11 Dynamic Panel Data

11.1 When Is It a Problem

Model setup: \[y_{i,t} = \gamma y_{i, t-1} + X_{i,t}' \beta + C_i + \epsilon_{i,t} \tag{11.1}\]

Pooled regression is biased and inconsistent. The problem is:

\[Cov( y_{i,t-1}, (C_i + \epsilon_{i,t})) \approx \frac{\sigma_c^2}{(1-\gamma)} \tag{11.2}\]

Random effects model is biased and inconsistent, for the same reason that $y_{i,t-1}$ is necessarily correlated with $C_i$, which is part of the composite error term in the random effect model.

Fixed effect model setup:

\[ y_{i,t} - \bar y_i = (X_{i,t} - \bar X_i)' \beta + \gamma (y_{i, t-1} - \overline{y_{i,-1}}) + (\epsilon_{i,t} - \bar \epsilon_i) \tag{11.3}\]

where $\overline{y_{i,-1}}$ is the unit mean of the lagged dependent variable. This is not exactly $\bar y_i$ (the mean of current $y_{i,t}$). Over the usable periods the two are $(y_{i2}+\cdots+y_{iT})/(T-1)$ and $(y_{i1}+\cdots+y_{i,T-1})/(T-1)$, so their difference is exactly

\[\frac{y_{iT}-y_{i1}}{T-1}, \tag{11.4}\]

the endpoints and nothing else. Balance does not make that vanish – only $T$ growing does, under stability. The distinction is asymptotically negligible and finite-sample real.

Nickell (1981) shows that (see also Anderson and Hsiao 1981)

\[Cov( (y_{i,t-1}- \overline{y_{i,-1}}), (\epsilon_{i,t} - \bar \epsilon_i )) \approx -\frac{\sigma_\epsilon^2}{T (1-\gamma)^2} [\frac{(T-1)-T \gamma + \gamma^T}{T}] \tag{11.5}\]

which indicates that the demeaned lagged $y$ is correlated with the demeaned error term. The correlation may be large if $T$ is small, but when $T$ is big, then the correlation goes down to near zero. When $T$ is big, we don’t have a problem with fixed effect model.

11.2 How Big Is the Bias

When $T$ is small, which is typically the case in many microeconomic settings, then we say the fixed effect model with a lagged dependent variable is inconsistent. But how bad is it?

The limit of $\hat \gamma - \gamma$ is approximately $-\frac{(1+\gamma)}{T-1}$. When $T=10$ and $\gamma=.5$, the bias is about $-0.167$ which is $1/3$ of the true value.

But a simulation study (Judson and Owen 1999) shows that the bias can be as big as $20\%$ of the true coefficient even when $T=30$.

11.3 Anderson and Hsiao estimator

Anderson and Hsiao (1981) suggested looking at the first difference estimator:

\[ y_{i,t} - y_{i,t-1} = (X_{i,t} - X_{i,t-1})' \beta + \gamma (y_{i,t-1} - y_{i,t-2}) + (\epsilon_{i,t} - \epsilon_{i,t-1}) \tag{11.6}\]

\[ \Delta y_{i,t}= \Delta X_{i,t} \beta + \gamma \Delta y_{i,t-1} + \Delta \epsilon_{i,t} \tag{11.7}\]

This does not solve the endogeneity problem, since $\Delta y_{i,t-1}$ is still correlated with the error term. AH’s idea is to instrument $\Delta y_{i,t-1}$ with the past level $y_{i,t-2}$, or past difference $y_{i,t-2}-y_{i,t-3}$. This estimator is consistent, since neither of these instruments is correlated with $\Delta \epsilon_{i,t}$ – provided the idiosyncratic error is serially uncorrelated. That proviso carries the argument. If $\epsilon_{i,t}$ is itself AR(1), then $y_{i,t-2}$ is correlated with $\Delta \epsilon_{i,t}$, the instrument fails, and one has to reach further back.

