Skip to content

Theory

Panel Cointegrating Regression

The model estimated by PyCupFM is:

\[y_{it} = \alpha_i + \beta' x_{it} + e_{it}, \quad i = 1,\ldots,N; \; t = 1,\ldots,T\]

where the error \(e_{it}\) has a common factor structure:

\[e_{it} = \lambda_i' F_t + u_{it}\]

Variables

  • \(y_{it}\): dependent variable (I(1))
  • \(x_{it}\): \(k \times 1\) vector of I(1) regressors with \(x_{it} = x_{i,t-1} + v_{it}\)
  • \(F_t\): \(r \times 1\) vector of common factors — I(1) global stochastic trends
  • \(\lambda_i\): \(r \times 1\) heterogeneous factor loadings
  • \(u_{it}\): idiosyncratic error (stationary, may be serially correlated)
  • \(\alpha_i\): unit-specific fixed effects
  • \(\beta\): \(k \times 1\) homogeneous cointegrating coefficients

The 5 Estimators

1. LSDV (Within / Fixed-Effects)

Standard panel within estimator. Eliminates \(\alpha_i\) by demeaning:

\[\hat\beta_{LSDV} = \left(\sum_i \tilde{X}_i' \tilde{X}_i\right)^{-1} \sum_i \tilde{X}_i' \tilde{y}_i\]

where \(\tilde{X}_i = X_i - \bar{X}_i\). Biased and inconsistent when \(F_t \sim I(1)\).

2. Bai FM (Two-Step Fully Modified)

From Bai & Kao (2005), Equations 7–8. Non-iterative:

  1. Estimate LSDV → get \(\hat\beta_0\), extract factors \(\hat{F}\), \(\hat\Lambda\)
  2. Construct FM correction using Bartlett long-run covariance \(\hat\Omega\)
  3. Apply bias correction once

From BKN (2009), Theorem 3, Equation 16. Iterates the Bai FM procedure:

\[\hat\beta^{(j)} \to \text{residuals} \to \text{PCA} \to \hat\Omega \to \text{FM correction} \to \hat\beta^{(j+1)}\]

Asymptotic distribution (BKN 2009, Theorem 3):

\[\sqrt{NT}\left(\hat\beta_{CupFM} - \beta\right) \xrightarrow{d} N\left(0, 6\Omega_{uu \cdot x} / \Sigma_{xx}\right)\]

Exhibits the smallest bias in all BKN Monte Carlo experiments.

4. CupFM-bar (Z-bar Variant)

Uses the instrument \(\bar{Z}_i = \bar{x}_i - \hat{F}\hat\delta_i\) instead of \(X_i\).

5. CupBC (Continuously-Updated Bias-Corrected)

From BKN (2009), Theorem 2. Iterates plain Cup-LS:

\[\hat\beta_{BC}^{(j)} = \left(\sum_i X_i' M_{\hat{F}} X_i\right)^{-1} \sum_i X_i' M_{\hat{F}} y_i\]

then applies bias correction at convergence.

Long-Run Covariance Estimation

The long-run covariance matrix is estimated using kernel methods:

\[\hat\Omega = \frac{1}{T}\sum_{j=-M}^{M} w_j \sum_t z_t z_{t-j}'\]

Available Kernels

Kernel Weight Function
Bartlett \(w_j = 1 - \|j\|/(M+1)\)
Parzen Smooth taper (cubic)
Quadratic Spectral Andrews (1991)

Bandwidth Selection

  • Manual: bandwidth=5 (BKN default)
  • Auto (Newey-West): \(M = \lfloor 4(T/100)^{2/9} \rfloor\)
  • Auto (Andrews): AR(1) plug-in rule

Factor Selection

The number of factors \(r\) is selected by minimizing the Bai & Ng (2002) information criterion:

\[IC_1(k) = \ln V(k, \hat{F}) + k \cdot \frac{N+T}{NT} \cdot \ln\frac{NT}{N+T}\]

where \(V(k, \hat{F}) = \frac{1}{NT}\sum_i\sum_t (e_{it} - \hat\lambda_i'\hat{F}_t)^2\).

References

  1. Bai, J., Kao, C. & Ng, S. (2009). Panel cointegration with global stochastic trends. Journal of Econometrics, 149(1), 82-99.
  2. Bai, J. & Kao, C. (2005). On the estimation and inference of a panel cointegration model with cross-sectional dependence. SSRN-1815227.
  3. Bai, J. & Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica, 70(1), 191-221.
  4. Andrews, D.W.K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817-858.