Where the method of least squares

method of least squares (OLS) allows to evaluate the different values ​​using the results of multiple measurements containing random errors.

Feature MNCs

The basic idea of ​​this method is that as a criterion for the accuracy of solving the problem is considered the sum of squared errors, which seek to minimize.When using this method can be used as a numerical and analytical approach.

In particular, as the numerical implementation of the method of least squares means carrying out the greatest possible number of dimensions unknown random variable.Moreover, the more calculations, the more accurate the solution.On this set of calculations (original data) get another set of alleged solutions from which then selected the best.If the solution set of parameterized, the least squares method reduces to the search for the optimal parameter values.

As an analytical approach to the implementation of the MNE on the set of input data (measurements) and the expected set of solutions is determined by some fun

ctional dependence (functional), which can be expressed by the formula obtained as a hypothesis requiring confirmation.In this case, the method of least squares is reduced to finding the minimum of this functional on the set of squares of the errors of initial data.

Note that no errors themselves, namely squares error.Why is that?The fact that often the deviation from the exact measurement values ​​are both positive and negative.In determining the average measurement error simple summation can lead to the wrong conclusion about the quality of assessment, as the mutual destruction of positive and negative values ​​lower power sample sets of measurements.And, consequently, the accuracy of the estimate.

To this did not happen, and summing the squares of the deviations.Even more, to align the dimension of the measured value and the final evaluation of the sum of squared errors of square roots.

Some applications MNCs

MNCs are widely used in various fields.For example, in the theory of probability and mathematical statistics method used to determine the characteristics of a random variable is the standard deviation, which determines the width of the range of values ​​of the random variable.

in mathematical analysis and various fields of physics, is used to display or confirmation of hypotheses this unit, OLS is used, in particular, to assess the approximate representation of functions defined on a numerical set, simpler functions, admits an analytic transformation.

Another application of this technique - the separation of the useful signal from the noise imposed on him in filtration problems.

Another field of application of the MNE - Econometrics.Here, this method is so widely used that it had identified some special modifications.

Most tasks econometrics, anyway, is reduced to solving systems of linear econometric equations describing the behavior of certain systems - structural models.The main element of each of these models - the time series, which is a set of certain characteristics, the values ​​of which depend on the time and a number of other factors.This may be a correspondence between the internal (endogenous) and external features of the model (exogenous) characteristics.This correspondence is usually expressed in the form of systems of linear equations economic.

characteristic feature of such systems is the existence of the relationship between the individual variables, which on the one hand, complicate it, the other - override.What is the cause of uncertainty in the choice of solutions of such systems.An additional factor that complicates the solution of such problems is the dependence of the model parameters from time to time.

main purpose of the tasks of econometrics - the identification of patterns, that is the definition of structural relationships in the model chosen, and the evaluation of a number of parameters.

Recovery dependencies in time series, model components can be performed, in particular, through both direct MNCs and some modifications, as well as a number of other methods.Special modifications MNCs in solving such problems specifically developed to resolve various problems arising in the process of solving systems of equations.

In particular, one of these problems associated with the presence of initial constraints on the parameters that must be evaluated.For example, the income of a private company can be spent on consumption or on its development.Consequently, the sum of these two parts of cost obviously equal to 1. The system of equations econometric these parts can include independently.Therefore, it is possible to evaluate the different types of spending by OLS, without limitation the source, and then adjust the result.This is called an indirect method for solving the least squares method.

indirect method of least squares (ILS) is used to accurately determine the structural model.ILS algorithm involves the following actions:

1) conversion of the structural model in a simple, reduced form by introducing additional dependence;

2) evaluation using conventional OLS reduced coefficients for each equation a simplified model;

3) obtained the coefficients of a simple shape model parameters are converted into the original structural model.

worth noting that sverhidentifitsiruemyh ILS systems are not used, as in this case, the job can not be definitive estimates of the parameters of the structural model.For such models can be used by another modification of the OLS - two-step method of least squares (KDOM).

KDOM following algorithm:

1) based on a simplified model to calculate the equation sverhidentifitsiruemogo values ​​of internal variables that are contained in the right part of the equation;

2) substitute the values ​​of the variables in place of actual relevant variables in the original model and once again use a normal MNC.

Detailed Description and indirect two-step method of least squares is given in many textbooks on econometrics.The peculiarity of these methods, as well as the usual OLS, in their versatility makes them suitable for estimating the coefficients of any structural model in whatever subject area.