Multicollinearity in cross-sectional regressions

Abstract

The paper examines robustness of results from cross-sectional regression paying attention to the impact of multicollinearity. It is well known that the reliability of estimators (least-squares or maximum-likelihood) gets worse as the linear relationships between the regressors become more acute. We resolve the discussion in a spatial context, looking closely into the behaviour shown, under several unfavourable conditions, by the most outstanding misspecification tests when collinear variables are added to the regression. A Monte Carlo simulation is performed. The conclusions point to the fact that these statistics react in different ways to the problems posed.

Appendix: Misspecification test used

The tests used always refer to the model of the null hypothesis; that is, of the static type such as: y =  Xβ +  u. This model has been estimated by LS, where $\hat{\sigma}^{2}$ and $\hat{\beta}$ are the corresponding LS estimations and $\hat{u}$ the residual series. The tests are the following (see Anselin and Florax 1995; or Florax and de Graaff 2004, for details):

Moran’s I test:

$$ I = \frac{R}{{S_{0}}}\frac{{\hat{u}'W\hat{u}}}{{\hat{u}'\hat{u}}}; \quad S_{0} = {\sum\limits_{r = 1}^R {{\sum\limits_{s = 1}^R {w_{{rs}}}}}} $$

(18)

LM-ERR test:

$$ \text{LM-ERR} = \frac{1}{{T_{1}}}\left(\frac{{\hat{u}W\hat{u}}}{{\hat{\sigma}^{2}}}\right)^{2}; \quad T_{1} = {\rm tr}(W'W + W^{2}) $$

(19)

LM-EL test:

$$ \text{LM-EL} = \frac{{\left(\frac{{\hat{u}'W\hat{u}}}{{\hat{\sigma}^{2}}} - \frac{{T_{1}}}{{R_{j}}}\frac{{\hat{u}'Wy}}{{\hat{\sigma}^{2}}}\right)^{2}}}{{T_{1} - \frac{{T^{2}_{1}}}{{R_{j}}}}} $$

(20)

KR test:

$$ {\rm KR} = h_{R} \frac{{\hat{\gamma}'Z'Z\hat{\gamma}}}{{\hat{e}`\hat{e}}} $$

(21)

LM-LAG test:

$$ \text{LM-LAG} = \frac{1}{{R_{j}}}\left(\frac{{\hat{u}'Wy}}{{\hat{\sigma}^{2}}}\right )^{2}$$

(22)

LM-LE test:

$$ \text{LM-LE} = \frac{{\left(\frac{{\hat{u}'Wy}}{{\hat{\sigma}^{2}}} - \frac{{\hat{u}'W\hat{u}}}{{\hat{\sigma}^{2}}}\right)^{2}}}{{R_{j} - T_{1}}} $$

(23)

SARMA test:

$$ \text{SARMA}= \frac{{\left(\frac{{\hat{u}'Wy}}{{\hat{\sigma}^{2}}} - \frac{{\hat{u}'W\hat{u}}}{{\hat{\sigma}^{2}}}\right)^{2}}}{{R_{j} - T_{1}}} + \frac{1}{{T_{1}}}\left(\frac{{\hat{u}W\hat{u}}}{{\hat{\sigma}^{2}}}\right)^{2} $$

(24)

Moreover, $R_{j} = T_{1} + \frac{{\hat{\beta}'X'W`MWX\hat{\beta}}}{{\hat{\sigma}^{2}}}$ and M =  I−X(X′X)^{− 1} X′. Furthermore, $\hat{e}$ is the vector of residuals from the auxiliary regression of the Kelejian–Robinson (KR) test, of order h _R ×  1, Z is the matrix of exogenous variables included in the last regression and $\hat{\gamma}$ the estimated coefficients obtained for the corresponding vector of parameters.

As is well-known, the asymptotic distribution of the standardised Moran’s I, obtained as $\frac{{I - E(I)}}{{{\sqrt {V(I)}}}}$, with $E(I) = \frac{R}{{S_{0} (R - k)}}{\rm tr}(MW)$ and

$$ V(I) = \left(\frac{R}{{S_{0}}}\right)^{2} \frac{{\{{\rm tr}(MWMW') + {\rm tr}(MWMW) + {\rm tr}^{2} (MW)\} - \{E(I)\} ^{2}}}{{(R - k)(R - k - 2)}}, $$

is an N(0,1); the two Lagrange Multipliers that follow, LM-ERR and LM-EL, have an asymptotic χ ² (1), the distribution of the KR test is a χ ²(m), with m being the number of regressors included in the auxiliary regression. The three final tests also have a chi-square distribution, with one degree of freedom in the first two, and two degrees of freedom in the SARMA test.