A new package for panel data analysis in R

It has been a long time coming, but my R package panelr is now on CRAN. Since I started work on it well over a year ago, it has become essential to my own workflow and I hope it can be useful for others.

panel_data object class

One key contribution, that I hope can help other developers, is the creation of a panel_data object class. It is a modified tibble, which is itself a modified data.frame. panel_data frames are grouped by entity, so many operations (e.g., mean(), cumsum()) performed by dplyr’s mutate() are groupwise operations. The panel_data frame also works very hard to stay in sequential order to ensure that lag and lead operations within mutate() make sense.

panel_data frames are in “long” format, in which each row is a unique combination of entity and time point. Let’s run through a quick example. First, the package includes the example “raw’ dataset called WageData, which comes from the Panel Study of Income Dynamics. This is what it looks like:

library(panelr)
data("WageData")
head(WageData)
  exp wks occ ind south smsa ms fem union ed blk   lwage t id
1   3  32   0   0     1    0  1   0     0  9   0 5.56068 1  1
2   4  43   0   0     1    0  1   0     0  9   0 5.72031 2  1
3   5  40   0   0     1    0  1   0     0  9   0 5.99645 3  1
4   6  39   0   0     1    0  1   0     0  9   0 5.99645 4  1
5   7  42   0   1     1    0  1   0     0  9   0 6.06146 5  1
6   8  35   0   1     1    0  1   0     0  9   0 6.17379 6  1

The key columns are id and t. They tell you which respondent and which time point the row refers to, respectively. Let’s convert it into a panel_data frame.

wages <- panel_data(WageData, id = id, wave = t)
wages
# Panel data:    4,165 x 14
# entities:      id [595]
# wave variable: t [1, 2, 3, ... (7 waves)]
   id        t   exp   wks   occ   ind south  smsa    ms   fem union    ed
   <fct> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
 1 1         1     3    32     0     0     1     0     1     0     0     9
 2 1         2     4    43     0     0     1     0     1     0     0     9
 3 1         3     5    40     0     0     1     0     1     0     0     9
 4 1         4     6    39     0     0     1     0     1     0     0     9
 5 1         5     7    42     0     1     1     0     1     0     0     9
 6 1         6     8    35     0     1     1     0     1     0     0     9
 7 1         7     9    32     0     1     1     0     1     0     0     9
 8 2         1    30    34     1     0     0     0     1     0     0    11
 9 2         2    31    27     1     0     0     0     1     0     0    11
10 2         3    32    33     1     1     0     0     1     0     1    11
# ... with 4,155 more rows, and 2 more variables: blk <dbl>, lwage <dbl>

panel_data() needs to know the ID and wave columns so that it can protect them (and you) against accidentally being dropped, re-ordered, and so on. It also allows other panel data functions in the package to know this information without you having to respecify every time.

Note that the wages data are grouped by id and sorted by t within each id. That means when you want to do things like calculate group means and create lagged variables, everything works correctly. A warning, though: this is only true within mutate() and transmute() from the dplyr package.

library(dplyr)
wages %>% 
  mutate(
    wks_mean = mean(wks), # this is the person-level mean
    wks_lag = lag(wks), # this will have a value of NA when t = 1
    cumu_wages = cumsum(exp(lwage)) # cumulative summation works within person
  ) %>%
  select(wks, wks_mean, wks_lag, lwage, cumu_wages) 
# Panel data:    4,165 x 7
# entities:      id [595]
# wave variable: t [1, 2, 3, ... (7 waves)]
   id        t   wks wks_mean wks_lag lwage cumu_wages
   <fct> <dbl> <dbl>    <dbl>   <dbl> <dbl>      <dbl>
 1 1         1    32     37.6      NA  5.56       260.
 2 1         2    43     37.6      32  5.72       565.
 3 1         3    40     37.6      43  6.00       967.
 4 1         4    39     37.6      40  6.00      1369.
 5 1         5    42     37.6      39  6.06      1798.
 6 1         6    35     37.6      42  6.17      2278.
 7 1         7    32     37.6      35  6.24      2793.
 8 2         1    34     31.6      NA  6.16       475.
 9 2         2    27     31.6      34  6.21       975.
10 2         3    33     31.6      27  6.26      1500.
# ... with 4,155 more rows