The assumption is testable, which is the point of the Arellano-Bond test for serial correlation in the differenced residuals. First-order correlation is expected by construction, since $\Delta \epsilon_{i,t}$ and $\Delta \epsilon_{i,t-1}$ share $\epsilon_{i,t-1}$; it is the second-order test that carries information. Rejecting AR(2) says the lag being used as an instrument does not sit far enough back.

The other regressors need classifying too, because that is what decides how they are instrumented. Strictly exogenous $X$ can serve as its own instrument. Predetermined $X$ – correlated with past shocks but not with current or future ones – is instrumented by its own lags from $t-1$ backwards. Endogenous $X$ is treated like the lagged dependent variable, with lags from $t-2$ backwards. Misclassifying a regressor here is one of the commoner ways a difference-GMM specification goes quietly invalid.

11.4 Arellano-Bond estimator

Arellano and Bond (1991) expanded the idea by using additional lags of the dependent variable as instruments. For example, both $y_{i,t-2}$ and $y_{i,t-3}$ can be used as instruments. In fact, as $t$ increases, the number of instruments available also increases. In period 3 only $y_{i,1}$ is available. In period 4 $y_{i,1}$ and $y_{i,2}$ are available. In period 5 $y_{i,1}$ and $y_{i,2}$ and $y_{i,3}$ are available, and so on. In other words, we’ll have an instrument matrix with one row for each time period that we are instrumenting:

\[Z_i = \begin{bmatrix} y_{i,1} & \ 0 & \ 0 & \ 0 & \ 0 & \ 0 & \ \cdots & \ 0 & \ 0 & \ 0 \\ 0 & \ y_{i,1} & \ y_{i,2} & \ 0 & \ 0 & \ 0 & \ \cdots & \ 0 & \ 0 & \ 0 \\ 0 & \ 0 & \ 0 & \ y_{i,1} & \ y_{i,2} & \ y_{i,3} & \ \cdots & \ 0 & \ 0 & \ 0 \\ \vdots & \ \vdots & \ \vdots & \ \vdots & \ \vdots & \ \ddots & \ \cdots & \ \vdots & \ \vdots & \ \vdots \\ 0 & \ 0 & \ 0 & \ 0 & \ 0 & \ \cdots & \ 0 & \ y_{i,1} & \ \cdots & \ y_{i,T-2} \end{bmatrix} \tag{11.8}\]

This is the so called difference GMM estimator.

11.5 Blundell and Bond estimator

AB model has a problem: when the instruments are weak, the estimator is not good. That will happen when $y$ follows random walk or near random walk. In that case the past levels won’t be a good predictor of the future changes.

BB add a second set of moment conditions in levels. The basic idea is that lagged differences $\Delta y_{i,t-1}$ are uncorrelated with the composite error $C_i + \epsilon_{i,t}$ under an additional initial-condition/stationarity restriction, so they can instrument the level equation.

This is implemented by so-called system GMM, which stacks two equations and estimates them jointly:

the differenced equation ($\Delta y_{i,t}$), instrumented by lagged levels $y_{i,t-2}, y_{i,t-3}, \dots$ (the Arellano–Bond moments); and
the level equation ($y_{i,t}$ in levels, keeping $C_i$), instrumented by lagged differences $\Delta y_{i,t-1}$.

The level-equation moments are valid only under the extra assumption that the initial deviations from the long-run mean are uncorrelated with $C_i$ (mean stationarity of the process). System GMM is more efficient than difference GMM when $y$ is persistent (near a random walk), and it allows time-invariant regressors. A practical warning: the instrument count grows with $T^2$, so instrument proliferation can overfit the endogenous regressors and weaken the Hansen $J$ test; lags should be collapsed or capped in practice.

11.6 Diagnostics in practice

The warning about instrument proliferation is easy to state and easy to ignore. It is worth seeing what it does to real estimates, and what the standard diagnostics look like when they are working.

