factor(gvkey)8148 4.111e+00 2.827e-01 14.540 < 2e-16 ***
## factor(gvkey)8240 7.798e+00 1.309e-01 59.591 < 2e-16 ***
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## factor(gvkey)8363 5.334e+00 1.224e-01 43.574 < 2e-16 ***
## factor(gvkey)8431 8.338e+00 1.095e-01 76.154 < 2e-16 ***
## factor(gvkey)8457 5.822e+00 1.264e-01 46.053 < 2e-16 ***
## factor(gvkey)8605 6.406e+00 1.309e-01 48.950 < 2e-16 ***
## factor(gvkey)8898 7.582e+00 1.095e-01 69.252 < 2e-16 ***
## factor(gvkey)9061 8.581e+00 1.095e-01 78.372 < 2e-16 ***
## factor(gvkey)9083 4.802e+00 1.358e-01 35.359 < 2e-16 ***
## factor(gvkey)9256 6.768e+00 1.632e-01 41.469 < 2e-16 ***
## factor(gvkey)9317 6.576e+00 1.095e-01 60.055 < 2e-16 ***
## factor(gvkey)9783 8.702e+00 1.095e-01 79.471 < 2e-16 ***
## factor(gvkey)10035 8.756e+00 1.095e-01 79.967 < 2e-16 ***
## factor(gvkey)10086 7.298e+00 1.123e-01 64.966 < 2e-16 ***
## factor(gvkey)10096 7.237e+00 1.264e-01 57.239 < 2e-16 ***
## factor(gvkey)10121 8.445e+00 1.096e-01 77.025 < 2e-16 ***
## factor(gvkey)10137 1.035e+01 1.414e-01 73.153 < 2e-16 ***
## factor(gvkey)10187 9.012e+00 1.095e-01 82.308 < 2e-16 ***
## factor(gvkey)10390 4.064e+00 1.999e-01 20.329 < 2e-16 ***
## factor(gvkey)10614 7.985e+00 1.095e-01 72.928 < 2e-16 ***
## factor(gvkey)10713 4.362e+00 2.448e-01 17.816 < 2e-16 ***
## factor(gvkey)10894 6.457e+00 1.154e-01 55.946 < 2e-16 ***
## factor(gvkey)10903 1.091e+01 1.224e-01 89.152 < 2e-16 ***
## factor(gvkey)10916 6.424e+00 1.095e-01 58.670 < 2e-16 ***
## factor(gvkey)10917 4.235e+00 2.827e-01 14.980 < 2e-16 ***
## factor(gvkey)11099 4.442e+00 1.414e-01 31.429 < 2e-16 ***
## factor(gvkey)11220 7.672e+00 1.224e-01 62.676 < 2e-16 ***
## factor(gvkey)11301 5.381e+00 1.224e-01 43.955 < 2e-16 ***
## factor(gvkey)11340 6.212e+00 1.224e-01 50.747 < 2e-16 ***
## factor(gvkey)11687 7.534e+00 1.095e-01 68.811 < 2e-16 ***
## factor(gvkey)11729 7.591e+00 1.095e-01 69.329 < 2e-16 ***
## factor(gvkey)11770 6.325e+00 1.224e-01 51.667 < 2e-16 ***
## factor(gvkey)11819 5.392e+00 1.999e-01 26.971 < 2e-16 ***
## factor(gvkey)11842 6.905e+00 1.095e-01 63.066 < 2e-16 ***
## factor(gvkey)11856 8.734e+00 1.095e-01 79.770 < 2e-16 ***
## factor(gvkey)11861 6.402e+00 1.095e-01 58.474 < 2e-16 ***
## factor(gvkey)11896 6.020e+00 1.095e-01 54.981 < 2e-16 ***
## factor(gvkey)12124 1.033e+01 1.071e-01 96.521 < 2e-16 ***
## factor(gvkey)12138 7.289e+00 1.095e-01 66.571 < 2e-16 ***
## factor(gvkey)12140 5.134e+00 1.188e-01 43.232 < 2e-16 ***
## factor(gvkey)12407 4.049e+00 3.462e-01 11.693 < 2e-16 ***
## factor(gvkey)12544 5.999e+00 1.224e-01 49.010 < 2e-16 ***
## factor(gvkey)12603 8.476e+00 1.095e-01 77.415 < 2e-16 ***
## factor(gvkey)12673 1.045e+01 1.097e-01 95.242 < 2e-16 ***
## factor(gvkey)12726 9.047e+00 1.095e-01 82.627 < 2e-16 ***
## factor(gvkey)12796 8.063e+00 1.095e-01 73.643 < 2e-16 ***
## factor(gvkey)12909 7.744e+00 1.264e-01 61.254 < 2e-16 ***
## factor(gvkey)13041 7.491e+00 1.095e-01 68.415 < 2e-16 ***
## factor(gvkey)13125 6.732e+00 1.264e-01 53.244 < 2e-16 ***
## factor(gvkey)13142 5.509e+00 1.264e-01 43.573 < 2e-16 ***
## factor(gvkey)13189 4.903e+00 1.476e-01 33.208 < 2e-16 ***
## factor(gvkey)13294 9.949e+00 2.190e-01 45.432 < 2e-16 ***
## factor(gvkey)13341 9.208e+00 1.095e-01 84.099 < 2e-16 ***
## factor(gvkey)13342 6.032e+00 1.095e-01 55.093 < 2e-16 ***
## factor(gvkey)13453 6.267e+00 1.095e-01 57.236 < 2e-16 ***
## factor(gvkey)13510 7.082e+00 1.224e-01 57.850 < 2e-16 ***
## factor(gvkey)13561 7.115e+00 1.188e-01 59.909 < 2e-16 ***
## factor(gvkey)13562 6.156e+00 1.224e-01 50.286 < 2e-16 ***
## factor(gvkey)13579 6.581e+00 1.264e-01 52.050 < 2e-16 ***
## factor(gvkey)13580 4.150e+00 2.448e-01 16.951 < 2e-16 ***
## factor(gvkey)13988 8.389e+00 1.095e-01 76.619 < 2e-16 ***
## factor(gvkey)14140 1.056e+01 1.103e-01 95.715 < 2e-16 ***
## factor(gvkey)14172 6.466e+00 1.095e-01 59.052 < 2e-16 ***
## factor(gvkey)14219 6.489e+00 1.095e-01 59.270 < 2e-16 ***
## factor(gvkey)14253 5.568e+00 1.095e-01 50.856 < 2e-16 ***
## factor(gvkey)14275 6.239e+00 1.095e-01 56.983 < 2e-16 ***
## factor(gvkey)14401 4.634e+00 2.190e-01 21.163 < 2e-16 ***
## factor(gvkey)14403 4.454e+00 2.190e-01 20.341 < 2e-16 ***
## factor(gvkey)14802 9.834e+00 1.096e-01 89.742 < 2e-16 ***
## factor(gvkey)14822 8.248e+00 1.188e-01 69.449 < 2e-16 ***
## factor(gvkey)14824 8.044e+00 1.095e-01 73.464 < 2e-16 ***
## factor(gvkey)14828 8.215e+00 1.264e-01 64.976 < 2e-16 ***
## factor(gvkey)15101 5.631e+00 1.309e-01 43.026 < 2e-16 ***
## factor(gvkey)15111 4.889e+00 1.224e-01 39.941 < 2e-16 ***
## factor(gvkey)15142 6.509e+00 1.264e-01 51.484 < 2e-16 ***
## factor(gvkey)15153 5.398e+00 1.224e-01 44.096 < 2e-16 ***
## factor(gvkey)15181 1.016e+01 1.096e-01 92.685 < 2e-16 ***
## factor(gvkey)15197 5.336e+00 1.095e-01 48.731 < 2e-16 ***
## factor(gvkey)15199 5.543e+00 1.095e-01 50.623 < 2e-16 ***
## factor(gvkey)15208 9.776e+00 1.216e-01 80.411 < 2e-16 ***
## factor(gvkey)15261 6.337e+00 1.264e-01 50.119 < 2e-16 ***
## factor(gvkey)15362 9.521e+00 1.096e-01 86.901 < 2e-16 ***
## factor(gvkey)15363 7.067e+00 1.095e-01 64.547 < 2e-16 ***
## factor(gvkey)15364 6.659e+00 1.224e-01 54.398 < 2e-16 ***
## factor(gvkey)15505 8.770e+00 1.154e-01 75.982 < 2e-16 ***
## factor(gvkey)15509 1.107e+01 1.191e-01 92.919 < 2e-16 ***
## factor(gvkey)15532 1.119e+01 1.321e-01 84.721 < 2e-16 ***
## factor(gvkey)15552 9.564e+00 1.316e-01 72.689 < 2e-16 ***
## factor(gvkey)15576 1.084e+01 1.100e-01 98.503 < 2e-16 ***
## factor(gvkey)15634 1.058e+01 1.228e-01 86.162 < 2e-16 ***
## factor(gvkey)15679 8.456e+00 1.309e-01 64.618 < 2e-16 ***
## factor(gvkey)15743 8.650e+00 1.309e-01 66.094 < 2e-16 ***
## factor(gvkey)15784 1.088e+01 1.270e-01 85.697 < 2e-16 ***
## factor(gvkey)15889 9.564e+00 1.124e-01 85.103 < 2e-16 ***
## factor(gvkey)15929 1.044e+01 1.270e-01 82.215 < 2e-16 ***
## factor(gvkey)16245 6.834e+00 1.095e-01 62.415 < 2e-16 ***
## factor(gvkey)16305 8.907e+00 1.309e-01 68.060 < 2e-16 ***
## factor(gvkey)16348 1.012e+01 2.832e-01 35.724 < 2e-16 ***
## factor(gvkey)16549 4.913e+00 2.827e-01 17.379 < 2e-16 ***
## factor(gvkey)16668 4.477e+00 1.095e-01 40.886 < 2e-16 ***
## factor(gvkey)16681 5.770e+00 1.123e-01 51.360 < 2e-16 ***
## factor(gvkey)16698 4.355e+00 1.309e-01 33.279 < 2e-16 ***
## factor(gvkey)16705 5.212e+00 1.095e-01 47.605 < 2e-16 ***
## factor(gvkey)16714 5.766e+00 1.224e-01 47.105 < 2e-16 ***
## factor(gvkey)16716 6.006e+00 1.224e-01 49.067 < 2e-16 ***
## factor(gvkey)16720 5.171e+00 3.462e-01 14.934 < 2e-16 ***
## factor(gvkey)16748 4.435e+00 