`EquiTrends`

is an R package for equivalence testing in
the context of Difference-in-Differences estimation. It allows users to
test if pre-treatment trends in the treated group are “equivalent” to
those in the control group. Here, “equivalence” means that rejection of
the null hypothesis implies that a function of the pre-treatment placebo
effects (maximum absolute, average or root mean squared value) does not
exceed a pre-specified threshold below which trend differences are
considered negligible. The package is based on the theory developed in
Dette & Schumann (2024).

The package contains the functions `maxEquivTest`

to
perform the testing procedure surrounding the maximum placebo
coefficient (see equation (3.1) of Dette & Schumann (2024)),
`meanEquivTest`

to perform the testing procedure surrounding
the mean placebo coefficient (see equation (3.2) of Dette & Schumann
(2024)) and
`rmsEquivTest`

to perform the testing procedure surrounding
the root mean squared placebo coefficient (see equation (3.3) and (3.4)
of Dette & Schumann (2024)).
Furthermore, the package contains the function
`sim_paneldata`

to simulate a paneldataset for such testing
purposes.

You can install the development version of `EquiTrends`

from GitHub
with:

```
# install.packages("devtools")
::install_github("TiesBos/EquiTrends") devtools
```

The `EquiTrends`

package contains a function to simulate
panel data, tailored to the Difference-in-Differences framework. The
function `sim_paneldata`

simulates a panel dataset with a
given number of individuals \(N\)
(`N`

), number of periods \(T+1\) (in the setting of this package,
indicating the number of pre-treatment periods. In
`sim_paneldata`

\(T+1\) is
referred to as `tt`

), number of covariates \(p\) (`p`

), and treatment
effects. Typically, period \(T+1\) is
referred to as the “base period”. The function also allows for the
simulation of heterogeneity in treatment effects (specified through
`eta`

) and time fixed effects (through `lambda`

).
Furthermore, the function allows for heteroscedasticty (specified
through the binary variable `het`

), serial correlation
(through the AR(1) coefficient `phi`

: \(u_{i,t} = \phi u_{i,t-1} + v_{i,t}\) where
\(v_{i,t}\) follows an i.i.d. \(N(0,\sigma^2)\) distribution and \(\sigma\) is specified through
`sd`

), and clustering in the model errors \(u_{i,t}\). The function returns a data
frame with the following columns: `ID`

(the cross-sectional
individual identifier), `period`

(the time identifier),
`Y`

(the dependent variable), `G`

(a binary vector
indicating if an individual receives treatment, indicated by 1, or not,
indicated by 0), and `X_1`

, `X_2`

, …,
`X_p`

(additional control variables). The construction of the
dependent variable follows the two-way fixed effect model, similar to
the model in equation (2.5) of Dette & Schumann (2024):

\[Y_{i,t} = \eta_i + \lambda_t + \sum_{l=1}^{T}{\beta_l}G_iD_l(t) + X_{1, i, t}\gamma_1+ \dots + X_{p,i,t}\gamma_p +u_{i,t} \quad \text{with} \ \ i=1,...,N, \ \ t=1,...,T+1\]

where \(D_l(t)\) is a dummy variable that equals 1 if \(t=l\) and 0 otherwise. The error-terms \(u_{i,t}\) are generated through a normal distribution with mean 0 and a variance-covariance structure depending on the user-specified parameters. In the following, the \(\beta_l\) coefficients are referred to as placebo coefficients, since they represent the difference in pre-trends between the treatment and control group before treatment has been assigned.

