Dynamic Panel Specifications And Their Moment Structure

Why Timing Determines the Moment Structure

A panel outcome often depends on its own past. For example, a region’s current unemployment rate may depend partly on its rate last quarter, even after accounting for observed covariates. This persistence creates an important distinction between a variable being dated in the past and being a valid instrument: validity depends on whether it is orthogonal to every error component in the equation being instrumented.

The central specification separates three components: a lagged dependent variable, an individual-specific effect that may be related to persistent characteristics, and an idiosyncratic error that varies across individuals and time. Once the individual effect is removed by differencing, the transformed error contains idiosyncratic errors from two dates. Timing assumptions must therefore be checked against both dates, not just the date of the current equation.

Definition: A first-order dynamic panel model is $$ y_{it}=\rho y_{i,t-1}+x_{it}'\beta+\alpha_i+\varepsilon_{it}, $$ where $i=1,\ldots,N$ indexes individuals and $t$ indexes time; $y_{it}$ is the outcome; $x_{it}$ is a vector of observed regressors; $\rho$ and $\beta$ are parameters; $\alpha_i$ is an unobserved, time-invariant individual effect; and $\varepsilon_{it}$ is a time-varying idiosyncratic error. The individual effect may be associated with $y_{i0}$ and the regressors. The idiosyncratic error is assumed to have mean zero and, for the standard lag-moment discussion, to be serially uncorrelated: $E[\varepsilon_{it}\varepsilon_{is}]=0$ for $t\ne s$.

Worked Example: Reading a Dynamic Panel Specification

Suppose the outcome is a region’s unemployment rate, measured in percentage points, and $x_{it}$ is a training-program exposure measure. A hypothetical specification is $$ y_{it}=0.65y_{i,t-1}+0.20x_{it}+\alpha_i+\varepsilon_{it}. $$ For one region and quarter, let $y_{i,t-1}=6.0$ percentage points, $x_{it}=2$ exposure units, $\alpha_i=0.4$ percentage points, and $\varepsilon_{it}=-0.1$ percentage points. Substitution gives $$ y_{it}=0.65(6.0)+0.20(2)+0.4-0.1=3.9+0.4+0.4-0.1=4.6 $$ percentage points.

The coefficient $0.65$ describes the model’s conditional association between last quarter’s outcome and this quarter’s outcome, holding the regressor and individual effect fixed. The regressor contribution is $0.20(2)=0.4$ percentage points for a two-unit increase in exposure, under the specified linear model. This arithmetic illustrates the specification; it does not establish that the timing or orthogonality assumptions hold in an actual panel.

The individual effect permits persistent differences between regions to enter the outcome. It need not be independent of the exposure or initial outcome. The idiosyncratic error represents time-specific unobserved influences, and its relationship to variables dated at different times is what determines potential instrument validity.

Timing Assumptions for Regressors

A timing label describes which idiosyncratic errors a regressor may be related to. It is useful to state the restrictions using dates explicitly. The conventions below concern orthogonality in expectation; they do not imply that a variable is statistically independent of every other variable.

Definition: A regressor $x_{it}$ is strictly exogenous with respect to the idiosyncratic errors if $E[x_{it}\varepsilon_{is}]=0$ for every error date $s$. It is predetermined if $E[x_{it}\varepsilon_{is}]=0$ for $s\ge t$, while association with earlier errors $s<t$ is allowed. It is contemporaneously endogenous under the usual timing convention if $E[x_{it}\varepsilon_{is}]=0$ for $s>t$, while association with the current or earlier errors is allowed.

These categories have different implications for instruments. Strict exogeneity rules out association with past, current, and future idiosyncratic errors. Predetermination rules out association with the current and future errors but permits feedback from past shocks into later regressor values. A contemporaneously endogenous regressor may respond to its current shock, so its current value cannot generally be treated as orthogonal to that shock.

For example, subtracting the model at $t-1$ from the model at $t$ gives an equation with transformed error $\varepsilon_{it}-\varepsilon_{i,t-1}$. A candidate instrument $z_{it}$ must satisfy $$ E[z_{it}(\varepsilon_{it}-\varepsilon_{i,t-1})]=0. $$ By linearity, this requires both error-date relationships to combine to zero. A simple sufficient condition is that $z_{it}$ be orthogonal to each of $\varepsilon_{it}$ and $\varepsilon_{i,t-1}$. Merely being uncorrelated with the current error is not enough.