1.188e-01 37.345 < 2e-16 ***
## factor(gvkey)16775 4.907e+00 1.224e-01 40.085 < 2e-16 ***
## factor(gvkey)16777 4.872e+00 1.264e-01 38.536 < 2e-16 ***
## factor(gvkey)16781 5.159e+00 1.095e-01 47.119 < 2e-16 ***
## factor(gvkey)16790 5.461e+00 1.095e-01 49.878 < 2e-16 ***
## factor(gvkey)16821 6.480e+00 1.095e-01 59.181 < 2e-16 ***
## factor(gvkey)16832 5.319e+00 1.154e-01 46.089 < 2e-16 ***
## factor(gvkey)16845 5.132e+00 1.095e-01 46.875 < 2e-16 ***
## factor(gvkey)16878 6.844e+00 1.414e-01 48.416 < 2e-16 ***
## factor(gvkey)16889 5.147e+00 1.095e-01 47.011 < 2e-16 ***
## factor(gvkey)16890 5.090e+00 1.123e-01 45.310 < 2e-16 ***
## factor(gvkey)16910 4.374e+00 1.154e-01 37.897 < 2e-16 ***
## factor(gvkey)16929 4.147e+00 1.095e-01 37.874 < 2e-16 ***
## factor(gvkey)16967 3.956e+00 2.448e-01 16.160 < 2e-16 ***
## factor(gvkey)16981 5.342e+00 1.095e-01 48.786 < 2e-16 ***
## factor(gvkey)16989 4.063e+00 1.548e-01 26.242 < 2e-16 ***
## factor(gvkey)17035 7.782e+00 1.224e-01 63.573 < 2e-16 ***
## factor(gvkey)17070 5.576e+00 1.095e-01 50.928 < 2e-16 ***
## factor(gvkey)17073 6.159e+00 1.224e-01 50.310 < 2e-16 ***
## factor(gvkey)17074 6.094e+00 1.224e-01 49.783 < 2e-16 ***
## factor(gvkey)17076 4.005e+00 2.190e-01 18.290 < 2e-16 ***
## factor(gvkey)17095 6.317e+00 1.095e-01 57.696 < 2e-16 ***
## factor(gvkey)17106 4.623e+00 1.123e-01 41.153 < 2e-16 ***
## factor(gvkey)17115 7.191e+00 1.095e-01 65.678 < 2e-16 ***
## factor(gvkey)17120 6.096e+00 1.095e-01 55.678 < 2e-16 ***
## factor(gvkey)17130 7.895e+00 1.095e-01 72.105 < 2e-16 ***
## factor(gvkey)17131 5.665e+00 1.224e-01 46.277 < 2e-16 ***
## factor(gvkey)17132 5.156e+00 1.095e-01 47.095 < 2e-16 ***
## factor(gvkey)17136 4.411e+00 1.095e-01 40.282 < 2e-16 ***
## factor(gvkey)17145 6.262e+00 1.095e-01 57.191 < 2e-16 ***
## factor(gvkey)17150 6.658e+00 1.095e-01 60.806 < 2e-16 ***
## factor(gvkey)17151 5.398e+00 1.095e-01 49.300 < 2e-16 ***
## factor(gvkey)17168 4.335e+00 1.476e-01 29.364 < 2e-16 ***
## factor(gvkey)17173 4.027e+00 1.731e-01 23.260 < 2e-16 ***
## factor(gvkey)17184 4.585e+00 1.095e-01 41.877 < 2e-16 ***
## factor(gvkey)17195 5.669e+00 1.095e-01 51.780 < 2e-16 ***
## factor(gvkey)17197 6.901e+00 1.123e-01 61.431 < 2e-16 ***
## factor(gvkey)17222 3.918e+00 3.462e-01 11.315 < 2e-16 ***
## factor(gvkey)17240 4.884e+00 1.123e-01 43.475 < 2e-16 ***
## factor(gvkey)17245 5.202e+00 1.095e-01 47.511 < 2e-16 ***
## factor(gvkey)17248 5.897e+00 1.095e-01 53.861 < 2e-16 ***
## factor(gvkey)17252 6.410e+00 1.224e-01 52.363 < 2e-16 ***
## factor(gvkey)17266 7.087e+00 1.224e-01 57.897 < 2e-16 ***
## factor(gvkey)17269 4.429e+00 1.095e-01 40.448 < 2e-16 ***
## factor(gvkey)17367 4.070e+00 1.851e-01 21.991 < 2e-16 ***
## factor(gvkey)17388 4.137e+00 1.414e-01 29.269 < 2e-16 ***
## factor(gvkey)17438 4.163e+00 2.190e-01 19.009 < 2e-16 ***
## factor(gvkey)17451 4.557e+00 2.448e-01 18.612 < 2e-16 ***
## factor(gvkey)17534 4.274e+00 1.632e-01 26.187 < 2e-16 ***
## factor(gvkey)17556 8.872e+00 2.827e-01 31.381 < 2e-16 ***
## factor(gvkey)17586 3.979e+00 4.897e-01 8.127 4.98e-16 ***
## factor(gvkey)17696 4.412e+00 2.448e-01 18.020 < 2e-16 ***
## factor(gvkey)17709 4.809e+00 2.448e-01 19.641 < 2e-16 ***
## factor(gvkey)17715 3.997e+00 2.190e-01 18.251 < 2e-16 ***
## factor(gvkey)17724 4.091e+00 1.548e-01 26.419 < 2e-16 ***
## factor(gvkey)17875 4.094e+00 3.462e-01 11.825 < 2e-16 ***
## factor(gvkey)17877 4.190e+00 1.731e-01 24.200 < 2e-16 ***
## factor(gvkey)18035 4.460e+00 3.462e-01 12.881 < 2e-16 ***
## factor(gvkey)18037 4.240e+00 1.264e-01 33.534 < 2e-16 ***
## factor(gvkey)18040 4.106e+00 2.448e-01 16.771 < 2e-16 ***
## factor(gvkey)18049 5.949e+00 1.095e-01 54.334 < 2e-16 ***
## factor(gvkey)18110 4.547e+00 1.095e-01 41.531 < 2e-16 ***
## factor(gvkey)18184 7.023e+00 2.827e-01 24.841 < 2e-16 ***
## factor(gvkey)18195 4.911e+00 2.827e-01 17.371 < 2e-16 ***
## factor(gvkey)18241 5.697e+00 1.095e-01 52.028 < 2e-16 ***
## factor(gvkey)18276 5.007e+00 1.095e-01 45.732 < 2e-16 ***
## factor(gvkey)18307 4.302e+00 1.632e-01 26.355 < 2e-16 ***
## factor(gvkey)18329 4.964e+00 1.154e-01 43.011 < 2e-16 ***
## factor(gvkey)18358 4.300e+00 1.188e-01 36.210 < 2e-16 ***
## factor(gvkey)18385 3.991e+00 2.190e-01 18.224 < 2e-16 ***
## factor(gvkey)18392 4.308e+00 2.827e-01 15.239 < 2e-16 ***
## factor(gvkey)18434 4.581e+00 1.632e-01 28.066 < 2e-16 ***
## factor(gvkey)18494 3.953e+00 4.897e-01 8.073 7.75e-16 ***
## factor(gvkey)18533 4.324e+00 1.309e-01 33.042 < 2e-16 ***
## factor(gvkey)18732 4.718e+00 1.224e-01 38.542 < 2e-16 ***
## factor(gvkey)18948 4.985e+00 2.827e-01 17.632 < 2e-16 ***
## factor(gvkey)19057 4.444e+00 1.264e-01 35.152 < 2e-16 ***
## factor(gvkey)19094 5.449e+00 1.095e-01 49.770 < 2e-16 ***
## factor(gvkey)19124 4.552e+00 1.095e-01 41.576 < 2e-16 ***
## factor(gvkey)19137 3.995e+00 1.548e-01 25.802 < 2e-16 ***
## factor(gvkey)19150 4.729e+00 1.095e-01 43.187 < 2e-16 ***
## factor(gvkey)19159 4.053e+00 3.462e-01 11.707 < 2e-16 ***
## factor(gvkey)19262 5.228e+00 1.095e-01 47.745 < 2e-16 ***
## factor(gvkey)19318 5.731e+00 1.095e-01 52.339 < 2e-16 ***
## factor(gvkey)19355 5.071e+00 1.095e-01 46.316 < 2e-16 ***
## factor(gvkey)19428 4.824e+00 1.095e-01 44.054 < 2e-16 ***
## factor(gvkey)19570 5.266e+00 1.095e-01 48.098 < 2e-16 ***
## factor(gvkey)19582 4.075e+00 4.897e-01 8.323 < 2e-16 ***
## factor(gvkey)19713 4.898e+00 1.188e-01 41.244 < 2e-16 ***
## factor(gvkey)19817 4.727e+00 1.224e-01 38.613 < 2e-16 ***
## factor(gvkey)19860 5.333e+00 3.462e-01 15.403 < 2e-16 ***
## factor(gvkey)19873 4.851e+00 1.123e-01 43.180 < 2e-16 ***
## factor(gvkey)19927 6.140e+00 3.462e-01 17.733 < 2e-16 ***
## factor(gvkey)20019 5.355e+00 1.095e-01 48.907 < 2e-16 ***
## factor(gvkey)20029 4.808e+00 1.731e-01 27.774 < 2e-16 ***
## factor(gvkey)20109 4.948e+00 1.095e-01 45.193 < 2e-16 ***
## factor(gvkey)20277 5.096e+00 2.827e-01 18.025 < 2e-16 ***
## factor(gvkey)20280 4.563e+00 3.462e-01 13.179 < 2e-16 ***
## factor(gvkey)20299 4.238e+00 3.462e-01 12.240 < 2e-16 ***
## factor(gvkey)20344 6.468e+00 2.827e-01 22.879 < 2e-16 ***
## factor(gvkey)20422 5.145e+00 3.462e-01 14.858 < 2e-16 ***
## factor(gvkey)20430 6.723e+00 4.897e-01 13.731 < 2e-16 ***
## factor(gvkey)20677 4.178e+00 3.462e-01 12.067 < 2e-16 ***
## factor(gvkey)20761 4.225e+00 3.462e-01 12.202 < 2e-16 ***
## factor(gvkey)20791 4.729e+00 1.358e-01 34.824 < 2e-16 ***
## factor(gvkey)20913 4.050e+00 4.897e-01 8.271 < 2e-16 ***
## factor(gvkey)21073 5.445e+00 3.462e-01 15.726 < 2e-16 ***
## factor(gvkey)21104 4.157e+00 3.462e-01 