xtusreg: Software for dynamic panel regression under irregular time spacing

Abstract

We introduce a new command, xtusreg, that estimates parameters of fixed-effects dynamic panel regression models under unequal time spacing. After reviewing the method, we examine the finite-sample performance of the command using simulated data. We also illustrate the command with the National Longitudinal Survey Original Cohorts: Older Men, whose personal interviews took place in the unequally spaced years of 1966, 1967, 1969, 1971, 1976, 1981, and 1990. The methods underlying xtusreg are those discussed by Sasaki and Xin (2017, Journal of Econometrics 196: 320–330).

Keywords

st0690 xtusreg dynamic panel regression fixed effect unequal time spacing

1 Introduction

Conventional methods to fit fixed-effects dynamic panel regression models require that a researcher observe three consecutive time periods or two pairs of two consecutive time periods. On the other hand, there are some panel datasets with irregularly spaced time intervals that do not satisfy these conventional time-spacing requirements for identification and estimation. For example, personal interviews were conducted in 1966, 1967, 1969, 1971, 1976, 1981, and 1990 for the National Longitudinal Survey (NLS) Original Cohorts: Older Men, and there are neither three consecutive time periods nor two pairs of two consecutive time periods in this list of years. Other examples with irregularly spaced panel data from the United States include Current Population Survey, Early Childhood Longitudinal Survey-K, NLS of Youth 1979, and Panel Survey of Income Dynamics—see section 2 for more details about specific survey periods of these datasets.

Even if panel data exhibit such irregular and unequal time spacing, Sasaki and Xin (2017) show that the fixed-effects dynamic panel regression parameters can be still identified as long as “two pairs of two consecutive time gaps” are available, which generalizes the conventional requirement of “two pairs of two consecutive time periods”. In the NLS Original Cohorts: Older Men, for instance, there is a time gap of 0 between 1966 and 1966, a time gap of 1 between 1966 and 1967, a time gap of 2 between 1967 and 1969, and a time gap of 3 between 1966 and 1969. Thus, there are two pairs, (0, 1) and (2, 3), of consecutive time gaps in this panel dataset. This requirement is also satisfied by the list of the aforementioned panel datasets: Current Population Survey, Early Childhood Longitudinal Survey-K, NLS of Youth 1979, and Panel Survey of Income Dynamics.

In this article, we introduce the xtusreg command, which executes estimation and inference for fixed-effects dynamic panel regression under irregular time spacing based on the method proposed in Sasaki and Xin (2017). After reviewing the method in section 2, we formally introduce the command in section 3. This is followed by simulation studies in section 4 and a real data illustration in section 5.

2 Review of the method

This section reviews a part of the method of identification and estimation of fixed-effects dynamic panel regression models under unequal time spacing proposed by Sasaki and Xin (2017). Consider the model

y_{i t} = γ y_{i, t - 1} + β x_{i t} + α_{i} + ε_{i t}

where y_it denotes an observed state variable, x_it denotes an observed covariate, α_i denotes an unobserved individual fixed effect, and ε_it denotes an unobserved idiosyncratic shock. A researcher is often interested in the autoregressive parameter γ and the regression parameter β in such a model. The following two examples illustrate concrete regressions in which economists would be interested.

Example 1. The seminal article by Ashenfelter (1978) proposes models of the form (1) as “earnings generating functions” to analyze the effect β of job training x_it on the subsequent earnings y_it , accounting for the dynamics of y_it . With equally spaced panel datasets, such as the one used in Ashenfelter (1978), standard methods such as xtabond may be used to estimate such an effect of interest. However, with unequally spaced panel datasets, such as the NLS Original Cohorts: Older Men (which also contains information about earnings and exposure to job training), those existing commands incur biased estimates in general.

Example 2. In economics of education, dynamic models of the form (1) are often used as “value-added models” to analyze persistent impact β of intervention x_it on the human capital y_it of children. The Early Childhood Longitudinal Survey-K is one of the most relevant datasets to study the value-added models, but this dataset is unequally spaced as T = 3, 4, 8, 12, and 18. With the unequal spacing, standard commands such as xtabond incur biased estimates of the value-added models in general.