Notice also that when you use select, the id and t columns ride along even though you didn’t explicitly ask for them. The idea here is that it isn’t a panel_data frame without them. It works the same way using base R subsetting:

wages["wks"]
# Panel data:    4,165 x 3
# entities:      id [595]
# wave variable: t [1, 2, 3, ... (7 waves)]
   id        t   wks
   <fct> <dbl> <dbl>
 1 1         1    32
 2 1         2    43
 3 1         3    40
 4 1         4    39
 5 1         5    42
 6 1         6    35
 7 1         7    32
 8 2         1    34
 9 2         2    27
10 2         3    33
# ... with 4,155 more rows

You can get just the one column using double brackets or the $ subsetting method. But note that using base R sub-assignment, you don’t need to sweat those extra columns:

wages["wage"] <- exp(wages[["lwage"]]) # note double brackets 

Describing panel data

I’m also working on building out some descriptive functionality just for panel data. panel_data objects have a summary() method, which works best when you have the skimr package installed. By default, it will provide descriptive statistics for each column in each wave. To shorten the output, you can choose columns using dplyr::select() style syntax.

summary(wages, union, lwage)

Variable type: numeric

skim_variable t missing complete n mean sd p0 p25 p50 p75 p100 hist
union 1 0 595 595 0.36 0.48 0.00 0.00 0.00 1.00 1.00 ▇▁▁▁▅
union 2 0 595 595 0.35 0.48 0.00 0.00 0.00 1.00 1.00 ▇▁▁▁▅
union 3 0 595 595 0.37 0.48 0.00 0.00 0.00 1.00 1.00 ▇▁▁▁▅
union 4 0 595 595 0.37 0.48 0.00 0.00 0.00 1.00 1.00 ▇▁▁▁▅
union 5 0 595 595 0.37 0.48 0.00 0.00 0.00 1.00 1.00 ▇▁▁▁▅
union 6 0 595 595 0.36 0.48 0.00 0.00 0.00 1.00 1.00 ▇▁▁▁▅
union 7 0 595 595 0.37 0.48 0.00 0.00 0.00 1.00 1.00 ▇▁▁▁▅
lwage 1 0 595 595 6.38 0.39 5.01 6.12 6.42 6.65 6.91 ▁▂▃▇▇
lwage 2 0 595 595 6.47 0.36 5.01 6.24 6.53 6.75 6.91 ▁▁▂▅▇
lwage 3 0 595 595 6.60 0.45 4.61 6.33 6.61 6.86 8.27 ▁▂▇▃▁
lwage 4 0 595 595 6.70 0.44 5.08 6.44 6.72 6.96 8.52 ▁▃▇▂▁
lwage 5 0 595 595 6.79 0.42 5.27 6.51 6.80 7.04 8.10 ▁▂▇▅▁
lwage 6 0 595 595 6.86 0.42 5.66 6.60 6.91 7.11 8.16 ▁▃▇▃▁
lwage 7 0 595 595 6.95 0.44 5.68 6.68 6.98 7.21 8.54 ▁▅▇▂▁

You can stop getting per-wave statistics by setting by.wave = FALSE. For panels with many fewer entities, you might also want per-entity statistics. You can achieve this by setting by.wave = FALSE and by.id = TRUE.

You can also visualize trends in your data using line_plot().

line_plot(wages, lwage)

Each line is an individual id in the data.

Let’s see what the mean trend looks like. While we’re at it, let’s make the individual lines a little more transparent using the alpha argument.

line_plot(wages, lwage, add.mean = TRUE, alpha = 0.2)

The blue line is the mean trend and we can see that nearly everyone increases over time.

Sometimes it is useful to isolate specific entities from your data. I’ll use a different example to illustrate. These data come from the Penn World Table and contain data about countries, their exchange rates, purchasing power parity, and related data. It is provided by Stata and discussed in its manual.

