Nickell Bias & LSDVC Correction

Key Takeaway

Fixed Effects estimation of dynamic panel models is inconsistent for fixed T due to the Nickell (1981) bias. The LSDVC estimator corrects this analytically using Kiviet (1995) bias formulas, with bootstrap inference following Bruno (2005). LSDVC is particularly useful when N is small to moderate, where GMM asymptotics may not hold.

The Dynamic Panel Model

Consider the standard dynamic panel data model with individual fixed effects:

\[ y_{it} = \rho \, y_{i,t-1} + X_{it}'\beta + \alpha_i + \varepsilon_{it} \]

where:

• \(y_{it}\) is the dependent variable for entity \(i\) at time \(t\)
• \(\rho\) is the autoregressive parameter (\(|\rho| < 1\) for stationarity)
• \(X_{it}\) is a vector of exogenous regressors
• \(\alpha_i\) is the unobserved entity-specific fixed effect
• \(\varepsilon_{it} \sim \text{iid}(0, \sigma^2_\varepsilon)\), independent of \(\alpha_i\)

For \(i = 1, \ldots, N\) entities and \(t = 1, \ldots, T\) time periods.

The Nickell (1981) Bias

Within Transformation

The standard Fixed Effects (within) estimator removes \(\alpha_i\) by demeaning:

\[ \ddot{y}_{it} = \rho \, \ddot{y}_{i,t-1} + \ddot{X}_{it}'\beta + \ddot{\varepsilon}_{it} \]

where \(\ddot{z}_{it} = z_{it} - \bar{z}_i\) and \(\bar{z}_i = \frac{1}{T}\sum_{t=1}^T z_{it}\).

Source of the Bias

The demeaned lagged dependent variable \(\ddot{y}_{i,t-1}\) is correlated with the demeaned error \(\ddot{\varepsilon}_{it}\), because:

\[ \ddot{y}_{i,t-1} = y_{i,t-1} - \bar{y}_i \]

contains \(y_{i,t-1}\), which depends on \(\varepsilon_{i,t-1}\), while:

\[ \ddot{\varepsilon}_{it} = \varepsilon_{it} - \bar{\varepsilon}_i = \varepsilon_{it} - \frac{1}{T}\sum_{s=1}^T \varepsilon_{is} \]

contains \(-\frac{1}{T}\varepsilon_{i,t-1}\) through the entity mean \(\bar{\varepsilon}_i\). This correlation does not vanish as \(N \to \infty\) with fixed \(T\).

Bias Formula

Nickell (1981) showed that for the AR(1) model without exogenous regressors:

\[ \text{plim}_{N \to \infty}(\hat{\rho}_{FE} - \rho) \approx -\frac{1 + \rho}{T - 1} \left[1 - \frac{1}{T(1-\rho)} \cdot \frac{1 - \rho^T}{1 - \rho}\right]^{-1} \]

For large \(T\), this simplifies to the well-known approximation:

\[ \text{plim}(\hat{\rho}_{FE} - \rho) \approx -\frac{1 + \rho}{T - 1} \]

Key Properties of the Nickell Bias

• Always downward for positive \(\rho\): the FE estimator underestimates persistence
• Does not vanish as \(N \to \infty\) with fixed \(T\)
• Magnitude depends on T: with \(\rho = 0.7\) and \(T = 10\), bias \(\approx -0.19\); with \(T = 5\), bias \(\approx -0.43\)
• Contaminates \(\hat{\beta}\): bias in \(\hat{\rho}\) propagates to all coefficient estimates

Numerical Examples

\(\rho\)	\(T = 5\)	\(T = 10\)	\(T = 20\)	\(T = 50\)
0.3	-0.325	-0.144	-0.068	-0.027
0.5	-0.375	-0.167	-0.079	-0.031
0.7	-0.425	-0.189	-0.089	-0.035
0.9	-0.475	-0.211	-0.100	-0.039

The table shows the approximate bias \(-\frac{1+\rho}{T-1}\) for different combinations of \(\rho\) and \(T\).

The LSDV Estimator

The LSDV (Least Squares Dummy Variable) estimator is algebraically equivalent to FE but explicitly includes entity dummies \(D_i\):

\[ y_{it} = \rho \, y_{i,t-1} + X_{it}'\beta + \sum_{i=1}^{N} \alpha_i D_i + \varepsilon_{it} \]

In matrix notation, define \(W = [y_{-1}, X]\) and \(D\) as the matrix of entity dummies. The LSDV estimator is:

\[ \hat{\delta}_{LSDV} = (W'A W)^{-1} W'A y \]

where \(A = I_{NT} - D(D'D)^{-1}D'\) is the within-transformation matrix that projects out entity means, and \(\delta = (\rho, \beta')'\) is the full parameter vector.

