This R package implements the dynamic panel data modeling framework described by Allison, Williams, and Moral-Benito (2017). This approach allows fitting models with fixed effects that do not assume strict exogeneity of predictors. That means you can simultaneously get the robustness to confounding offered by fixed effects models and account for reciprocal causation between the predictors and the outcome variable. The estimating approach from Allison et al. provides better finite sample performance in terms of both bias and efficiency than other popular methods (e.g., the Arellano-Bond estimator).
These models are fit using structural equation models, using maximum likelihood estimation and offering the missing data handling and flexibility afforded by SEM. This package will reshape your data, specify the model properly, and fit it with lavaan
.
Note: This software is in development. It is best to cross-reference all results with xtdpdml for Stata. Go to https://www3.nd.edu/~rwilliam/dynamic/ to learn about xtdpdml and the underlying method. You may also be interested in the article by Paul Allison, Richard Williams, and Enrique Moral-Benito in Socius, accessible here.
You will need to the devtools
package installed to install this package from Github as well as its companion package, panelr
.
install.packages("devtools")
devtools::install_github("jacob-long/panelr")
devtools::install_github("jacob-long/dpm")
Also note this package’s other dependencies: lavaan
, stringr
, rlang
, and crayon
, in addition to the dependencies of panelr
and the aforementioned packages.
This package assumes your data are in long format, with each row representing a single observation of a single participant. Contrast this with wide format in which each row contains all observations of a single participant. Better compatibility with wide data will be implemented in a future release, but in the meantime you may use the long_panel
function implemented in panelr
to reshape to long format in a relatively pain-free way.
First we load the package and the WageData
from panelr
.
This next line of code converts the data to class panel_data
, which is a class specific to the panelr
that helps to simplify the treatment of the long-form panel data. You don’t have to do this, but it saves you from providing id
and wave
arguments to the model fitting function each time you use it.
wages <- panel_data(WageData, id = id, wave = t)
The formula syntax used in this package is meant to be as similar to a typical regression model as possible.
The most basic model can be specified like any other: y ~ x
, where y
is the dependent variable and x
is a time-varying predictor. If you would like to include time-invariant predictors, you will make the formula consist of two parts, separated with a bar (|
) like so: y ~ x | z
where z is a time invariant predictor, like ethnicity.
One of the innovations of the method, however, is the notion of predetermined, or sequentially exogenous, predictors. To specify a model with a predetermined variable, put the variable within a pre
function, y ~ pre(x1) + x2 | z
. This tells the function that x1
is predetermined while x2
is strictly exogenous by assumption. You could have multiple predetermined predictors as well (e.g., y ~ pre(x1) + pre(x2) | z
).
As implied by the “cross-lagged” terminology, you may also fit models with lagged predictors. Simply apply the lag function to the lagged predictors in the formula: y ~ pre(lag(x1)) + lag(x2) | z
. To specify more than 1 lag, just provide it as an argument. For instance, y ~ pre(lag(x1, 2)) + lag(x2) | z
will use 2 lags of the x1
variable.
This will replicate the analysis of the wages data in the Socius article that describes these models.
Note that to get matching standard errors, set information = "observed"
to override lavaan
’s default, information = "expected"
.
fit <- dpm(wks ~ pre(lag(union)) + lag(lwage) | ed, data = wages,
error.inv = TRUE, information = "observed")
summary(fit)
MODEL INFO:
Dependent variable:
Total observations: 595
Complete observations: 595
Time periods: 2 - 7
MODEL FIT:
<U+0001D6D8>²(76) = 138.476
RMSEA = 0.037, 90% CI [0.027, 0.047]
p(RMSEA < .05) = 0.986
SRMR = 0.025
| | Est. | S.E. | z val. | p |
|:------------------|-------:|------:|-------:|------:|
| union (t - 1) | -1.206 | 0.522 | -2.309 | 0.021 |
| lwage (t - 1) | 0.588 | 0.488 | 1.204 | 0.229 |
| ed | -0.107 | 0.056 | -1.893 | 0.058 |
| wks (t - 1) | 0.188 | 0.020 | 9.586 | 0.000 |
Model converged after 603 iterations
Any arguments supplied other than those that are documented within the dpm
function are passed on to sem
from the lavaan
package.
The following arguments allow you to make changes to the default model specification:
y.lag
: By default the lag 1 value of the DV is included as a predictor (this is why they are dynamic models). You may choose a different value or multiple values instead, including 0 (no lagged DV at all).fixed.effects
: By default, the model is specified as a fixed effects model. If you set this to FALSE, you get a random effects specification instead.error.inv
: This constrains error variances to be equal in each wave. It is FALSE by default.const.inv
: This constrains the constants to be equal in each wave. It is FALSE by default, but if TRUE it eliminates cross-sectional dependence.y.free
: This allows the regression coefficent of the lagged DV to vary across time. It is FALSE by default and you can either set it to TRUE or to the specific lag number(s).alpha.free
: If TRUE, relaxes the constraint that the fixed effects are equal across time. Default is FALSE to be consistent with how fixed effects models normally work.You have most of the options available to you via lavaan
’s summary method.
