In both cases, the estimates of the coefficients can be obtained by ols with covariates:

  ,,

.

The design matrix contains adjacent columns revenues lagging in one time unit, therefore, adjacent columns are almost identical, and this leads to multicollinearity and poor reversibility of the matrix , .

Therefore, starting from the substantive nature of the simulated dependence (),postulate with a small number of parameters: ; problem reduces to the evaluation of their: . For example, it is implemented in a polynomial lag structure Almon S. (1965).

The lag polynomial structure Almon

Approximate dependence of β on lag l low degree polynomial M (M <= 3):

,

or

(17)

Substituting (17) into (15) and collecting similar, we obtain

Variables not multicollinearity, and OLS is no problem. Selection of L is carried out on the basis of the length of the available time series empirically.

Example :using data

xt	10	18	18	16	18	20	24	24	20	21	22	25	27	27	30	28	32	32	30
yt						28	30	31	32	33	32	33	34	36	37	38	40	41	41

estimate the model parameters (15) with L = 5 for structure Almon with M = 3. Redenoting , have a multiple regression model (18)

Matrix , .

OLS coefficient vector :

.

Substituting this in (17), we find that the desired estimates of the coefficients of the model (15):и

.

The coefficient of determination:

. R²= 0.9931.

Forecast by the model. Suppose we know the new value of the independent variable x20 = 24. Obviously, the prognosis of the corresponding value of the dependent variable is given by. .

The geometric structure of the lag cot

Example: study the dependence of – introduction of fixed assets in the t-th year of

_{- Capital investments in previous years. Built all long, so we assume the existence of arbitrarily large lags.}

.

L. Koike (1954) suggested:

, l = 0, 1, 2... ,

(0 < λ < 1) – denominator exponentially

, (19)

,

, (20)

.

(19) – (20): ,

. (21)

Definition:model like

, (22)

including explaining the endogenous variable lags , called the model of Koyk.

Note: model often correlates with and with (autocorrelation), for example, see. (21):. This makes it impossible to use in the OLS estimation of 3 coefficients in (22), and methods of identifying the model cot depend on the properties.

Partial adjustment model

Example: study the dependence of the release some product on demand it. Obviously demand determines – optimum production of goods;

; (23)

and the actual output: ,

where – the power of intuition and / or capabilities of the enterprise.

put in(23):

.

This is a special case of the model Koyk with an error at one time is not correlated with Y^t-1. Estimates of OLS is asymptotically unbiased.

Example ([2], table. 6.9, с. 198):

The economic data from the United States ($ billion)

–housing costs;

–personal income;

–house price index as a percentage of 1972.

If you build a regression on и , we get:

,

–autocorrelated errors, estimates are inefficient.

If you build a regression on , and , we get:

errors are not autocorrelated.

Thus , the estimate of the correction factor(ie, the population of the USA slowly adjusts investments in housing when changing personal income and housing prices); estimation of optimum investment in housing:

,

where

Note: partial adjustment model should be used when a clear connection with the idealized characteristic factors and the observed characteristics of the resulting partially corrected to idealized.

Model of adaptive expectations

Example: Let study the dependence of investment bank-rate . In fact, investors are based on the expected bank-rate in the future year.

; (24)

very magnitude they define as:

;

;.Substituting this in (24):

–this leads to a model Koyk with errors correlated with lagged .

The model is nonlinear in the parameters, so instead of the usual OLS should use a nonlinear OLS:

(модели );

;

, (25)

.

Example: Cagan model of hyperinflation F. (1956 г.).

Investigated the 7 periods of hyperinflation in the United States (1921–1956 гг.),

time step was 1 month., – the expected inflation rate for the next month. Kagan is reasonably believed that the higher the level of inflation in the next month, the lower the demand for money.,

where – index changes in the volume of money in circulation;

–price index.

unobserved, then it used a model of adaptive expectations, then used the nonlinear least squares. result:

  ,.

If (10 %), the demand for money will be 37.4% less than with stable prices ():

.

Consumption pattern Friedman

M. Friedman conjectured that individual consumption in period t th

proportionally feel it sustainable, permanent income in this period:

. (26)

If – real income, then

, – parameter adjustment.

, and substituting it in (26), we obtain a model of Koyk.

Using data on real per capita consumption and real per capita income in the United States in the 1905 - 1951 period. To estimate the coefficients used nonlinear least squares preserving 17 terms in (24). It turned .

Dynamic properties of the model to better analyze Friedman after conversion Koyk:

.

Short-term marginal propensity to consume:

, .

In the long term (equilibrium): ;

; ;

–long-term propensity to consume. The difference between these two values ​​was first explained Friedman.

Main literature

1. Айвазян С.А. Прикладная статистика и основы эконометрики / С.А. Айвазян, В.С. Мхитарян. М.:ЮНИТИ, 2001.

2. Доугерти К. Введение в эконометрику / К. Доугерти. М. : Инфра – М, 2001. 402 с.

3. Тихомиров Н.П. Эконометрика / Н.П. Тихомиров, Е.Ю. Дорохина. М.: Экзамен, 2003. 512 с.

4. Магнус Я.Р. Эконометрика. Начальный курс / Я.Р. Магнус, П.К. Катышев, А.А. Пересецкий. М. : Дело, 2005.

5. Берндт Э.Р. Практика эконометрики / Э.Р. Берндт. М. : Юнити-Дана, 2005.

Additional literature:

1. Handbook of Econometrics. Volumes 1, 2 and 3 / preface by Z. Griliches and M.D. Intriligator. Режим доступа: http://www.elsevier.com/hes/books/02/preface/preface.htm.

2. Тюрин Ю.Н. Статистический анализ данных на компь­ютере /

Ю.Н. Тюрин, А.А. Макаров. М. : Инфра – М, 1998.

3. Campbell G. The Econometrics of Financial Markets / G. Campbell, R. Lo,

J. MacKinlay. N. Y., 1997.

4. Уотшем Т.Дж. Количественные методы в финансах / Т.Дж. Уотшем,

К.М. Паррамоу. М. : ЮНИТИ, 1999.

5. Пугачев В.С. Теория вероятностей и математическая статистика /

В.С. Пугачев. М. : ФИЗМАТЛИТ, 2002.

6. Пащенко Ф.Ф. Введение в состоятельные методы моделирования систем / Ф.Ф. Пащенко. М. : Финансы и статистика, 2006.

7. Секей Г. Парадоксы в теории вероятностей и математической статистике / Г. Секей. М. : Мир, 1990.

8. Руководство пользователя пакета программ SPSS v. 7.

9. Руководство пользователя пакета программ STATISTICA v. 6.

10. Справочник по прикладной статистике: в 2 т. / под ред. Э. Ллойда,

У. Ледермана. М. : Финансы и статистика, 1990.

11. Колемаев В.А. Теория вероят­ностей и математическая статистика / В.А. Колемаев, О.В. Староверов, В.Б. Турундаевский. М. : Высшая школа, 1991.

12. Кейн Э. Экономическая статистика и эконометрия. Вып.1, 2 / Э. Кейн. М. : Статистика, 1977.

13. Breiman L. Estimating Optimal Transformations for Multiple Regression and Correlation / L. Breiman, J.H. Friedman // Journal of the American Statistical Association. 1985. V. 80. N 391.

14. Ширяев А.Н. Основы стохастической финансовой математики: в 2 т. / А.Н. Ширяев. М. : ФАЗИС, 1998.

15. Tsay R.S. Analysis of financial time series / R.S. Tsay. John Wiley & Sons, 2002.

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