Kiviet (1995) Bias Correction

The Idea

Rather than using instruments (as in GMM), Kiviet (1995) derives an analytical expression for the bias of \(\hat{\delta}_{LSDV}\) and subtracts it:

\[ \hat{\delta}_{LSDVC} = \hat{\delta}_{LSDV} - \widehat{\text{Bias}}(\hat{\delta}_{LSDV}) \]

The bias depends on unknown parameters \((\rho, \beta, \sigma^2)\), which are estimated iteratively.

Bias Decomposition

The bias of the LSDV estimator can be decomposed as:

\[ E[\hat{\delta}_{LSDV} - \delta] = c_1(\delta, \sigma^2) + c_2(\delta, \sigma^2) + c_3(\delta, \sigma^2) + O(N^{-1}T^{-2}) \]

where \(c_1\), \(c_2\), \(c_3\) are bias terms of increasing order.

Order 1: \(O(T^{-1})\)

The leading bias term is:

\[ c_1 = \sigma^2 (W'AW)^{-1} \left[\text{tr}(AF) \cdot q_0 + \text{tr}(AF^2) \cdot q_1\right] \]

where:

• \(F = \rho L + X \cdot g(\beta)\) captures the dynamic structure (with \(L\) being the lag operator matrix)
• \(q_0, q_1\) are functions of the data matrices
• \(\text{tr}(\cdot)\) denotes the matrix trace

More specifically, for the AR(1) model with exogenous regressors:

\[ c_1 = \sigma^2 (W'AW)^{-1} \sum_{j=0}^{T-2} \text{tr}(A \, C^j) \cdot (W'A)_{(\cdot, j+1)} \]

where \(C\) is the companion matrix encoding the AR dynamics.

Order 2: \(O(N^{-1}T^{-1})\)

The second-order term corrects for the interaction between the cross-sectional and time-series dimensions:

\[ c_2 = \frac{\sigma^2}{N} (W'AW)^{-1} \left[\text{tr}(B_1) \cdot h_1 + \text{tr}(B_2) \cdot h_2\right] \]

where \(B_1, B_2\) are functions of \(A\), the companion matrix, and the data.

Order 3: \(O(N^{-1}T^{-2})\)

Bun & Kiviet (2003) derive the third-order term, which provides a further refinement:

\[ c_3 = \frac{\sigma^2}{NT} (W'AW)^{-1} \left[\text{tr}(G_1) \cdot r_1 + \text{tr}(G_2) \cdot r_2 + \text{tr}(G_3) \cdot r_3\right] \]

Practical Guidance on Bias Order

• Order 1 (bias_order=1): Simplest; captures the dominant bias from the within transformation. Sufficient when T is moderate.
• Order 2 (bias_order=2): Recommended default. Adds the \(O(N^{-1}T^{-1})\) correction, which matters when both N and T are small.
• Order 3 (bias_order=3): Bun & Kiviet (2003) refinement. Gains over order 2 are typically small; use when maximum accuracy is needed.

Iterated Bias Correction

Since the bias formulas depend on the unknown parameters \(\delta\) and \(\sigma^2\), LSDVC uses an iterative procedure:

1. Initialize: Obtain a consistent estimate \(\hat{\rho}_0\) from an initial estimator (Anderson-Hsiao, Arellano-Bond, or Blundell-Bond)
2. LSDV: Compute \(\hat{\delta}_{LSDV}\) and \(\hat{\sigma}^2\) from Fixed Effects
3. Correct: \(\hat{\delta}^{(k+1)} = \hat{\delta}_{LSDV} - \widehat{\text{Bias}}(\hat{\delta}^{(k)}, \hat{\sigma}^2)\)
4. Iterate: Repeat step 3 until \(\|\hat{\delta}^{(k+1)} - \hat{\delta}^{(k)}\| < \text{tol}\)

The initial consistent estimator is needed only to start the iteration. The final LSDVC estimates are typically robust to the choice of initial estimator, though convergence may differ.

Bootstrap Inference

Why Bootstrap?

The analytical bias correction produces corrected point estimates, but the asymptotic distribution of \(\hat{\delta}_{LSDVC}\) is not straightforward. Standard errors from the LSDV variance-covariance matrix are not valid for the corrected estimator. Bruno (2005) proposes parametric bootstrap for inference.

