LECTURELECTURE 7 7

CONSTRUCTION CONSTRUCTION OF ECONOMETRIC OF ECONOMETRIC MODELS WITH MODELS WITH AUTOCORRELATED AUTOCORRELATED RESIDUESRESIDUES

PlanPlan7.1. The nature and consequences of

autocorrelation.7.2. Methods for determining

autocorrelation. Durbin-Watson criterion. Von Neumann criterion.

7.3. Autocorrelation coefficients and their applications.

7.4. Models with autocorrelated residues.7.5. The method of for Aitken parameter

estimation.7.6. Kochren-Orkatt method. 7.7. The method of converting the output

information. Durbin method (self-directed learning).

After proposed for the algebraic dependence of the investigated process are conducted factor analysis; investigated for relevant and notrelevant factors; researched, are there factors that affect the constant of the free member variance; the value of the determination coefficient is received more than 0,6 - 0,7; conducted a study of multicolinearity factors, the next step of the research model is verification of the third Gauss-Markov precondition about absence of correlation between a free member in the i-th observation and in the j-th observation.

Why the question arises:1) If residues, or in other words the set of

residues in n step are not correlated among themselves, in other words, the correlation coefficient between the residues

ε1, ε2, …, εn

ε1+k, ε2+k, …, εn-k

are taken a value of less than 0.5 or more than -0.5, then it is suggested that we could use the algebraic view of the model for forecasting y.

2) If the value of the correlation coefficient between two series, we will provide statistically significant, this will talk about the presence of the k-th order autocorrelation.

Examples: There are economic processes, in

the study of which we assume that the autocorrelation can take place.

This is very clearly we can observe in forecasting sales of ice-cream, beer sales, sales of drinking water, i.e. in these processes, we can assume that selling is affected by ambient temperature

Temperature increases - sales of beer, ice cream increases, the temperature is decreasing - sales goes down

In the sale of some drugs, for example, anti-inflammatory action, when the temperature drops, the sales of such products is growing

Winter

Summer

100 packes

This is an example of positive first-order autocorrelationWe gave an example, when we could assumeBut there are studied processes, where it is very difficult to do it

So if the quality of the model does not suit us, we have to test for autocorrelation of residuals without any assumptions

Example, flowers sales: is depend the growth of flowers sales of on Friday from sales on Thursday.

In Sumy, we obtained results on the example of statistical information in the context of flower shops

We conclude that the increase in sales of flowers on Friday does not depend on the previous days of weeks, and is explained by the fact that in Sumy as practically in all cities of Ukraine, on Friday are marriage registrations

And this is an example of factor analysis, and not autocorrelation research.

7.1. 7.1. The nature and The nature and consequences of consequences of autocorrelationautocorrelation

(7.1)

(7.2)

(7.3)

Let us consider the classical linear multifactor model:

or in matrix form

Y – a column vector of the dependent variable of dimension (n1); X – a matrix of independent variables of dimension (n(m + 1)); a – a column vector of unknown parameters of dimension ((m + 1)1); ε – a column vector of random errors of dimension (n1);

mm xxxy ...22110

XY

Gauss-Marcov Assumption

3. Absence of systematic relation between the values of the random errors in any two observations

4. Random errors must be distributed independently of explanatory variables

7.1. 7.1. The nature and The nature and consequences of consequences of autocorrelationautocorrelation

Autocorrelation of residues – a Autocorrelation of residues – a phenomenon, which occurs in phenomenon, which occurs in case of violation of the case of violation of the assumption for the classical assumption for the classical regression analysis on the regression analysis on the independence of random independence of random variables (although the variables (although the variance of residuals is variance of residuals is constant there is the constant there is the homoscedasticity of residues).homoscedasticity of residues).

0),cov( ji ji

Causes of autocorrelationCauses of autocorrelation1. Autocorrelation of residues occurs

when the econometric model is based on time series.

2. Autocorrelation occurs in the context of the inertia and the cyclical nature of many economic processes.

3. Autocorrelation occurs due to specification of functional dependence in regression models incorrectly.

(7.4)

(7.5)

(7.6)

IM 2)'(

2)'( M

Assume that the model has autocorrelated residues, that the random variables εi dependent among themselves

So, as in the case of heteroscedasticity, dispersion of residues equals to:

Note that the presence of residual autocorrelation, as in the presence of heteroscedasticity, dispersion residues has the form

jiM ,0)'(

First order Autoregressive model

(7.7)

For example, if you note first order autoregressive model

ρ characterizes the strength of residues connection in t period from residues values in t-1 period

ttt vp 1

Table 7.1 - Comparative analysis of Gauss-Table 7.1 - Comparative analysis of Gauss-Markov assumptions violations Markov assumptions violations in the case ofin the case of heteroskedasticity and autocorrelation of heteroskedasticity and autocorrelation of residuesresidues

Gauss-Markov Condition

heteroskedasticity

autocorrelation

Variance of residuals

change const

Covariance of residues

absence presence

The consequences of ignoring the matrix Ω when determining residual variances by estimating the

parameters of the model by OLS

The estimates of the model parameters can be unbiased, but inefficient, that is the sample variance estimation vector can be unnecessarily large

Statistical criteria for t - and F-statistics obtained from the classical linear model cannot be used for analysis of variance, because their calculation does not consider the presence of residues covariance.

