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# Testing and evaluation of multivariable demand function

## 1.

Construction of multivariable demand functionTesting and evaluation of

multivariable demand function

1

## 2.

Step 1. Testing the suitability of the modelSigns of the coefficients

Parameter values

2

## 3.

Are marks of b1 and b2 consistent with the theory?Q = 3,45 + 0,5 X1 + 0,009 X2

The sign of the parameter indicates the direction

Income

capita to

of change of the demand variable

withper

respect

changes

in of

the independent variable

Quantity

prospective

consumers (1000)

Demand Variable changes in the same

direction

as the independent

variable

Root-mean-square

error

Variable

№

The positive

sign:

of regression coef.

The demand variable and the independent

variable are changing in opposite directions

The negative sign:

0,009

Dispersion analysis

sum of

squares

coefficient of determination

Root-mean-square error of regression

3

## 4.

Parameter valuesThis is parameter validation on

economic sense

The generally accepted limits do not exist, but most

economists subjectively limit values of each parameter

] aggregate demand = a function of prices and disposable income:

Cd = b0 + b1 X1 + b2 X2

] b1 = 2 b2 = 1,3

Do these parameters have

sense?

In accordance with b2, the consumer must spend 1,3 $

per each additional 1$ income

4

## 5.

Step 2. Statistical tests and evaluationCommon tests

Plural coefficient of determination, R^2

Corrected plural coefficient of determination, R^2

Root-mean-square error of estimation for the regression

I'm still waiting for the day when I

will need to know the solution of

in real life

5

## 6.

Multiple regression describes the regression plane and theobserved points lie above, below, and on this plane

Step 2. Statistical tests and evaluation

Plural coefficient of determination, R^2

Is a measure of how well the plane described by the regression

equation, satisfies the experimental data

^

^

(

Q

Q

)

(

Q

Q

)

(

Q

Q

)

i

i

i

2

Full variation

=

Explainable variation

2

2

i

+ Unexplained variation

Variation is the sum of the squared deviations of observed values from the regression line

The factor has only mathematical sense and does not determine any

causal relationships

R^2 = Explainable variation /Full variation =

^

(Qi Q)2

(Qi Q)2

6

## 7.

0 < R^2 < 1R^2 = 0 – there is no relationship between demand and other variables

R^2 = 1 –all changes in demand are explained by simultaneous changes

of the independent variables

0,009

Dispersion analysis

SSR Explainable variation

SSЕ

Unexplained variation

SSТ

Full variation

coefficient of determination

sum of

squares

This means that 99.89 per

cent changes in sales are

explained by changes in the

size of the target population

and per capita income

7

## 8.

Step 2. Statistical tests and evaluationCorrected plural coefficient of determination, R^2

To get useful results, the number of observations should be sufficient

Pays due attention to the degrees of freedom determined by the

number of observations and number of parameters

k 2

R

R

1

R

n

k

1

2

2

number of observations

The number of independent variables

8

8

## 9.

Acceptable values ofR2 ?

Usually if the number of observations is three or four times

more than the number of independent variables, it is

considered that acceptable value is

R2 0,75

9

## 10.

Step 2. Statistical tests and evaluationRoot-mean-square error of estimation for the regression

Characterizes the dispersion of the observed points from the theoretical

regression line (determines the random scatter of the observed values of

^

Q, relative to the estimated values of Q)

The observed value of the dependent

demand variable in the i-th point

(

Q

Q

)

S

^

i

e

Root-mean-square error of

estimation

2

i

n

k

1Estimated value of the

The number of

independent

variables

dependent demand variable,

calculated for the i-th point on

the regression equation

10

## 11.

0,009Root-mean-square error of regression

11