Worked Example: Checking a Predetermined Regressor Against Both Dates

Consider the differenced equation at date $t=5$, whose error is $\varepsilon_{i5}-\varepsilon_{i4}$. Suppose a regressor $x_{i5}$ is predetermined according to the definition above. The assumption gives $E[x_{i5}\varepsilon_{i5}]=0$, because $5\ge5$. It does not restrict $E[x_{i5}\varepsilon_{i4}]$, because $4<5$.

Suppose, hypothetically, that $E[x_{i5}\varepsilon_{i4}]=0.30$ in the relevant units. Then $$ E[x_{i5}(\varepsilon_{i5}-\varepsilon_{i4})] =E[x_{i5}\varepsilon_{i5}]-E[x_{i5}\varepsilon_{i4}] =0-0.30=-0.30. $$ The moment is not zero, so the current value $x_{i5}$ is not justified as an instrument for this differenced equation by predetermination alone. It is orthogonal to $\varepsilon_{i5}$, but may be associated with the earlier error $\varepsilon_{i4}$.

Now consider $x_{i4}$, which is predetermined at date 4. The definition implies $E[x_{i4}\varepsilon_{i4}]=0$, and also $E[x_{i4}\varepsilon_{i5}]=0$, since $5\ge4$. Thus $$ E[x_{i4}(\varepsilon_{i5}-\varepsilon_{i4})]=0-0=0. $$ Under these stated timing restrictions, $x_{i4}$ satisfies the required orthogonality condition for the date-5 differenced error. This conclusion depends on the timing assumptions, not simply on $x_{i4}$ having an earlier date.

Lagged Outcomes and Initial Conditions

The lagged dependent variable requires separate care. It is not an externally assigned regressor: earlier outcomes contain earlier shocks, and the outcome process also contains the individual effect. Consequently, serially uncorrelated idiosyncratic errors alone do not establish the moment conditions for lagged outcomes.

For the differenced equation at date $t$, a commonly relevant candidate is $y_{i,t-2}$. Its validity requires $$ E[y_{i,t-2}\varepsilon_{it}]=0 \quad\text{and}\quad E[y_{i,t-2}\varepsilon_{i,t-1}]=0, $$ so that its expectation when multiplied by $\varepsilon_{it}-\varepsilon_{i,t-1}$ is zero. These are outcome-error restrictions, not consequences of the date labels alone.

Key condition: A direct sufficient condition for using $y_{i,t-2}$ with the date-$t$ differenced error is that it be orthogonal to both $\varepsilon_{it}$ and $\varepsilon_{i,t-1}$. One set of structural assumptions supporting such restrictions includes serially uncorrelated idiosyncratic errors, orthogonality of $\alpha_i$ to future idiosyncratic errors, initial outcomes orthogonal to future idiosyncratic errors, and regressor timing restrictions that make earlier regressors orthogonal to later errors. Alternatively, the required outcome-error moments can be imposed directly. Allowing $\alpha_i$ to be correlated with $y_{i0}$ does not by itself violate these conditions; allowing $\alpha_i$ to be correlated with future idiosyncratic errors can.

The role of these assumptions can be seen from the outcome recursion. In the simple model without time-varying regressors, repeated substitution gives $$ y_{i3}=\rho^3y_{i0}+(1+\rho+\rho^2)\alpha_i+\rho^2\varepsilon_{i1}+\rho\varepsilon_{i2}+\varepsilon_{i3}. $$ This expression shows why the conditions must cover more than serial correlation among the $\varepsilon$'s: $y_{i3}$ also contains the initial outcome and the individual effect. If either is associated with a later idiosyncratic error, the desired outcome-error moment may fail.

Worked Example: Verifying a Lagged-Outcome Moment

Consider a hypothetical panel with $\rho=0.5$ and no time-varying regressors. Suppose the idiosyncratic errors have mean zero, variance 1, and zero covariance across dates. Also suppose $y_{i0}$ and $\alpha_i$ are each orthogonal to $\varepsilon_{i4}$ and $\varepsilon_{i5}$. The assumptions allow $y_{i0}$ and $\alpha_i$ to be associated with each other.

