10B11PD311 Economics REGRESSION ANALYSIS

10B11PD311 Economics Regression Techniques and Demand Estimation Some important questions before a firm are : How much will the revenue of the firm change after increasing the price of the commodity? How much will the quantity demanded of the commodity increase if consumers’ income increase by a certain amount? How much the demand of the commodity will change if the firm double its advertising expenditure? Answers for these questions are important for a firm to achieve its objective.

10B11PD311 Economics To Solve these problems, we have a tool -- REGRESSION ANALYSIS which helps to predict the value of one variable for the given value of the other variable. In Managerial Economics, regression analysis helps for estimating the demand of a commodity in different conditions. Regression Techniques : Y = a + bX Dependent variable Independent variable Vertical intercept Slope i.e., marginal change

10B11PD311 Economics Y X a b = dY/dX

10B11PD311 Economics Scatter Diagram

10B11PD311 Economics  Regression Line: Line of Best Fit  Regression Line: Minimizes the sum of the squared vertical deviations (e t ) of each point from the regression line.  Ordinary Least Squares (OLS) Method

10B11PD311 Economics

Model:

10B11PD311 Economics  Let f(x) = a + bx be the straight line to be fitted to the given data points (x i, y i ), i = 1,2, ….,n.  At x = x i, the experimental value of the ordinate is y i and the corresponding value of the fitting curve is f(x i ).  Then, e i = y i - f(x i ) is the error of approximation at x = x i.  The best possible approximation is given by minimizing the sum of squares of errors called the Least square approximation. (x i, y i ) f(x)=a+bx xixi yiyi

10B11PD311 Economics  The sum of squares of errors is e i = y i - f(x i ) f(x) = a + bx

10B11PD311 Economics  The weights of a calf taken at weekly intervals are supplied below. Fit a straight line and calculate the average rate of growth per week. Age x : Weight y : Solve the above two equations and get the values of a and b.

10B11PD311 Economics x i y i xy

10B11PD311 Economics a = 46.1 and b = 6.06 Straight line is a + bx = x The average rate of growth per week is 6.06.

10B11PD311 Economics Q. Fit a straight line to the following data : x:01234 y: y = x

10B11PD311 Economics The strength of relationship between the dependent variable and the independent variables can be measured in two ways : 1.The t-statistic : - used to test the strength of the relationship between an independent variable and the dependent variable. 2. The Coefficient of Determination (R 2 ) : - used to measure how well the overall equation explains changes in the dependent variable. Testing Regression Estimates :

10B11PD311 Economics Calculation of t Statistic : The t-test is used to determine if there is significant relationship between the dependent variable and the independent variable.

10B11PD311 Economics Standard Error of the Slope Estimate where k is the no. of independent variables in the equation. The value of “n – k-1” is called as degree of freedom. Here k = 1; And n is number of observations.

10B11PD311 Economics Calculation of the t Statistic Degrees of Freedom = (n-k-1) = (10-2) = 8 Critical Value at 5% level =2.306

10B11PD311 Economics

If the absolute value of t > tabular value or The value of standard error < tabular value, there exists a statistically significant relationship between the two parameters

10B11PD311 Economics Confidence interval b *(s b ) *(0.52)

10B11PD311 Economics Determination of “Coefficient of Determination (R 2 ) Total variation = ∑ ( Y t – Y ) 2 Total Variation = Explained Variation + Unexplained Variation t

10B11PD311 Economics. YtYt Y X Y Total Variation ( Y t – Y ) Ŷ = â + bX Unexplained variation ( Y t – Ŷ t ) Explained variation (Ŷ t – Y ) ˆ XtXt

10B11PD311 Economics The value of R 2 ranges from 0 to 1. If there is no relationship between the independent variables and the dependent variable R 2 = zero. When relationship is high the equation is said to fit the data well and a low value would be indicative of rather poor fit. t

10B11PD311 Economics Coefficient of Determination

10B11PD311 Economics Coefficient of Correlation

10B11PD311 Economics The regression equation can be used to predict or estimate the value of independent variable. But the predicted value would not be 100 percent correct. The standard error of the estimate (Se) is a measure of probable error in the predicted values. It can be calculated as – S e = ∑ (Y – Y) 2 – b ∑ (X – X)(Y – Y) (n – k-1) The predicted value of Ŷ is called as a point estimate of the value of the dependent variable to distinguish that estimate from a confidence interval estimate. Prediction Using Regression Equations: tt t

10B11PD311 Economics The range of values of Ŷ at a particular confidence interval, can be calculated as – Ŷ ± t n-k-1 S e This means the actual value of Y will fall within this range.