library(haven)
penn <- read_dta("http://www.stata-press.com/data/r13/pennxrate.dta")
penn <- panel_data(penn, id = country, wave = year)
penn
# Panel data:    5,134 x 10
# entities:      country [151]
# wave variable: year [1970, 1971, 1972, ... (34 waves)]
   country  year xrate   ppp    id  capt realxrate lnrxrate  oecd    g7
   <fct>   <dbl> <dbl> <dbl> <dbl> <dbl>     <dbl>    <dbl> <dbl> <dbl>
 1 AFG      1970  45   10.8      1    34     1         0        0     0
 2 AFG      1971  45   11.2      1    34     0.250    -1.39     0     0
 3 AFG      1972  45    9.58     1    34     0.213    -1.55     0     0
 4 AFG      1973  45    8.94     1    34     0.199    -1.62     0     0
 5 AFG      1974  45    9.52     1    34     0.211    -1.55     0     0
 6 AFG      1975  45    9.12     1    34     0.203    -1.60     0     0
 7 AFG      1976  45    8.97     1    34     0.199    -1.61     0     0
 8 AFG      1977  45    9.33     1    34     0.207    -1.57     0     0
 9 AFG      1978  45    9.44     1    34     0.210    -1.56     0     0
10 AFG      1979  43.7  9.54     1    34     0.218    -1.52     0     0
# ... with 5,124 more rows

We’ll look at trends in the real exchange rate with the United States (realxrate).

line_plot(penn, realxrate)

We can also look at each country separately by setting overlay = FALSE. Since there are so many, we will want to look at just a subset. I’ll look at members of the “G7” countries, minus the USA.

line_plot(penn, realxrate, overlay = FALSE, 
          subset.ids = filter(penn, g7 == 1)$country, add.mean = TRUE)

We can see some heterogeneity in the trends. You may also want to fit a trend line that isn’t strictly linear, which is doable via the mean.function argument.

line_plot(penn, realxrate, overlay = FALSE, 
          subset.ids = filter(penn, g7 == 1)$country,
          add.mean = TRUE, mean.function = "loess")

Tools for reshaping data

Although you can get a much more detailed walk-through in the package’s tutorial vignette, I also want to mention some tools I created to help people get their data into the long format demanded by panel_data() (and most methods of analysis) as well as out of long format into a wide format in which there is just 1 row per entity.

There are a number of tools that can do this, most notably base R’s reshape() function. The problem with reshape() is that it can be a real pain to use, especially if you have a lot of time-varying variables and/or they aren’t labeled in a way congenial to what the function is looking for. The tidyr package is also designed to help with problems like these, but I (and apparently many others) struggle with the featured spread() and gather() functions, which in the case of panel data have a tendency to make the data longer than you actually want it unless you’re careful. They are great general tools, but my goal was to make a specific tool to make life easier in this particular situation.

Going from long to wide format is fairly straightforward. Let’s take our wages data. As a reminder, it looks like this:

wages
# Panel data:    4,165 x 15
# entities:      id [595]
# wave variable: t [1, 2, 3, ... (7 waves)]
   id        t   exp   wks   occ   ind south  smsa    ms   fem union    ed
   <fct> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
 1 1         1     3    32     0     0     1     0     1     0     0     9
 2 1         2     4    43     0     0     1     0     1     0     0     9
 3 1         3     5    40     0     0     1     0     1     0     0     9
 4 1         4     6    39     0     0     1     0     1     0     0     9
 5 1         5     7    42     0     1     1     0     1     0     0     9
 6 1         6     8    35     0     1     1     0     1     0     0     9
 7 1         7     9    32     0     1     1     0     1     0     0     9
 8 2         1    30    34     1     0     0     0     1     0     0    11
 9 2         2    31    27     1     0     0     0     1     0     0    11
10 2         3    32    33     1     1     0     0     1     0     1    11
# ... with 4,155 more rows, and 3 more variables: blk <dbl>, lwage <dbl>,
#   wage <dbl>

Let’s widen it, which will leave us with one row for each id.