Parametric Bootstrap Procedure

For \(b = 1, \ldots, B\) bootstrap replications:

1. Generate errors: \(\varepsilon^*_{it} \sim N(0, \hat{\sigma}^2)\)

2. Simulate panel data from the estimated DGP:

   \[ y^*_{it} = \hat{\rho} \, y^*_{i,t-1} + X_{it}'\hat{\beta} + \hat{\alpha}_i + \varepsilon^*_{it} \]

   using the original \(X_{it}\) and estimated fixed effects \(\hat{\alpha}_i\)

3. Re-estimate LSDVC on the simulated panel → \(\hat{\delta}^*_b\)

The bootstrap distribution \(\{\hat{\delta}^*_1, \ldots, \hat{\delta}^*_B\}\) provides:

• Standard errors: \(\text{SE}(\hat{\delta}_k) = \text{sd}(\hat{\delta}^*_{1,k}, \ldots, \hat{\delta}^*_{B,k})\)
• Confidence intervals: Percentile method using the \(\alpha/2\) and \(1-\alpha/2\) quantiles
• z-statistics and p-values: Using bootstrap standard errors with normal approximation

Number of Bootstrap Replications

• 200-500 replications are sufficient for standard errors
• 1000+ replications recommended for accurate confidence interval endpoints
• Use seed parameter for reproducibility

Comparison with GMM

Both LSDVC and GMM address the Nickell bias, but they take fundamentally different approaches:

Feature	LSDVC	Difference GMM	System GMM
Approach	Analytical bias correction	Moment conditions + instruments	Moment conditions + instruments
Asymptotics	Fixed N, fixed T	N → ∞, fixed T	N → ∞, fixed T
Small N	Works well (N < 100)	Unreliable	Unreliable
Instruments	Not needed (uses initial estimator)	Lagged levels	Lagged levels + differences
Exogenous regressors	Assumed strictly exogenous	Can handle predetermined/endogenous	Can handle predetermined/endogenous
Inference	Bootstrap	Asymptotic (Windmeijer-corrected)	Asymptotic (Windmeijer-corrected)
Overidentification test	Not applicable	Hansen J / Sargan	Hansen J / Sargan
Persistent series	Works with any \(\rho\)	Weak instruments for high \(\rho\)	Better for high \(\rho\)

When to Use Each

• LSDVC: Small to moderate N (N < 100), strictly exogenous regressors, preference for simplicity
• Difference GMM: Large N, predetermined or endogenous regressors, need for overidentification tests
• System GMM: Large N, highly persistent processes, near-unit-root data

Key limitation of LSDVC

LSDVC assumes that all regressors in \(X_{it}\) are strictly exogenous — they cannot be correlated with past, present, or future errors. If you have predetermined or endogenous regressors, use GMM instead.

Extension to Unbalanced Panels

Bruno (2005) extends the Kiviet bias correction to unbalanced panels. The key modifications are:

• Entity-specific time dimensions \(T_i\) replace the common \(T\)
• The within-transformation matrix \(A\) accounts for varying \(T_i\)
• Bias formulas sum over entity-specific contributions weighted by \(T_i\)

The PanelBox implementation handles unbalanced panels automatically.

References

1. Nickell, S. (1981). "Biases in Dynamic Models with Fixed Effects." Econometrica, 49(6), 1417-1426.

2. Anderson, T. W., & Hsiao, C. (1982). "Formulation and Estimation of Dynamic Models Using Panel Data." Journal of Econometrics, 18(1), 47-82.

3. Kiviet, J. F. (1995). "On Bias, Inconsistency, and Efficiency of Various Estimators in Dynamic Panel Data Models." Journal of Econometrics, 68(1), 53-78.

4. Bun, M. J. G., & Kiviet, J. F. (2003). "On the Diminishing Returns of Higher-Order Terms in Asymptotic Expansions of Bias." Economics Letters, 79(2), 145-152.

5. Bruno, G. S. F. (2005). "Approximating the Bias of the LSDV Estimator for Dynamic Unbalanced Panel Data Models." Economics Letters, 87(3), 361-366.

6. Arellano, M., & Bond, S. (1991). "Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations." Review of Economic Studies, 58(2), 277-297.

7. Blundell, R., & Bond, S. (1998). "Initial Conditions and Moment Restrictions in Dynamic Panel Data Models." Journal of Econometrics, 87(1), 115-143.