An example of the `sim_paneldata`

function is provided
below:

```
library(EquiTrends)
# Simulate a panel dataset with 500 individuals, 5 periods, 2 additional
# regressors, and a binary treatment variable without heteroscedasticity,
# serial correlation, and clustering. Furthermore, there are no fixed effects or
# pre-trends in the model (since all values in beta are 0).
<- sim_paneldata(N = 500, tt = 5, p = 2, beta = rep(0, 5),
sim_data gamma = rep(1, 2), het = 0, phi = 0, sd = 1,
burnins = 50)
head(sim_data)
#> ID period Y G X_1 X_2
#> 1 1 1 -0.8123777 0 -0.4407095 -0.655157012
#> 2 1 2 -1.8888861 0 -0.2212108 -0.349846262
#> 3 1 3 -2.2912561 0 -0.9741446 -0.000781637
#> 4 1 4 -0.5314161 0 0.2259398 -1.557426790
#> 5 1 5 -1.5528134 0 -0.1413597 -1.590501621
#> 6 2 1 1.5202663 1 -0.1386675 1.074761245
```

The `EquiTrends`

package contains functions to test for
equivalence of pre-trends in Difference-in-Differences estimation. The
functions `rmsEquivTest`

, `meanEquivTest`

, and
`maxEquivTest`

are used to test for equivalence of pre-trends
in Difference-in-Differences estimation using the placebo coefficients
\(\beta_{l} \ (l=1,...,T)\) estimates.
The functions are based on the work of Dette & Schumann (2024).

`rmsEquivTest`

function`rmsEquivTest`

implements the equivalence testing
procedure surrounding the root mean squared placebo coefficient as
described in section 4.2.3 of Dette & Schumann (2024). The
function tests the null hypothesis that the root mean squared placebo
coefficient is larger than or equal to a user-specified equivalence
threshold \(\delta\). That is, if

\[\beta_{RMS} = \sqrt{\frac{1}{T}\sum_{l=1}^{T} \beta_l^2},\]

the tested hypotheses can be represented as

\[H_0: \beta_{RMS} \geq \delta \quad \text{vs.} \quad H_1: \beta_{RMS} < \delta.\]

The null and alternative hypothesis can therefore be seen as
non-negligible and negligible differences in pre-trends, respectively.
The function returns an object of class `rmsEquivTest`

containing

`placebo_coefficients`

: A numeric vector of the estimated placebo coefficients,`rms_placebo_coefs`

: The root mean squared value of the placebo coefficients,`significance_level`

: The significance level of the test,`base_period`

: The base period used in the testing procedure,`num_individuals`

: The number of cross-sectional individuals in the panel used for testing,`num_periods`

: The number of pre-treatment periods in the panel used for testing (if the panel is unbalanced,`num_periods`

represents the range in the number of time periods covered by different individuals),`num_observations`

: The total number of observations in the panel used for testing,`is_panel_balanced`

: A logical value indicating whether the used panel is balanced,`equiv_threshold_specified`

: A logical value indicating whether an equivalence threshold was specified.- If
`equiv_threshold_specified = TRUE`

, then additionally:`rms_critical_value`

: The critical value at the chosen significance level,`reject_null_hypothesis`

: A logical value indicating whether to reject the null hypothesis,`equiv_threshold`

: The equivalence threshold specified.

- If
`equiv_threshold_specified = FALSE`

, then additionally:`minimum_equiv_threshold`

: The minimum equivalence threshold for which the null hypothesis of non-negligible trend-differences can be rejected.

One should note that rows containing `NA`

values are
removed from the panel before the testing procedure is performed.

