Common mistake 1 with Dynamic panel GMM

Instrument proliferation. Using every available lag can generate hundreds of instruments; cap them (Roodman 2009).

Common mistake 2 with Dynamic panel GMM

Reporting two-step standard errors without the Windmeijer correction - they are severely too small.

Common mistake 3 with Dynamic panel GMM

Using GMM on a long panel where fixed effects would have been fine and far more transparent.

Common mistake 4 with Dynamic panel GMM

Ignoring a rejected AR(2) test because the coefficients look plausible.

EconStatsCraft › Dynamic panel GMM

Dynamic panel GMM

When the past of the outcome belongs on the right-hand side, fixed effects is biased (Nickell) and OLS is biased the other way. Difference and system GMM instrument the lagged dependent variable with its own deeper lags.

What it needs

Entity identifier
Time identifier
Dependent variable
Other regressors - Optional.

The intuition

Put last period's outcome on the right-hand side of a panel and both standard estimators break, in opposite directions. Pooled OLS confuses persistence with the fixed effect and reads too high. Fixed effects demeans the data, which drags the lagged term into the error and reads too low. The truth is trapped between two biased numbers, and GMM instruments the lag with deeper lags to get at it.

When to use it

Short, wide panels - many entities, few periods - where the outcome is persistent. If T is large the Nickell bias is small and plain fixed effects is the better choice.

Where the estimator comes from

For y_{it}=\rho y_{i,t-1}+x_{it}'\beta+\alpha_i+\varepsilon_{it}, the within transformation subtracts the time mean, so the transformed regressor y_{i,t-1}-\bar{y}_{i,-1} contains \varepsilon_{it} through \bar{y} and the transformed error contains \varepsilon_{i,t-1}. They are correlated by construction, and Nickell (1981) showed the bias is O(1/T) and NEGATIVE. Arellano and Bond difference instead: \Delta y_{it}=\rho\Delta y_{i,t-1}+\Delta x_{it}'\beta+\Delta\varepsilon_{it} removes \alpha_i, and levels dated t-2 and earlier are valid instruments for \Delta y_{i,t-1} because E[y_{i,t-2}\Delta\varepsilon_{it}]=0. Blundell and Bond add the level equation instrumented by lagged differences, which is far more informative when \rho is near one and differences are nearly uninformative.

The model

\Delta y_{it}=\rho\,\Delta y_{i,t-1}+\Delta \mathbf{x}_{it}^{\prime}\boldsymbol{\beta}+\Delta\varepsilon_{it},\qquad E\!\left[y_{i,t-s}\,\Delta\varepsilon_{it}\right]=0\ \ \text{for } s\ge 2

What it assumes

No second-order serial correlation in the differenced errors - this is what makes lags from t−2 back valid, and the AR(2) test checks it directly.
The instruments are jointly valid (Hansen/Sargan).
System GMM assumes ADDITIONALLY that the lagged differences are uncorrelated with the entity effect - a real extra restriction, not a free upgrade.
No cross-sectional dependence.
Short T relative to N. GMM is not designed for long panels.

Statistical properties

The two naive estimators bracket the truth, and Bond (2002) turns that into a diagnostic: a credible GMM estimate should lie between the fixed-effects and pooled-OLS values. One outside the bracket is a warning, not a discovery.
Nickell bias is of order 1/T, so it is a small-T problem. It is severe for the three to ten periods typical of firm panels and negligible by T = 50.
The instrument count grows with the SQUARE of T. With many instruments the two-step weight matrix is badly estimated, overidentification tests lose all power, and the estimator is biased towards the fixed-effects value. Collapse or restrict the lag depth.
The Hansen J test should not be too comfortable. A p-value near 1.00 signals instrument proliferation rather than a well-specified model, and Roodman (2009) treats anything above about 0.25 with suspicion.
AR(1) in differences is expected by construction and means nothing. AR(2) is the test that matters: if it rejects, lag 2 is not a valid instrument and deeper lags are needed.
Two-step standard errors are severely biased downward in finite samples. Windmeijer (2005) corrects them, and the correction should always be applied.

