# Local Sensitivity in the Inequality Restricted Linear Model

Diagnostic testing has traditionally been an important aspect of statistical modeling, but in recent years, sensitivity analysis has also been drawing increasing attention from econometricians and statisticians. Essentially, a diagnostic test ascertains if the model coincides with the assumed data generating process, while sensitivity analysis investigates if it matters at all that the model deviates from what is being assumed. That is, sensitivity analysis answers the question of whether a wrong model is still useful for certain purposes, and if so, it matters little that the model may be incorrect. For example, Banerjee and Magnus (1999) pointed out that the ordinary least squares (OLS) estimator of the coefficients in a linear regression model is in fact not very sensitive to disturbances’ deviation from the white noise assumption. Consequently, it is quite usual to find the estimates of the parameters not changing much after fitting the model with a more general covariance structure. However, the *F* and *t* tests based on the OLS residuals are sensitive to covariance misspecification in the sense that a small stepping away from white noise disturbances is likely to cause a substantial distortion in the significance levels of the tests (Banerjee and Magnus (2000)).

The current paper continues this line of research. We are concerned with a linear regression with a possibly incorrect inequality restriction (as opposed to strict equality restrictions as in Wan, Zou and Qin (2007)) on the coefficients. In econometric applications inequality restrictions frequently arise on the parameters. Finite sample properties of the inequality constraint least squares (ICLS) estimator have been investigated by Thomson (1982), Judge and Yancey (1986), Wan (1994a), Wan (1994b), among others. Judge and Yancey (1986), Wan (1994a), Wan (1994b), Wan (1995) and Wan (1996) considered the properties of the so-called inequality pre-test (IPT) estimator which chooses between the inequality restricted and OLS estimators depending on the outcome of a one-sided t test. In this paper, we investigate the sensitivity of the ICLS and IPT estimators to deviations of the disturbances from the white noise assumption. In the spirit of Banerjee and Magnus (1999), we propose sensitivity measures on the ICLS and IPT estimators to covariance misspecification and investigate the properties of these measures allowing for both correctly and incorrectly specified constraints.

The rest of this paper is organized as follows. Section 2 gives some preliminary results and defines sensitivity statistics to measure the sensitivity of the ICLS and IPT coefficient and variance estimators to covariance misspecification. Section 3 emphasizes the case of AR(1) disturbances and derives results concerning the limiting behavior of the sensitivity statistics when the AR(1) parameter is near the unitroot. Section 4 presents numerical findings on the sensitivity of the estimators under a variety of AR(1) and MA(1) settings and Section 5 concludes. Proofs of theorems are contained in Appendix A.

## Keywords

Ordinary Little Square Generalize Little Square Sensitivity Statistic Local Sensitivity Sensitivity Curve## Preview

Unable to display preview. Download preview PDF.

## References

- Banerjee A, Magnus JR. (1999) The sensitivity of OLS when variance matrix is (partially) unknown. Journal of Econometrics 92:295-323MATHCrossRefGoogle Scholar
- Banerjee A, Magnus JR (2000) On the sensitivity of the usual t- and F - tests to covariance misspecification. Journal of Econometrics 95:157-176MATHCrossRefMathSciNetGoogle Scholar
- Judge GG, Yancey TA (1986) Improved methods of inference in econometrics. North-Holland, AmsterdamMATHGoogle Scholar
- Magnus JR (1999) The traditional pre-test estimators. Theory of Probability and Its Applications 44:293-308MATHCrossRefMathSciNetGoogle Scholar
- Magnus JR, Vasnev AL (2007) Local sensitivity and diagnostic tests. Econometrics Journal 10:166-192MATHCrossRefMathSciNetGoogle Scholar
- Qin HZ, Wan ATK, Zou GH (2007) On the sensitivity of the one-sided t-test to covariance misspecification. mimeo., City University of Hong KongGoogle Scholar
- Thomson M (1982) Some results on the statistical properties of an inequality constrained least squares estimator in a linear model with two regressors. Journal of Econometrics 19:215-231CrossRefMathSciNetGoogle Scholar
- Wan ATK (1994a) The sampling performance of inequality restricted and pre-test estimators in a mis-specified linear model. Australian Journal of Statistics 36:313-325MATHCrossRefGoogle Scholar
- Wan ATK (1994b) Risk comparison of the inequality constrained least squares and other related estimators under balanced loss. Economics Letters 46:203-210CrossRefGoogle Scholar
- Wan ATK (1995) The optimal critical value of a pre-test for an inequality restriction in a mis-specified regression model. Australian Journal of Statistics 37:73-82MATHCrossRefGoogle Scholar
- Wan ATK (1996) Estimating the error variance after a pre-test for an inequality restriction on the coefficients. Journal of Statistical Planning and Inference 52:197-213MATHCrossRefMathSciNetGoogle Scholar
- Wan ATK, Zou GH (2003) Optimal critical values of pre-tests when estimating the regression error variance: analytical findings under a general loss structure. Journal of Econometrics 114:165-196MATHCrossRefMathSciNetGoogle Scholar
- Wan ATK, Zou GH, Ohtani K (2006) Further results on optimal critical values of pre-test when estimating the regression error variance. Econometrics Journal 9:159-176MATHCrossRefMathSciNetGoogle Scholar
- Wan ATK, Zou GH, Qin HZ (2007) On the sensitivity of restricted least squares estimators to covariance misspecification. Econometrics Journal 10:471-487MATHCrossRefMathSciNetGoogle Scholar