Please be aware that the equivalence test based on the root mean squared placebo coefficient applies a randomization technique (as described by Dette & Schumann (2024)), leading to a stochastic critical value and minimum equivalence threshold. Therefore, the results may vary between different runs of the function.

```
# Perform the equivalence test using an equivalence threshold of 1 with periods
# 1-4 as pre-treatment periods based on the RMS testing procedure:
# - option 1: using column names in the panel
# One can use the names of the columns in the panel to specify the variables:
rmsEquivTest(Y = "Y", ID = "ID", G = "G", period = "period", X = c("X_1", "X_2"),
data = sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4)
#>
#> ==================================================
#> Equivalence Tests for Pre-trends in DiD Estimation
#> ==================================================
#> Type: Root Mean Squared Placebo Effect
#> Significance level: 0.05
#> Alternative hypothesis: the root mean squared placebo effect does not exceed the equivalence threshold of 1 .
#> ---
#> RMS Placebo Effect Simulated Crit. Val. Reject H0
#> 0.1835 0.9558 TRUE
#> ---
#> No. placebo coefficients estimated: 3
#> Base period: 4
#>
#> Balanced Panel:
#> + No. pre-treatment periods: 4
#> + No. individuals: 500
#> + Total no. observations: 2000
```

```
# - option 2: using column numbers in the panel
# Alternatively, one can use the column numbers in the panel to specify the variables:
rmsEquivTest(Y = 3, ID = 1, G = 4, period = 2, X = c(5, 6),
data = sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4)
# - option 3: using separate variables
# One can also use the variables directly without specifying the data variable:
<- sim_data$Y
data_Y <- sim_data$ID
data_ID <- sim_data$G
data_G <- sim_data$period
data_period <- cbind(sim_data$X_1, sim_data$X_2)
data_X
rmsEquivTest(Y = data_Y, ID = data_ID, G = data_G, period = data_period, X = data_X,
equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4)
```

The testing procedures can also be performed without specifying the equivalence threshold. Then, the minimum equivalence threshold is returned for which the null hypothesis of non-negligible trend-differences can be rejected. Again, the three possible ways of entering the data as above can be used.

```
rmsEquivTest(Y = "Y", ID = "ID", G = "G", period = "period", X = c("X_1", "X_2"),
data = sim_data, equiv_threshold = NULL, pretreatment_period = 1:4,
base_period = 4)
#>
#> ==================================================
#> Equivalence Tests for Pre-trends in DiD Estimation
#> ==================================================
#> Type: Root Mean Squared Placebo Effect
#> Significance level: 0.05
#> Alternative hypothesis: the root mean squared placebo effect does not exceed the equivalence threshold.
#> ---
#> RMS Placebo Effect Min. Equiv. Threshold
#> 0.1835 0.2558
#> ---
#> No. placebo coefficients estimated: 3
#> Base period: 4
#>
#> Balanced Panel:
#> + No. pre-treatment periods: 4
#> + No. individuals: 500
#> + Total no. observations: 2000
```

Finally, one should note that the test procedure also works for unbalanced panels.

```
# To illustrate this, we generate an unbalanced panel dataset by randomly selecting
# 70% of the observations from the balanced panel dataset:
<- sample(nrow(sim_data), 0.7*nrow(sim_data))
random_indices <- sim_data[random_indices, ]
unbalanced_sim_data # With Equivalence Threshold:
rmsEquivTest(Y = 3, ID = 1, G = 4, period = 2, X = c(5, 6),
data = unbalanced_sim_data, equiv_threshold = 1,
pretreatment_period = 1:4, base_period = 4)
# Without Equivalence Threshold:
rmsEquivTest(Y = 3, ID = 1, G = 4, period = 2, X = c(5, 6),
data = unbalanced_sim_data, equiv_threshold = NULL,
pretreatment_period = 1:4, base_period = 4)
```

`maxEquivTest`

functionThe `maxEquivTest`

function tests the null hypothesis that
the maximum placebo coefficient is larger than or equal to a
user-specified equivalence threshold \(\delta\). That is, if

\[\lVert\beta\rVert_\infty = \max_{l=1,...T} |\beta_l|,\]

the tested hypotheses can be represented as

\[H_0: \lVert\beta\rVert_\infty \geq \delta \quad \text{vs.} \quad H_1: \lVert\beta\rVert_\infty < \delta.\]

The null and alternative hypothesis can therefore be seen as non-negligible and negligible differences in pre-trends, respectively.