How to read the output

AR(1) SHOULD reject - differencing induces it mechanically. AR(1) not rejecting is the surprise.
AR(2) must NOT reject. If it does, the instruments are invalid and the estimates are not usable.
Hansen/Sargan should not reject - but a p-value near 1.00 is a warning of too many instruments, not reassurance.
Sanity-check with the Bond bracket: the GMM coefficient on the lag should sit between pooled OLS (biased up) and fixed effects (biased down).
Watch the instrument count. More instruments than entities over-fits and silently breaks the Hansen test.

A worked reading

A production function for 120 firms over 10 periods. The three estimators bracket the persistence exactly as the theory says they must: pooled OLS returns 0.6317 on the lagged output term, biased UP because it absorbs the firm effect; fixed effects returns 0.0023, biased DOWN by the Nickell term of order 1/T; and Arellano-Bond two-step GMM returns 0.0355, inside the bracket. That bracket is the single most useful diagnostic in this module - an estimate outside [0.0023, 0.6317] would be the first thing to distrust, whatever its standard error said. Here the persistence is not distinguishable from zero (p = 0.60), which is itself the finding: output in this panel is driven by capital (0.3670) and labour (0.5582), not by its own past. The instruments survive their tests - AR(2) does not reject at p = 0.57, and Hansen gives p = 0.6882 on 25 degrees of freedom, comfortably short of the suspicious 1.00 that too many instruments produce.

Symptom → diagnosis → what to do

If you see this	It means	Do this
GMM estimate above pooled OLS	Outside the Bond bracket	Something is wrong. Check the instrument set
GMM estimate at the fixed-effects value	Too many weak instruments	Collapse the instrument matrix
Hansen p-value of 1.00	Instrument proliferation, not a good fit	Reduce the instrument count and re-test
AR(2) rejecting	Lag 2 is not exogenous	Start instruments at lag 3
Instruments outnumbering groups	Guaranteed overfitting	Collapse or limit the lag range
rho estimated above 1	A unit root, or weak identification	Difference GMM is very weak here. Use system GMM

Common mistakes

Instrument proliferation. Using every available lag can generate hundreds of instruments; cap them (Roodman 2009).
Reporting two-step standard errors without the Windmeijer correction - they are severely too small.
Using GMM on a long panel where fixed effects would have been fine and far more transparent.
Ignoring a rejected AR(2) test because the coefficients look plausible.

Where to read this properly

Nickell (1981), "Biases in Dynamic Models with Fixed Effects", Econometrica 49(6).
Arellano and Bond (1991), Review of Economic Studies 58(2); Blundell and Bond (1998), Journal of Econometrics 87(1).
Roodman (2009), "A Note on the Theme of Too Many Instruments", Oxford Bulletin of Economics and Statistics 71(1).
Windmeijer (2005), Journal of Econometrics 126(1), for the two-step standard error correction.

Related Panel data

Panel regression · Panel cointegration tests · Heterogeneous panel: Mean Group & PMG · Gravity model of trade · Panel diagnostics · Panel unit-root tests

All analyses

Data & variables

Each example is simulated from a known model, so you can check whether the estimator recovers the truth - and nothing here is anyone's private data.

Filter variables

Drag a variable into a slot on the right - or just click it.

Choose an example dataset above to begin.

Choose a transformation - it opens in the data editor, with a live preview.

Apply to

Multiply by

Columns ctrl-click for several

Group by ctrl-click for several

Within groups

Statistic

Bands

How many

Cut points

Keep rows where

Value

and

Fill with

Value

Also mark which rows were filled

Rules

Base period = 100

Which part

Percent at each tail

Threshold

Survey weight optional

Direction

Names column

Values column

Largest first

Delete them, not just count them

Read this column as

Tidy the text

New name optional

EconStatsCraft rigorous, plain-spoken econometrics

Econometrics that explains itself.

Regression, panel data, time series and the whole battery of tests. Every result is followed by a verdict you can defend.

Load a dataset to see it here.

Dynamic panel GMM

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

Related Panel data

All analyses

Charts

Maps

Explore

Regression

Time series - one series

Time series - several series

Panel data

Multivariate statistics

Marketing research

Causal

Finance & risk

Actuarial

Tests & breaks

Machine learning

Econometrics that explains itself.