Common mistake 1 with Heterogeneous panel: Mean Group & PMG

Running this on a short wide panel. With T = 8 the individual regressions are noise and their average is noise with a confident-looking standard error attached.

Common mistake 2 with Heterogeneous panel: Mean Group & PMG

Reporting PMG because it has the smaller standard errors. If the long run is not common, PMG is inconsistent and the small standard error is the problem, not the point.

Common mistake 3 with Heterogeneous panel: Mean Group & PMG

Ignoring a positive φ. It is the model telling you the series are not cointegrated.

Common mistake 4 with Heterogeneous panel: Mean Group & PMG

Forgetting the dynamic-fixed-effects column is there as a warning. It pools the adjustment speed too, and when the speeds genuinely differ it biases the long run - Pesaran and Smith showed the bias does not vanish as T grows.

Common mistake 5 with Heterogeneous panel: Mean Group & PMG

One wild entity dragging the Mean Group average. Look at the histogram before trusting the mean; the outlier-robust alternative is to report the median.

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EconStatsCraft › Heterogeneous panel: Mean Group & PMG

Heterogeneous panel: Mean Group & PMG

Long-run elasticities from a panel where every entity is allowed its own dynamics. Mean Group runs a separate error-correction model per entity and averages; Pooled Mean Group imposes one common long run while leaving the short run and the speed of adjustment free; the Hausman test says which of the two the data supports.

What it needs

Entity identifier - Country, state, firm…
Time identifier - These estimators need a long time dimension - twenty periods and up.
Dependent variable - In logs if you want the coefficients to read as elasticities.
Long-run regressors

The intuition

Suppose you want the income elasticity of import demand and you have forty years for each of twenty-four countries. You could pool everything and run one regression, but countries plainly do not adjust at the same speed - a small open economy absorbs a shock in a year, a large closed one takes five. Pesaran and Smith showed that pooling those different speeds does not merely lose detail, it BIASES the long run, and the bias does not go away as the sample grows. So you run a separate regression per country and average. Pesaran, Shin and Smith then noticed you can have it both ways: the long run is often a structural quantity that really should be shared, while the short run obviously is not. Pooled Mean Group imposes the first and frees the second.

When to use it

Macro panels where T is large: import demand elasticities across countries, growth and investment, energy demand, the exchange-rate pass-through. Anywhere the long run is a structural quantity you expect to be shared while the adjustment path obviously is not.

Where the estimator comes from

Write each entity as an ARDL(1,1) and rearrange into error-correction form: \Delta y_{it}=\phi_i\left(y_{i,t-1}-\theta_i'x_{i,t-1}\right)+\gamma_i'\Delta x_{it}+\mu_i+\varepsilon_{it}, where \phi_i=-(1-\lambda_i) is the speed of adjustment and \theta_i the long-run coefficients. For MG, estimate this separately for each i by OLS and set \hat\theta_{MG}=N^{-1}\sum_i\hat\theta_i with \widehat{\operatorname{Var}}(\hat\theta_{MG})=\frac{1}{N(N-1)}\sum_i(\hat\theta_i-\hat\theta_{MG})^2 - the variance comes from the DISPERSION across entities, not from any one regression. For PMG, impose \theta_i=\theta and maximise the concentrated likelihood; the score for \theta is \sum_i\frac{\phi_i^2}{\sigma_i^2}x_{i,t-1}'\left(\Delta y_{it}/\phi_i-y_{i,t-1}+\ldots\right), which is why \theta enters weighted by the squared adjustment speed: entities that adjust fast carry more information about where the relationship settles. The short-run terms must be partialled out of both sides of that score, or the information matrix credits the long run with variation the constant and \Delta x already explain.

The model

\Delta y_{it}=\phi_i\big(y_{i,t-1}-\theta_i' x_{i,t-1}\big)+\gamma_i'\Delta x_{it}+\mu_i+\varepsilon_{it}\qquad \hat{\theta}_{MG}=\frac{1}{N}\sum_i\hat{\theta}_i

What it assumes

T is large. Every entity gets its own regression, so each one needs enough periods to estimate it - twenty at the very least, and these are asymptotics in T, not in N.
The variables are I(1) and cointegrated within each entity, so that φ is negative and the error-correction term means something.
The errors are serially uncorrelated once enough lags of Δy and Δx are in, which is what justifies treating the regressors as weakly exogenous.
For Pooled Mean Group only: the long-run coefficients really are the same everywhere. That is the restriction the Hausman test is checking.
Entities are cross-sectionally independent, or the averaging picks up a common factor rather than a structural elasticity. Test with Pesaran CD; if it rejects, the common-correlated-effects version is the next step.