The recursion becomes $$ y_{i3}=0.125y_{i0}+1.75\alpha_i+0.25\varepsilon_{i1}+0.5\varepsilon_{i2}+\varepsilon_{i3}. $$ Taking covariance with $\varepsilon_{i4}$, each term has zero covariance under the stated assumptions: the initial outcome and individual effect are orthogonal to this error, and the errors dated 1, 2, and 3 are serially uncorrelated with it. Therefore, $$ E[y_{i3}\varepsilon_{i4}]=0. $$ The same reasoning gives $E[y_{i3}\varepsilon_{i5}]=0$. Hence for the differenced error at date 5, $$ E[y_{i3}(\varepsilon_{i5}-\varepsilon_{i4})] =E[y_{i3}\varepsilon_{i5}]-E[y_{i3}\varepsilon_{i4}] =0-0=0. $$ Thus $y_{i3}$, which is $y_{i,5-2}$, satisfies this moment under the assumptions stated. Without the restrictions involving $\alpha_i$ and initial outcomes, the conclusion would not follow from serial uncorrelatedness alone.

A Practical Audit of Candidate Moments

A moment condition is a population restriction, not a label attached automatically to a variable. Before using a proposed internal instrument, write down the transformed error and check the instrument against each dated component. For a first-differenced equation, cancellation of $\alpha_i$ follows from its time invariance, but the idiosyncratic error becomes a difference. Removing the individual effect therefore does not itself guarantee instrument orthogonality.

Write the equation and its transformation.
For first differences, identify the exact error combination $\varepsilon_{it}-\varepsilon_{i,t-1}$.

State the candidate’s timing restriction.
Specify which error dates the variable may be associated with; do not infer this solely from its date.

Check every component error.
Establish orthogonality to both $\varepsilon_{it}$ and $\varepsilon_{i,t-1}$, or directly establish that the required combined moment is zero.

For outcomes, audit their history.
Check assumptions about the individual effect, initial outcome, and earlier regressors as well as serial correlation of idiosyncratic errors.

Worked Example: Rejecting an Unsupported Outcome Moment

Suppose someone argues that $y_{i3}$ must be orthogonal to $\varepsilon_{i4}$ because the errors are serially uncorrelated. The argument overlooks the individual effect. Let $\rho=0$, let $y_{i3}=\alpha_i+\varepsilon_{i3}$, and suppose the idiosyncratic errors are independent across dates with variance 1. Set $\alpha_i=\varepsilon_{i4}$. Then $$ E[y_{i3}\varepsilon_{i4}] =E[\alpha_i\varepsilon_{i4}]+E[\varepsilon_{i3}\varepsilon_{i4}] =E[\varepsilon_{i4}^2]+0=1. $$ The proposed moment is not zero, even though the idiosyncratic errors are serially uncorrelated. This specification violates orthogonality between the individual effect and the future idiosyncratic error. To justify the moment, one must add suitable restrictions on the individual effect and initial conditions, or state the required outcome-error orthogonality directly.

Why the Distinctions Matter

Incorrect timing assumptions can make a plausible-looking internal instrument invalid. If a regressor responds to a past shock, its later value may carry information about an error included in a transformed equation. If the individual effect is related to future idiosyncratic errors, lagged outcomes may carry that relationship into candidate moments. The resulting problem is not fixed merely by choosing a variable with an older date.

A useful habit is to separate three claims: the model’s structural specification, the timing assumptions on each regressor, and the resulting moment restrictions. The first describes how outcomes are generated; the second states which variables can respond to which shocks; the third follows only when the relevant orthogonality has actually been established. Keeping these claims separate makes assumptions auditable and prevents date-based guesses from being mistaken for valid moments.

Common pitfall: “This variable is dated before the current outcome, so it is a valid instrument.” That reasoning ignores the error dates in the transformed equation and may ignore the individual effect embedded in past outcomes. Instead, write the transformed error explicitly and verify orthogonality to each component, including the additional outcome-history assumptions when the candidate is a lagged dependent variable.

Check Your Understanding

For each question, state the relevant error dates and assumptions rather than relying only on whether a candidate is lagged.

For a differenced equation at date $t$, which two idiosyncratic errors appear in its transformed error?
Under the stated definition of a predetermined regressor, may $x_{it}$ be associated with $\varepsilon_{i,t-1}$? What about $\varepsilon_{it}$?
Why does orthogonality of $y_{i,t-2}$ to $\varepsilon_{it}$ alone not establish its moment with the differenced error?
Name two features of the outcome history, besides serial correlation of idiosyncratic errors, that may matter for lagged-outcome moments.
If $E[x_{i5}\varepsilon_{i5}]=0$ but $E[x_{i5}\varepsilon_{i4}]=0.2$, calculate $E[x_{i5}(\varepsilon_{i5}-\varepsilon_{i4})]$. Is the moment zero?