For convenience of writing, we first define a few shorthand and auxiliary notations. Write E_i (·) := E(·| α_i ) for the expectation conditional on individual i’s specific fixed effect α_i . With this notation, we in turn define the auxiliary random variables

\begin{array}{r} Z_{i τ} : = E_{i} (y_{i t} y_{i, t + τ}) & z_{i τ} : = E_{i} (x_{i t} x_{i, t + τ}) \\ ζ_{i τ} : = E_{i} (y_{i t} x_{i, t + τ}) & ζ_{i, - τ} : = E_{i} (x_{i t} y_{i, t + τ}) \end{array}

for each time t and time gap τ. Furthermore, we denote their cross-sectional means by Z_τ := E(Z_i _τ ), z_τ := E(z_i _τ ), ζ_τ := E(ζ_i _τ ), and ζ_−τ := E(ζ_i, _−τ ).

Let T be the set of unequally spaced time periods t for which a researcher observes the panel data ${(y_{i t}, x_{i t})}_{i = 1}^{N}$ . We define the set of survey gaps by

T = \{|t_{1} - t_{2}| : t_{1}, t_{2} \in T\}

We also define the set of gap-associated survey years by

T (τ) = {t \in T : t + τ \in T}

for each time gap τ ∊ T and let T(τ) = ∅ if τ ∉ T. For the NLS Original Cohorts Older Men, for example, personal interviews were conducted in 1966, 1967, 1969, 1971, 1976, 1981, and 1990. In this case, we can write T = {1966, 1967, 1969, 1971, 1976, 1981, 1990}, T = {0, 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 19, 21, 23, 24}, T(0) = T, T(1) = {1966}, T(2) = {1967, 1969}, T(3) = {1966}, T(4) = {1967}, T(5) = {1966, 1971, 1976}, and so on.

If panel data with unequal spacing have T(1) ≠ ∅, T(Δt) ≠ ∅, and T(Δt + 1) ≠ ∅ for some gap Δt in the set of natural numbers, then we call its spacing structure the “U.S. spacing” (compare Sasaki and Xin [2017, def. 2]). For instance, one can verify that each of the following datasets has this U.S. spacing structure:

See example 2 in Sasaki and Xin (2017) for detailed discussions.

Under the U.S. spacing, Sasaki and Xin [2017, corollary 1 (ii)] show that (γ, β)′ can be identified by

(\begin{array}{l} γ \\ β \end{array}) = \frac{1}{| Δ |} (\begin{matrix} (z_{Δ t + 1} - z_{1}) (Z_{Δ t + 1} - Z_{1}) + (ζ_{1} - ζ_{Δ t + 1}) (ζ_{- (Δ t + 1)} - ζ_{- 1}) \\ (ζ_{0} - ζ_{- Δ t}) (Z_{Δ t + 1} - Z_{1}) + (Z_{Δ t} - Z_{0}) (ζ_{- (Δ t + 1)} - ζ_{- 1}) \end{matrix})

with

| Δ | : = (Z_{Δ t} - Z_{0}) (z_{Δ t + 1} - z_{1}) - (ζ_{- Δ t} - ζ_{0}) (ζ_{Δ t + 1} - ζ_{1})

provided that | Δ| ≠ 0 holds.

Given the identification result, one can now take the sample counterpart of it to obtain an estimator. The auxiliary random variables, Z_τ , z_τ , ζ_τ , and ζ_−τ , may be estimated, respectively, by

{\hat{Z}}_{τ} = \frac{1}{N} \sum_{i = 1}^{N} {\bar{Z}}_{i τ} {\hat{z}}_{τ} = \frac{1}{N} \sum_{i = 1}^{N} {\bar{z}}_{i τ} {\hat{ζ}}_{τ} = \frac{1}{N} \sum_{i = 1}^{N} {\bar{ζ}}_{i τ} {\hat{ζ}}_{- τ} = \frac{1}{N} \sum_{i = 1}^{N} {\bar{ζ}}_{i, - τ}

