Extracting the Influential Commodities in Stochastic Model of Simple Laspeyre Price Index Numbers with AR(2) Errors

This paper, on the first hand, deals with the problem of estimation of Laspeyre price index number when the errors are assumed to be generated from AR(2) process. The general expression of hat matrix and DFBETA measure to find the influential consumer commodities in stochastic Laspeyre price model with AR(2) errors are developed on the other. The hat values show the noteworthy findings that the corresponding weights of consumer items have large influence on the parameter estimates for simple Laspeyre price index number and are not affected by the parameter of autoregressive process of order two. While, DFBETA measures are the functions of both weights and autocorrelation parameters. Lastly, an example is presented with reference to price data of Pakistan, and shows its practical importance in financial time series.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Maqsood, A. and Burney, S. (2014) Extracting the Influential Commodities in Stochastic Model of Simple Laspeyre Price Index Numbers with AR(2) Errors. Open Journal of Statistics, 4, 220-229. doi: 10.4236/ojs.2014.43021.

 [1] Maqsood, A. and Burney, S.M.A. (2008) Study of Inflation in Pakistan Using Statistical Approach. 4th International Statistical (ISSOS), University of Gujrat, Hafiz Hayat Campus, Pakistan. [2] Clements, K.W. and Izan, H.Y. (1981) A Note on Estimating Divisia Index Numbers. International Economic Review, 22, 745-747. http://dx.doi.org/10.2307/2526174 [3] Clements, K.W. and Izan, H.Y. (1987) The Measurement of Inflation: A Stochastic Approach. Journal of Business and Economic Statistics, 5, 339-350. [4] Selvanathan, E.A. (1991) Standard Errors for Laspeyers and Paasche Index Numbers. Economics Letters, 35, 35-38.http://dx.doi.org/10.1016/0165-1765(91)90101-P [5] Selvanathan, E.A. (1993) More on the Laspeyers Price Index. Economics Letters, 43, 157-162.http://dx.doi.org/10.1016/0165-1765(93)90029-C [6] Selvanathan, E.A. (2003) Extending the Stochastic Approach to Index Numbers: A Comment. Applied Economics Letters, 10, 213-215. http://dx.doi.org/10.1080/1350435022000043986 [7] Burney, S.M.A. and Maqsood, A. (2013) Extending the Stochastic Approach to Paasches Price Index Numbers. Pakistan Journal of Engineering Technology & Science, 3, 1-17. [8] Prais, G.J. and Winsten, C.B. (1954) Trend Estimates and Serial Correlation. Cowles Commission Discussion Paper, Stat. No. 383, University of Chicago, Chicago. [9] Kadiyala, K.R. (1968) A Transformation Used to Circumvent the Problem of Autocorrelation. Econometrica, 36, 9396. http://dx.doi.org/10.2307/1909605 [10] Griliches, Z. and Rao, P. (1969) Small-Sample Properties of Several Two-Stage Regression Methods in the Context of Autocorrelated Disturbances. Journal of American Statistical Association, 64, 253-272. http://dx.doi.org/10.1080/01621459.1969.10500968 [11] Maeshiro, A. (1979) On the Retention of the First Observation in Serial Correlation Adjustment of Regression Models. International Economic Review, 20, 259-265. http://dx.doi.org/10.2307/2526430 [12] Park, R.E. and Mitchell, B.M. (1980) Estimating the Autocorrelated Error Model with Trended Data. Journal of Econometrics, 13, 185-201. http://dx.doi.org/10.1016/0304-4076(80)90014-7 [13] Belsley, P.A., Kuh, E. and Welsch, R.E. (1980) Regression Diagnostics. John Wiley, New York.http://dx.doi.org/10.1002/0471725153 [14] Cook, R.D. (1977) Detection of Influential Observations in Linear Regression. Technometrics, 19, 15-18.http://dx.doi.org/10.2307/1268249 [15] Cook, R.D. (1979) Influential Observations in Linear Regression. Journal of American Statistical Association, 74, 169174. http://dx.doi.org/10.1080/01621459.1979.10481634 [16] Cook, R.D. and Weisberg, S. (1982) Residuals and Influence in Regression. Chapman and Hall, New York. [17] Draper, N.R. and John, J.A. (1981) Influential Observations and Outliers in Regression. Technometrics, 23, 21-26. http://dx.doi.org/10.1080/00401706.1981.10486232 [18] Draper, N.R. and Smith, H. (1998) Applied Regression Analysis. 3rd Edition, John Wiley, New York. [19] Puterman, M.L. (1988) Leverage and Influence in Autocorrelated Regression Model. Journal of the Royal Statistical Society, 37, 76-86. [20] Cochrane, D. and Orcutt, G.H. (1949) Application of Least Squares Regression to Relationships Containing Auto-Correlated Error Terms. Journal of the American Statistical Association, 44, 32-61. [21] Stemann, D. and Trenkler, G. (1993) Leverage and Cochrane-Orcutt Estimation in Linear Regression. Communications in Statistics-Theory and Methods, 22, 1315-1333. http://dx.doi.org/10.1080/03610929308831088 [22] Pena, D. (2005) A New Statistic for Influence in Linear Regression. Technometrics, 47, 1-12. http://dx.doi.org/10.1198/004017004000000662 [23] Turkan, S. and Toktamis, O. (2012) Detection of Influential Observations in Ridge Regression and Modified Ridge Regression. Model Assisted Statistics and Applications, 7, 91-97. [24] Turkan, S. and Toktamis, O. (2013) Detection of Influential Observations in Semiparametric Regression Model. Revista Colombiana de Estadistica, 36, 91-97. [25] Barry, A.M., Burney, S.M.A. and Bhatti, M.I. (1997) Optimum Influence of Initial Observations in Regression Models with AR(2) Errors. Applied Mathematics and Computation, 82, 57-65. http://dx.doi.org/10.1016/S0096-3003(96)00024-0 [26] Burney, S.M.A. and Maqsood, A. (2014) Influential Observations in Stochastic Model of Divisia Index Numbers with AR(1) Errors. Applied Mathematics, 5, 975-982. http://dx.doi.org/10.4236/am.2014.56093 [27] Wise, J. (1955) The Autocorrelation Function and the Spectral Density Function. Biometrika, 42, 151-159. http://dx.doi.org/10.2307/2333432 [28] Pakistan Bureau of Statistics. Monthly Bulletin of Statistics. http://www.pbs.gov.pk/sites/default/files/tables/Monthly%20Bulletin%20Of%20Statistics.pdf