ISSN:2249-7137 Vol. 6 Issue 8, August 2016 Impact ...

ISSN:2249-7137

Vol. 6 Issue 8, August 2016

Impact Factor: SJIF 2013=5.099

South Asian Academic Research Journals http://www.saarj.com

ISSN:2249-7137




ISSN:2249-7137



P ublis he d b y: S out h A s ia n A c ade m ic R es e arc h J our nals

ACADEMICIA:

Special Issue

An International Multidisciplinary Research Journal ( A D o u b le B l i n d R e fe r e e d & R e v ie we d I nt e r na t io na l J o ur na l)

SR. PARTICULAR NO. SOCIO – ECONOMIC CONDITION OF RURAL INDIA 1.

PAGE NO

DOI NUMBER

1-7

10.5958/2249-7137.2016.00035.5

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2.

EFFECT OF EMOTIONAL MATURITY ON ACADEMIC CHEATING AMONG SENIOR SECONDARY STUDENTS

Dr. Umender Malik & Rahul Kant

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EXCESSIVE LIGHT IS ANOTHER FORM OF POLLUTION ON THE ENVIRONMENT

P Muralidhar & V Srihari

4.

ESTIMATING THE IMPACT OF SIZE OF THE FIRM ON NET OPERATING CYCLE AND ITS ELEMENTS IN THE CONTEXT OF INDIAN MANUFACTURING INDUSTRIES

Dr.Vikas Kumar Choubey 5.

VITAMIN HEALTH

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AND

NON-SKELETAL

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A STUDY ON CUSTOMER PERCPETION OF POSTAL SAVINGS OF ERNAKULAM DISTRICT 85-94

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Naseema C M 7.

ESTIMATION OF MULTIPLE MODELS THROUGH A SINGLE REGRESSION

95-100

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M.Venkataramana,Dr.M.Subbarayudu South Asian Academic Research Journals http://www.saarj.com

ISSN:2249-7137



M.Rajani & Dr. K.N. Sreenivasulu

8.

SPECTRUM SENSING IN COGNITIVE RADIO NETWORKS USING THE ENERGY DETECTION TECHNIQUE 101-108

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WORKPLACE SPIRITUALITYTRANSFORMING THE WORKPLACE INTO A SPIRITUAL ONE 109-113

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EXPLORING THE TALENT AND EMPOWERING THE ENTREPRENUERIAL TOUCH OF THE WOMEN: A STUDY OF 114-122 KUDUMBASHREE UNITS IN KERALA

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COOPERATIVE SPECTRUM SENSING IN COGNITIVE RADIO NETWORKS USING RANDOM ADAPTIVE ACCESS 133-142

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B.Ramadasu & Suresh Pabboju

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IMPACT OF FAMILY CLIMATE ON THE ACADEMIC ACHIEVEMENT OF SENIOR SECONDARY STUDENTS 143-153

10.5958/2249-7137.2016.00047

Dr. Sadaf Jafri


ISSN:2249-7137



P ublis he d b y: S out h A s ia n A c ade m ic R es e arc h J our nals

ACADEMICIA: An International Multidisciplinary Research Journal ( A D o u b le B l i n d R e fe r e e d & R e v ie we d I nt e r na t io na l J o ur na l)

DOI NUMBER: 10.5958/2249-7137.2016.00041.0 ESTIMATION OF MULTIPLE MODELS THROUGH A SINGLE REGRESSION M. Venkataramana *; Dr.M.Subbarayudu**; M.Rajani***; Dr. K.N. Sreenivasulu**** * Research Scholar, Department of Statistics, S.V. University, T irupati-India. **Professor and Head, Department of Statistics, S.V. University, Tirupati-India, ***Research Scholar Department of Statistics, S.V. University, Tirupati-India. ****Assistant Professor & Head, Department of Statistics, A. N. G. R. Ag. University, Ag. College, Mahanandi, Kurnool (Dt), India. _____________________________________________________________________________ ABSTRACT Regression analysis with dummy variables has interesting and useful relationship among qualitative and quantitative variables under study. Qualitative variables are used to classify the data into different groups. The aim of this paper is to suggest a method to estimate two variable linear model, three variable linear model and three variable non linear model in a single regression using dummy variables with a numerical example.