The function `maxEquivTest`

contains three testing
procedures for this test, as described in Section 4.2.1. of Dette &
Schumann (2024). The
function allows for the testing of the equivalence of pre-trends using a
bootstrap for spherical errors (`type = "Boot"`

), a wild
bootstrap for clustered standard errors (`type = "Wild"`

),
and an Intersection Union approach (`type = "IU"`

) that
rejects the null if all estimates for \(\beta_1,...,\beta_{T}\) are smaller than
their individual critical values. The function returns an object of
class `maxEquivTestBoot`

if `type = "Boot"`

or
`type = "Wild"`

or `maxEquivTestIU`

if
`type = "IU"`

. If no type is specified,
`maxEquivTest`

applies the Intersection Union procedure for
efficiency reasons.

`maxEquivTest`

function with
`type = "IU"`

Examples of implementing the Intersection unit test with different
possible variance-covariance matrices (required to perform the test) are
provided below (for more information on the possible variance-covariance
matrices, see the documentation of the `maxEquivTest`

function). If an equivalence threshold is supplied, the function will
test the previous hypothesis. If no equivalence threshold is supplied,
the function finds the minimum equivalence threshold for which the null
of non-negligible trend-differences can be reject using the Intersection
Union test. The function returns an object of class
`maxEquivTestIU`

containing the following information:

`placebo_coefficients`

: A numeric vector of the estimated placebo coefficients,`abs_placebo_coefficients`

: A numeric vector with the absolute values of estimated placebo coefficients,`placebo_coefficients_se`

: A numeric vector with the standard errors of the placebo coefficients,`significance_level`

: The chosen significance level of the test,`base_period`

: The base period used in the testing procedure,`placebo_names`

: The names corresponding to the placebo coefficients,`num_individuals`

: The number of cross-sectional individuals in the panel used for testing,`num_periods`

: The number of pre-treatment periods in the panel used for testing (if the panel is unbalanced,`num_periods`

represents the range in the number of time periods covered by different individuals),`num_observations`

: The number of observations in the panel used for testing,`is_panel_balanced`

: A logical value indicating whether the panel data is balanced,`equiv_threshold_specified`

: A logical value indicating whether an equivalence threshold was specified.- If
`equiv_threshold_specified = TRUE`

, then additionally:`IU_critical_values`

: A numeric vector with the individual critical values for each of the placebo coefficients,`reject_null_hypothesis`

: A logical value indicating whether the null hypothesis of negligible pre-trend differences can be rejected at the specified significance level,`equiv_threshold`

: The equivalence threshold employed.

- If
`equiv_threshold_specified = FALSE`

, then additionally:`minimum_equiv_thresholds`

: A numeric vector including for each placebo coefficient the minimum equivalence threshold for which the null hypothesis of negligible pre-trend differences can be rejected for the corresponding placebo coefficient individually,`minimum_equiv_threshold`

: A numeric scalar minimum equivalence threshold for which the null hypothesis of negligible pre-trend differences can be rejected for all placebo coefficients.

