- 1. Regression Analysis Regression analysis is a mathematical
measure of the averages relationship between two or more variable
in terms of the original units of data. Types of Regression (i)
Simple Regression (Two Variable at a time) (ii) Multiple Regression
(More than two variable at a time) Linear Regression: If the
regression curve is a straight line then there is a linear
regression between the variables . Non-linear Regression/
Curvilinear Regression: If the regression curve is not a straight
line then there is a non-linear regression between the
variables.
2. Simple Linear Regression Model & its Estimation A simple
linear regression model is based on a single independent variable
and its general form is:Yt X t t HereInterceptsYt Xtt= dependent
variable or regressands = independent variable or regressor =
random error or disturbance termImportance of error (i)Slope/
Regression Coefficients t term:It captures the effect of on the
dependent variable of all variable not included in the model. (ii)
It captures any specification error related to assumed linear
functional form. (iii) It captures the effects of unpredictable
random componenets present in the dependent variable. 3. Estimation
of the Model Yt Xt Sales Adver Exp (thousands (million of of Unit)
Rs.) Yt X=309/ 736/7txt2y t Y t Yt xt X t X
txtyt-7.14286-0.642864.5918370.413265374.5486.53.8571431.3571435.2346941.841837453.50.857143-1.64286-1.408162.69898363-8.14286-2.1428617.448984.591837252.5-19.1429-2.6428650.591846.984694558.510.857143.35714336.4489811.27041637.518.857142.35714344.448985.55612244.1428Yt
=309Xt = 365.1428xt yt =157.37xt 2 = 33.354 4. Estimation of the
Model xy x t2 tt157 . 357 4 . 71733 . 354 Y X 44 . 143 ( 4 . 717 )(
5 . 143 ) 19 . 882 Then the estimated simple linear regression
model is Y t 19 . 882 4 . 717 X t 5. 2 x y x 22 tt157 . 357 4 .
71733 . 354 Y X 44 . 143 ( 4 . 717 )( 5 . 143 ) 19 . 882 Y t 19 .
882 4 . 717 X t 6. General Formula for First Order Coefficients rYX
.W rXY rXW rYW (1 rXW )(1 rYW ) 22General Formula for Second Order
CoefficientsrYX .WO rXY .O rXW .O rYW .O (1 r2 XW . O)(1 r2 YW . O)
7. Partial Correlation Remarks: 1. Partial correlation coefficients
lies between -1 & 1 2. Correlation coefficients are calculated
on the bases of zero order coefficients or simple correlation where
no variable is kept constant. Limitation: 1. In the calculation of
partial correlation coefficients, it is presumed that there exists
a linear relation between variables. In real situation, this
condition lacks in some cases. 2. The reliability of the partial
correlation coefficient decreases as their order goes up. This
means that the second order partial coefficients are not as
dependable as the first order ones are. Therefore, it is necessary
that the size of the items in the gross correlation should be
large. 3. It involves a lot of calculation work and its analysis is
not easy. 8. Partial Correlation Example: From the following data
calculate 12.3 x1 : 4 0 1 1 1 3 x2 : 2 0 2 4 2 3 x3 : 1 4 2 2 3 0
Solution: X116 2 2,X216 2 2andX316 2 2413040 9. Partial Correlation
10. Multiple Correlation The fluctuation in given series are not
usually dependent upon a single factor or cause. For example wheat
yields is not only dependent upon rain but also on the fertilizer
used, sunshine etc. The association between such series and several
variable causing these fluctuation is known as multiple
correlation. It is also defined as the correlation between several
variable.Co-efficient of Multiple Correlation: Let there be three
variable X1, X2 and X3. Let X1 be dependent variable, depending
upon independent variable , X2 and X3. The multiple correlation
coefficient are defined as follows: R1.23 = Multiple correlation
with X1 as dependent variable and X2. and X3. , as independent
variable R2.13 = Multiple correlation with X2 as dependent variable
and X1. and X3. , as independent variable R3.12 = Multiple
correlation with X3 as dependent variable and X1. and X2 , as
independent variable 11. Calculation of Multiple Correlation
Coefficient General FormulaFor example 12. Remarks Multiple
correlation coefficient is a non-negative coefficient. It is value
ranges between 0 and 1. It cannot assume a minus value. If R1.23 =
0, then r12 = 0 and r13=0 R1.23 r12 and R1.23 r13 R1.23 is the same
as R1.32 (R1.23 )2 = Coefficient of multiple determination. If
there are 3 independent variable and one dependent variable the
formula for finding out the multiple correlation isR1 .234 1 (1
r214 )(1 r212 . 3 )(1 r2 12 . 34) 13. Limitation 14. Advantages of
Multiple Correlation 15. Example Given the following data X1: 3 5
X2: 16 10 X3: 90 726 7 548 4 4212 3 30Compute coefficients of
correlation of X3 on X1 and X214 2 12 16. Example 17. Example 18.
Types of Correlation r12.3 is the correlation between variables 1
and 2 with variable 3 removed from both variables. To illustrate
this, run separate regressions using X3 as the independent variable
and X1 and X2 as dependent variables. Next, compute residuals for
regression...X