ISSN:2249-7137



KEYWORDS: Regression model, OLS estimation, dummy variables. _____________________________________________________________________________ INTRODUCTION: Regression models plays a vital role in framing the relationship between a dependent variable and one or more independent variables based on significance of the estimates of the model parameters and model adequacy. Method of least squares is the most frequent technique used for estimation of linear regression model. Even some non linear models are estimated by converting into log linear models. Introduction of qualitative explanatory variables called dummy variables makes the linear model an extremely flexible tool that is capable of many interesting problems encountered in empirical results. The aim of this research article is to estimate different regression models through a single regression for the same data set using dummy variables. METHOD: Let Y be the response variable (R); X1, X2 are two explanatory variables (E) with n observations on each of the variables Y, X1 and X2. And let Model 1:

Y   0  1 X 1  u1

(1)

Model 2: Y  0  1 X1  2 X 2  u2

(2)

Model 3: Y   0 X11 X 2 2 u3

(3)

be the two variable linear model , three variable linear model and three variable non linear (Cobb Douglas) model fitted to the same data set. For compatibility let us consider logarithm of equation (3) i.e log Y  log  0   1 log X1   2 log X 2  log u3

(4)

be the log linear model of equation (3). The purpose of this note is to estimate three regression models (1), (2) and (3) at a time using dummy variable technique. Hence we need two dummy variables to differentiate three regressions. In order to achieve the required objective we may club the equations (1), (2) and (4) into a single regression equation using dummy variables as R = a0 + a1D1 + a2D2 + a3E1 + a4D1E1 + a5D2E1 + a6E2 + a7D1E2 + a8D2E2 + u Where R : Response variable, E1: Explanatory variable 1 = X1, E2: Explanatory variable 2 = X2 D1 = 1, if the data belongs model 2 = 0, other wise South Asian Academic Research Journals http://www.saarj.com

(5)

ISSN:2249-7137



D2 = 1, if the data belongs model 4 = 0, other wise Also a0: Intercept for the model 1 a1, a2 : Differential intercept for model 2 and model 4 respectively a3: Slope coefficient of R w.r.to E1 for model 1 a4: Differential slope coefficient of R w.r.to E1 for model 2 a5: Differential slope coefficient of R w.r.to E1 for model 4 a6: Slope coefficient of R w.r.to E2 for model 1 a7: Differential slope coefficient of R w.r.to E2 for model 2 a8 : Differential slope coefficient of R w.r.to E 2 for model 4 u : Error variable which follows usual assumptions of linear model. From the estimated equation (5) we can derive the actual estimates of model 1, model 2 and model 4 by substituting D1 = D2 = 0; D1 = 1, D2 = 0 and D1 = 0, D2 = 1 respectively as follows: Model 1: Y = a0+a3X1

(6)

Model 2: Y = (a0+a1) + (a3+a4) X1+ (a6+a7) X2

(7)

Model 4: log Y = (a0+a2) + (a3+a5) log X1 + (a6+a8) log X2

(8)

It is to be noted that for model 1: R = Y, E1 = X1, E2 = 0 ; for model 2: R = Y, E1 = X1, E2 = X2 and for model 4: R = log Y, E1 = log X1, E2 = log X2. Since E2 = 0 for model 1, the corresponding coefficient a6 = 0. The matrix representation of data corresponding to equation (5) presented as given in table (1) S.No 1 2 . . . n n+1 n+2

R y1 y2 . . .

yn y1 y2

D1 0 0 . . . 0 1 1

D2 0 0 . . . 0 0 0

Table (1) E1 x11 x12 . . .

x1n x11 x12

D1 E1 0 0 . . . 0 x11 x12

D2 E1 0 0 . . . 0 0 0


E2 0 0 . . . 0 x21 x22

D1 E2 0 0 . . . 0 x21 x22

D2 E2 0 0 . . . 0 0 0

ISSN:2249-7137 . . . 2n 2n+1 2n+2 . . . 3n

. . . yn log y1 log y2 . . . log yn

Vol. 6 Issue 8, August 2016 . . . 1 0 0 . . . 0

. . . 0 1 1 . . . 1

.

.

.

.

.

.

x1n log x11 log x12

x1n 0 0 . . . 0

. . .

log x1n

Impact Factor: SJIF 2013=5.099 . . . 0 log x11 log x12

.

.

x2n log x21 log x22

.

.

.

.

x2n 0 0 . . . 0

.

.

.

.

.

.

log x1n

log x2n

. . . 0 log x21 log x22 . . .

log x2n

EXAMPLE: For illustration purpose we consider 20 data points on Y, X1, X2 as a sample data set. Then we construct the required data matrix to run the regression (5) as given in table (2).The source of data is Damodar Gujarati (Dec-1970), which is originally taken from Snedecor and Cochran (1968). Table (2) R 1.4 1.79 1.72 1.47 1.26 1.28 1.34 1.55 1.57 1.26 1.61 1.31 1.12 1.35 1.29 1.24 1.29 1.43 1.29 1.26 1.4 1.79 1.72 1.47 1.26 1.28