The data are Arellano and Bond’s own: an unbalanced panel of 140 UK firms observed between 1976 and 1984, shipped with plm as EmplUK. The specification is theirs too – log employment on its own first two lags, current and lagged log wages, three lags of log capital, current and lagged log output, with firm and year effects removed and two-step weighting. Only the instrument set changes across the three fits: every available lag of log employment from $t-2$ back, the same capped at three lags, and then the system-GMM version that adds the level-equation moments.

Code

library(plm)
data("EmplUK", package = "plm")

dif_full <- pgmm(log(emp) ~ lag(log(emp), 1:2) + lag(log(wage), 0:1) +
                   lag(log(capital), 0:2) + lag(log(output), 0:1) |
                   lag(log(emp), 2:99),
                 data = EmplUK, effect = "twoways", model = "twosteps")

dif_cap  <- pgmm(log(emp) ~ lag(log(emp), 1:2) + lag(log(wage), 0:1) +
                   lag(log(capital), 0:2) + lag(log(output), 0:1) |
                   lag(log(emp), 2:4),
                 data = EmplUK, effect = "twoways", model = "twosteps")

sys      <- pgmm(log(emp) ~ lag(log(emp), 1:2) + lag(log(wage), 0:1) +
                   lag(log(capital), 0:2) + lag(log(output), 0:1) |
                   lag(log(emp), 2:99),
                 data = EmplUK, effect = "twoways", model = "twosteps",
                 transformation = "ld")

diag_row <- function(fit, label) {
  co <- summary(fit, robust = TRUE)$coefficients
  sg <- sargan(fit)
  data.frame(specification = label,
             moments  = ncol(fit$W[[1]]),
             lag1     = round(co[1, 1], 3),
             se       = round(co[1, 2], 3),
             sargan_p = round(sg$p.value, 3),
             AR1_p    = round(mtest(fit, order = 1)$p.value, 3),
             AR2_p    = round(mtest(fit, order = 2)$p.value, 3))
}

rbind(diag_row(dif_full, "difference GMM, lags 2+"),
      diag_row(dif_cap,  "difference GMM, lags 2-4"),
      diag_row(sys,      "system GMM"))

                  specification moments  lag1    se sargan_p AR1_p AR2_p
chisq   difference GMM, lags 2+      40 0.645 0.194    0.162 0.002 0.596
chisq1 difference GMM, lags 2-4      30 0.437 0.359    0.166 0.065 0.848
chisq2               system GMM      55 1.114 0.051    0.121 0.028 0.689

Read the AR columns first, because they are the ones that can invalidate everything else. First-order serial correlation in the differenced residuals is expected: $\Delta \epsilon_{i,t}$ and $\Delta \epsilon_{i,t-1}$ both contain $\epsilon_{i,t-1}$, so a small AR(1) $p$-value is the normal state of affairs and says nothing against the specification. The AR(2) test is the informative one, and none of the three fits shows evidence that the instrumenting lag sits too close.

The Sargan column passes in all three cases, which is exactly where the instrument-count warning earns its keep. The full difference-GMM fit imposes 40 moment conditions on 15 parameters across 140 firms; the system-GMM fit imposes 55. A Sargan or Hansen test with many moments and few groups has little power, so a comfortable $p$-value is weak reassurance rather than strong evidence. The test is being asked to detect a violation using an instrument set flexible enough to fit almost anything.

The coefficients show what that costs. Capping the instruments at three lags cuts the moment conditions from 40 to 30, moves the coefficient on lagged log employment from about $0.65$ to about $0.44$, and nearly doubles its standard error. That is the honest trade: fewer, better-behaved instruments buy less precision. System GMM goes the other way and returns a coefficient above $1$ with a standard error near $0.05$, which should stop the reader rather than reassure them. A persistence parameter above one implies an explosive employment process, and the tight interval around it comes from 55 moment conditions on 140 firms. Every diagnostic in the table passes. The estimate is still not credible.

