Forward Orthogonal Deviations For Dynamic Panels

Why Transform Toward the Future?

A dynamic-panel model can contain a persistent individual effect that is correlated with the regressors. Removing that effect is necessary before using the transformed equations for GMM. First differences remove a constant individual effect, but a difference at date $t$ requires observations at both $t$ and $t-1$. When panels have gaps, that requirement can discard an otherwise observed equation. Forward orthogonal deviations (FOD) offer another transformation: each observation is compared with the average of its available future observations.

FOD does not create data where observations are missing, nor does it by itself guarantee valid instruments. Its practical value is that it removes a time-constant effect while retaining an observed equation before a gap, provided that there is at least one later usable equation for comparison. Instrument validity still depends on the timing assumptions for the idiosyncratic errors and regressors.

Definition and Derivation

Consider a dynamic-panel equation for individual $i$ at date $t$:

y_{it}=\rho y_{i,t-1}+x_{it}'\beta+\eta_i+u_{it}.

Here $y_{it}$ is the outcome, $x_{it}$ is a vector of regressors, $\eta_i$ is an individual-specific effect that does not vary over time, and $u_{it}$ is the idiosyncratic error. The parameter $\rho$ describes the lagged outcome's coefficient. The purpose of the transformation is to remove $\eta_i$, not to remove all sources of endogeneity.

Definition: For a variable $z_{it}$, let $R_i$ be the ordered set of dates at which the equation and all variables being transformed are observed. At date $t\in R_i$, let $m_{it}$ be the number of later dates in $R_i$. If $m_{it}\geq 1$, its forward orthogonal deviation is $$ z_{it}^{*}=\sqrt{\frac{m_{it}}{m_{it}+1}}\left(z_{it}-\frac{1}{m_{it}}\sum_{\substack{s\in R_i\\s>t}}z_{is}\right). $$ There is no transformed observation for the last date in $R_i$, because it has no later observed value to average.

The formula applies to each variable in an equation, including the outcome, regressors, lagged outcome, and error. In a balanced panel observed at every date $1,\ldots,T$, $R_i=\{1,\ldots,T\}$ and $m_{it}=T-t$. Thus, for $t=1,\ldots,T-1$, the transformation is the current value minus the mean of all later values, multiplied by the stated normalization.

The individual effect disappears because it is constant across the current and future observations. The difference inside the parentheses is $\eta_i-\eta_i=0$. The normalization does not change that cancellation.

Worked Example: Transforming a Balanced Outcome

Suppose a hypothetical panel has four observed outcome values for one individual: $y_{i1}=10$, $y_{i2}=13$, $y_{i3}=12$, and $y_{i4}=16$. Compute the first three FOD values.

At $t=1$, there are three future observations. Therefore, $$ y_{i1}^{*}=\sqrt{\frac{3}{4}}\left(10-\frac{13+12+16}{3}\right) =\sqrt{0.75}(10-13.6667) \approx -3.175. $$

At $t=2$, there are two future observations: $$ y_{i2}^{*}=\sqrt{\frac{2}{3}}\left(13-\frac{12+16}{2}\right) =\sqrt{\frac{2}{3}}(13-14) \approx -0.8165. $$

At $t=3$, only date 4 is in the future: $$ y_{i3}^{*}=\sqrt{\frac{1}{2}}(12-16)\approx -2.828. $$

There is no FOD value at date 4. If a constant $\eta_i=5$ were added to all four outcomes, it would add 5 to both the current value and the future average in each calculation, leaving all three transformed values unchanged.

Transformed Equations and the Role of the Normalization

Applying the same linear transformation to every term in the model gives, for each usable transformed date,

y_{it}^{*}=\rho y_{i,t-1}^{*}+{x_{it}^{*}}'\beta+u_{it}^{*}.

This expression uses the same set of equation dates to transform every term. In particular, $y_{i,t-1}^{*}$ means the FOD of the lagged-outcome variable across equation rows; it is not simply the original value $y_{i,t-1}$. The transformed error is

u_{it}^{*}=\sqrt{\frac{m_{it}}{m_{it}+1}}\left(u_{it}-\frac{1}{m_{it}}\sum_{\substack{s\in R_i\\s>t}}u_{is}\right).