One should note that rows containing `NA`

values are
removed from the panel before the testing procedure is performed.

```
# Perform the test with equivalent threshold specified as 1 based on
# pre-treatment periods 1-4 and homoscedastic error-terms:
# To select variables, one can use the column names / numbers in the panel data
maxEquivTest(Y = "Y", ID = "ID", G = "G", period = 2, X= c(5,6),
data = sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, type = "IU")
#>
#> ==================================================
#> Equivalence Tests for Pre-trends in DiD Estimation
#> ==================================================
#> Type: Intersection Union
#> Alternative hypothesis: the maximum placebo effect does not exceed the equivalence threshold of 1 .
#> Reject null hypothesis: TRUE
#> ( Critical values are printed for the significance level: 0.05 )
#> ---
#> Abs. Estimate Std. Error Critical Value
#> 0.09848 0.1221 0.7992
#> 0.27253 0.1221 0.7992
#> 0.13041 0.1220 0.7993
#> ---
#> No. placebo coefficients estimated: 3
#> Base period: 4
#>
#> Balanced Panel:
#> + No. pre-treatment periods: 4
#> + No. individuals: 500
#> + Total no. observations: 2000
# Alternatively, one can enter the variables separately:
<- sim_data$Y
data_Y <- sim_data$ID
data_ID <- sim_data$G
data_G <- sim_data$period
data_period <- sim_data[, c(5, 6)]
data_X maxEquivTest(Y = data_Y, ID = data_ID, G = data_G, period = data_period, X = data_X,
equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, type = "IU")
#>
#> ==================================================
#> Equivalence Tests for Pre-trends in DiD Estimation
#> ==================================================
#> Type: Intersection Union
#> Alternative hypothesis: the maximum placebo effect does not exceed the equivalence threshold of 1 .
#> Reject null hypothesis: TRUE
#> ( Critical values are printed for the significance level: 0.05 )
#> ---
#> Abs. Estimate Std. Error Critical Value
#> 0.09848 0.1221 0.7992
#> 0.27253 0.1221 0.7992
#> 0.13041 0.1220 0.7993
#> ---
#> No. placebo coefficients estimated: 3
#> Base period: 4
#>
#> Balanced Panel:
#> + No. pre-treatment periods: 4
#> + No. individuals: 500
#> + Total no. observations: 2000
```

```
# Perform the test without specifying the equivalence threshold with heteroscedastic
# and autocorrelation robust variance-covariance matrix estimator:
maxEquivTest(Y = 3, ID = 1, G = 4, period = 2,
data = sim_data, equiv_threshold = NULL, pretreatment_period = 1:4,
base_period = 4, type = "IU", vcov = "HAC")
#>
#> ==================================================
#> Equivalence Tests for Pre-trends in DiD Estimation
#> ==================================================
#> Type: Intersection Union
#> Significance level: 0.05
#> Alternative hypothesis: the maximum placebo effect does not exceed the equivalence threshold.
#> Minimum equivalence threshold to accept the alternative: 0.4974
#> ---
#> Estimate Std. Error Minimum Equivalence Threshold
#> 0.1028 0.2172 0.4489
#> 0.1558 0.2088 0.4974
#> 0.1257 0.2131 0.4711
#> ---
#> No. placebo coefficients estimated: 3
#> Base period: 4
#>
#> Balanced Panel:
#> + No. pre-treatment periods: 4
#> + No. individuals: 500
#> + Total no. observations: 2000
```

```
# Perform the test without specifying the equivalence threshold with a custom
# variance-covariance matrix estimator:
<- function(x) {plm::vcovHC(x, method = "white1", type = "HC2")}
vcov_func maxEquivTest(Y = "Y", ID = "ID", G = "G", period = "period",
data = sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, type = "IU", vcov = vcov_func)
# Perform the test using clustered standard errors based on a vector indicating
# the cluster. For instance, two clusters with the following rule: all
# individuals with an ID below 250 are in the same cluster.
<- ifelse(sim_data$ID < 250, 1, 2)
cluster_ind maxEquivTest(Y = data_Y, ID = data_ID, G = data_G, period = data_period, X = data_X,
equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, type = "IU", vcov = "CL", cluster = cluster_ind)
#> Registered S3 method overwritten by 'clubSandwich':
#> method from
#> bread.mlm sandwich
```

Note that the testing procedure can also handle unbalanced panels.

```
# To illustrate this, we generate an unbalanced panel dataset by randomly selecting
# 70% of the observations from the balanced panel dataset:
<- sample(nrow(sim_data), 0.7*nrow(sim_data))
random_indices <- sim_data[random_indices, ]
unbalanced_sim_data maxEquivTest(Y = "Y", ID = "ID", G = "G", period = "period", X = c(5, 6),
data = unbalanced_sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, type = "IU", vcov = "HAC")
```