So what to report: the AR(2) test, the Sargan or Hansen statistic with its degrees of freedom, the instrument count next to the number of groups, and the estimate under at least one narrower instrument set. The last of those does most of the work, because it is the one thing in the list that a flexible instrument set cannot make look good.

# Dynamic Panel Data ## When Is It a Problem Model setup: $$y_{i,t} = \gamma y_{i, t-1} + X_{i,t}' \beta + C_i + \epsilon_{i,t}$$ {#eq-dynamic-panel-1} Pooled regression is biased and inconsistent. The problem is: $$Cov( y_{i,t-1}, (C_i + \epsilon_{i,t})) \approx \frac{\sigma_c^2}{(1-\gamma)}$$ {#eq-dynamic-panel-2} Random effects model is biased and inconsistent, for the same reason that $y_{i,t-1}$ is necessarily correlated with $C_i$, which is part of the composite error term in the random effect model. Fixed effect model setup: $$ y_{i,t} - \bar y_i = (X_{i,t} - \bar X_i)' \beta + \gamma (y_{i, t-1} - \overline{y_{i,-1}}) + (\epsilon_{i,t} - \bar \epsilon_i)$$ {#eq-dynamic-panel-3} where $\overline{y_{i,-1}}$ is the unit mean of the *lagged* dependent variable. This is not exactly $\bar y_i$ (the mean of current $y_{i,t}$). Over the usable periods the two are $(y_{i2}+\cdots+y_{iT})/(T-1)$ and $(y_{i1}+\cdots+y_{i,T-1})/(T-1)$, so their difference is exactly $$\frac{y_{iT}-y_{i1}}{T-1},$$ {#eq-dynamic-panel-mean-gap} the endpoints and nothing else. Balance does not make that vanish -- only $T$ growing does, under stability. The distinction is asymptotically negligible and finite-sample real. Nickell (1981) shows that (see also Anderson and Hsiao 1981) $$Cov( (y_{i,t-1}- \overline{y_{i,-1}}), (\epsilon_{i,t} - \bar \epsilon_i )) \approx -\frac{\sigma_\epsilon^2}{T (1-\gamma)^2} [\frac{(T-1)-T \gamma + \gamma^T}{T}]$$ {#eq-dynamic-panel-4} which indicates that the demeaned lagged $y$ is correlated with the demeaned error term. The correlation may be large if $T$ is small, but when $T$ is big, then the correlation goes down to near zero. When $T$ is big, we don't have a problem with fixed effect model. ## How Big Is the Bias When $T$ is small, which is typically the case in many microeconomic settings, then we say the fixed effect model with a lagged dependent variable is inconsistent. But how bad is it? The limit of $\hat \gamma - \gamma$ is approximately $-\frac{(1+\gamma)}{T-1}$. When $T=10$ and $\gamma=.5$, the bias is about $-0.167$ which is $1/3$ of the true value. But a simulation study (Judson and Owen 1999) shows that the bias can be as big as $20\%$ of the true coefficient even when $T=30$. ## Anderson and Hsiao estimator Anderson and Hsiao (1981) suggested looking at the first difference estimator: $$ y_{i,t} - y_{i,t-1} = (X_{i,t} - X_{i,t-1})' \beta + \gamma (y_{i,t-1} - y_{i,t-2}) + (\epsilon_{i,t} - \epsilon_{i,t-1}) $$ {#eq-dynamic-panel-5} or $$ \Delta y_{i,t}= \Delta X_{i,t} \beta + \gamma \Delta y_{i,t-1} + \Delta \epsilon_{i,t} $$ {#eq-dynamic-panel-6} This does not solve the endogeneity problem, since $\Delta y_{i,t-1}$ is still correlated with the error term. AH's idea is to instrument $\Delta y_{i,t-1}$ with the past level $y_{i,t-2}$, or past difference $y_{i,t-2}-y_{i,t-3}$. This estimator is consistent, since neither of these instruments is correlated with $\Delta \epsilon_{i,t}$ -- provided the idiosyncratic error is serially uncorrelated. That proviso carries the argument. If $\epsilon_{i,t}$ is itself AR(1), then $y_{i,t-2}$ is correlated with $\Delta \epsilon_{i,t}$, the instrument fails, and one has to reach further back. The assumption is testable, which is the point of the Arellano-Bond test for serial correlation in the differenced