Under serially uncorrelated errors with common variance $\sigma_u^2$, the normalization gives each transformed error variance $\sigma_u^2$. To see this, the current error and the $m_{it}$ future errors are mutually uncorrelated, so the variance inside the parentheses is $\sigma_u^2+\sigma_u^2/m_{it}$. Multiplying by $m_{it}/(m_{it}+1)$ yields $\sigma_u^2$. This variance result relies on the stated error conditions; it is not a general property under arbitrary serial dependence or heteroskedasticity.

With a complete sequence of observed equation rows and serially uncorrelated errors, distinct FOD errors are also uncorrelated. For an earlier transformed row, its coefficients on all later errors are equal. A later transformed row has coefficients summing to zero across its own date and later dates. The two rows therefore have coefficient vectors with zero inner product. More generally, the FOD rows are orthonormal transformations of the ordered observed error sequence under the common-variance, serially uncorrelated error assumption.

Justification: For any earlier row $r$ and later row $q$, all error dates used by row $q$ are future dates relative to row $r$. Row $r$ assigns the same coefficient to each of those dates. Row $q$'s coefficients sum to zero, since its current coefficient is $\sqrt{m/(m+1)}$ and its $m$ future coefficients are each $-\sqrt{m/(m+1)}/m$. The dot product is therefore zero. The squared length of a row's coefficient vector is $\frac{m}{m+1}(1+\frac{1}{m})=1$.

Gaps: Transform the Observed Sequence

When equation dates are missing, use the ordered set $R_i$, not the calendar distance between dates, to count future observations. A usable equation at date $t$ is retained whenever at least one later usable equation exists. The formula averages over those later observed equation dates; it does not insert zeros or impute values for missing dates. The last usable equation still has no FOD row.

Worked Example: Retaining an Equation Before a Gap

Suppose the usable equation dates for a hypothetical individual are $R_i=\{1,3,4,6\}$. The gaps at dates 2 and 5 mean there are no usable equations at those dates. Suppose the corresponding error values are $u_{i1}=1$, $u_{i3}=-1$, $u_{i4}=2$, and $u_{i6}=0$.

At date 1, three later equation rows are observed, so $$ u_{i1}^{*}=\sqrt{\frac{3}{4}}\left(1-\frac{-1+2+0}{3}\right) =\sqrt{0.75}\left(\frac{2}{3}\right) \approx 0.5774. $$

At date 3, the later observed rows are dates 4 and 6: $$ u_{i3}^{*}=\sqrt{\frac{2}{3}}\left(-1-\frac{2+0}{2}\right) =\sqrt{\frac{2}{3}}(-2) \approx -1.633. $$

At date 4, only date 6 is later: $$ u_{i4}^{*}=\sqrt{\frac{1}{2}}(2-0)\approx 1.414. $$

The usable equation at date 1 is retained even though date 2 is missing: its comparison uses dates 3, 4, and 6. First differencing could not form the adjacent difference at date 1 to date 2. FOD does not, however, retain a row if its current equation variables are missing, and it cannot transform the last usable row at date 6.

The transformed model is constructed by applying this observed-row transformation to the whole equation. If a row cannot be formed because a required outcome, lagged outcome, or regressor is unavailable, it is not a usable row in $R_i$. The pattern of missingness must also be compatible with the moment conditions used for estimation: if observation or attrition depends on unobserved errors in a way that destroys orthogonality, the transformation alone does not repair that problem.

Internal Instruments and Timing Conditions

FOD changes the error in a useful way for instrument timing. The transformed error at date $t$ is a weighted combination of $u_{it}$ and errors at later usable dates; it contains no error dated before $t$. Under a no-serial-correlation assumption and suitable sequential exogeneity, information dated before $t$ can therefore be uncorrelated with $u_{it}^{*}$.

Moment condition: An internal instrument $z_{it}$ is valid for a transformed equation at date $t$ when its timing and the error assumptions imply $E[z_{it}u_{it}^{*}]=0$. For example, if $y_{i,t-1}$ is determined using information through date $t-1$, and errors at dates $t$ and later are mean-independent of that information, then $E[y_{i,t-1}u_{it}^{*}]=0$. This conclusion requires the stated timing conditions; it does not follow from the FOD formula alone.