for any ${\bar{Z}}_{i τ}$ , ${\bar{z}}_{i τ}$ , ${\bar{ζ}}_{i τ}$ , and ${\bar{ζ}}_{i, - τ}$ such that $Z_{τ} = E ({\bar{Z}}_{i τ})$ , $z_{τ} = E ({\bar{z}}_{i τ})$ , $ζ_{τ} = E ({\bar{ζ}}_{i τ})$ , and $ζ_{- τ} = E ({\bar{ζ}}_{i, - τ})$ . Under a time-invariant moment assumption (assumption 2 of Sasaki and Xin [2017]), ${\bar{Z}}_{i τ}$ , ${\bar{z}}_{i τ}$ , ${\bar{ζ}}_{i τ}$ , and ${\bar{ζ}}_{i, - τ}$ can be constructed by any linear combination of the form

\begin{array}{r} {\bar{Z}}_{i τ} = \sum_{t \in T (τ)} a_{t}^{τ} y_{i t} y_{i, t + τ} & {\bar{z}}_{i τ} = \sum_{t \in T (τ)} b_{t}^{τ} x_{i t} x_{i, t + τ} \\ {\bar{ζ}}_{i τ} = \sum_{t \in T (τ)} c_{t}^{τ} y_{i t} x_{i, t + τ} & {\bar{ζ}}_{i, - τ} = \sum_{t \in T (τ)} d_{t}^{τ} x_{i t} y_{i, t + τ} \end{array}

where $a^{τ} = {(a_{t}^{τ})}_{t \in T (τ)}, b^{τ} = {(b_{t}^{τ})}_{t \in T (τ)}, c^{τ} = {(c_{t}^{τ})}_{t \in T (τ)},$ and $d^{τ} = {(d_{t}^{τ})}_{t \in T (τ)}$ satisfy, respectively, $\sum_{t \in T (τ)} a_{t}^{τ} = 1, \sum_{t \in T (τ)} b_{t}^{τ} = 1, \sum_{t \in T (τ)} c_{t}^{τ} = 1,$ and $\sum_{t \in T (τ)} d_{t}^{τ} = 1$ . The xtusreg command uses the simple arithmetic means. The sample counterpart of the identifying formula yields the explicit estimator

(\begin{array}{l} \hat{γ} \\ \hat{β} \end{array}) = \frac{1}{| \hat{Δ} |} (\begin{matrix} ({\hat{z}}_{Δ t + 1} - {\hat{z}}_{1}) ({\hat{Z}}_{Δ t + 1} - {\hat{Z}}_{1}) + ({\hat{ζ}}_{1} - {\hat{ζ}}_{Δ t + 1}) ({\hat{ζ}}_{- (Δ t + 1)} - {\hat{ζ}}_{- 1}) \\ ({\hat{ζ}}_{0} - {\hat{ζ}}_{- Δ t}) ({\hat{Z}}_{Δ t + 1} - {\hat{Z}}_{1}) + ({\hat{Z}}_{Δ t} - {\hat{Z}}_{0}) ({\hat{ζ}}_{- (Δ t + 1)} - {\hat{ζ}}_{- 1}) \end{matrix})

where

| \hat{Δ} | = ({\hat{Z}}_{Δ t} - {\hat{Z}}_{0}) ({\hat{z}}_{Δ t + 1} - {\hat{z}}_{1}) - ({\hat{ζ}}_{- (Δ t)} - {\hat{ζ}}_{0}) ({\hat{ζ}}_{Δ t + 1} - {\hat{ζ}}_{1})

While the above procedure focuses on the just-identified case for the sake of clarity, the generic generalized method of moments (GMM) restriction is provided by

E \{g (w_{i}, θ_{0})\} = 0

where θ = (γ, β), w _i = (x_it, y_it ) _t _∊ _T , and the rows of the moment function g ( w _i, θ) consist of

\begin{array}{l} g_{1, t t^{'} t^{''} t^{'''}} (w_{i}, θ) = (y_{i, t^{'''}} y_{i, t^{'''}} + Δ t + 1 - y_{i, t^{'}} y_{i, t^{'}} + 1) \\ - γ (y_{i, t^{''}} y_{i, t^{''}} + Δ t - y_{i t} y_{i t}) - β (y_{i, t^{'''}} x_{i, t^{'''}} + Δ t + 1 - y_{i, t^{'}} x_{i, t^{'} + 1}) \\ g_{2, t t^{'} t^{''} t^{'''}} (w_{i}, θ) = (x_{i, t^{'''}} y_{i, t^{'''}} + Δ t + 1 - x_{i, t^{'}} y_{i, t^{'} + 1}) \\ - γ (x_{i, t^{''}} y_{i, t^{''} + Δ t} - x_{i t} y_{i t}) - β (x_{i, t^{'''}} x_{i, t^{'''} + Δ t + 1} - x_{i, t^{'}} x_{i, t^{'} + 1}) \end{array}

for all (t, t^′, t′′, t′′′) ∊ T(0) × T(1) × T(Δt) × T(Δt + 1). The xtusreg command implements the GMM based on these moment restrictions, so it can handle both the just-identified and overidentified cases. Specifically, the estimator is given by