12.006 < 2e-16 ***
## factor(gvkey)21326 4.320e+00 2.827e-01 15.280 < 2e-16 ***
## factor(gvkey)21382 9.494e+00 1.266e-01 75.002 < 2e-16 ***
## factor(gvkey)21503 7.757e+00 1.095e-01 70.842 < 2e-16 ***
## factor(gvkey)21593 4.095e+00 1.632e-01 25.090 < 2e-16 ***
## factor(gvkey)21616 5.996e+00 2.827e-01 21.209 < 2e-16 ***
## factor(gvkey)21815 4.599e+00 4.897e-01 9.392 < 2e-16 ***
## factor(gvkey)21825 8.555e+00 2.827e-01 30.262 < 2e-16 ***
## factor(gvkey)22025 4.992e+00 3.462e-01 14.417 < 2e-16 ***
## factor(gvkey)22086 6.219e+00 1.095e-01 56.802 < 2e-16 ***
## factor(gvkey)22459 5.743e+00 3.462e-01 16.588 < 2e-16 ***
## factor(gvkey)22717 4.584e+00 1.188e-01 38.600 < 2e-16 ***
## factor(gvkey)22983 6.804e+00 3.462e-01 19.651 < 2e-16 ***
## factor(gvkey)23025 8.223e+00 1.358e-01 60.549 < 2e-16 ***
## factor(gvkey)23111 4.781e+00 1.188e-01 40.254 < 2e-16 ***
## factor(gvkey)23147 4.362e+00 4.897e-01 8.909 < 2e-16 ***
## factor(gvkey)23432 6.541e+00 3.462e-01 18.891 < 2e-16 ***
## factor(gvkey)23450 4.120e+00 1.264e-01 32.584 < 2e-16 ***
## factor(gvkey)23485 5.212e+00 2.190e-01 23.800 < 2e-16 ***
## factor(gvkey)23500 5.612e+00 1.095e-01 51.254 < 2e-16 ***
## factor(gvkey)23698 4.195e+00 3.462e-01 12.115 < 2e-16 ***
## factor(gvkey)23768 4.982e+00 1.851e-01 26.920 < 2e-16 ***
## factor(gvkey)23793 5.507e+00 1.224e-01 44.989 < 2e-16 ***
## factor(gvkey)23848 8.324e+00 1.731e-01 48.082 < 2e-16 ***
## factor(gvkey)24098 6.943e+00 3.462e-01 20.054 < 2e-16 ***
## factor(gvkey)24232 5.953e+00 1.095e-01 54.369 < 2e-16 ***
## factor(gvkey)24233 6.907e+00 1.224e-01 56.421 < 2e-16 ***
## factor(gvkey)24287 6.808e+00 1.309e-01 52.022 < 2e-16 ***
## factor(gvkey)24318 8.518e+00 1.123e-01 75.828 < 2e-16 ***
## factor(gvkey)24379 7.123e+00 1.095e-01 65.054 < 2e-16 ***
## factor(gvkey)24440 6.436e+00 1.264e-01 50.908 < 2e-16 ***
## factor(gvkey)24447 6.857e+00 1.095e-01 62.627 < 2e-16 ***
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## factor(gvkey)24533 4.805e+00 1.264e-01 38.009 < 2e-16 ***
## factor(gvkey)24578 9.096e+00 1.853e-01 49.098 < 2e-16 ***
## factor(gvkey)24678 6.759e+00 1.095e-01 61.730 < 2e-16 ***
## factor(gvkey)24725 5.551e+00 1.224e-01 45.346 < 2e-16 ***
## factor(gvkey)24731 6.736e+00 1.264e-01 53.276 < 2e-16 ***
## factor(gvkey)24750 4.709e+00 2.448e-01 19.236 < 2e-16 ***
## factor(gvkey)24825 5.872e+00 1.264e-01 46.445 < 2e-16 ***
## factor(gvkey)25030 5.431e+00 1.731e-01 31.374 < 2e-16 ***
## factor(gvkey)25140 4.123e+00 1.154e-01 35.725 < 2e-16 ***
## factor(gvkey)25173 5.000e+00 1.095e-01 45.667 < 2e-16 ***
## factor(gvkey)25236 4.175e+00 1.999e-01 20.884 < 2e-16 ***
## factor(gvkey)25296 6.132e+00 1.731e-01 35.419 < 2e-16 ***
## factor(gvkey)25339 5.593e+00 1.224e-01 45.691 < 2e-16 ***
## factor(gvkey)25376 5.065e+00 1.154e-01 43.882 < 2e-16 ***
## factor(gvkey)25425 5.542e+00 3.462e-01 16.007 < 2e-16 ***
## factor(gvkey)25462 4.693e+00 2.190e-01 21.433 < 2e-16 ***
## factor(gvkey)25481 4.030e+00 1.476e-01 27.298 < 2e-16 ***
## factor(gvkey)25570 5.313e+00 1.069e-01 49.721 < 2e-16 ***
## factor(gvkey)25632 4.859e+00 1.999e-01 24.307 < 2e-16 ***
## factor(gvkey)25633 5.322e+00 1.264e-01 42.092 < 2e-16 ***
## factor(gvkey)25665 4.381e+00 1.264e-01 34.650 < 2e-16 ***
## factor(gvkey)25714 9.321e+00 1.358e-01 68.622 < 2e-16 ***
## factor(gvkey)25751 5.438e+00 1.095e-01 49.667 < 2e-16 ***
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## factor(gvkey)25895 6.570e+00 1.095e-01 60.008 < 2e-16 ***
## factor(gvkey)25950 6.384e+00 1.264e-01 50.492 < 2e-16 ***
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## factor(gvkey)26071 4.816e+00 1.264e-01 38.093 < 2e-16 ***
## factor(gvkey)26270 4.008e+00 4.897e-01 8.184 3.10e-16 ***
## factor(gvkey)26347 4.017e+00 3.462e-01 11.603 < 2e-16 ***
## factor(gvkey)27780 6.514e+00 1.224e-01 53.210 < 2e-16 ***
## factor(gvkey)27875 6.064e+00 1.264e-01 47.965 < 2e-16 ***
## factor(gvkey)27914 9.947e+00 1.224e-01 81.255 < 2e-16 ***
## factor(gvkey)27932 5.579e+00 1.548e-01 36.027 < 2e-16 ***
## factor(gvkey)28022 5.328e+00 1.095e-01 48.657 < 2e-16 ***
## factor(gvkey)28034 8.979e+00 1.123e-01 79.930 < 2e-16 ***
## factor(gvkey)28119 7.509e+00 1.188e-01 63.222 < 2e-16 ***
## factor(gvkey)28152 4.267e+00 1.548e-01 27.556 < 2e-16 ***
## factor(gvkey)28155 4.663e+00 1.095e-01 42.589 < 2e-16 ***
## factor(gvkey)28216 8.472e+00 1.224e-01 69.204 < 2e-16 ***
## factor(gvkey)28278 4.391e+00 1.224e-01 35.866 < 2e-16 ***
## factor(gvkey)28322 5.529e+00 1.264e-01 43.731 < 2e-16 ***
## factor(gvkey)28323 4.789e+00 1.224e-01 39.119 < 2e-16 ***
## factor(gvkey)28340 5.334e+00 1.224e-01 43.570 < 2e-16 ***
## factor(gvkey)28349 1.033e+01 1.095e-01 94.306 < 2e-16 ***
## factor(gvkey)28409 7.343e+00 2.190e-01 33.534 < 2e-16 ***
## factor(gvkey)28608 5.858e+00 1.264e-01 46.338 < 2e-16 ***
## factor(gvkey)28620 4.604e+00 1.188e-01 38.771 < 2e-16 ***
## factor(gvkey)28629 6.407e+00 1.264e-01 50.676 < 2e-16 ***
## factor(gvkey)28733 7.498e+00 1.123e-01 66.747 < 2e-16 ***
## factor(gvkey)28769 4.900e+00 1.224e-01 40.031 < 2e-16 ***
## factor(gvkey)28866 4.490e+00 1.632e-01 27.511 < 2e-16 ***
## factor(gvkey)28967 4.523e+00 1.548e-01 29.209 < 2e-16 ***
## factor(gvkey)29052 5.321e+00 1.188e-01 44.809 < 2e-16 ***
## factor(gvkey)29055 5.123e+00 1.358e-01 37.726 < 2e-16 ***
## factor(gvkey)29061 4.083e+00 1.999e-01 20.423 < 2e-16 ***
## factor(gvkey)29082 6.727e+00 1.224e-01 54.956 < 2e-16 ***
## factor(gvkey)29097 8.523e+00 1.548e-01 55.039 < 2e-16 ***
## factor(gvkey)29099 6.172e+00 1.264e-01 48.821 < 2e-16 ***
## factor(gvkey)29101 5.085e+00 1.095e-01 46.441 < 2e-16 ***
## factor(gvkey)29211 5.545e+00 1.224e-01 45.297 < 2e-16 ***
## factor(gvkey)29246 6.854e+00 1.095e-01 62.602 < 2e-16 ***
## factor(gvkey)29282 6.523e+00 1.095e-01 59.572 < 2e-16 ***
## factor(gvkey)29286 7.076e+00 1.095e-01 64.631 < 2e-16 ***
## factor(gvkey)29359 5.442e+00 1.224e-01 44.453 < 2e-16 ***
## factor(gvkey)29389 8.148e+00 1.264e-01 64.447 < 2e-16 ***
## factor(gvkey)29446 4.719e+00 1.095e-01 43.097 < 2e-16 ***
## factor(gvkey)29613 4.554e+00 1.414e-01 32.219 < 2e-16 ***
## factor(gvkey)29649 5.895e+00 1.264e-01 46.623 < 2e-16 ***
## factor(gvkey)29710 9.380e+00 1.224e-01 76.624 < 2e-16 ***
## factor(gvkey)29804 8.199e+00 1.476e-01 55.533 < 2e-16 ***
## factor(gvkey)29855 4.048e+00 1.999e-01 20.250 < 2e-16 ***
## factor(gvkey)29868 6.674e+00 1.224e-01 54.519 < 2e-16 ***
## factor(gvkey)29875 6.708e+00 1.188e-01 56.480 < 2e-16 ***
## factor(gvkey)29984 7.097e+00 1.188e-01 59.760 < 2e-16 ***
## factor(gvkey)29994 