Непараметрические методы в экономической статистике (на английском яз.)
Chapter 1. The history of development of nonparametric methods
Chapter 2. Nonparametric methods in economics
Chapter 3. Practice of nonparametric methods in the economical problems
The list of abbreviations
The list of formulas
1.Afriat, S. (1972): Efficiency Estimation of Production Functions, International Economic Review 13, 568-598.
2.Caves, D.W., L.R. Christensen, and W.E. Diewert (1982): The Economic Theory of Index Numbers and the Measurement of Input, Output and Productivity, Econometrica 50, 1393-1414
3.Charnes, A., W. Cooper, and E. Rhodes (1978): Measuring the Efficiency of Decision Making Units, European Journal of Operational Research 2, 429-444.
4.Corder, G.W. & Foreman, D.I, "Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach", Wiley (2009) ISBN: 9780470454619, 125-135.
5.Fere, R., S. Grosskopf, M. Norris, and Z. Zhang (1994): Productivity Growth, Technical Progress, and Efficiency Change in Industrialized Countries, American Economic Review 84(1), 66-83.
Показать все(1957): The Measurement of Productive Efficiency, Journal of the Royal Statistical Society Series A 120, 253-281.
7.Gibbons, Jean Dickinson and Chakraborti, Subhabrata, "Nonparametric Statistical Inference", 4th Ed. CRC (2003) (ISBN: 0824740521)? 2007, 24-25.
8.Hanoch, G. and M. Rothschild (1972), Testing Assumptions of Production Theory: A Nonparametric Approach, Journal of Political Economy 80, 256-275.
9.Maarit Kallio NONPARAMETRIC METHODS FOR EVALUATING ECONOMIC EFFICIENCY AND IMPERFECT COMPETITION Markku KallioFinnish Forest Research Institute, Finland 2002 ISSN 0895-562X 171-189
10.Samuelson, P.A. (1948): Consumption Theory in Terms of Revealed Preference, Economica 15, 243-253.
11.Shephard. R.W. (1953): Cost and Production Functions, Princeton University Press, Princeton.Varian, H.R. (1984): The Non-Parametric Approach to Production Analysis, Econometrica 52, 279-297.
12.Springer Semiparametric and Nonparametric Methods in Econometrics 2009 ISBN: 0387928693, 276.
13.Varian, H.R. (1985): Non-Parametric Tests of Optimizing Behavior with Measurement Error, Journal of Econometrics 30, 445-458. Скрыть
Nonparametric methods provide a way to reduce the strength of the assumptions required for estimation and inference, thereby reducing the opportunities for obtaining misleading results. These methods are applicable to a wide variety of estimation problems in empirical economics and other fields, and they are being used in applied research with increasing frequency.
Chapter 3. Practice of nonparametric methods in the economical problems
Nonparametric methods of confidence estimation of the basic characteristics of number of random variables - the expectation, median, variance, standard deviation, coefficient of variation.
Let the initial data is a sample x1... xn, where n — sample volume. Selective values x1, x2... xn are considered as realization of independent, equally distributed rand
Показать всеom variables X1, X2... Xn with the general function of distribution F (x) =P (Xi <x), i = 1, 2... n. As distribution function is any (to within conditions of a regularity of type of existence of the moments) considered problems of confidential estimation of characteristics of distribution are nonparametric. Existence of the moments is faster mathematical restriction, than real, as almost all real statistical data are limited from above and from below, for example, by a device scale. 8
This study employs monthly stock price indices from 1988M1 to 2002M8 for five major ASEAN stock markets, namely Jakarta Composite Index (JCI), Kuala Lumpur Composite Index (KLCI), Philippines Composite Price (PCOMP), Strait Times Index (STI) and Stock Exchange of Thailand (SET). All the data are collected from Kuala Lumpur Stock Exchange (KLSE). The indices are denominated in local currency units. Prior to the analysis, all stock price indices are transformed into logarithm form, and the graphical depiction is provided in
Bierens’s (1997) Non-parametric Cointegration Test
The Bierens non-parametric cointegration test considers the general framework as:
where ( 1) 0 * q × and ( 1) 1 * q × are optimal mean and trend terms, and t y is a zero-mean unobservable process such that t y is stationary and ergodic. The general framework assumes that t z is observable q-variate process for t = 0, 1, 2, …, n.
Apart from some mild regularity conditions, or estimation of structural and/or nuisance parameters, further specification of the data-generating process for t z are not required and thus this test is completely non-parametric.