\hat{θ} = \arg \min_{θ \in Θ} {\{\frac{1}{N} \sum_{i = 1}^{N} g (w_{i}, θ)\}}^{'} W_{N} \{\frac{1}{N} \sum_{i = 1}^{N} g (w_{i}, θ)\}

where W _N is a weighting matrix. The xtusreg command uses the identity matrix in the first step and uses the estimated optimal weighting matrix in the second step. In the just-identified case, the GMM estimator coincides with the closed-form estimator (2). Even with this GMM framework, we emphasize that it is still necessary for panel data to exhibit the U.S. spacing in order for the command to run.

The variance matrix of the GMM estimator is calculated based on the asymptotic normality

\sqrt{N} (\hat{θ} - θ_{0}) \overset{d}{\to} N \{0, {(G^{'} W G)}^{- 1} G^{'} W S W W {(G^{'} W G)}^{- 1}\}

where S is the variance matrix of g ( w _i, θ₀) and G is given by

G = E [\begin{matrix} ⋮ & ⋮ \\ - (y_{i, t^{''}} y_{i, t^{''} + Δ t} - y_{i t} y_{i t}) & - (y_{i, t^{''}} x_{i, t^{'''} + Δ t + 1} - y_{i, t^{'}} x_{i, t^{'} + 1}) \\ ⋮ & ⋮ \\ - (x_{i, t^{''}} y_{i, t^{''} + Δ t} - x_{i t} y_{i t}) & - (x_{i, t^{''}} x_{i, t^{'''} + Δ t + 1} - x_{i, t^{'}} x_{i, t^{'} + 1}) \\ ⋮ & ⋮ \end{matrix}]

See theorem 2 of Sasaki and Xin (2017).

In addition to the U.S. spacing, there is another spacing pattern, called the “U.K.spacing” (Sasaki and Xin 2017, ex. 1). However, the xtusreg command introduced in this article does not handle the U.K. spacing, because it tends to incur a weak identification as reported in Sasaki and Xin (2017, sec. 5). A weak-identification–robust approach to inference is recommended under the U.K. spacing, and this suggests a direction for future research in development of a Stata program.

The xtusreg command introduced in this article does not use all the combinations of two pairs of two consecutive time gaps. Instead, it uses two pairs of the two smallest consecutive time gaps. This is because larger time gaps tend to incur larger finite-sample biases in practice.

Before forming the moment restrictions presented above, the xtusreg command introduced in this article automatically drops observations with missing values. Further, it uses the rectangle of a balanced portion of the panel data.

3 The xtusreg command

3.1 Syntax

The syntax of the xtusreg command is as follows:

xtusreg depvar [indepvars] [if] [in] [, onestep stationary sweight(var) gamma(real) beta(real)]

Here depvar stands for the dependent variable y, and indepvars include independent variables x. Exactly one depvar variable should be included to run the command, while indepvars are optional and may include multiple variables.

3.2 Options

The xtusreg command has five options.

onestep sets an indicator for the one-step GMM estimation. By default, two-step efficient GMM estimation is set. This option will not make a difference in the results if parameters are just identified.

stationary sets an indicator for not executing the location-scale normalization of variables. By default, implementation of the location-scale normalization is set. Appendix C.1 of Sasaki and Xin (2017) recommends the implementation of the location-scale normalization under nonstationary distributions.

sweight(var) sets sampling weights. By default, uniform weights for all the observations are set.

gamma(real) sets the initial value of the autoregressive coefficient for a numerical optimization in the GMM estimation. The default is gamma(0).

beta(real) sets the initial value of the regression coefficients for a numerical optimization in the GMM estimation. The default is beta(0).