For a lagged dependent variable, the level $y_{i,t-1}$ is a natural internal instrument for the FOD equation at date $t$ under sequentially uncorrelated errors and no anticipation of future shocks. A predetermined regressor dated $t-1$ can likewise be used when it is uncorrelated with current and future errors. For a contemporaneously endogenous regressor $x_{it}$, a lagged value such as $x_{i,t-1}$ may be valid if it is uncorrelated with the current and future errors; if the variable's timing or serial dependence violates that condition, deeper lags or different instruments are needed. Strictly exogenous regressors can provide instruments at their own dates under the corresponding orthogonality assumption.

These are date-specific moments: the instrument must be valid for the particular transformed error it is paired with. Missing rows alter which transformed equations and instruments are available, but do not relax the orthogonality requirement. In practice, instrument sets should be chosen using the timing assumptions and observed dates, not by mechanically assigning every available lag to every row.

Worked Example: Matching Instruments to Transformed Dates

Consider a hypothetical panel with complete equation rows through date 5. Suppose the errors have no serial correlation, and past outcomes are mean-independent of current and future errors. The FOD error at date 3 is $$ u_{i3}^{*}=\sqrt{\frac{2}{3}}\left(u_{i3}-\frac{u_{i4}+u_{i5}}{2}\right). $$

The level $y_{i2}$ is dated before every error appearing in this expression. Under the stated assumptions, $E[y_{i2}u_{i3}]=E[y_{i2}u_{i4}]=E[y_{i2}u_{i5}]=0$. Linearity then gives $$ E[y_{i2}u_{i3}^{*}] =\sqrt{\frac{2}{3}}\left(E[y_{i2}u_{i3}]-\frac{E[y_{i2}u_{i4}]+E[y_{i2}u_{i5}]}{2}\right)=0. $$

Thus $y_{i2}$ is a valid internal instrument for the transformed equation at date 3 under these assumptions. The conclusion is about an orthogonality condition, not proof that the assumptions hold in an application. If errors are serially correlated, or if future shocks affect $y_{i2}$, the displayed moments need not be zero.

When FOD Is Useful—and What It Does Not Do

The main practical distinction from first differencing is how a gap affects usable equations. First differences require adjacent observations. FOD uses the later observed equation rows as a reference, so it can retain an earlier row even when its next calendar period is absent. In a complete panel, FOD also changes the transformed error structure in a way that can make the transformed disturbances orthogonal under the standard serially uncorrelated, common-variance error specification.

FOD is not automatically superior. It loses the final usable equation because no future value exists; it can have no transformed rows for a unit with only one usable equation; and its moment conditions can fail under serially correlated errors, invalid regressor timing, or informative missingness. The normalization has a specific variance motivation under particular error assumptions, but it does not make an incorrectly specified model correct.

Feature	Forward orthogonal deviations	First differences
Removes a time-constant individual effect	Yes, by subtracting the future mean	Yes, by subtracting the preceding value
Comparison required	At least one later usable equation	An adjacent preceding observation
Effect of a gap	Can retain an observed row before the gap	Cannot form a difference across a missing adjacent value
Last usable equation	No transformed row	Can be differenced if its preceding value is observed

Common Mistakes

Counting calendar periods instead of observed future rows. With gaps, $m_{it}$ counts later usable equation observations, not the number of calendar dates between $t$ and the panel endpoint.
Transforming only the outcome. The entire equation must be transformed with the same row set, including the lagged outcome, regressors, and error.
Calling a lag valid without stating timing assumptions. FOD removes the individual effect, but instrument validity still requires orthogonality between each instrument and the transformed error.
Assuming gaps are harmless. FOD may retain rows before gaps, but it neither imputes missing variables nor guarantees that the missingness process preserves the required moments.

Key takeaway: FOD subtracts the mean of later observed equation values and applies a normalization that removes a constant individual effect. Its gap advantage is conditional on having later usable rows; its GMM instruments remain valid only under explicit timing and error assumptions.

Check Your Understanding

Answer each question by referring to the transformation formula and its assumptions.

For observed equation dates $2,4,5$, how many later observations enter the transformation at date 2?
Why does the last usable equation date have no FOD-transformed row?
Explain why applying FOD to the outcome but not to the regressors does not produce the stated transformed model.
Under what timing condition can $y_{i,t-1}$ be a valid instrument for the FOD equation at date $t$?
Does retaining a row before a gap establish that the missingness process is ignorable for the moment conditions? Explain.