Statistical properties

MG is consistent for the average long-run coefficient whether or not the long run is common, provided T is large. It is asymptotics in T first: the individual estimates are biased in small T and averaging does not remove that.
PMG is consistent AND efficient when the long run is genuinely common, and inconsistent when it is not. That is the whole trade-off, and the Hausman test is the arbiter.
Dynamic fixed effects pools the adjustment speed as well. Pesaran and Smith (1995) showed this biases the long-run coefficients when the speeds differ, and that the bias survives T going to infinity - which is the result that made MG necessary in the first place.
A negative, significant phi is itself an error-correction test for cointegration within each entity. A positive one says the series drift apart and the long run reported above it is spurious.
These are large-T estimators. With T below about fifteen the per-entity regressions are too noisy for their average to be informative, and the standard error - built from their dispersion - can look reassuring precisely because the dispersion is sampling noise.
Everything here assumes entities are independent. If a common factor drives them all, the cross-entity average absorbs it; Pesaran's common correlated effects estimator adds cross-sectional means of y and x to each regression to strip it out.

How to read the output

The long-run coefficient on a logged regressor is an elasticity: how far y ends up moving, not how far it moves this year.
φ is the fraction of last period's gap closed this period. −0.4 means 40% a year, so about two years to work off half a shock.
φ must be negative and significant. A positive one means there is no long-run relationship at all and the coefficients above it describe nothing.
A Hausman p-value above 0.05 says pooling is defensible, so report PMG - it is far more precise. Below 0.05, report MG.
MG standard errors come from the SPREAD of the individual estimates, not from any single regression. That is why the histogram matters: it is the actual sampling distribution being averaged.

A worked reading

On twenty-four countries of import demand over forty-five years, with a true income elasticity of 1.20 and price elasticity of -0.60 built in and heterogeneous adjustment speeds between -0.18 and -0.55: Mean Group returns 1.2135 (se 0.0175) and -0.5952 (se 0.0830); Pooled Mean Group returns 1.1991 (se 0.0065) and -0.5552 (se 0.0311). Both find the truth; PMG does it with a standard error under half the size, which is what the pooling bought. The Hausman statistic is 1.06 on 2 degrees of freedom, p = 0.59 - no evidence against the restriction, so the precision is free. Repeat the experiment and the test rejects a genuinely common long run 4% of the time, which is the size it is supposed to have; make the long run differ across countries and it rejects 60% of the time while PMG's error grows eighteen-fold and MG's does not. That asymmetry is the entire case for running both.

Symptom → diagnosis → what to do

If you see this	It means	Do this
phi is positive	No cointegration in that entity	The long run is spurious; test the variables for a unit root and for cointegration first
Hausman rejects	The long run is not common	Report MG. PMG is inconsistent here whatever its standard errors say
MG standard errors enormous	The individual estimates are scattered	Check the histogram for one wild entity; consider the median or a longer T
MG and DFE far apart	Heterogeneous adjustment speeds	This is the Pesaran-Smith bias, and MG is the one to believe
A single entity fails to estimate	Too few periods after differencing and lagging	It is dropped from the average; check it was not the interesting one
Pesaran CD rejects on the residuals	A common factor across entities	Move to CCEMG or augment with cross-sectional averages

Common mistakes

Running this on a short wide panel. With T = 8 the individual regressions are noise and their average is noise with a confident-looking standard error attached.
Reporting PMG because it has the smaller standard errors. If the long run is not common, PMG is inconsistent and the small standard error is the problem, not the point.
Ignoring a positive φ. It is the model telling you the series are not cointegrated.
Forgetting the dynamic-fixed-effects column is there as a warning. It pools the adjustment speed too, and when the speeds genuinely differ it biases the long run - Pesaran and Smith showed the bias does not vanish as T grows.
One wild entity dragging the Mean Group average. Look at the histogram before trusting the mean; the outlier-robust alternative is to report the median.

Where to read this properly

Pesaran and Smith (1995), "Estimating Long-Run Relationships from Dynamic Heterogeneous Panels", Journal of Econometrics 68(1).
Pesaran, Shin and Smith (1999), "Pooled Mean Group Estimation of Dynamic Heterogeneous Panels", JASA 94(446).
Blackburne and Frank (2007), "Estimation of Nonstationary Heterogeneous Panels", Stata Journal 7(2) - the xtpmg command.
Pesaran (2006), "Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure", Econometrica 74(4).
Eberhardt and Teal (2011), "Econometrics for Grumblers", Journal of Economic Surveys 25(1).

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Heterogeneous panel: Mean Group & PMG

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

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