4.249e+00 1.999e-01 21.257 < 2e-16 ***
## factor(gvkey)30067 3.976e+00 4.897e-01 8.119 5.30e-16 ***
## factor(gvkey)30146 5.878e+00 1.095e-01 53.681 < 2e-16 ***
## factor(gvkey)30188 4.295e+00 1.632e-01 26.315 < 2e-16 ***
## factor(gvkey)30218 7.876e+00 1.264e-01 62.293 < 2e-16 ***
## factor(gvkey)30222 4.679e+00 1.123e-01 41.655 < 2e-16 ***
## factor(gvkey)30293 5.774e+00 1.154e-01 50.033 < 2e-16 ***
## factor(gvkey)30298 6.246e+00 1.188e-01 52.596 < 2e-16 ***
## factor(gvkey)30328 4.548e+00 1.414e-01 32.177 < 2e-16 ***
## factor(gvkey)30354 6.444e+00 1.154e-01 55.838 < 2e-16 ***
## factor(gvkey)30384 5.963e+00 1.264e-01 47.168 < 2e-16 ***
## factor(gvkey)30416 4.119e+00 1.414e-01 29.141 < 2e-16 ***
## factor(gvkey)30452 1.040e+01 1.264e-01 82.217 < 2e-16 ***
## factor(gvkey)30490 7.102e+00 1.188e-01 59.805 < 2e-16 ***
## factor(gvkey)30495 6.968e+00 1.309e-01 53.243 < 2e-16 ***
## factor(gvkey)30501 9.381e+00 1.224e-01 76.623 < 2e-16 ***
## factor(gvkey)30539 6.148e+00 1.224e-01 50.224 < 2e-16 ***
## factor(gvkey)30580 9.965e+00 1.265e-01 78.792 < 2e-16 ***
## factor(gvkey)30582 8.728e+00 1.224e-01 71.302 < 2e-16 ***
## factor(gvkey)30637 6.504e+00 1.188e-01 54.765 < 2e-16 ***
## factor(gvkey)30640 4.720e+00 1.414e-01 33.392 < 2e-16 ***
## factor(gvkey)30647 4.006e+00 3.462e-01 11.570 < 2e-16 ***
## factor(gvkey)30679 6.229e+00 4.897e-01 12.722 < 2e-16 ***
## factor(gvkey)30822 5.638e+00 1.264e-01 44.597 < 2e-16 ***
## factor(gvkey)30857 4.025e+00 4.897e-01 8.220 2.30e-16 ***
## factor(gvkey)30888 4.660e+00 1.095e-01 42.559 < 2e-16 ***
## factor(gvkey)30932 5.688e+00 1.095e-01 51.947 < 2e-16 ***
## factor(gvkey)30990 9.045e+00 1.095e-01 82.606 < 2e-16 ***
## factor(gvkey)31242 4.068e+00 1.731e-01 23.499 < 2e-16 ***
## factor(gvkey)31358 4.493e+00 1.309e-01 34.329 < 2e-16 ***
## factor(gvkey)31368 4.970e+00 1.188e-01 41.846 < 2e-16 ***
## factor(gvkey)31521 9.755e+00 1.632e-01 59.764 < 2e-16 ***
## factor(gvkey)31692 5.639e+00 1.154e-01 48.862 < 2e-16 ***
## factor(gvkey)31702 3.961e+00 2.827e-01 14.012 < 2e-16 ***
## factor(gvkey)31718 3.965e+00 3.462e-01 11.451 < 2e-16 ***
## factor(gvkey)31764 4.209e+00 1.548e-01 27.181 < 2e-16 ***
## factor(gvkey)31895 4.067e+00 1.632e-01 24.916 < 2e-16 ***
## factor(gvkey)60914 5.068e+00 1.224e-01 41.402 < 2e-16 ***
## factor(gvkey)60990 4.065e+00 1.731e-01 23.478 < 2e-16 ***
## factor(gvkey)61019 4.197e+00 3.462e-01 12.120 < 2e-16 ***
## factor(gvkey)61067 7.422e+00 1.123e-01 66.074 < 2e-16 ***
## factor(gvkey)61129 5.680e+00 1.309e-01 43.399 < 2e-16 ***
## factor(gvkey)61163 6.651e+00 1.414e-01 47.054 < 2e-16 ***
## factor(gvkey)61188 6.331e+00 1.123e-01 56.362 < 2e-16 ***
## factor(gvkey)61302 7.976e+00 1.358e-01 58.731 < 2e-16 ***
## factor(gvkey)61380 8.156e+00 1.851e-01 44.067 < 2e-16 ***
## factor(gvkey)61388 7.984e+00 1.095e-01 72.916 < 2e-16 ***
## factor(gvkey)61406 7.269e+00 1.548e-01 46.946 < 2e-16 ***
## factor(gvkey)61408 8.114e+00 1.123e-01 72.229 < 2e-16 ***
## factor(gvkey)61452 7.286e+00 1.095e-01 66.543 < 2e-16 ***
## factor(gvkey)61487 5.224e+00 1.123e-01 46.504 < 2e-16 ***
## factor(gvkey)61544 4.358e+00 2.190e-01 19.903 < 2e-16 ***
## factor(gvkey)61585 4.953e+00 1.123e-01 44.089 < 2e-16 ***
## factor(gvkey)61586 6.296e+00 1.224e-01 51.428 < 2e-16 ***
## factor(gvkey)61693 5.335e+00 1.264e-01 42.194 < 2e-16 ***
## factor(gvkey)61739 9.811e+00 1.095e-01 89.604 < 2e-16 ***
## factor(gvkey)61939 4.426e+00 1.548e-01 28.586 < 2e-16 ***
## factor(gvkey)62243 3.963e+00 2.448e-01 16.189 < 2e-16 ***
## factor(gvkey)62365 3.999e+00 3.462e-01 11.550 < 2e-16 ***
## factor(gvkey)62646 4.034e+00 2.190e-01 18.419 < 2e-16 ***
## factor(gvkey)62654 4.981e+00 1.095e-01 45.492 < 2e-16 ***
## factor(gvkey)62689 9.725e+00 1.095e-01 88.816 < 2e-16 ***
## factor(gvkey)62755 4.011e+00 1.851e-01 21.671 < 2e-16 ***
## factor(gvkey)62895 4.104e+00 1.632e-01 25.146 < 2e-16 ***
## factor(gvkey)62919 4.033e+00 2.448e-01 16.475 < 2e-16 ***
## factor(gvkey)62976 4.742e+00 1.999e-01 23.720 < 2e-16 ***
## factor(gvkey)62979 4.804e+00 1.224e-01 39.243 < 2e-16 ***
## factor(gvkey)63058 4.095e+00 1.632e-01 25.087 < 2e-16 ***
## factor(gvkey)63069 5.872e+00 2.190e-01 26.815 < 2e-16 ***
## factor(gvkey)63135 5.115e+00 1.123e-01 45.530 < 2e-16 ***
## factor(gvkey)63178 4.511e+00 1.188e-01 37.981 < 2e-16 ***
## factor(gvkey)63232 4.715e+00 1.123e-01 41.975 < 2e-16 ***
## factor(gvkey)63244 4.231e+00 1.632e-01 25.923 < 2e-16 ***
## factor(gvkey)63288 6.548e+00 1.264e-01 51.791 < 2e-16 ***
## factor(gvkey)63501 7.264e+00 1.123e-01 64.663 < 2e-16 ***
## factor(gvkey)63538 5.018e+00 1.224e-01 40.992 < 2e-16 ***
## factor(gvkey)63590 8.748e+00 1.188e-01 73.645 < 2e-16 ***
## factor(gvkey)63639 6.081e+00 1.123e-01 54.136 < 2e-16 ***
## factor(gvkey)63687 4.153e+00 1.476e-01 28.131 < 2e-16 ***
## factor(gvkey)63725 4.233e+00 1.358e-01 31.166 < 2e-16 ***
## factor(gvkey)63781 6.023e+00 1.188e-01 50.713 < 2e-16 ***
## factor(gvkey)64189 4.457e+00 1.548e-01 28.785 < 2e-16 ***
## factor(gvkey)64218 4.966e+00 1.414e-01 35.134 < 2e-16 ***
## factor(gvkey)64228 4.062e+00 2.827e-01 14.369 < 2e-16 ***
## factor(gvkey)64306 5.544e+00 1.154e-01 48.040 < 2e-16 ***
## factor(gvkey)64536 7.005e+00 1.632e-01 42.919 < 2e-16 ***
## factor(gvkey)64547 4.932e+00 1.188e-01 41.527 < 2e-16 ***
## factor(gvkey)64552 7.122e+00 1.188e-01 59.973 < 2e-16 ***
## factor(gvkey)64584 4.194e+00 1.999e-01 20.981 < 2e-16 ***
## factor(gvkey)64628 4.278e+00 2.448e-01 17.475 < 2e-16 ***
## factor(gvkey)64699 6.475e+00 1.123e-01 57.637 < 2e-16 ***
## factor(gvkey)64794 5.830e+00 1.264e-01 46.116 < 2e-16 ***
## factor(gvkey)64821 5.779e+00 1.224e-01 47.212 < 2e-16 ***
## factor(gvkey)64925 7.286e+00 1.188e-01 61.351 < 2e-16 ***
## factor(gvkey)64934 4.110e+00 3.462e-01 11.870 < 2e-16 ***
## factor(gvkey)64979 4.122e+00 1.851e-01 22.272 < 2e-16 ***
## factor(gvkey)65106 4.984e+00 1.224e-01 40.714 < 2e-16 ***
## factor(gvkey)65108 7.612e+00 1.264e-01 60.207 < 2e-16 ***
## factor(gvkey)65214 3.912e+00 4.897e-01 7.990 1.51e-15 ***
## factor(gvkey)65228 5.836e+00 1.264e-01 46.159 < 2e-16 ***
## factor(gvkey)65230 4.142e+00 1.999e-01 20.718 < 2e-16 ***
## factor(gvkey)65235 4.289e+00 1.999e-01 21.455 < 2e-16 ***
## factor(gvkey)65290 6.599e+00 1.264e-01 52.196 < 2e-16 ***
## factor(gvkey)65345 6.068e+00 1.358e-01 44.679 < 2e-16 ***
## factor(gvkey)65365 4.198e+00 1.999e-01 21.001 < 2e-16 ***
## factor(gvkey)65532 4.215e+00 1.632e-01 25.822 < 2e-16 ***
## factor(gvkey)65540 5.539e+00 1.414e-01 39.184 < 2e-16 ***
## factor(gvkey)65548 4.612e+00 