The inefficiency of the estimation of the econometric models parameters, as a rule, leads to inefficient forecasts, so the expected value will have greate sample variance

7.2. 7.2. Durbin-Watson Durbin-Watson criterioncriterionStep 1. Calculation the d-

statistics value

(7.8)

n

tt

n

ttt

dDW

1

2

2

21)(

StepStep 2. 2. We set the significance level We set the significance level . . With use of Durbin-Watson table for a With use of Durbin-Watson table for a given significance levelgiven significance level , , the number of the number of factors factors mm andand nn number of observations number of observations we have to find two values we have to find two values DW1DW1 і і DW2DW2::

Positive Autocorrelation is absent Negative

Zone of uncertainty Zone of uncertainty

0 DW1 DW2 2 4- DW2 4- DW1 4

7.2. 7.2. Von Neumann Von Neumann criterioncriterion

(7.9)

In von Neumann criterion we have to calculate the actual value of the criterion

Hence

Consequently,

In case if there is a positive autocorrelation

1

)(

1

2

2

21

n

nQQ n

tt

n

ttt

факт

Table of critical values for the ratio of von Neumann

77.3 .3 Autocorrelation Autocorrelation coefficients and their coefficients and their applicationsapplicationsLet's calculate residues, i.e. the

deviation of actual and theoretical values for each of the i-th observation, which are located in the following sequence

Let’s shift values of random deviations by one item and receive the following sequence

m ,...,,, 321

1131211 ,...,,, m

On the basis of application of random residues sequences let’s calculate the correlation coefficient between their values, which is called the first order autocorrelation coefficient, because it determines the relationship between the values of random deviations and values of the same variance, but shifted by one element

1, iir

i

i

ii

ii

iiiir

1

1 2

12

1

11,

Let’s shift values of random deviations by two elements, we obtain the following sequence

On the basis of application of random residues sequences let’s calculate the correlation coefficient , which is called the second order autocorrelation coefficient, because it determines the relationship between the values of random deviations and values of the same variance, but shifted by two elements:

2232221 ,...,,, m

2, iir

i

i

ii

ii

iiiir

2

2 2

22

2

22,

k-th order autocorrelation k-th order autocorrelation coefficientcoefficient

kmkkk ,...,,, 321

i

ki

kii

kiki

kiikiir

22,

Noncyclical autocorrelation Noncyclical autocorrelation coefficientcoefficient

It reflects the degree of correlation of the series , and is calculated by the formula

121 ,...,, n

21

2

2 21

21

21

2 2

2

2 21

21

1

1

1

1

1

1

n

t

n

ttt

n

t

n

ttt

n

t

n

tt

n

tttt

nn

nr

Cyclical autocorrelation Cyclical autocorrelation coefficientcoefficientSince it is difficult to establish the

probability distribution of r*, in practice is calculated the cyclical autocorrelation coefficient r0

n

t

n

ttt

n

t

n

ttntt

n

nr

1

2

1

2

2

2 111

0

1

1

Cyclical autocorrelation Cyclical autocorrelation coefficientcoefficientIf the last member of a series

equals to the first one, that is , noncyclical autocorrelation coefficient equal to cyclic autocorrelation coefficient

n 1

n

tt

n

ttt

r

1

2

21

)1,1(r

7.4 7.4 Models with the Models with the autocorelated residuesautocorelated residues

1) Aitken (Generalized OLS) method;

2) converting the original information;

3) Kochren-Orkatt method;

4) Durbin method.

7.4 7.4 Models with the Models with the autocorelated residuesautocorelated residuesThe first two methods appropriate to apply

when the residues describes by the first order autoregressive model

Iterative Kochrane-Orcutt and Darbin method can be applied to estimate the parameters of econometric models, when the residues describes by autoregressive model of the highest order:

ttt v 1

tttt v 2211

ttttt v 332211

7.5 7.5 The method of AitkenThe method of Aitken parameter estimation parameter estimation

YXXXa 111 )(ˆ

YSXXSXa 111 )(ˆ the matrix inverse to dispersion-covariance matrix of residuals Ω

the matrix inverse to the matrix

11S

2S

0

1...0000

.....................

0...010

0...001

0...0001

1

1 2

2

21

n

tt

n

ttt

r

1

2

22

1

1

2

21

n

nr

n

tt

n

ttt

7.6 7.6 Kochren-Orkatt Kochren-Orkatt methodmethod

Steps of Kochren-Orcutt method realization

1 step. Choice arbitrarily the parameter ρ value, for example, ρ=r1. Putting it in

we obtain

n

t

n

tttttt xxaayy

2 2

21101

2 1

)1(0a )1(

1a

Step 2Put and ,

substituting them into equation of the previous step, we can calculate

Step 3Putting into equation of the first

step the value , we can calculate and

)1(00 ˆˆ aa )1(

11 ˆˆ aa

1r

2r)2(

0a )2(1a

Step 4We can use and to

minimize the sum of squared residuals in the equation of the first step in the context of unknown parameter . Repeat the procedure, until the following parameter values and do not differ by less than a specified amount.

)2(00 ˆˆ aa )2(

11 ˆˆ aa

3r

10 ˆ,ˆ aa

Advantages of Advantages of Kochren-Orcutt Kochren-Orcutt methodmethod1.Give an opportunity to find a global optimum;

2.Have relatively good convergence.

7.7. The method of 7.7. The method of converting the output converting the output informationinformationAlternative approach to model

parameters estimatesStep 1. Transformation of the

input information with use of parameter ρ (covariance of each residues value with previous one).

Step 2. Application OLS for parameters estimation on the basis of the conversed data.

TTXATY

1...0000

..................

0...100

0...010

0...001

0...0001 2

T

1

34

23

12

12

...

1

nn yy

yy

yy

yy

y

TY

Durbin methodDurbin method

Thank you for your attention!