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`

.

If a result doesn’t seem right, it would be a good idea to cross-reference it 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.

`dpm`

will soon be on CRAN. In the meantime, you can get it from Github.

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. For help on converting data from wide to long format, check out the tutorial that accompanies the `panelr`

package.

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.

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 pre-determined, or sequentially exogenous, predictors. To specify a model with a pre-determined variable, put the variable within a `pre`

function, `y ~ pre(x1) + x2 | z`

. This tells the function that `x1`

is pre-determined while `x2`

is strictly exogenous by assumption. You could have multiple pre-determined predictors as well (e.g., `y ~ pre(x1) + pre(x2) | z`

).

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: wks
Total observations: 595
Complete observations: 595
Time periods: 2 - 7
MODEL FIT:
𝛘²(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 600 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 coefficient 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).`x.free`

: This allows the regression coefficients for the predictors to vary across time. It is FALSE by default and you can either set it to TRUE to set all predictors’ coefficients free over time or else pass a vector of strings of the predictors whose coefficients should be set free over time.`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.`partial.pre`

: If TRUE (FALSE by default), predetermined lagged predictors will also be allowed to correlate with the contemporaneous error term as suggested by Paul Allison for scenarios when it’s not clear whether you have chosen the right lag structure.

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.

```
## 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.

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()`

.

- CFI/TLI fit measures are much different than Stata’s and consistently more optimistic. For now, they are not printed with the summary because they are probably misleading.
~~You cannot use multiple lags of the same predictor (e.g.,~~(Fixed in`y ~ x + lag(x)`

).`1.0.0`

)- The function does not yet support input data that is already in wide format. (Not planning to fix)
~~You cannot apply arbitrary functions to variables in the formula like you can with regression models. For instance, a specification like~~(Works as of`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:

- [x]
`alphafree`

(as`alpha.free`

) - [x]
`xfree`

(as`x.free`

) - [x]
`xfree(varlist)`

(as`x.free`

) - [x]
`yfree`

(added as`y.free`

argument in`1.0.0`

) - [ ]
`yfree(numlist)`

- [x]
`re`

(added via`fixed.effects`

argument in`1.0.0`

) - [x]
`errorinv`

(as`error.inv`

) - [x]
`nocsd`

/`constinv`

(as`const.inv`

) - [x]
`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) - [x]
`dryrun`

(as`print.only`

)

Many and perhaps more SEM fitting options are implemented by virtue of accepting any `lavaan::sem()`

argument.

- [ ] Get proper CFI/TLI statistics — this is a
`lavaan`

problem. - [x] Allow full use of formula syntax, e.g.
`y ~ scale(x)`

(fixed in`1.1.0`

) - [x] Add
`broom`

methods (`tidy`

,`glance`

) (added`tidy`

in`1.1.0`

) - [ ] Create a
`predict`

method and perhaps some ability to plot predictions - [x] Add
`x.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. (2022, October 24). Getting the lags right – a new solution. *Statistical Horizons*. https://statisticalhorizons.com/getting-the-lags-right-a-new-solution/

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. (2022). How to deal with reverse causality using panel data? Recommendations for researchers based on a simulation study. *Sociological Methods & Research*, *51*(2), 837–865. https://doi.org/10.1177/0049124119882473

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