Examples of implementing the bootstrap based test are provided below.
For both `type = "Boot"`

and `type = "Wild"`

, an
equivalence threshold is required to perform the test. Furthermore, both
testing procedures return an object of class “maxEquivTestBoot”
containing

`placebo_coefficients`

: A numeric vector of the estimated placebo coefficients,`abs_placebo_coefficients`

: A numeric vector with the absolute values of estimated placebo coefficients,`max_abs_coefficient`

: The maximum absolute estimated placebo coefficient,`B`

: The number of bootstrap samples used to find the critical value,`significance_level`

: The chosen significance level of the test,`base_period`

: The base period used in the testing procedure,`placebo_names`

: The names corresponding to the placebo coefficients,`num_individuals`

: The number of cross-sectional individuals in the panel used for testing,`num_periods`

: The number of pre-treatment periods in the panel used for testing (if the panel is unbalanced,`num_periods`

represents the range in the number of time periods covered by different individuals),`num_observations`

: The total number of observations in the panel used for testing,`is_panel_balanced`

: A logical value indicating whether the panel data is balanced,`equiv_threshold_specified`

: A logical value indicating whether an equivalence threshold was specified.- If
`equiv_threshold_specified = TRUE`

, then additionally:`bootstrap_critical_value`

: The by bootstrap found critical value for the equivalence test based on the maximum absolute placebo coefficient,`reject_null_hypothesis`

: A logical value indicating whether the null hypothesis of negligible pre-trend differences can be rejected at the specified significance level,

- If
`equiv_threshold_specified = FALSE`

, then additionally:`minimum_equiv_threshold`

: The minimum equivalence threshold for which the null hypothesis of negligible pre-trend differences can be rejected for the bootstrap procedure.

One should note that rows containing `NA`

values are
removed from the panel before the testing procedure is performed.

On top of that, please be aware that the bootstrap procedures for the equivalence test based on the maximum absolute placebo coefficient apply a bootstrap procedure (as described by Dette & Schumann (2024)), leading to a stochastic critical value and minimum equivalence threshold. Therefore, the results may vary slightly between different runs of the function.

The bootstrap for spherical errors with 1000 bootstrap iterations:

```
# Perform the test with equivalence threshold specified as 1 based on
# pre-treatment periods 1:4 (with base period 4) with the general bootstrap procedure:
maxEquivTest(Y = "Y", ID = "ID", G = "G", period = "period",
data = sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, type = "Boot", B = 1000)
#>
#> ==================================================
#> Equivalence Tests for Pre-trends in DiD Estimation
#> ==================================================
#> Type: Bootstrap for Spherical Errors (Based on 1000 bootstrap samples)
#> Significance level: 0.05
#> Alternative hypothesis: the maximum placebo effect does not exceed the equivalence threshold of 1 .
#> ---
#> Max. Abs. Coefficient Bootstrap Critical Value Reject H0
#> 0.1558 0.6586 TRUE
#> ---
#> No. placebo coefficients estimated: 3
#> Base period: 4
#>
#> Balanced Panel:
#> + No. pre-treatment periods: 4
#> + No. individuals: 500
#> + Total no. observations: 2000
```

The Wild boostrap with 1000 bootstrap iterations:

```
# Perform the test with the equivalence threshold specified as 1 based on
# pre-treatment periods 1:4 (with base period 4) with the wild bootstrap procedure:
maxEquivTest(Y = "Y", ID = "ID", G = "G", period = "period",
data = sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, type = "Wild")
#>
#> ==================================================
#> Equivalence Tests for Pre-trends in DiD Estimation
#> ==================================================
#> Type: Cluster Wild Bootstrap (Based on 1000 bootstrap samples)
#> Significance level: 0.05
#> Alternative hypothesis: the maximum placebo effect does not exceed the equivalence threshold of 1 .
#> ---
#> Max. Abs. Coefficient Bootstrap Critical Value Reject H0
#> 0.1558 0.6642 TRUE
#> ---
#> No. placebo coefficients estimated: 3
#> Base period: 4
#>
#> Balanced Panel:
#> + No. pre-treatment periods: 4
#> + No. individuals: 500
#> + Total no. observations: 2000
```