1.188e-01 38.836 < 2e-16 ***
## factor(gvkey)65556 6.872e+00 1.224e-01 56.138 < 2e-16 ***
## factor(gvkey)65581 7.157e+00 2.190e-01 32.681 < 2e-16 ***
## factor(gvkey)65610 4.214e+00 1.632e-01 25.815 < 2e-16 ***
## factor(gvkey)65640 5.108e+00 1.264e-01 40.402 < 2e-16 ***
## factor(gvkey)65689 4.497e+00 1.358e-01 33.110 < 2e-16 ***
## factor(gvkey)65710 5.201e+00 1.264e-01 41.136 < 2e-16 ***
## factor(gvkey)65717 5.370e+00 1.548e-01 34.679 < 2e-16 ***
## factor(gvkey)65737 5.204e+00 1.264e-01 41.163 < 2e-16 ***
## factor(gvkey)65796 4.318e+00 1.632e-01 26.458 < 2e-16 ***
## factor(gvkey)65886 6.904e+00 1.224e-01 56.401 < 2e-16 ***
## factor(gvkey)65958 4.459e+00 1.358e-01 32.835 < 2e-16 ***
## factor(gvkey)66085 5.795e+00 1.548e-01 37.426 < 2e-16 ***
## factor(gvkey)66148 4.088e+00 1.999e-01 20.448 < 2e-16 ***
## factor(gvkey)66161 3.967e+00 3.462e-01 11.457 < 2e-16 ***
## factor(gvkey)66235 4.989e+00 1.264e-01 39.463 < 2e-16 ***
## factor(gvkey)66281 4.904e+00 1.476e-01 33.218 < 2e-16 ***
## factor(gvkey)66285 4.259e+00 1.476e-01 28.850 < 2e-16 ***
## factor(gvkey)66313 4.041e+00 2.190e-01 18.454 < 2e-16 ***
## factor(gvkey)66394 4.372e+00 4.897e-01 8.929 < 2e-16 ***
## factor(gvkey)66503 4.370e+00 1.476e-01 29.599 < 2e-16 ***
## factor(gvkey)66504 4.170e+00 1.309e-01 31.863 < 2e-16 ***
## factor(gvkey)66597 4.012e+00 1.414e-01 28.380 < 2e-16 ***
## factor(gvkey)66599 6.503e+00 1.154e-01 56.345 < 2e-16 ***
## factor(gvkey)66654 5.249e+00 1.358e-01 38.653 < 2e-16 ***
## factor(gvkey)66731 6.753e+00 1.188e-01 56.859 < 2e-16 ***
## factor(gvkey)105538 4.259e+00 1.264e-01 33.690 < 2e-16 ***
## factor(gvkey)105547 1.076e+01 3.465e-01 31.052 < 2e-16 ***
## factor(gvkey)105670 4.650e+00 1.095e-01 42.467 < 2e-16 ***
## factor(gvkey)106853 4.058e+00 1.999e-01 20.299 < 2e-16 ***
## factor(gvkey)107325 4.528e+00 1.358e-01 33.342 < 2e-16 ***
## factor(gvkey)107817 4.264e+00 1.414e-01 30.165 < 2e-16 ***
## factor(gvkey)108979 4.127e+00 2.448e-01 16.858 < 2e-16 ***
## factor(gvkey)109183 4.282e+00 1.548e-01 27.657 < 2e-16 ***
## factor(gvkey)109318 4.090e+00 2.448e-01 16.706 < 2e-16 ***
## factor(gvkey)109599 4.225e+00 1.264e-01 33.415 < 2e-16 ***
## factor(gvkey)109621 4.703e+00 1.224e-01 38.418 < 2e-16 ***
## factor(gvkey)109683 6.020e+00 1.154e-01 52.162 < 2e-16 ***
## factor(gvkey)109699 4.646e+00 1.632e-01 28.464 < 2e-16 ***
## factor(gvkey)109926 6.070e+00 1.264e-01 48.009 < 2e-16 ***
## factor(gvkey)110065 6.933e+00 1.264e-01 54.834 < 2e-16 ***
## factor(gvkey)110179 6.418e+00 1.224e-01 52.428 < 2e-16 ***
## factor(gvkey)110250 4.393e+00 1.476e-01 29.755 < 2e-16 ***
## factor(gvkey)110359 4.065e+00 2.190e-01 18.565 < 2e-16 ***
## factor(gvkey)110382 5.476e+00 1.309e-01 41.844 < 2e-16 ***
## factor(gvkey)110979 4.231e+00 2.190e-01 19.323 < 2e-16 ***
## factor(gvkey)111179 4.170e+00 1.414e-01 29.503 < 2e-16 ***
## factor(gvkey)111721 4.677e+00 1.632e-01 28.652 < 2e-16 ***
## factor(gvkey)111779 4.786e+00 1.264e-01 37.858 < 2e-16 ***
## factor(gvkey)111819 4.203e+00 1.632e-01 25.750 < 2e-16 ***
## factor(gvkey)111940 4.229e+00 1.548e-01 27.311 < 2e-16 ***
## factor(gvkey)112112 4.048e+00 2.827e-01 14.318 < 2e-16 ***
## factor(gvkey)112254 4.073e+00 2.827e-01 14.407 < 2e-16 ***
## factor(gvkey)112410 4.855e+00 1.154e-01 42.062 < 2e-16 ***
## factor(gvkey)112542 4.701e+00 1.632e-01 28.800 < 2e-16 ***
## factor(gvkey)112626 4.831e+00 1.548e-01 31.199 < 2e-16 ***
## factor(gvkey)112721 5.147e+00 1.123e-01 45.818 < 2e-16 ***
## factor(gvkey)113978 4.178e+00 1.851e-01 22.577 < 2e-16 ***
## factor(gvkey)114628 1.041e+01 1.157e-01 89.905 < 2e-16 ***
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## factor(gvkey)114956 4.628e+00 1.851e-01 25.005 < 2e-16 ***
## factor(gvkey)115766 4.778e+00 2.827e-01 16.899 < 2e-16 ***
## factor(gvkey)115876 5.641e+00 1.309e-01 43.106 < 2e-16 ***
## factor(gvkey)117020 3.980e+00 2.827e-01 14.079 < 2e-16 ***
## factor(gvkey)117141 4.598e+00 1.264e-01 36.368 < 2e-16 ***
## factor(gvkey)117161 4.715e+00 1.188e-01 39.700 < 2e-16 ***
## factor(gvkey)117260 5.368e+00 1.476e-01 36.362 < 2e-16 ***
## factor(gvkey)118042 6.041e+00 1.154e-01 52.344 < 2e-16 ***
## factor(gvkey)118264 5.658e+00 1.224e-01 46.216 < 2e-16 ***
## factor(gvkey)118525 4.311e+00 1.548e-01 27.841 < 2e-16 ***
## factor(gvkey)119414 6.037e+00 1.154e-01 52.311 < 2e-16 ***
## factor(gvkey)119714 5.911e+00 1.264e-01 46.752 < 2e-16 ***
## factor(gvkey)119756 7.696e+00 1.188e-01 64.805 < 2e-16 ***
## factor(gvkey)120193 4.024e+00 3.462e-01 11.621 < 2e-16 ***
## factor(gvkey)120318 5.349e+00 1.548e-01 34.543 < 2e-16 ***
## factor(gvkey)120413 4.407e+00 1.999e-01 22.047 < 2e-16 ***
## factor(gvkey)120458 4.723e+00 2.827e-01 16.706 < 2e-16 ***
## factor(gvkey)121381 4.132e+00 1.851e-01 22.327 < 2e-16 ***
## factor(gvkey)121713 4.723e+00 1.188e-01 39.771 < 2e-16 ***
## factor(gvkey)121815 5.719e+00 1.358e-01 42.115 < 2e-16 ***
## factor(gvkey)121816 5.557e+00 1.264e-01 43.953 < 2e-16 ***
## factor(gvkey)122015 7.131e+00 1.188e-01 60.050 < 2e-16 ***
## factor(gvkey)122515 4.240e+00 1.731e-01 24.490 < 2e-16 ***
## factor(gvkey)124046 5.492e+00 1.224e-01 44.862 < 2e-16 ***
## factor(gvkey)124434 7.753e+00 1.224e-01 63.334 < 2e-16 ***
## factor(gvkey)127377 7.021e+00 1.731e-01 40.553 < 2e-16 ***
## factor(gvkey)127797 4.172e+00 1.309e-01 31.881 < 2e-16 ***
## factor(gvkey)133768 1.072e+01 1.188e-01 90.237 < 2e-16 ***
## factor(gvkey)136265 5.587e+00 1.358e-01 41.141 < 2e-16 ***
## factor(gvkey)137230 4.295e+00 2.448e-01 17.542 < 2e-16 ***
## factor(gvkey)137232 6.985e+00 1.188e-01 58.816 < 2e-16 ***
## factor(gvkey)137351 4.818e+00 1.224e-01 39.358 < 2e-16 ***
## factor(gvkey)137944 7.386e+00 1.309e-01 56.436 < 2e-16 ***
## factor(gvkey)138541 4.253e+00 2.190e-01 19.420 < 2e-16 ***
## factor(gvkey)139025 5.228e+00 1.548e-01 33.765 < 2e-16 ***
## factor(gvkey)140983 5.034e+00 1.224e-01 41.122 < 2e-16 ***
## factor(gvkey)142088 4.703e+00 1.632e-01 28.812 < 2e-16 ***
## factor(gvkey)142462 7.729e+00 1.224e-01 63.140 < 2e-16 ***
## factor(gvkey)143153 4.219e+00 1.548e-01 27.247 < 2e-16 ***
## factor(gvkey)143356 1.049e+01 1.264e-01 82.988 < 2e-16 ***
## factor(gvkey)143689 5.214e+00 1.264e-01 41.241 < 2e-16 ***
## factor(gvkey)144122 4.068e+00 3.462e-01 11.749 < 2e-16 ***
## factor(gvkey)144535 7.401e+00 1.264e-01 58.539 < 2e-16 ***
## factor(gvkey)145046 1.050e+01 1.264e-01 83.071 < 2e-16 ***
## factor(gvkey)145552 7.700e+00 1.264e-01 60.902 < 2e-16 ***
## factor(gvkey)145636 4.139e+00 1.548e-01 26.729 < 2e-16 ***
## factor(gvkey)145701 9.147e+00 