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Managerial EconomicsRegression and Multiple Regression

It is a basic human instinct to relate many different happenings all around. Political analysts

always try to find out causes of the events taking place in a country or internationally.Stock market analysts point out reasons for a rise or fall in the price of a single share or

share index. These analysts and others always rely on some qualified information or

observation that they have, about the factors contributing change in a particular situation.

More formally, we can say, they (the analysts) are sure about the independent and

dependent variables involved in a specific situation. They try to keep their discussion

focused around chosen independent and dependent variables. It is generally found that

an analysis that involves quantifiable variables is more conclusive than the one using

qualitative variables.Regression Analysis is an endeavor to establish quantitative relationship between a cause

(independent variable) and its effect on something (dependent variable) [SIMPLE

REGRESSION] or the individual1 and collective2 impact of a number of variables on

something (dependent variable) [MULTIPLE REGRESSION].

The process of regression analysis starts with a simple step of choosing variables, about

which, you want to estimate a relationship. It could be the impact of global warming

(measured in terms of centigrade) on rainfall (measured in terms of millimeters). Lets

assume that you are given with the information about average global temperature (T) and

average global rainfall (R). If you want to estimate relationship between the two variables,

or more simply, you want to estimate the impact of global warming on global rainfall, you

need to process the information with you. Following is the detail of global temperature &

rainfall3.

Years 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006T 22 20 25 22 23 27 19 21 23 23

R 1022 1100 1232 1435 1407 1385 980 1355 1387 1416

We can describe the relationship as Rt = f (Tt)4

We can build a model as R = a5 + b T +

Where a stands for the impact of everything else on global rainfall (except temperature); b

stands for the impact change in temperature on global rainfall; and is termed as the error

term (a provision for the limitation of our estimation).

Now the next step is to find out the values ofa & b6.The formulas of a & b are given as

under:

b =( )( )

( ) 2tT

TT

RRT

t

t

& a = TbR

For this purpose we need to find out certain other things. Firstly, we shall find out an

average of all the T and R values for all the years ( T = T / n ; R = R / n n stands for

the number of years). Then, secondly, we need ( tT -T ) & ( )RRt ---- this means wehave to subtract each value of T & R from their respective average values. Then thirdly,

1 With the help of beta coefficients2 with coefficient of determination3 Data is fictitious and is given only for the practice purposes.4 t here refers to the time period5 a is called the interceptand measures the impact of everything else (besides global temperature) on theglobal rainfall.6 the estimated values of a & b (called theslope) are quoted as a & b

Handout Managerial Economics

M. Salahuddin

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we shall generate the values for ( tT - T ) * ( )RRt -- the multiplication of the values ofeach of the two multiplying terms. In order to make use of these values in the formula of

b , we shall add all these multiplied values -- ( tT - T ) * ( )RRt . In the fourth step,we shall take squares of all the values for ( tT - T ) and add them to quote against

( )2

TTt in the denominator of the formula for b

. At this stage, we shall have all thevalues needed for finding a . At this stage, we can express the regression equation (with

the help of estimated values of a & b ) as:

=R 297.14 + 43.32 tT

Now, having the estimated regression, we need to interpret this equation. The equation says

that one degree increase in global temperature brings about 42.32 mm more global rainfall.

Moreover, of the total global rainfall 297.14 mm is because of all the factors other than

global temperature.

The next logical question is to check whether the estimates of this regression equation are

true or not. For this purpose, we shall conduct a test of our parametric 7 estimates. Usually,

the value of intercept is not tested and to test the significance ofb

, we compare twodifferent values t statistic & critical value of t distribution.

A t statistic is a ratio ofb

sb

8. The denominator of the formula can be found as:

( )( ) ( ) 2

2

TTkn

RRs

t

t

b

=

9 where k stands for number estimated parameters ( a & b )

At this stage we need estimated values of global rainfall for each value of the global

temperature given in the table. After having all tR values for each value of temperature in

the table we shall deduct all these values from their respective table values. Thereafter, we

shall square all the results of this subtraction process to get ( )2RR

t

. The sum total of all

the squared terms would be used in the numerator of the above mentioned highlighted

formula. By furnishing the remaining terms of the formula as well, we will have the value

of bs for using it in the formula for t-statistic.

The critical value of t distribution in the t-table has two points of reference the degree

of freedom (d/f) & the level of significance10. The d/f is quoted in the extreme left column

of the table. For any d/f there are eight different values quoted in the corresponding row.

Each value corresponds to a specific level of significance (quoted on the top row of the

table). Most tests are performed at 95% level of significance (.05 in the table). Thus, if the

critical table value of t is found less than the t statistic we declare our estimates a & b as

significant and reject our estimates if the results are otherwise. If the value of t-statistic inour example is 1.92 and the critical value in the table for 8 d/f and 95% significance is

2.306 we shall accept the null hypothesis. However at 90% significance the critical value is

1.86 (which is less than the t-statistic) and we shall reject null hypothesis,

7 we call the estimates as parameters because there values dont remain the same in the face of increase or

decrease in values and other changes.8

Termed as standard error of b

9 (n - k) is termed as degree of freedom & ( RRt

is termed as error term

10 For this see the slides in the shares folder