D1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1

D2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

E1 78 90 94 71 99 80 83 75 62 67 78 99 80 75 94 91 75 63 62 67 78 90 94 71 99 80

D1 E1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 78 90 94 71 99 80

D2 E1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

E2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 61 59 76 50 61 54


D1 E2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 61 59 76 50 61 54

D2 E2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

ISSN:2249-7137 1.34 1.55 1.57 1.26 1.61 1.31 1.12 1.35 1.29 1.24 1.29 1.43 1.29 1.26 0.1461 0.2529 0.2355 0.1673 0.1004 0.1072 0.1271 0.1903 0.1959 0.1004 0.2068 0.1173 0.0492 0.1303 0.1106 0.0934 0.1106 0.1553 0.1106 0.1004

Vol. 6 Issue 8, August 2016 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

83 75 62 67 78 99 80 75 94 91 75 63 62 67 1.8921 1.9542 1.9731 1.8513 1.9956 1.9031 1.9191 1.8751 1.7924 1.8261 1.8921 1.9956 1.9031 1.8751 1.9731 1.9590 1.8751 1.7993 1.7924 1.8261

83 75 62 67 78 99 80 75 94 91 75 63 62 67 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Impact Factor: SJIF 2013=5.099 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.8921 1.9542 1.9731 1.8513 1.9956 1.9031 1.9191 1.8751 1.7924 1.8261 1.8921 1.9956 1.9031 1.8751 1.9731 1.9590 1.8751 1.7993 1.7924 1.8261

57 45 41 40 74 75 64 48 62 42 52 43 50 40 1.7853 1.7709 1.8808 1.6990 1.7853 1.7324 1.7559 1.6532 1.6128 1.6021 1.8692 1.8751 1.8062 1.6812 1.7924 1.6232 1.7160 1.6335 1.6990 1.6021

57 45 41 40 74 75 64 48 62 42 52 43 50 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.7853 1.7709 1.8808 1.6990 1.7853 1.7324 1.7559 1.6532 1.6128 1.6021 1.8692 1.8751 1.8062 1.6812 1.7924 1.6232 1.7160 1.6335 1.6990 1.6021

Source of data : Damodar Gujarati (Dec-1970), which is originally taken from Snedecor and Cochran (1968). The regression results through least squares estimation for the data in table (2) are as follows R = 1.3505 – 0.0166 D1 – 1.2174 D2 + 0.0005 E1 – 0.0039 D1E1 – 0.1853 D2E1 (0.2274) (0.3218)

(1.0067)

+ 0.0059 D1E2 + 0.2066 D2E2 (0.0040) (0.5170)

(0.0028)

,

(0.0048)

(0.7128)

R2 = 0.9477

(9)

Let us derive the regressions for individual models from equation (9) as shown in equations (6) to (8) and given by South Asian Academic Research Journals http://www.saarj.com

ISSN:2249-7137



Model 1: Yˆ = 1.3505 + 0.0005 X1

(10)

Model 2: Yˆ = 1.3338 – 0.0034 X1 + 0.0059 X2

(11)

^

Model 4: logY = 0.1331 – 0.1848 log X1 + 0.2066 log X2

(12)

In terms of non linear relationship the estimated equation (12) may be written as Yˆ = (1.3586) X 10.1848X 20.2066

(13)

It is observed that the derived individual models (10), (11) and (13) are equivalent to the directly estimated models (1), (2) and (3) respectively with corresponding R 2 values as 0.001, 0.085 and 0.068. Hence in this case it is to be concluded that, instead of estimation of different models for same data set independently, it is preferable to estimate the models using dummy variables by running a single regression in view of R2 values and consumption time.

REFERENCES:

1. Damodar Gujarati (Dec-1970): Use of dummy variables for equality between sets of coefficients in linear regression: A generalisation, The American Statistician, Vol.24, No.5, PP: 18-22. 2. Damodar N.Gujarati (1995): Basic Econometrics, 3rd edition McGraw-Hill, Inc. 3. Snedecor , George W and Cochran, Willam G (1968) : Statistical methods , 6 th edition.


ISSN:2249-7137



0816/07

15-08-2016

M. Venkataramana, Dr.M.Subbarayudu, M.Rajani & Dr. K.N. Sreenivasulu

I am very pleased to inform you that your research paper titled ESTIMATION OF MULTIPLE MODELS THROUGH A SINGLE REGRESSION

has been published in ACADEMICIA: An International Multidisciplinary Research Journal (ISSN:2249-7137) (Impact Factor:SJIF 2013=5.099) Vol.6, Issue-8, (August, 2016). The scholarly paper provided invaluable insights on the topic. It gives me immense pleasure in conveying to your good self that our Editorial Board has highly appreciated your esteemed piece of work. We look forward to receive your other articles/research work for publication in the ensuing issues of our journal and hope to make our association everlasting.


ISSN:2249-7137




ISSN:2249-7137