1.264e-01 72.347 < 2e-16 ***
## factor(gvkey)146140 6.336e+00 1.414e-01 44.822 < 2e-16 ***
## factor(gvkey)146607 3.988e+00 1.999e-01 19.952 < 2e-16 ***
## factor(gvkey)146734 4.615e+00 1.999e-01 23.087 < 2e-16 ***
## factor(gvkey)147303 5.697e+00 1.414e-01 40.305 < 2e-16 ***
## factor(gvkey)147312 4.076e+00 2.448e-01 16.648 < 2e-16 ***
## factor(gvkey)147792 4.145e+00 1.999e-01 20.735 < 2e-16 ***
## factor(gvkey)148469 6.482e+00 1.358e-01 47.727 < 2e-16 ***
## factor(gvkey)148669 4.519e+00 1.476e-01 30.606 < 2e-16 ***
## factor(gvkey)149070 7.200e+00 1.264e-01 56.948 < 2e-16 ***
## factor(gvkey)149082 4.369e+00 2.190e-01 19.952 < 2e-16 ***
## factor(gvkey)149337 7.502e+00 1.309e-01 57.323 < 2e-16 ***
## factor(gvkey)149618 5.172e+00 2.827e-01 18.293 < 2e-16 ***
## factor(gvkey)149738 8.477e+00 1.224e-01 69.236 < 2e-16 ***
## factor(gvkey)150279 5.622e+00 1.264e-01 44.468 < 2e-16 ***
## factor(gvkey)150306 5.339e+00 1.264e-01 42.229 < 2e-16 ***
## factor(gvkey)152149 7.784e+00 1.309e-01 59.481 < 2e-16 ***
## factor(gvkey)152249 7.599e+00 1.548e-01 49.073 < 2e-16 ***
## factor(gvkey)153130 7.916e+00 1.358e-01 58.287 < 2e-16 ***
## factor(gvkey)154595 4.512e+00 1.851e-01 24.377 < 2e-16 ***
## factor(gvkey)154739 4.125e+00 2.827e-01 14.593 < 2e-16 ***
## factor(gvkey)154759 4.090e+00 2.827e-01 14.468 < 2e-16 ***
## factor(gvkey)155174 4.943e+00 1.548e-01 31.924 < 2e-16 ***
## factor(gvkey)155738 6.939e+00 1.358e-01 51.093 < 2e-16 ***
## factor(gvkey)155754 6.511e+00 1.476e-01 44.104 < 2e-16 ***
## factor(gvkey)156156 5.869e+00 1.851e-01 31.714 < 2e-16 ***
## factor(gvkey)156157 4.402e+00 1.414e-01 31.145 < 2e-16 ***
## factor(gvkey)156176 4.310e+00 1.632e-01 26.408 < 2e-16 ***
## factor(gvkey)156383 6.606e+00 1.309e-01 50.478 < 2e-16 ***
## factor(gvkey)156384 6.033e+00 1.548e-01 38.960 < 2e-16 ***
## factor(gvkey)156653 6.070e+00 1.414e-01 42.940 < 2e-16 ***
## factor(gvkey)156953 5.430e+00 1.358e-01 39.985 < 2e-16 ***
## factor(gvkey)157057 9.010e+00 1.358e-01 66.341 < 2e-16 ***
## factor(gvkey)157307 4.189e+00 1.999e-01 20.957 < 2e-16 ***
## factor(gvkey)157353 7.643e+00 1.548e-01 49.357 < 2e-16 ***
## factor(gvkey)157452 4.535e+00 2.827e-01 16.042 < 2e-16 ***
## factor(gvkey)157679 5.564e+00 1.999e-01 27.832 < 2e-16 ***
## factor(gvkey)157955 6.581e+00 1.414e-01 46.560 < 2e-16 ***
## factor(gvkey)158053 5.653e+00 2.827e-01 19.996 < 2e-16 ***
## factor(gvkey)158354 9.247e+00 1.358e-01 68.087 < 2e-16 ***
## factor(gvkey)158587 4.682e+00 1.548e-01 30.235 < 2e-16 ***
## factor(gvkey)158742 4.645e+00 1.548e-01 29.997 < 2e-16 ***
## factor(gvkey)160173 4.904e+00 1.476e-01 33.216 < 2e-16 ***
## factor(gvkey)160181 5.572e+00 1.358e-01 41.032 < 2e-16 ***
## factor(gvkey)160225 8.470e+00 1.999e-01 42.370 < 2e-16 ***
## factor(gvkey)160233 5.150e+00 1.851e-01 27.824 < 2e-16 ***
## factor(gvkey)160293 5.398e+00 1.548e-01 34.861 < 2e-16 ***
## factor(gvkey)160312 5.632e+00 2.190e-01 25.718 < 2e-16 ***
## factor(gvkey)160378 5.447e+00 1.632e-01 33.372 < 2e-16 ***
## factor(gvkey)160417 5.060e+00 1.358e-01 37.257 < 2e-16 ***
## factor(gvkey)160479 5.629e+00 1.476e-01 38.128 < 2e-16 ***
## factor(gvkey)160541 4.552e+00 1.999e-01 22.770 < 2e-16 ***
## factor(gvkey)160546 4.045e+00 4.897e-01 8.261 < 2e-16 ***
## factor(gvkey)160621 6.011e+00 1.851e-01 32.477 < 2e-16 ***
## factor(gvkey)160667 3.981e+00 2.448e-01 16.260 < 2e-16 ***
## factor(gvkey)160706 6.419e+00 1.548e-01 41.452 < 2e-16 ***
## factor(gvkey)160719 6.586e+00 1.414e-01 46.596 < 2e-16 ***
## factor(gvkey)160776 5.984e+00 1.476e-01 40.530 < 2e-16 ***
## factor(gvkey)160891 4.263e+00 1.548e-01 27.530 < 2e-16 ***
## factor(gvkey)160989 5.719e+00 1.414e-01 40.457 < 2e-16 ***
## factor(gvkey)160990 5.315e+00 1.414e-01 37.599 < 2e-16 ***
## factor(gvkey)160991 6.192e+00 1.414e-01 43.805 < 2e-16 ***
## factor(gvkey)161000 6.356e+00 1.632e-01 38.939 < 2e-16 ***
## factor(gvkey)161013 5.937e+00 1.548e-01 38.342 < 2e-16 ***
## factor(gvkey)161040 7.293e+00 1.476e-01 49.395 < 2e-16 ***
## factor(gvkey)161048 5.305e+00 1.358e-01 39.063 < 2e-16 ***
## factor(gvkey)161065 5.298e+00 2.448e-01 21.641 < 2e-16 ***
## factor(gvkey)161853 4.577e+00 1.414e-01 32.381 < 2e-16 ***
## factor(gvkey)161942 4.304e+00 1.414e-01 30.450 < 2e-16 ***
## factor(gvkey)161952 5.753e+00 1.414e-01 40.697 < 2e-16 ***
## factor(gvkey)161953 4.950e+00 1.548e-01 31.966 < 2e-16 ***
## factor(gvkey)161966 5.963e+00 1.632e-01 36.531 < 2e-16 ***
## factor(gvkey)162160 4.960e+00 1.632e-01 30.391 < 2e-16 ***
## factor(gvkey)162385 6.119e+00 1.632e-01 37.487 < 2e-16 ***
## factor(gvkey)162489 4.665e+00 1.476e-01 31.598 < 2e-16 ***
## factor(gvkey)162557 4.441e+00 1.548e-01 28.679 < 2e-16 ***
## factor(gvkey)162559 7.458e+00 1.414e-01 52.764 < 2e-16 ***
## factor(gvkey)162560 4.692e+00 1.851e-01 25.352 < 2e-16 ***
## factor(gvkey)162925 4.894e+00 1.632e-01 29.983 < 2e-16 ***
## factor(gvkey)163049 4.528e+00 3.462e-01 13.077 < 2e-16 ***
## factor(gvkey)163610 6.451e+00 1.414e-01 45.640 < 2e-16 ***
## factor(gvkey)163678 6.297e+00 1.414e-01 44.547 < 2e-16 ***
## factor(gvkey)163680 5.613e+00 1.999e-01 28.077 < 2e-16 ***
## factor(gvkey)163863 4.559e+00 2.827e-01 16.126 < 2e-16 ***
## factor(gvkey)163867 4.207e+00 4.897e-01 8.593 < 2e-16 ***
## factor(gvkey)163872 4.641e+00 2.827e-01 16.415 < 2e-16 ***
## factor(gvkey)163884 4.895e+00 1.414e-01 34.629 < 2e-16 ***
## factor(gvkey)163920 5.430e+00 1.414e-01 38.413 < 2e-16 ***
## factor(gvkey)163924 4.904e+00 2.827e-01 17.347 < 2e-16 ***
## factor(gvkey)163963 6.393e+00 2.827e-01 22.614 < 2e-16 ***
## factor(gvkey)164059 4.095e+00 3.462e-01 11.828 < 2e-16 ***
## factor(gvkey)164132 6.409e+00 2.190e-01 29.267 < 2e-16 ***
## factor(gvkey)164364 5.844e+00 1.414e-01 41.347 < 2e-16 ***
## factor(gvkey)164365 4.685e+00 1.414e-01 33.143 < 2e-16 ***
## factor(gvkey)164368 4.296e+00 1.851e-01 23.210 < 2e-16 ***
## factor(gvkey)164404 4.932e+00 1.851e-01 26.648 < 2e-16 ***
## factor(gvkey)164555 4.170e+00 2.827e-01 14.750 < 2e-16 ***
## factor(gvkey)164572 5.745e+00 1.548e-01 37.100 < 2e-16 ***
## factor(gvkey)164633 5.116e+00 1.476e-01 34.655 < 2e-16 ***
## factor(gvkey)164708 9.074e+00 1.414e-01 64.194 < 2e-16 ***
## factor(gvkey)165264 8.751e+00 1.414e-01 61.912 < 2e-16 ***
## factor(gvkey)165284 4.134e+00 1.632e-01 25.327 < 2e-16 ***
## factor(gvkey)166005 6.887e+00 1.548e-01 44.478 < 2e-16 ***
## factor(gvkey)166368 6.728e+00 1.476e-01 45.569 < 2e-16 ***
## factor(gvkey)166582 4.378e+00 2.190e-01 19.990 < 2e-16 ***
## factor(gvkey)166705 7.566e+00 1.851e-01 40.880 < 2e-16 ***