widen_panel(wages)
# A tibble: 595 x 74
   id      fem    ed   blk exp_1 wks_1 occ_1 ind_1 south_1 smsa_1  ms_1
   <fct> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl> <dbl>
 1 1         0     9     0     3    32     0     0       1      0     1
 2 2         0    11     0    30    34     1     0       0      0     1
 3 3         0    12     0     6    50     1     1       0      0     1
 4 4         1    10     1    31    52     1     0       0      1     0
 5 5         0    16     0    10    50     1     0       0      0     1
 6 6         0    12     0    26    44     1     1       0      1     1
 7 7         0    12     0    15    46     1     0       0      0     1
 8 8         0    10     0    23    51     1     1       1      0     1
 9 9         0    16     0     3    50     0     0       1      1     1
10 10        0    16     0     3    49     0     0       1      1     1
# ... with 585 more rows, and 63 more variables: union_1 <dbl>,
#   lwage_1 <dbl>, wage_1 <dbl>, exp_2 <dbl>, wks_2 <dbl>, occ_2 <dbl>,
#   ind_2 <dbl>, south_2 <dbl>, smsa_2 <dbl>, ms_2 <dbl>, union_2 <dbl>,
#   lwage_2 <dbl>, wage_2 <dbl>, exp_3 <dbl>, wks_3 <dbl>, occ_3 <dbl>,
#   ind_3 <dbl>, south_3 <dbl>, smsa_3 <dbl>, ms_3 <dbl>, union_3 <dbl>,
#   lwage_3 <dbl>, wage_3 <dbl>, exp_4 <dbl>, wks_4 <dbl>, occ_4 <dbl>,
#   ind_4 <dbl>, south_4 <dbl>, smsa_4 <dbl>, ms_4 <dbl>, union_4 <dbl>,
#   lwage_4 <dbl>, wage_4 <dbl>, exp_5 <dbl>, wks_5 <dbl>, occ_5 <dbl>,
#   ind_5 <dbl>, south_5 <dbl>, smsa_5 <dbl>, ms_5 <dbl>, union_5 <dbl>,
#   lwage_5 <dbl>, wage_5 <dbl>, exp_6 <dbl>, wks_6 <dbl>, occ_6 <dbl>,
#   ind_6 <dbl>, south_6 <dbl>, smsa_6 <dbl>, ms_6 <dbl>, union_6 <dbl>,
#   lwage_6 <dbl>, wage_6 <dbl>, exp_7 <dbl>, wks_7 <dbl>, occ_7 <dbl>,
#   ind_7 <dbl>, south_7 <dbl>, smsa_7 <dbl>, ms_7 <dbl>, union_7 <dbl>,
#   lwage_7 <dbl>, wage_7 <dbl>

Notice that for variables that vary over time, there is now a column for each wave.

Going from wide to long is a bit more complicated because you need to automate the process of knowing how many waves there are, which variables change over time, and how the time-varying variables are labeled to reflect the time of the measurement. We’ll use another example dataset from this package, called teen_poverty, that starts in the wide format.

data("teen_poverty")
teen_poverty
# A tibble: 1,151 x 28
      id  pov1 mother1 spouse1 inschool1 hours1  pov2 mother2 spouse2
   <dbl> <dbl>   <dbl>   <dbl>     <dbl>  <dbl> <dbl>   <dbl>   <dbl>
 1    22     1       0       0         1     21     0       0       0
 2    75     0       0       0         1      8     0       0       0
 3    92     0       0       0         1     30     0       0       0
 4    96     0       0       0         0     19     1       1       0
 5   141     0       0       0         1      0     0       0       0
 6   161     0       0       0         1      0     0       0       0
 7   220     0       0       0         1      6     0       0       0
 8   229     0       0       0         1      0     1       0       0
 9   236     0       0       0         1      0     0       0       0
10   240     0       0       0         1     18     1       0       0
# ... with 1,141 more rows, and 19 more variables: inschool2 <dbl>,
#   hours2 <dbl>, pov3 <dbl>, mother3 <dbl>, spouse3 <dbl>,
#   inschool3 <dbl>, hours3 <dbl>, pov4 <dbl>, mother4 <dbl>,
#   spouse4 <dbl>, inschool4 <dbl>, hours4 <dbl>, age <dbl>, black <dbl>,
#   pov5 <dbl>, mother5 <dbl>, spouse5 <dbl>, inschool5 <dbl>,
#   hours5 <dbl>

We have some variables that don’t change over time (like whether the respondent is black) and a number that do, like whether the respondent is married (spouse).

long_panel() needs to know what the waves are called (1, 2, 3, …), where the wave label is in the variable name (beginning or end), and whether the label has prefixes or suffixes (e.g., “W1_variable” has a “W” prefix and “_” suffix). In this case, we have no prefix/suffix, the label is at the end, and the labels go from 1 to 5.