The Bierens’s method is based on the generalized eigenvalues of matrices m A and ( + −2 −1) m m B n A , where m A and m B are defined in the following matrices: which are computed as sums of outer-products of weighted means of t z and Δ t z , and n is the sample size. To ensure invariance of the test statistics to drift terms, the weight functions of cos(2kπ(t – 0.5)/n) are recommended here.
Similar to the properties of the Johansen and Juselius likelihood ratio method, the ordered generalized eigenvalues of this non-parametric method are obtained as solution to the problem det [ n P – λ n Q ] = 0 when the pair of random matrices n P = m A and n Q = ( + −2 −1) m m B n A are defined. Thus, it can be used to test hypothesis on the cointegration rank r.
To estimate r, two test statistics are used. First, Bierens (1997) derived the ‘lambda-min’
which corresponds to the Johansen’s maximum likelihood procedure, to test for the hypothesis of H ( r) 0 r against H ( 1) 1 r +1 . The critical values for this test are tabulated in the same article. Second, Bierens’s approach also provides the g (r) m , which is computed from the Bierens’s generalized eigenvalues:
This statistic uses the tabulated optimal values (see Bierens, 1997, Table 1) for r, provided r > q, and m = q is chosen when r = n. Then gˆ m(r ) converges in probability to infinity if the true number of cointegrating vector is unequal to r, and gˆ m(r ) = O (1) p if the true number of cointegrating vector is equal to r. Therefore, we have , when useful as a tool to double-check on the determination of r.. Thus, this test statistic is
Finally, a linear restriction on the cointegrating vectors is needed because not all of the series will enter the cointegrating vector system. To address this issue, Bierens (1997) proposed the trace and lambda-max statistics. The critical values of trace (m = 2q, F (x) k = cos(2k*x) ) and lambda-max tests (m = 2q, F (x) k = cos(2k*x)) are given in Bierens (1997, Tables 3 and 4).9
Instead of using parametric assumptions on the functions and distributions in an economic model, the methods use the restrictions that can be derived from the model. Examples of such restrictions are the concavity and monotonicity of functions, equality conditions, and exclusion restrictions.
For synthesis of integrated indicators of economic potential, regional efficiency, an investment situation pertinently to use nonparametric methods of the statistical analysis. The basic advantage of application of nonparametric methods consists in decrease in dimension of matrixes initial given by "compression" of the initial information. Thus nonparametric methods of multidimensional statistical comparisons possess insignificant sensitivity to distortions of the statistical data, are applicable to small samples and maintenance of comparability of units of measurements of private indicators doesn't demand.10
A promising area of application of nonparametrics is finance. Varian (1983b) suggested this application long ago. It may be particularly fruitful given the large data series available in finance. It may also be a way to supplement traditional stochastic dominance analysis. Sengupta has provided a way to estimate the portfolio efficiency frontier using this method. It may also be useful in analyzing models with risk. For example, one could determine the level of risk aversion consistent with a particular set of choices. Another application might be to measure the risk premium in financial markets. While this literature is just starting, it may be an important application in the future.11
At processing of the real economic data received in the course of supervision, measurements, the calculations, sometimes one or several results of supervision are sharply allocated, i.e. will far defend from a great bulk of the data. Such sharply allocated results of supervision often consider containing rough errors, accordingly naming their misses or emissions. In considered cases there is a natural thought that similar supervision don't concern studied set as contain a rough error, and are received as a result of an error, a miss. In metrology about this phenomenon speak so: «Rough errors and misses arise because of errors or wrong actions of the operator (its psychophysiological condition, incorrect readout, errors in records or calculations, wrong inclusion of devices, etc.).12 And also at short-term sharp changes of conditions of carrying out of measurements (because of vibration, receipt of cold air, a push of the device the operator, etc.). If rough errors and misses find out in the course of measurements the results containing them, reject. However more often them reveal only at definitive processing of results of measurements by means of special criteria of an estimation of rough errors».
Traditional nonparametric methods, such as order statistics, are underutilized in much of economic research and could be fruitfully exploited in the future, particularly for preliminary data analysis and in analyzing the results of economic models.
Series and kernel estimators provide an important way to analyze large data sets. They avoid unnecessary functional specifications but may be of limited use for many data sets and time periods.
They will also require more emphasis on the statistical basis of estimates. Most researchers tend to be spoiled by the assumptions and strong properties of least squares, and have forgotten most of the probability and inference they ever learned.
The nonparametric procedures cataloged under the terms data envelopment analysis and revealed preference suffer from a lack of statistical basis. They are nevertheless extremely powerful ways to analyze data in a general framework. Скрыть
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