## factor(gvkey)170375 4.942e+00 2.190e-01 22.568 < 2e-16 ***
## factor(gvkey)170396 4.884e+00 2.448e-01 19.947 < 2e-16 ***
## factor(gvkey)170419 4.312e+00 2.190e-01 19.693 < 2e-16 ***
## factor(gvkey)171023 4.371e+00 2.827e-01 15.461 < 2e-16 ***
## factor(gvkey)174022 8.552e+00 1.476e-01 57.928 < 2e-16 ***
## factor(gvkey)174053 7.456e+00 1.548e-01 48.152 < 2e-16 ***
## factor(gvkey)174159 5.757e+00 2.827e-01 20.363 < 2e-16 ***
## factor(gvkey)174301 4.403e+00 1.632e-01 26.977 < 2e-16 ***
## factor(gvkey)174313 4.836e+00 1.476e-01 32.759 < 2e-16 ***
## factor(gvkey)174647 7.594e+00 1.632e-01 46.529 < 2e-16 ***
## factor(gvkey)174729 8.351e+00 1.548e-01 53.933 < 2e-16 ***
## factor(gvkey)174744 6.083e+00 1.632e-01 37.267 < 2e-16 ***
## factor(gvkey)175131 5.507e+00 1.632e-01 33.740 < 2e-16 ***
## factor(gvkey)175263 8.506e+00 1.476e-01 57.612 < 2e-16 ***
## factor(gvkey)175272 3.970e+00 4.897e-01 8.108 5.79e-16 ***
## factor(gvkey)175307 6.275e+00 1.476e-01 42.500 < 2e-16 ***
## factor(gvkey)175575 4.531e+00 2.448e-01 18.508 < 2e-16 ***
## factor(gvkey)175646 3.992e+00 3.462e-01 11.529 < 2e-16 ***
## factor(gvkey)175674 4.746e+00 1.731e-01 27.415 < 2e-16 ***
## factor(gvkey)175688 7.412e+00 1.548e-01 47.871 < 2e-16 ***
## factor(gvkey)176239 4.980e+00 1.632e-01 30.511 < 2e-16 ***
## factor(gvkey)176268 5.537e+00 1.548e-01 35.758 < 2e-16 ***
## factor(gvkey)176351 4.580e+00 1.632e-01 28.062 < 2e-16 ***
## factor(gvkey)176375 4.456e+00 2.827e-01 15.763 < 2e-16 ***
## factor(gvkey)176591 6.241e+00 1.476e-01 42.275 < 2e-16 ***
## factor(gvkey)176592 5.362e+00 1.476e-01 36.318 < 2e-16 ***
## factor(gvkey)176595 7.071e+00 1.632e-01 43.321 < 2e-16 ***
## factor(gvkey)176637 5.641e+00 2.448e-01 23.041 < 2e-16 ***
## factor(gvkey)176701 7.213e+00 1.548e-01 46.580 < 2e-16 ***
## factor(gvkey)176703 4.017e+00 1.731e-01 23.201 < 2e-16 ***
## factor(gvkey)176725 4.505e+00 2.448e-01 18.402 < 2e-16 ***
## factor(gvkey)176766 6.181e+00 1.548e-01 39.919 < 2e-16 ***
## factor(gvkey)176828 5.096e+00 1.999e-01 25.493 < 2e-16 ***
## factor(gvkey)176973 5.960e+00 1.731e-01 34.427 < 2e-16 ***
## factor(gvkey)177088 8.513e+00 1.731e-01 49.176 < 2e-16 ***
## factor(gvkey)177216 4.780e+00 2.448e-01 19.522 < 2e-16 ***
## factor(gvkey)177255 4.976e+00 1.731e-01 28.745 < 2e-16 ***
## factor(gvkey)177300 5.198e+00 1.548e-01 33.569 < 2e-16 ***
## factor(gvkey)177376 8.886e+00 1.548e-01 57.387 < 2e-16 ***
## factor(gvkey)177640 5.004e+00 3.462e-01 14.452 < 2e-16 ***
## factor(gvkey)177782 4.770e+00 1.731e-01 27.553 < 2e-16 ***
## factor(gvkey)177996 7.279e+00 1.632e-01 44.594 < 2e-16 ***
## factor(gvkey)178371 7.210e+00 1.851e-01 38.959 < 2e-16 ***
## factor(gvkey)178529 4.705e+00 2.190e-01 21.485 < 2e-16 ***
## factor(gvkey)178539 4.585e+00 2.448e-01 18.726 < 2e-16 ***
## factor(gvkey)178545 4.126e+00 1.999e-01 20.640 < 2e-16 ***
## factor(gvkey)178610 5.491e+00 1.731e-01 31.720 < 2e-16 ***
## factor(gvkey)178703 4.526e+00 1.632e-01 27.732 < 2e-16 ***
## factor(gvkey)178811 5.719e+00 1.851e-01 30.902 < 2e-16 ***
## factor(gvkey)178823 5.286e+00 2.827e-01 18.698 < 2e-16 ***
## factor(gvkey)178834 4.461e+00 1.548e-01 28.808 < 2e-16 ***
## factor(gvkey)178862 6.247e+00 1.999e-01 31.248 < 2e-16 ***
## factor(gvkey)179077 5.360e+00 1.548e-01 34.617 < 2e-16 ***
## factor(gvkey)179298 4.436e+00 1.731e-01 25.622 < 2e-16 ***
## factor(gvkey)179361 3.971e+00 2.827e-01 14.048 < 2e-16 ***
## factor(gvkey)179395 4.311e+00 4.897e-01 8.805 < 2e-16 ***
## factor(gvkey)179534 8.901e+00 1.632e-01 54.531 < 2e-16 ***
## factor(gvkey)179889 6.490e+00 1.999e-01 32.464 < 2e-16 ***
## factor(gvkey)179974 4.987e+00 2.190e-01 22.775 < 2e-16 ***
## factor(gvkey)180183 4.019e+00 2.827e-01 14.215 < 2e-16 ***
## factor(gvkey)180193 5.112e+00 2.448e-01 20.878 < 2e-16 ***
## factor(gvkey)180228 5.695e+00 2.190e-01 26.009 < 2e-16 ***
## factor(gvkey)180272 7.583e+00 1.851e-01 40.975 < 2e-16 ***
## factor(gvkey)180423 5.044e+00 1.731e-01 29.136 < 2e-16 ***
## factor(gvkey)182701 4.998e+00 1.851e-01 27.007 < 2e-16 ***
## factor(gvkey)182788 6.146e+00 1.851e-01 33.210 < 2e-16 ***
## factor(gvkey)183247 4.174e+00 1.731e-01 24.113 < 2e-16 ***
## factor(gvkey)183324 5.801e+00 2.448e-01 23.694 < 2e-16 ***
## factor(gvkey)183388 5.849e+00 2.190e-01 26.710 < 2e-16 ***
## factor(gvkey)183603 4.942e+00 2.448e-01 20.187 < 2e-16 ***
## factor(gvkey)183606 4.315e+00 2.448e-01 17.625 < 2e-16 ***
## factor(gvkey)183780 5.904e+00 2.448e-01 24.116 < 2e-16 ***
## factor(gvkey)183797 5.689e+00 1.999e-01 28.460 < 2e-16 ***
## factor(gvkey)183822 5.733e+00 4.897e-01 11.708 < 2e-16 ***
## factor(gvkey)183826 4.936e+00 1.999e-01 24.691 < 2e-16 ***
## factor(gvkey)183830 5.674e+00 2.190e-01 25.909 < 2e-16 ***
## factor(gvkey)183833 4.236e+00 4.897e-01 8.652 < 2e-16 ***
## factor(gvkey)183963 5.399e+00 2.190e-01 24.657 < 2e-16 ***
## factor(gvkey)184009 5.080e+00 1.999e-01 25.413 < 2e-16 ***
## factor(gvkey)184167 5.991e+00 1.851e-01 32.373 < 2e-16 ***
## factor(gvkey)184287 4.964e+00 1.999e-01 24.834 < 2e-16 ***
## factor(gvkey)184498 6.841e+00 2.190e-01 31.240 < 2e-16 ***
## factor(gvkey)184500 6.227e+00 1.999e-01 31.150 < 2e-16 ***
## factor(gvkey)184571 5.092e+00 1.999e-01 25.473 < 2e-16 ***
## factor(gvkey)184688 4.741e+00 2.448e-01 19.363 < 2e-16 ***
## factor(gvkey)184689 4.450e+00 2.827e-01 15.739 < 2e-16 ***
## factor(gvkey)184735 4.227e+00 2.827e-01 14.951 < 2e-16 ***
## factor(gvkey)184899 8.369e+00 1.851e-01 45.221 < 2e-16 ***
## factor(gvkey)185177 5.402e+00 2.448e-01 22.065 < 2e-16 ***
## factor(gvkey)185339 4.637e+00 2.448e-01 18.939 < 2e-16 ***
## factor(gvkey)185345 4.035e+00 4.897e-01 8.239 < 2e-16 ***
## factor(gvkey)185370 5.827e+00 3.462e-01 16.828 < 2e-16 ***
## factor(gvkey)185396 5.651e+00 2.827e-01 19.989 < 2e-16 ***
## factor(gvkey)185453 4.207e+00 3.462e-01 12.152 < 2e-16 ***
## factor(gvkey)185518 6.153e+00 3.462e-01 17.771 < 2e-16 ***
## factor(gvkey)185549 5.260e+00 1.999e-01 26.313 < 2e-16 ***
## factor(gvkey)185585 4.658e+00 2.190e-01 21.270 < 2e-16 ***
## factor(gvkey)185618 4.806e+00 2.448e-01 19.631 < 2e-16 ***
## factor(gvkey)185824 6.711e+00 2.190e-01 30.645 < 2e-16 ***
## factor(gvkey)186230 4.468e+00 3.462e-01 12.905 < 2e-16 ***
## factor(gvkey)186344 4.250e+00 1.999e-01 21.259 < 2e-16 ***
## factor(gvkey)186363 6.629e+00 1.999e-01 33.159 < 2e-16 ***
## factor(gvkey)186428 6.924e+00 2.190e-01 31.619 < 2e-16 ***
## factor(gvkey)186990 4.165e+00 4.897e-01 8.507 < 2e-16 ***
## factor(gvkey)186993 4.186e+00 4.897e-01 8.550 < 2e-16 ***
## factor(gvkey)187164 5.441e+00 