long_panel(teen_poverty, label_location = "end", periods = 1:5)
# Panel data:    5,755 x 9
# entities:      id [1151]
# wave variable: wave [1, 2, 3, ... (5 waves)]
   id     wave   age black   pov mother spouse inschool hours
   <fct> <dbl> <dbl> <dbl> <dbl>  <dbl>  <dbl>    <dbl> <dbl>
 1 22        1    16     0     1      0      0        1    21
 2 22        2    16     0     0      0      0        1    15
 3 22        3    16     0     0      0      0        1     3
 4 22        4    16     0     0      0      0        1     0
 5 22        5    16     0     0      0      0        1     0
 6 75        1    17     0     0      0      0        1     8
 7 75        2    17     0     0      0      0        1     0
 8 75        3    17     0     0      0      0        1     0
 9 75        4    17     0     0      0      0        1     4
10 75        5    17     0     1      0      0        1     0
# ... with 5,745 more rows

Perfect! As a note, long_panel() does fairly well in more complicated situations, like when time-varying variables are only measured in some waves and not others. See the vignette for more details.

Regression models

The other main contribution of the panelr package is that it provides a straightforward way to fit some panel data regression models. These are, by and large, doable via other common packages. The reason for implementing them in panelr is that they typically require some programming that would be difficult for novice and maybe even intermediate R users and even for the best of us, can be error-prone.

The first and most important of these is what is often called the “within-between” or sometimes “between-within” and “hybrid” model, which separates within-entity and between-entity variance. The within-entity portion is equivalent to what econometricians called the “fixed effects” model. People like these models because they are robust to confounding by individual differences. You don’t have to measure income, or personality, or whatever it may be and it is automatically controlled for because each person serves as their own control. Unlike fixed effects models, however, you can still include stable variables if you’re interested in their effects.

And because the models are estimated via multilevel models, you can take advantage of the specification flexibility afforded by them with random slopes and so on.

You can learn in more detail what these models are all about in the package’s introductory vignette.

These models are implemented via the wbm() function (within-between model). Let’s run through an example with the teen_poverty data. First we’ll transform it to long format like in the earlier example, then we’ll predict hours worked (hours) using indicators of whether the respondent’s marital status changed (spouse), they became a mother (mother), or have enrolled in school (inschool).

teen <- long_panel(teen_poverty, label_location = "end", periods = 1:5)
model <- wbm(hours ~ spouse + mother + inschool, data = teen)
summary(model)
MODEL INFO:
Entities: 1151
Time periods: 1-5
Dependent variable: hours
Model type: Linear mixed effects
Specification: within-between

MODEL FIT:
AIC = 45755.31, BIC = 45815.23
Pseudo-R² (fixed effects) = 0.15
Pseudo-R² (total) = 0.35
Entity ICC = 0.23

WITHIN EFFECTS:
--------------------------------------------------------
                   Est.   S.E.   t val.      d.f.      p
-------------- -------- ------ -------- --------- ------
spouse            -1.22   0.83    -1.47   4601.00   0.14
mother            -6.52   0.74    -8.76   4601.00   0.00
inschool         -11.09   0.47   -23.65   4601.00   0.00
--------------------------------------------------------

BETWEEN EFFECTS:
---------------------------------------------------------------
                          Est.   S.E.   t val.      d.f.      p
--------------------- -------- ------ -------- --------- ------
(Intercept)              20.38   0.76    26.87   1147.00   0.00
imean(spouse)            -1.53   1.29    -1.18   1147.00   0.24
imean(mother)            -9.83   0.90   -10.95   1147.00   0.00
imean(inschool)         -15.23   0.94   -16.27   1147.00   0.00
---------------------------------------------------------------

p values calculated using Satterthwaite d.f.
 
RANDOM EFFECTS:
------------------------------------
  Group      Parameter    Std. Dev. 
---------- ------------- -----------
    id      (Intercept)     6.504   
 Residual                   11.74   
------------------------------------

We have within- and between-subject effects here. The within effects can be interpreted as the effects of changes in spouse, mother, and inschool on hours worked. The between effects (which are the individual-level means, hence imean()) reflect how the overall level of the variables correspond with the overall level of hours worked, but don’t tell us much about change in either one.

From the output, we can see the within and between effects are quite similar. Unsurprisingly, starting school corresponds with a substantial decrease in hours worked as does becoming a mother.