residuals. First-order correlation is expected by construction, since $\Delta \epsilon_{i,t}$ and $\Delta \epsilon_{i,t-1}$ share $\epsilon_{i,t-1}$; it is the *second*-order test that carries information. Rejecting AR(2) says the lag being used as an instrument does not sit far enough back. The other regressors need classifying too, because that is what decides how they are instrumented. Strictly exogenous $X$ can serve as its own instrument. Predetermined $X$ -- correlated with past shocks but not with current or future ones -- is instrumented by its own lags from $t-1$ backwards. Endogenous $X$ is treated like the lagged dependent variable, with lags from $t-2$ backwards. Misclassifying a regressor here is one of the commoner ways a difference-GMM specification goes quietly invalid. ## Arellano-Bond estimator Arellano and Bond (1991) expanded the idea by using additional lags of the dependent variable as instruments. For example, both $y_{i,t-2}$ and $y_{i,t-3}$ can be used as instruments. In fact, as $t$ increases, the number of instruments available also increases. In period 3 only $y_{i,1}$ is available. In period 4 $y_{i,1}$ and $y_{i,2}$ are available. In period 5 $y_{i,1}$ and $y_{i,2}$ and $y_{i,3}$ are available, and so on. In other words, we'll have an instrument matrix with one row for each time period that we are instrumenting: $$Z_i = \begin{bmatrix} y_{i,1} & \ 0 & \ 0 & \ 0 & \ 0 & \ 0 & \ \cdots & \ 0 & \ 0 & \ 0 \\ 0 & \ y_{i,1} & \ y_{i,2} & \ 0 & \ 0 & \ 0 & \ \cdots & \ 0 & \ 0 & \ 0 \\ 0 & \ 0 & \ 0 & \ y_{i,1} & \ y_{i,2} & \ y_{i,3} & \ \cdots & \ 0 & \ 0 & \ 0 \\ \vdots & \ \vdots & \ \vdots & \ \vdots & \ \vdots & \ \ddots & \ \cdots & \ \vdots & \ \vdots & \ \vdots \\ 0 & \ 0 & \ 0 & \ 0 & \ 0 & \ \cdots & \ 0 & \ y_{i,1} & \ \cdots & \ y_{i,T-2} \end{bmatrix} $$ {#eq-dynamic-panel-7} This is the so called difference GMM estimator. ## Blundell and Bond estimator AB model has a problem: when the instruments are weak, the estimator is not good. That will happen when $y$ follows random walk or near random walk. In that case the past levels won't be a good predictor of the future changes. BB add a second set of moment conditions in *levels*. The basic idea is that lagged differences $\Delta y_{i,t-1}$ are uncorrelated with the composite error $C_i + \epsilon_{i,t}$ under an additional initial-condition/stationarity restriction, so they can instrument the level equation. This is implemented by so-called *system GMM*, which stacks two equations and estimates them jointly: - the **differenced equation** ($\Delta y_{i,t}$), instrumented by lagged *levels* $y_{i,t-2}, y_{i,t-3}, \dots$ (the Arellano--Bond moments); and - the **level equation** ($y_{i,t}$ in levels, keeping $C_i$), instrumented by lagged *differences* $\Delta y_{i,t-1}$. The level-equation moments are valid only under the extra assumption that the initial deviations from the long-run mean are uncorrelated with $C_i$ (mean stationarity of the process). System GMM is more efficient than difference GMM when $y$ is persistent (near a random walk), and it allows time-invariant regressors. A practical warning: the instrument count grows with $T^2$, so *instrument proliferation* can overfit the endogenous regressors and weaken the Hansen $J$ test; lags should be collapsed or capped in practice. ## Diagnostics in practice The warning about instrument proliferation is easy to state and easy to ignore. It is worth seeing what it does to real estimates, and what the standard diagnostics look like when they are working. The data are Arellano and Bond's own: an unbalanced panel of 140 UK firms observed between 1976 and 1984, shipped with `plm` as `EmplUK`. The specification is theirs too -- log employment on its own first two lags, current and lagged log wages, three lags of log capital, current and lagged log output, with firm and year effects removed and two-step weighting. Only the instrument set changes across the three fits: every available lag of log employment from $t-2$ back, the same capped at three lags, and then the system-GMM version that adds the level-equation moments. ```{r} #| label: dynpanel-diagnostics #| warning: false #| message: false library(plm) data("EmplUK", package = "plm") dif_full <- pgmm(log(emp) ~ lag(log(emp), 1:2) + lag(log(wage), 0:1) + lag(log(capital), 0:2) + lag(log(output), 0:1) | lag(log(emp), 2:99), data = EmplUK, effect = "twoways", model = "twosteps") dif_cap <- pgmm(log(emp) ~ lag(log(emp), 1:2) + lag(log(wage), 0:1) + lag(log(capital), 0:2) + lag(log(output), 0:1) | lag(log(emp), 2:4), data = EmplUK, effect = "twoways", model = "twosteps") sys <- pgmm(log(emp) ~ lag(log(emp), 1:2) + lag(log(wage), 0:1) + lag(log(capital), 0:2) + lag(log(output), 0:1) | lag(log(emp), 2:99), data = EmplUK, effect = "twoways", model = "twosteps", transformation = "ld") diag_row <- function(fit, label) { co <- summary(fit, robust = TRUE)$coefficients sg <- sargan(fit) data.frame(specification = label, moments = ncol(fit$W[[1]]), lag1 = round(co[1, 1], 3), se = round(co[1, 2], 3), sargan_p = round(sg$p.value, 3), AR1_p = round(mtest(fit, order = 1)$p.value, 3), AR2_p = round(mtest(fit, order = 2)$p.value, 3)) } rbind(diag_row(dif_full, "difference GMM, lags 2+"), diag_row(dif_cap, "difference GMM, lags 2-4"), diag_row(sys, "system GMM")) ``` Read the AR columns first, because they are the ones that can invalidate everything else. First-order serial correlation in the differenced residuals is expected: $\Delta \epsilon_{i,t}$ and $\Delta \epsilon_{i,t-1}$ both contain $\epsilon_{i,t-1}$, so a small AR(1) $p$-value is the normal state of affairs and says nothing against the specification. The AR(2) test is the informative one, and none of the three fits shows evidence that the instrumenting lag sits too close. The Sargan column passes in all three cases, which is exactly where the instrument-count warning earns its keep. The full difference-GMM fit imposes 40 moment conditions on 15 parameters across 140 firms; the system-GMM fit imposes 55. A Sargan or Hansen test with many moments and few groups has little power, so a comfortable $p$-value is weak reassurance rather than strong evidence. The test is being asked to detect a violation using an instrument set flexible enough to fit almost anything. The coefficients show what that costs. Capping the instruments at three lags cuts the moment conditions from 40 to 30, moves the coefficient on lagged log employment from about $0.65$ to about $0.44$, and nearly doubles its standard error. That is the honest trade: fewer, better-behaved instruments buy less precision. System GMM goes the other way and returns a coefficient above $1$ with a standard error near $0.05$, which should stop the reader rather than reassure them. A persistence parameter above one implies an explosive employment process, and the tight interval around it comes from 55 moment conditions on 140 firms. Every diagnostic in the table passes. The estimate is still not credible. So what to report: the AR(2) test, the Sargan or Hansen statistic with its degrees of freedom, the instrument count next to the number of groups, and the estimate under at least one narrower instrument set. The last of those does most of the work, because it is the one thing in the list that a flexible instrument set cannot make look good.