3.3 Stored results

xtusreg stores the following in e() :

4 Simulation studies

In this section, we present the finite sample performance of the xtusreg command using simulated data. We provide do-files to implement all the simulation results presented below.

We generate data following the time-spacing pattern of the NLS Original Cohorts: Older Men. First, we consider a simple data-generating process with just a state variable y and without an observed predictor x. Independently generate

\begin{array}{r} y_{i 1} ~ N (0, 1^{2}) & α_{i} ~ N (0, 1^{2}) \end{array}

for individuals i = 1,…, N, where N = 1000. For the subsequent time periods t = 2,…, 70, we iteratively and autoregressively generate

\begin{array}{r} y_{i t} = γ y_{i, t - 1} + α_{i} + ε_{i t} & ε_{i t} ~ N (0, 1^{2}) \end{array}

where we set γ = 0.5 for the autoregressive coefficient. After we have accumulated the whole data [(y_it ) : i ∊ {1,…, N}, t ∊ {1,…, 70}] for 70 periods, we drop all the time periods except for t = 1966, 1967, and 1969. This leaves us with [(y_it ) : i ∊ {1,…, N}, t ∊ {1966, 1967, 1969}], similarly to the first three survey years for the NLS Original Cohorts: Older Men. The first row of table 1 shows simulation results based on 2,000 Monte Carlo iterations.

Table 1.

Baseline simulation results

Unequally spaced	γ
time periods	Bias	SD	RMSE	95%
{1966, 1967, 1969}	0.004	0.093	0.093	0.941
{1966, 1967, 1970}	0.000	0.045	0.045	0.937
{1966, 1967, 1971}	−0.000	0.043	0.043	0.946

The displayed statistics include the bias, standard deviation (SD), root mean squared error (RMSE), and 95% coverage frequency for the parameter, γ. The bias, SD, and RMSE are all small relative to the magnitude of the parameter value γ = 0.5, and the coverage frequencies by the 95% confidence interval are close to the nominal probability of 0.95. We repeat this simulation analysis using extended lists of unequally spaced time periods. The results based on T = {1966, 1967, 1970} are displayed in the second row in table 1. The results based on T = {1966, 1967, 1971} are displayed in the third row in table 1. The results are very similar to those found in the first row, and therefore the same conclusion applies.

Next, we consider a data-generating process with an observed predictor x as well as the state variable y. Independently generate

\begin{array}{r} y_{i 1} ~ N (0, 1^{2}) & x_{i 1} ~ N (0, 1^{2}) & α_{i} ~ N (0, {0.5}^{2}) \end{array}

for individuals i = 1,…, N, where N = 1000. For the subsequent time periods t = 2,…, 70, we iteratively and autoregressively generate

\begin{array}{l} y_{i t} = γ y_{i, t - 1} + β x_{i t} + α_{i} + ε_{i t} & ε_{i t} ~ N (0, 1^{2}) \\ x_{i t} = ρ x_{i, t - 1} + α_{i} + η_{i t} & η_{i t} ~ N (0, 5^{2}) \end{array}

where we set γ = β = ρ = 0.5. Again, we drop all the time periods except for t = 1966, 1967, and 1969, leaving us with [(y_it, x_it ) : i ∊ {1,…, N}, t ∊ {1966, 1967, 1969}].

The first row of table 2 shows simulation results based on 2,000 Monte Carlo iterations. We repeat this simulation analysis using extended lists of unequally spaced time periods. The results based on T = {1966, 1967, 1970} are displayed in the second row in table 2. The results based on T = {1966, 1967, 1971} are displayed in the third row in table 2.

Table 2.

Simulation results for a model with a covariate

Unequally spaced	γ				β
timeperiods	Bias	SD	RMSE	95%	Bias	SD	RMSE	95%
{1966, 1967, 1969}	−0.003	0.115	0.115	0.920	0.011	0.083	0.084	0.905
{1966, 1967, 1970}	−0.009	0.070	0.070	0.913	0.019	0.082	0.084	0.934
{1966, 1967, 1971}	−0.009	0.066	0.067	0.924	0.021	0.090	0.092	0.932

These unequally spaced time periods cannot be handled by conventional commands, such as xtabond, for fixed-effects dynamic panel regressions. On the other hand, the simulation results demonstrate that xtusreg can still produce estimates along with their valid standard errors.