What if we want to know about the effect of race? wbm() uses a multi-part formula to allow you to explicitly specify stable variables. You separate the within- and between-entity variables with a bar (|). For example:

model <- wbm(hours ~ spouse + mother + inschool | black, data = teen)
summary(model)
MODEL INFO:
Entities: 1151
Time periods: 1-5
Dependent variable: hours
Model type: Linear mixed effects
Specification: within-between

MODEL FIT:
AIC = 45755.79, BIC = 45822.37
Pseudo-R² (fixed effects) = 0.15
Pseudo-R² (total) = 0.35
Entity ICC = 0.23

WITHIN EFFECTS:
--------------------------------------------------------
                   Est.   S.E.   t val.      d.f.      p
-------------- -------- ------ -------- --------- ------
spouse            -1.22   0.83    -1.47   4601.00   0.14
mother            -6.52   0.74    -8.76   4601.00   0.00
inschool         -11.09   0.47   -23.65   4601.00   0.00
--------------------------------------------------------

BETWEEN EFFECTS:
---------------------------------------------------------------
                          Est.   S.E.   t val.      d.f.      p
--------------------- -------- ------ -------- --------- ------
(Intercept)              20.60   0.79    26.07   1146.00   0.00
imean(spouse)            -1.67   1.30    -1.29   1146.00   0.20
imean(mother)            -9.65   0.92   -10.54   1146.00   0.00
imean(inschool)         -15.15   0.94   -16.13   1146.00   0.00
black                    -0.52   0.51    -1.01   1146.00   0.31
---------------------------------------------------------------

p values calculated using Satterthwaite d.f.
 
RANDOM EFFECTS:
------------------------------------
  Group      Parameter    Std. Dev. 
---------- ------------- -----------
    id      (Intercept)     6.504   
 Residual                   11.74   
------------------------------------

There does not seem to be a difference in hours worked between black and non-black respondents, at least after accounting for these other factors.

You can use a third part of the formula as well, where you can specify cross-level interactions (i.e., within by between interactions) as well as use the lme4 syntax for random effects (by default, (1 | id) is included without you putting it into the formula). Here’s we will see if the effect of becoming a mother is different for black and non-black respondents.

model <- wbm(hours ~ spouse + mother + inschool | black | black * mother, data = teen)
summary(model)
MODEL INFO:
Entities: 1151
Time periods: 1-5
Dependent variable: hours
Model type: Linear mixed effects
Specification: within-between

MODEL FIT:
AIC = 45735.34, BIC = 45808.58
Pseudo-R² (fixed effects) = 0.15
Pseudo-R² (total) = 0.35
Entity ICC = 0.24

WITHIN EFFECTS:
--------------------------------------------------------
                   Est.   S.E.   t val.      d.f.      p
-------------- -------- ------ -------- --------- ------
spouse            -0.87   0.83    -1.05   4600.00   0.30
mother           -10.78   1.21    -8.92   4600.00   0.00
inschool         -11.01   0.47   -23.51   4600.00   0.00
--------------------------------------------------------

BETWEEN EFFECTS:
---------------------------------------------------------------
                          Est.   S.E.   t val.      d.f.      p
--------------------- -------- ------ -------- --------- ------
(Intercept)              20.60   0.79    26.07   1146.00   0.00
imean(spouse)            -1.67   1.30    -1.29   1146.00   0.20
imean(mother)            -9.65   0.92   -10.54   1146.00   0.00
imean(inschool)         -15.15   0.94   -16.13   1146.00   0.00
black                    -0.52   0.51    -1.01   1146.00   0.31
---------------------------------------------------------------

CROSS-LEVEL INTERACTIONS:
----------------------------------------------------------
                     Est.   S.E.   t val.      d.f.      p
------------------ ------ ------ -------- --------- ------
mother:black         6.34   1.42     4.47   4600.00   0.00
----------------------------------------------------------

p values calculated using Satterthwaite d.f.
 
RANDOM EFFECTS:
------------------------------------
  Group      Parameter    Std. Dev. 
---------- ------------- -----------
    id      (Intercept)     6.512   
 Residual                   11.72   
------------------------------------

Indeed, there seems to be.

There are a number of other things available for regression modeling of panel data that I will not cover in detail here — see the introductory vignette for more info. These include detrending variables in the within-between model, estimating within-between models with generalized estimating equations (GEE), first differences models, and asymmetric effects models in which increases and decreases over time are expected to have different effects.

Jacob A. Long
PhD Candidate, Communication

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