The bootstrap procedures can handle unspecified equivalence thresholds:

```
maxEquivTest(Y = "Y", ID = "ID", G = "G", period = "period",
data = sim_data, equiv_threshold = NULL, pretreatment_period = 1:4,
base_period = 4, type = "Boot", B = 1000)
maxEquivTest(Y = "Y", ID = "ID", G = "G", period = "period",
data = sim_data, equiv_threshold = NULL, pretreatment_period = 1:4,
base_period = 4, type = "Wild", B = 1000)
```

The bootstrap procedures can handle unbalanced panels:

```
maxEquivTest(Y = "Y", ID = "ID", G = "G", period = "period",
data = unbalanced_sim_data, equiv_threshold = 1,
pretreatment_period = 1:4,
base_period = 4, type = "Boot", B = 1000)
maxEquivTest(Y = "Y", ID = "ID", G = "G", period = "period",
data = unbalanced_sim_data, equiv_threshold = 1,
pretreatment_period = 1:4,
base_period = 4, type = "Wild", B = 1000)
```

`meanEquivTest`

functionThe `meanEquivTest`

implements the equivalence testing
procedure surrounding the mean placebo coefficient, as described in
Section 4.2.2. of Dette & Schumann (2024). The
function tests the null hypothesis that the absolute mean placebo
coefficient is larger than or equal to a user-specified equivalence
threshold, \(\delta\). That is, if

\[\bar{\beta} = \frac{1}{T}\sum_{l=1}^{T} \beta_l,\]

the tested hypotheses can be represented as

\[H_0: |\bar{\beta}| \geq \delta \quad \text{vs.} \quad H_1: |\bar{\beta}| < \delta.\]

The null and alternative hypothesis can therefore be seen as
non-negligible and negligible differences in pre-trends, respectively.
Implementation of the test is similar to the `maxEquivTest`

function in terms of the possible variance-covariance matrices (for more
information on the possible variance-covariance matrices, see the
documentation of the `meanEquivTest`

function). The function
returns an object of class `meanEquivTest`

containing

`placebo_coefficients`

: A numeric vector of the estimated placebo coefficients,`abs_mean_placebo_coefs`

: The absolute value of the mean of the placebo coefficients,`var_mean_placebo_coef`

: The estimated variance of the mean placebo coefficient,`significance_level`

: The significance level of the test,`base_period`

: The base period used in the testing procedure,`num_individuals`

: The number of cross-sectional individuals in the panel used for testing,`num_periods`

: The number of pre-treatment periods in the panel used for testing (if the panel is unbalanced,`num_periods`

represents the range in the number of time periods covered by different individuals)`num_observations`

: The total number of observations in the panel used for testing,`is_panel_balanced`

: A logical value indicating whether the panel is balanced,`equiv_threshold_specified`

: A logical value indicating whether an equivalence threshold was specified.- If
`equiv_threshold_specified = TRUE`

, then additionally:`mean_critical_value`

: The critical value at the chosen significance level,`p_value`

: The p-value of the test,`reject_null_hypothesis`

: A logical value indicating whether to reject the null hypothesis,`equiv_threshold`

: The equivalence threshold specified.

- If
`equiv_threshold_specified = FALSE`

, then additionally:`minimum_equiv_threshold`

: The minimum equivalence threshold for which the null hypothesis of non-negligible (based on the equivalence threshold) trend-differences can be rejected.

