Spatial Nonstationarity and Spurious Regression: the Case with a Row-normalized Spatial Weights Matrix

Abstract

This paper investigates the spurious regression in the spatial setting where the regressant and regressors may be generated from possible nonstationary spatial autoregressive processes. Under the near unit root specification with a row-normalized spatial weights matrix, it is shown that the possible spurious regression phenomena in the spatial setting are relatively weaker than those in the nonstationary time series scenario. The regression estimates might or might not converge to 0. The divergence might occur only when the regressant has a near unit root much closer to unity than that of the regressor. For the t and F statistics, there could be over-rejection of the null of uncorrelatedness under certain situations, but they do not diverge. However, the coefficient of determination R ² converges to 0, which provides strong evidence of the spurious regression even when t and F statistics are large. Simulation results about different statistics are in line with the theoretical results we derive in this paper.

Non-stationnarité spatiale et fausse régression: l'argument pour la matrice de pondération spatiale à normalisation ‘row-normalized’

RÉSUMÉ La présente communication se penche sur la fausse régression dans les cadres spatiaux, o[ugrave] des variables dépendantes et des variables explicatives peuvent être produites par d’éventuels procédés autorégressifs spatiaux non stationnaires. Dans le cadre de la spécification de la racine quasi-unitaire, avec une matrice de pondération spatiale normalisée ‘row-normalized’, il est démontré que les phénomènes de fausse régression dans les cadres spatiaux sont relativement plus faibles que ceux du scénario à série chronologique non stationnaire. Pour les statistiques t et F, on pourra assister à une sur-réjection du néant de la non corrélation dans certaines circonstances, mais aucune divergence. Toutefois, le coefficient de détermination R² converge vers 0, en apportant ainsi une preuve substantielle de la fausse, même en présence de statistiques t et F élevées. Les résultats des simulations sur différentes statistiques sont en accord avec les résultats théoriques que nous dérivons dans la présente communication.

No estacionariedad espacial y regresión falsa: el caso con la matriz de pesos espaciales standardizada por filas

RÉSUMÉ Este trabajo investiga la regresión falsa en el ámbito espacial donde la variable dependiente y las variables independientes pueden generarse a partir de posibles procesos autorregresivos espaciales no estacionarios. Bajo la especificación de raíz unitaria con una matriz de pesos espaciales estandarizada por filas, se muestra que los posibles fenómenos de regresión falsa son relativamente más débiles que los del caso de la serie de tiempo no estacionario. En las estadísticas t y F, podría producirse un sobrerrechazo de la hipótesis nula de incorrelación bajo ciertas situaciones, pero no son divergentes. No obstante, el coeficiente de determinación R² converge a 0, lo que ofrece una evidencia fuerte de la regresión falsa incluso cuando las estadísticas t y F son amplias. Los resultados de simulación sobre diferentes estadísticas se mantienen en línea con los resultados teóricos que obtenemos en este trabajo.

Keywords:

• Near unit root
• spatial nonstationarity
• spurious regression

JEL CLASSIFICATION:

• C13
• C23
• R15

Notes

1. The estimation and testing for spatial dependence in cross sectional data can be found in Anselin (1988, 1992), Kelejian & Robinson (1993), Cressie (1993), Anselin & Florax (1995), Anselin & Rey (1997), Anselin & Bera (1998), Kelejian & Prucha (1998, 2001, 2007) and Lee (2003, 2004, 2007), among others.

2. When W _jn is row normalized from a symmetric matrix, W _jn is diagonalizable. See Lemma A.1 in Yu et al. (2007). A weights matrix row normalized has real eigenvalues, with all its eigenvalues less than or equal to 1 in absolute value and its largest eigenvalue always 1 (see Ord, 1975). Hence, W _jn being diagonalizable with the specified eigenvalues is a slight generalization of W _jn being row-normalized from a symmetric matrix.

3. In a time series near unit root model, the deviation from the unit root is measured through a noncentrality parameter, where the AR(1) coeffi cient is usually specified as exp(c/n) or 1 − c/n, with c being the noncentrality parameter (see Phillips, 1987). For the near unit root in the SAR model, ψ _n can take a general form provided it is increasing in n, which does not need to be specified in empirical applications.

4. This property might be useful for the estimation. However, we do not explore the use of this spatial difference operator, I _n −W _jn , in this paper.

5. We say a (sequence of n×n) matrix P _n is uniformly bounded in row and column sum norms if and , where is the row sum norm and is the column sum norm.

6. As our asymptotic analyses below can allow both the near unit root and stable cases, this generality provides a unified framework for our study.

7. When both λ _1n and λ _jn are near unit root, one can easily evaluate the variance of in (13) via and , and the t-statistic can be adjusted to be asymptotically N(0,1) distributed. For a general case, one needs to estimate λ values in order to adjust such a variance.

8. Equivalently, it is asymptotically a weighted sum of (m−1) independent χ ²(l) random variables.

9. For the estimates of α, the values are large, and they diverge when ψ _1n =1,000, while the t-statistics do not diverge but have a fat tail compared to the standard normal distribution. As the inference of α is not of much interest, we do not report the relevant statistics.

10. In an additional simulation where we set ψ _1n =2 so that λ _1n is not close to 1, I _Moran is smaller with a mean value of 0.4831 for ψ _2n =2 and 0.4828 for ψ _2n =1,000. Hence, the Moran I statistic is close to 1 only when ψ _1n is large, which is implied by (16).

11. Using the 4×4 queen matrix, we can increase the number of blocks from 11 to 125 to observe the effect of the sample size. Prom the results (due to space limitations, the tables are not presented), the effect of the sample size on , t and Moran I statistics are not apparent. However, and the LM statistic are increasing in n and R ² is decreasing in n.

12. The values of are 3.9656, 3.9656 and 4, respectively when λ _1n =0.95 and λ _2n =0.999, λ _1n =0.999 and λ _2n =0.95, and λ _1n =λ _2n =0.999.

13. The other results are similar. Due to space limitations, we do not report them in the figures.

14. For example, R ² for his MC results shows some relations between unrelated Y _n and X _n . However, this might be due to smaller sample sizes than ours.