#### Date of Submission

2-28-1981

#### Date of Award

2-28-1982

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Economic Research Unit (ERU-Kolkata)

#### Supervisor

Bhattacharya, Nikhilesh (ERU-Kolkata; ISI)

#### Abstract (Summary of the Work)

In many econometric investigations, the 'errors-in variables' (EIVs) are not negligible (Morgenstern, 1963). Examination of 25 series relating to national accounts by Langaskens and Rijekeghan (1974) showed that the standard deviations of the errors ranged from 5 to 77 per cent of the average value of the corresponding variable. Such errors may vitiate least-squares (LS) estimation of regression coefficients (Johnaton, 1972). The well-known methods (ML; IV, including grouping method) proposed for handling classical EIV model (EMM) in regression analysis muffer from serious linitations. Same of them make strong distributional assumptions about the errors (and the regressors) and/or assume prier knowledge about the values of the error variancen; othera need auxiliary variables called inatrumental-variables (IVs) which are supposed to be uncorrelated with the error tems, but strongly correlated with the true regressors. The IVs are thus not always handy and, in any case, one can never check the assumptions.y1 = a + 8 X i- 1, 2, .., n .(0.1) where a and 8 are paraneters to be estimated; e, is the disturbance term distributed normally wi th mean zero and variance o for all and X1, and Yi, are norobservable true values of the regressor and the coy of CV 2 regres and respectively. The e's are assumed to be independent of X'e where X is stochastic. The observed values x, and y, are written as (0.2) where u, and vá»‹ are the EIVe which are independent of each other and of the true values Xi, and Yi,. u, s and v,s have neans zero and variances o and o respectively for all i. Fori- 1, 2, ..., n, we assume that (X, Y, u, v, ) are i.i.d. random variables.one finds that 'ordinary least aquares' (OLS) regression of y on x gives an inconsistent estimator of 8 essentially because eov(xi, wi)0 (Johns ton, 1972, p.262). Extension of this model to more than one and to regressor is obvious. Other important extensions allow u, be correlated or the distribution of ui, to depend on the valIhe of Xi,. Various alternative methods of estimation have been suggested by previous researchers. These are based on different sets of assump- tions. Thas, some assume X to be stochastic while others do not. Chapter 1 makes a critical survey of the different assumptions made in the literature on the distribution of errors and of the regressor X and reviews the different methods of estimation Å›uggested so far. There are, of course, some models which oan not be fully identified at all (vide Section 1.6 of Chapter 1 see also Appendix 4.2 of Chapter 4). It may be mentioned here that some good review articles on EVM already exist in the literature (Durbin, 1954; Madansky, 1959; Cochran, 1968; Moran, 1971; Pal, 1980a). Among other things, this chapter diecusses how one can ob tain consistent estimators of 6 if (i) one has prior knowledge about the value of the error variances or of their ratio or if (ii) IVare available. Introduction of lagged values of regressors/ regressand may also be helpful in finding consistent estimates of the parameters. Sometimes in the laboratory experiments repeated meagure- ments are available for the same value of the variable. This may help in finding oÃ³nsistent estimates. The problem becomes more difficult if instead of one relation we have many relations in the model, but the variables are affected by EIV apart from economists, sociologists have long beon applying such simul taneous equations models in path analysis and multiple indicator analysis.

#### Control Number

ISILib-TH57

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Pal, Manoranjan Dr., "Estimation in Errors-In-Variables Models." (1982). *Doctoral Theses*. 295.

https://digitalcommons.isical.ac.in/doctoral-theses/295

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843331