`NA`

values are
removed from the panel before the testing procedure is performed.

```
# Perform the test with equivalent threshold specified as 1 based on
# pre-treatment periods 1-4 and assuming homoscedastic error-terms:
# To select variables, one can use the column names / column numbers in the panel data:
meanEquivTest(Y = "Y", ID = "ID", G = "G", period = 2, X = c(5, 6),
data = sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4)
#>
#> ==================================================
#> Equivalence Tests for Pre-trends in DiD Estimation
#> ==================================================
#> Type: Mean Placebo Effect
#> Alternative hypothesis: the mean placebo effect does not exceed the equivalence threshold of 1 .
#> ---
#> Abs. Mean Placebo Effect Std. Error p-value Reject H0
#> 0.1671 0.09965 <2e-16 TRUE
#> ---
#> No. placebo coefficients estimated: 3
#> Base period: 4
#>
#> Balanced Panel:
#> + No. pre-treatment periods: 4
#> + No. individuals: 500
#> + Total no. observations: 2000
```

```
# Alternatively, one can use separate variables:
<- sim_data$Y
data_Y <- sim_data$ID
data_ID <- sim_data$G
data_G <- sim_data$period
data_period <- sim_data[, c(5, 6)]
data_X meanEquivTest(Y = data_Y, ID = data_ID, G = data_G, period = data_period, X = data_X,
equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4)
```

```
# Perform the test with a heteroscedastic and autocorrelation robust
# variance-covariance matrix estimator, and without specifying the equivalence threshold:
meanEquivTest(Y = "Y", ID = "ID", G = "G", period = "period", X = c(5, 6),
data = sim_data, equiv_threshold = NULL, pretreatment_period = 1:4,
base_period = 4, vcov = "HAC")
#>
#> ==================================================
#> Equivalence Tests for Pre-trends in DiD Estimation
#> ==================================================
#> Type: Mean Placebo Effect
#> Significance level: 0.05
#> Alternative hypothesis: the mean placebo effect does not exceed the equivalence threshold.
#> ---
#> Abs. Mean Placebo Effect Std. Error Min. Equiv. Threshold
#> 0.1671 0.09691 0.3265
#> ---
#> No. placebo coefficients estimated: 3
#> Base period: 4
#>
#> Balanced Panel:
#> + No. pre-treatment periods: 4
#> + No. individuals: 500
#> + Total no. observations: 2000
```

```
# Perform the test with an equivalence threshold of 1 and a custom
# variance-covariance matrix estimator:
<- function(x) {plm::vcovHC(x, method = "white1", type = "HC2")}
vcov_func meanEquivTest(Y = "Y", ID = "ID", G = "G", period = "period",
data = sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, vcov = vcov_func)
# Perform the test using clustered standard errors based on a vector indicating
# the cluster. For instance, two clusters with the following rule: all
# individuals with an ID below 250 are in the same cluster:
<- ifelse(sim_data$ID < 250, 1, 2)
cluster_ind meanEquivTest(Y = data_Y, ID = data_ID, G = data_G, period = data_period, X = data_X,
equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, vcov = "CL", cluster = cluster_ind)
```

Note that the testing procedure can also handle unbalanced panels:

```
# Finally, one should note that the test procedure also works for unbalanced panels.
# To illustrate this, we generate an unbalanced panel dataset by randomly selecting
# 70% of the observations from the balanced panel dataset:
<- sample(nrow(sim_data), 0.7*nrow(sim_data))
random_indices <- sim_data[random_indices, ]
unbalanced_sim_data meanEquivTest(Y = "Y", ID = "ID", G = "G", period = "period", X = c(5, 6),
data = unbalanced_sim_data, equiv_threshold = 1, pretreatment_period = 1:4,
base_period = 4, vcov = "HAC")
```

Dette H., & Schumann M. (2024). “Testing for Equivalence of
Pre-Trends in Difference-in-Differences Estimation.” *Journal of
Business & Economic Statistics*, 1–13. DOI: 10.1080/07350015.2024.2308121