You can choose to omit any of: the z statistics (zstat = FALSE
), the standard errors (se = FALSE
), or the p values (pvalue = FALSE
). You may also add confidence intervals (ci = TRUE
) at any specified level (ci.level = .95
). If you used bootstrapping for uncertainty intervals, you can also specify the method (boot.ci.type = "perc"
).
The number of digits to print can be set via digits
or with the option dpm-digits
. You may also standardize coefficients via lavaan
’s method using standardize = TRUE
.
If you just want the lavaan
model specification and don’t want this package to fit the model for you, you can set print.only = TRUE
. To reduce the amount of output, I’m condensing wages
to 4 waves here.
dpm(wks ~ pre(lag(union)) + lag(lwage) | ed, data = wages[wages$t < 5,],
print.only = TRUE)
## Main regressions
wks_2 ~ en1 * union_1 + ex1 * lwage_1 + c1 * ed + p1 * wks_1
wks_3 ~ en1 * union_2 + ex1 * lwage_2 + c1 * ed + p1 * wks_2
wks_4 ~ en1 * union_3 + ex1 * lwage_3 + c1 * ed + p1 * wks_3
## Alpha latent variable (random intercept)
alpha =~ 1 * wks_2 + 1 * wks_3 + 1 * wks_4
## Alpha free to covary with observed variables (fixed effects)
alpha ~~ union_1 + union_2 + union_3 + lwage_1 + lwage_2 + lwage_3 + wks_1
## Correlating DV errors with future values of predetermined predictors
wks_2 ~~ union_3
## Predetermined predictors covariances
union_1 ~~ ed + lwage_1 + lwage_2 + lwage_3 + wks_1
union_2 ~~ ed + lwage_1 + lwage_2 + lwage_3 + union_1 + wks_1
union_3 ~~ ed + lwage_1 + lwage_2 + lwage_3 + union_1 + union_2 + wks_1
## Exogenous (time varying and invariant) predictors covariances
lwage_1 ~~ ed + wks_1
lwage_2 ~~ ed + lwage_1 + wks_1
lwage_3 ~~ ed + lwage_1 + lwage_2 + wks_1
ed ~~ wks_1
## DV error variance free to vary across waves
wks_2 ~~ wks_2
wks_3 ~~ wks_3
wks_4 ~~ wks_4
## Let DV variance vary across waves
wks_2 ~ 1
wks_3 ~ 1
wks_4 ~ 1
Alternately, you can extract the lavaan
model syntax and wide-formatted data from the fitted model object to do your own fitting and tweaking.
get_wide_data(fit)
get_syntax(fit)
The model is a special type of lavaan
object. This means most methods implemented for lavaan
objects will work on these. You can also convert the fitted model into a typical lavaan
object:
lavaan
summaryWhile you could convert the model to lavaan
model and apply any of lavaan
’s functions to it (and you should!), as a convenience you can use lav_summary()
to get lavaan
’s summary of the model.
Take advantage of lavaan
’s missing data handling by using the missing = "fiml"
argument as well as any other arguments accepted by lavaan::sem()
.
y ~ x + lag(x)
).1.0.0
)y ~ scale(x)
will cause an error.1.1.0
)Feature parity with xtdpdml
(Stata) is a goal. Here’s how we are doing in terms of matching relevant xtdpdml
options:
alphafree
(as alpha.free
)xfree
(as x.free
)xfree(varlist)
(as x.free
)yfree
(added as y.free
argument in 1.0.0
)yfree(numlist)
re
(added via fixed.effects
argument in 1.0.0
)errorinv
(as error.inv
)nocsd
/constinv
(as const.inv
)ylag(numlist)
(added as y.lag
argument in 1.0.0
; option to specify as 0 — no lagged DV — added in 1.1.0
)std
(but standardize
argument of summary
may suffice)dryrun
(as print.only
)Many and perhaps more SEM fitting options are implemented by virtue of accepting any lavaan::sem
argument.
lavaan
problem.y ~ scale(x)
(fixed in 1.1.0
)broom
methods (tidy
, glance
) (added tidy
in 1.1.0
)predict
method and perhaps some ability to plot predictionsx.free
option to allow the coefficients of all predictors to vary across periods. This will make the summary
output a pain, so it will take some time to implement. (added in 1.1.1
)Allison, P. D., Williams, R., & Moral-Benito, E. (2017). Maximum likelihood for cross-lagged panel models with fixed effects. Socius, 3, 1–17. https://doi.org/10.1177/2378023117710578
Leszczensky, L., & Wolbring, T. (2018, August 30). How to deal with reverse causality using panel data? Recommendations for researchers based on a simulation study. Working paper. https://doi.org/10.31235/osf.io/8xb4z
Moral-Benito, E., Allison, P., & Williams, R. (2019). Dynamic panel data modelling using maximum likelihood: An alternative to Arellano-Bond. Applied Economics, 51, 2221–2232. https://doi.org/10.1080/00036846.2018.1540854
Williams, R., Allison, P. D., & Moral-Benito, E. (2018). Linear dynamic panel-data estimation using maximum likelihood and structural equation modeling. The Stata Journal, 18, 293–326. https://doi.org/10.1177/1536867X1801800201