Bose, Arup; Mukherjee, Kanchan Estimating the ARCH parameters by solving linear equations. (English) Zbl 1113.62095 J. Time Ser. Anal. 24, No. 2, 127-136 (2003). The authors deal with the ARCH processes \(\{X_i,1-p\leq i\leq n\}\) determined by the equation \(X_i=\sigma_{i-1}(\beta)\varepsilon_i, 1\leq i\leq n,\) where \(\beta' = [\beta_0,\beta_1,\dots,\beta_p]\) is the unknown parameter with \(\beta_0>0,\beta_j\geq0,1\leq j\leq p\); \(\sigma_{i-1}(\beta)= \{\beta_0+\beta_1X^2_{i-1}+\cdots+\beta_pX^2_{i-p}\}^{1/2}\), \(\{\varepsilon_i:1\leq i\leq n\}\) are independent and identically distributed with zero mean, unit variance and finite fourth moment (independent of \(\{X_i:1-p\leq i\leq0\}\)). They are concerned with the problem of estimating the parameter \(\beta\). It is assumed that the process \(\{X_i:1-p\leq i\}\) is stationary and ergodic. One of the most commonly used estimation procedures for ARCH models is the Gaussian likelihood approach where the estimator is obtained as a maximizer of the logarithm of a Gaussian likelihood function. The resulting estimator is called quasi-maximum likelihood estimator (QMLE). This yields a consistent estimator even when the conditional error density is not normal. The authors of this paper propose another approach to estimating the parameter \(\beta\). Let \(Y_i=X_i^2\), \(Z_{i-1}=[1,Y_{i-1},\dots,Y_{i-p}]'\), and let \(\eta_i=\varepsilon_i^2-1\). Then \(\sigma^2_{i-1}(\beta)=Z'_{i-1}\beta\), \(Y_i=Z'_{i-1}\beta+\sigma^2_{i-1}(\beta )\eta_i\). Ignoring the randomness of \(\sigma^2_{i-1}(\beta)\) and also the presence of \(\beta\) in it, one can obtain a (preliminary) least squares estimator of \(\beta\) as \(\widehat{\beta}_{pr}=(ZZ')^{-1}Z'Y\), where \(Z\) is the matrix of order \(n\times(1+p)\) with i-th row equal to \(Z'_{i-1}\), and \(Y\) is the vector with the \(i\)th entry \(Y_i,i\leq i\leq n\). Next \(\widehat{\beta}_{pr}\) is used to construct an improved estimator \(\widehat{\beta}\) of \(\beta\) as \[ \widehat{\beta}= \left[\sum_{i=1}^n \left\{(Z_{i-1}Z'_{i-1})/\sigma^4_{i-1}(\widehat{\beta}_{pr})\right\}\right]^{-1} \left[\sum_{i=1}^n \left\{(Z_{i-1}Y_{i})/\sigma^4_{i-1}(\widehat{\beta}_{pr})\right\}\right]. \] The authors discuss the asymptotics of this two-stage least squares estimator of the parameters of ARCH models. They show that the estimator has the same asymptotic efficiency as that of the quasi-maximum likelihood estimator. Reviewer: Mikhail P. Moklyachuk (Kyïv) Cited in 1 ReviewCited in 22 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F12 Asymptotic properties of parametric estimators Keywords:ARCH model; quasi-maximum likelihood estimate; asymptotic PDFBibTeX XMLCite \textit{A. Bose} and \textit{K. Mukherjee}, J. Time Ser. Anal. 24, No. 2, 127--136 (2003; Zbl 1113.62095) Full Text: DOI References: [1] Bera A. K., J. Economic Surveys 7 pp 305– (1993) [2] DOI: 10.1016/0304-4076(92)90064-X · Zbl 0825.90057 [3] Engle R. F., Handbook of Econometrics (1994) · Zbl 0982.62503 [4] DOI: 10.1016/0304-4076(92)90067-2 · Zbl 0746.62087 [5] Engle R. F., Econometrica 50 pp 987– (1982) [6] Gourieroux C., ARCH Models and Financial Applications (1997) [7] Hall P., Martingale Limit Theory and its Applications (1980) · Zbl 0462.60045 [8] Nelson D. B., Econometric Theory 6 pp 318– (1990) [9] Pantula S. G., Sankhya, Ser B 50 pp 119– (1988) [10] N. Shephard, D. R. Cox, D. V. Hinkley, and O. E. Barndorff-Nielsen (1996 ) Statistical aspects of ARCH and stochastic volatility . InTime Series Models in Econometric, Finance and Other Fields , (eds), London: Chapman & Hall, 1 -67 . [11] Weiss A. A., Econometric Theory 2 pp 107– (1986) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.