1.999e-01 27.220 < 2e-16 ***
## factor(gvkey)187252 5.190e+00 2.827e-01 18.360 < 2e-16 ***
## factor(gvkey)187253 4.899e+00 2.190e-01 22.370 < 2e-16 ***
## factor(gvkey)187549 5.763e+00 2.448e-01 23.537 < 2e-16 ***
## factor(gvkey)189516 4.571e+00 4.897e-01 9.336 < 2e-16 ***
## factor(gvkey)189517 6.610e+00 2.827e-01 23.381 < 2e-16 ***
## factor(gvkey)190963 5.401e+00 2.448e-01 22.062 < 2e-16 ***
## factor(gvkey)192458 5.438e+00 2.448e-01 22.210 < 2e-16 ***
## factor(gvkey)200664 6.759e+00 2.190e-01 30.866 < 2e-16 ***
## factor(gvkey)211732 6.894e+00 1.476e-01 46.695 < 2e-16 ***
## factor(gvkey)223148 8.544e+00 1.264e-01 67.572 < 2e-16 ***
## factor(gvkey)241366 7.577e+00 1.224e-01 61.894 < 2e-16 ***
## factor(gvkey)241388 8.903e+00 2.827e-01 31.494 < 2e-16 ***
## factor(gvkey)243588 8.551e+00 2.448e-01 34.925 < 2e-16 ***
## factor(gvkey)248136 1.017e+01 1.557e-01 65.349 < 2e-16 ***
## factor(gvkey)252819 9.296e+00 1.309e-01 71.026 < 2e-16 ***
## factor(gvkey)252940 1.041e+01 1.360e-01 76.498 < 2e-16 ***
## factor(gvkey)258664 6.926e+00 1.476e-01 46.914 < 2e-16 ***
## factor(gvkey)260774 8.325e+00 1.358e-01 61.302 < 2e-16 ***
## factor(gvkey)260778 8.297e+00 1.358e-01 61.094 < 2e-16 ***
## factor(gvkey)260779 6.604e+00 1.358e-01 48.629 < 2e-16 ***
## factor(gvkey)264395 5.455e+00 1.358e-01 40.170 < 2e-16 ***
## factor(gvkey)264510 5.402e+00 3.462e-01 15.603 < 2e-16 ***
## factor(gvkey)266214 5.135e+00 1.358e-01 37.808 < 2e-16 ***
## factor(gvkey)266216 4.057e+00 2.448e-01 16.571 < 2e-16 ***
## factor(gvkey)266257 5.386e+00 1.632e-01 33.001 < 2e-16 ***
## factor(gvkey)266315 4.917e+00 1.548e-01 31.755 < 2e-16 ***
## factor(gvkey)275661 6.204e+00 3.462e-01 17.917 < 2e-16 ***
## factor(gvkey)285313 9.816e+00 1.731e-01 56.696 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4897 on 9148 degrees of freedom
## (15 observations deleted due to missingness)
## Multiple R-squared: 0.995, Adjusted R-squared: 0.9946
## F-statistic: 2169 on 846 and 9148 DF, p-value: < 2.2e-16Each firm’s dummy variable is time invariant—it never changes for each firm’s \(t's\). That means that for each \(t\), we’re only getting the variance in x that is associated with that particular \(i\) over time.
We got rid of all of the between-firm differences by holding them constant. What we have left is the average within-firm effect over time for, effectively, an average firm in the sample.
The challenge with this approach is that it’s not all that efficient, and there may be other things going on.
So we can use what’s called the within transformation approach, by subtracting out the time invariant component—the between effect—of each term.
This is what most fixed effect estimators do for you, although you can do it by hand.
We’ll use the within model in plm to estimate our fixed effect specification.
fixed.model <- plm(log.revt ~ dltt, data = panel.plm.df,
index=c("gvkey", "fyear"), model="within")
summary(fixed.model)## Oneway (individual) effect Within Model
##
## Call:
## plm(formula = log.revt ~ dltt, data = panel.plm.df, model = "within",
## index = c("gvkey", "fyear"))
##
## Unbalanced Panel: n=845, T=1-21, N=9994
##
## Residuals :
## Min. 1st Qu. Median 3rd Qu. Max.
## -5.1200 -0.2090 0.0287 0.2600 2.0200
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## dltt 9.788e-07 7.807e-08 12.537 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 2231.1
## Residual Sum of Squares: 2193.4
## R-Squared: 0.016893
## Adj. R-Squared: -0.073917
## F-statistic: 157.189 on 1 and 9148 DF, p-value: < 2.22e-16That display is a lot easier to make sense of! Notice though the coefficient estimate of our OLS model with dummy variables and our fixed effect model. Look similar?
Theoretical considerations aside, a random effect model is more efficient, because it uses within and between variance. So ideally, we would like to use that one.
But if \(i's\) correlate with their \(t's\), or if there is unobserved heterogeneity, then wanting to use a random effect model is immaterial—the estimate of \(\beta\) will be inconsistent.
Fortunately, we can use a workhorse of economics, the Hausman test, to determine whether there is a systematic difference in \(\beta\) as a function of the unobserved variance in the \(i\) effect.
phtest(fixed.model, random.model)##
## Hausman Test
##
## data: log.revt ~ dltt
## chisq = 404.53, df = 1, p-value < 2.2e-16
## alternative hypothesis: one model is inconsistentOur null is that there is no systematic difference, and we start from the assumption that our fixed effect model is consistent. So, how do we interpret this finding?
So, looks like we need to use a fixed effect model, which isn’t all that surprising, right?
Remember, our coefficient estimate then is the expected change in an average firm’s increase in revenue for a change in the firm’s long-term debt over time.
Now, another factor in panel models is time. Specifically, does time—or more accurately, the time period specified in the panel structure—result in systematic differences in the effect of \(x\) on \(y\).
If it does, than we need to include a separate fixed effect for time. In this case, the firm’s fiscal year.
plmtest(fixed.model, c("time"), type=("bp"))##
## Lagrange Multiplier Test - time effects (Breusch-Pagan) for
## unbalanced panels
##
## data: log.revt ~ dltt
## chisq = 4.3185, df = 1, p-value = 0.0377
## alternative hypothesis: significant effectsSo it looks like there is a small, but statistically significant effect of time on our model. As such, we incorporate a time fixed effect into our model.
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-12Purely FWIW, but running OLS using dummy variables for firms and for fiscal year yields the same estimate as the preceding fixed effect model…
summary(lm(log.revt ~ dltt + factor(fyear)-1 + factor(gvkey)-1, data = my.panel.df))
Lastly, we need to correct our standard errors. There are several options for this, but this one is a good go-to…
library(sandwich)
library(lmtest)
coeftest(fixed.year.model,
vcov.=vcovHC(fixed.year.model, type="HC1", cluster="group"))##
## t test of coefficients:
##
## Estimate Std. Error t value Pr(>|t|)
## dltt 4.3075e-07 1.2750e-07 3.3784 0.0007323 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Notice that after the standard errors correction, we see a demonstrably lower t-value.
T-stat looks too good
Try clustered standard errors—
Significance gone.
~ Keisuke Hirano
Wrap-up and what’s next.
Lab Today – Panel model assessment
Seminar 17 April – Panel data and causal effects