5 Illustration of the command

In this section, we illustrate the xtusreg command with the real data of NLS Original Cohorts: Older Men.

In the software package, a subsample of this dataset can be loaded by the following command line:

These data contain six variables: id, year, logincome, age, edu, and white. We first set the individual and time indices to refer to id and year, respectively, by the following command line:

Having set the panel-data structure, we first run the simple autoregression of y = logincome by the following line:

The estimate of the autoregressive parameter γ is a significant positive below one value, implying that the log income is positively autocorrelated and follows a stationary process. (Note that income was received in the years {1965, 1966, 1968}, while personal interviews were conducted in years {1966, 1967, 1969}.)

We next include age as a control x by the following line:

The inclusion of the control affects the point estimate of the autoregressive parameter γ, but the level of statistical significance is larger than before. Furthermore, the coefficient β of age is significant.

The dataset has two remaining candidates, white and edu, of controls x. However, because both of these two variables are generally time invariant, the parameters of fixed-effects panel regression models may not be identified if these variables are included as controls. (This also applies to conventional commands to estimate fixed-effects panel regressions.) To account for the observed demographic heterogeneity, white, one can instead run a regression for each of the ethnic categories.

The autoregressive parameter γ is larger for white individuals than for nonwhite individuals, implying that log income is more persistent for white individuals than for nonwhite individuals.

Finally, we also run the fixed-effects dynamic panel regression controlling for edu. Similarly to white, we may not include this time-invariant variable edu as a control x, and hence we run a regression for the following two education level categories:

The autoregressive parameter γ is larger for those individuals with 12 years of education or higher, implying that log income is more persistent for this group of individuals.

Existing commands could not have produced these results, because of the irregular time spacing of the NLS Original Cohorts: Older Men. The new command, xtusreg, allows researchers to make new discoveries with these types of panel datasets, to which existing commands did not apply.

6 Conclusion

Conventional methods to fit fixed-effects dynamic panel regression models require an observation of three consecutive time periods or two pairs of two consecutive time periods. Sasaki and Xin (2017) show that the availability of “two pairs of two consecutive time gaps”, which generalizes the conventional requirement of “two pairs of two consecutive time periods”, suffices for the identification of the model parameters. In this article, we introduced the xtusreg command, which executes estimation and inference for fixed-effects dynamic panel regression under unequal time spacing based on the methods proposed in Sasaki and Xin (2017). The command was illustrated with an analysis of income dynamics using the NLS Original Cohorts: Older Men, for which personal interviews were conducted in 1966, 1967, 1969, and so on.

Finally, we discussed a limitation of the xtusreg command and a direction for future research. The current command accommodates only a certain pattern of unequal time spacing in panel data, namely, the U.S. spacing (Sasaki and Xin 2017, def. 2). Another important class of spacing pattern, called U.K. spacing (Sasaki and Xin 2017, ex. 1), tends to incur a weak identification as reported in Sasaki and Xin (2017, sec. 5). Hence, an implementation of the standard estimation and inference is not preferable under the U.K. spacing. An alternative is to conduct inference robust to weak identification. We leave development of a command to meet this goal for future research.

7 Programs and supplemental materials

Supplemental Material, sj-zip-1-stj-10.1177_1536867X221124567 - xtusreg: Software for dynamic panel regression under irregular time spacing

Supplemental Material, sj-zip-1-stj-10.1177_1536867X221124567 for xtusreg: Software for dynamic panel regression under irregular time spacing by Yuya Sasaki and Yi Xin in The Stata Journal

Footnotes

7 Programs and supplemental materials

To install a snapshot of the corresponding software files as they existed at the time of publication of this article, type

xtusreg can also be downloaded from the Statistical Software Components Archive by typing

References

Ashenfelter

1978. Estimating the effect of training programs on earnings. Review of Economics and Statistics 60: 47–57. https://doi.org/10.2307/1924332.

Sasaki

Xin

2017. Unequal spacing in dynamic panel data: Identification and estimation. Journal of Econometrics 196: 320–330. https://doi.org/10.1016/j.jeconom.2016.10.002.