Chapter 4 Basic Estimation Techniques Mc GrawHillIrwin Copyright

Basic Estimation • Parameters • The coefficients in an equation that determine the exact mathematical relation among the variables • Parameter estimation • The process of finding estimates of the numerical values of the parameters of an equation 4 -2

Regression Analysis • Regression analysis • A statistical technique for estimating the parameters of an equation and testing for statistical significance 4 -3

Simple Linear Regression • Simple linear regression model relates dependent variable Y to one independent (or explanatory) variable X • Intercept parameter (a) gives value of Y where regression line crosses Y-axis(value of Y when X is zero) • Slope parameter (b) gives the change in Y associated with a one-unit change in X: 4 -4

Random Effect Firm expects $10, 000 in sales from each agency plus an additional $5 in sales from each additional $1 of advertising. 4 -5

Simple Linear Regression • Parameter estimates are obtained by choosing values of a & b that minimize the sum of squared residuals • The residual is the difference between the actual and fitted values of Y: Yi – Ŷi • The sample regression line is an estimate of the true regression line 4 -6

Sample Data • Time series data – values taken by a variable over time • Cross sectional data – values for multiple occurrences of a variable at a point in time 4 -7

Sample Regression Line (Figure 4. 2) S Population regression line – true regression line Sample regression line – estimate of the true regression line Si = 60, 000 70, 000 Sales (dollars) 60, 000 ei 50, 000 20, 000 10, 000 • • 40, 000 30, 000 • Sample regression line Ŝi = 11, 573 + 4. 9719 A • Ŝi = 46, 376 • • • A 0 2, 000 4, 000 6, 000 8, 000 10, 000 Advertising expenditures (dollars) 4 -8

The Method of Least Squares 4 -9

Statistical Output - Excel Y=11573. 0 + 4. 97191 X 4 -10

Three Kinds of Correlation 4 -11

Unbiased Estimators • The estimates â & do not generally equal the true values of a & b • â & are random variables computed using data from a random sample • The distribution of values the estimates might take is centered around the true value of the parameter 4 -12

Unbiased Estimators • An estimator is unbiased if its average value (or expected value) is equal to the true value of the parameter 4 -13

Example of an Unbiased Estimate You blindly draw 5 balls from a pot containing 80 red balls and 20 blue balls. What is the probability of drawing a sample that proportionately replicates the percentage of red and blue balls in the pot? You might draw all red balls and inaccurately predict that there are no blue balls in the pot. Now place the balls back in the pot and draw another sample of 5 balls. As you repeat this exercise and average the percentage of red and blue balls in your samples, the average should approach the population average. The average percentages in the samples is an unbiased estimate of the population average. The greater the number of draws, the closer you will likely come to accurately predicting the characteristics of the population. 4 -14

Statistical Significance • Must determine if there is sufficient statistical evidence to indicate that Y is truly related to X (i. e. , b 0) • Even if b = 0, it is possible that the sample will produce an estimate that is different from zero • Test for statistical significance using -tests or p-values t 4 -15

Relative Frequency Distribution* (Figure 4. 3) 1 0 1 2 3 4 5 *Also called a probability density function (pdf) 6 7 8 9 10 4 -16

Errors Around Regression Line f(e ) Y X X 1 X 2 4 -17

Statistical Significance • Confidence interval • An estimate of a population parameter that consists of a range of values bounded by statistics called upper and lower confidence limits, within which the value of the parameter is expected to be located. • Probability Density Function (PDF) • The statistical function that shows how the density of possible observations in a population is distributed. w Areas under the PDF measure probabilities 4 -18

Relative Frequency Distribution* (Figure 4. 3) Relative frequency of estimated b when true b is zero 1 Estimated b Probability of Type I error- finding parameter significant when it is not True b -5 -4 -3 -2 -1 0 1 *Also called a probability density function (pdf) 2 3 4 5 4 -19

Test for Statistical Significance • To test for statistical significance we need a statistic for measuring deviations from the mean value • The standard error of the estimate provides that measure • t-value measures how many standard errors we are from the mean • The t-test indicates whether the slope parameter is statistically significant 4 -20

Statistical Significance 4 -21

Statistical Output - Excel Y=11573. 0 + 4. 97191 X 4 -22

Performing a t-Test • First determine the level of significance • Probability of finding a parameter estimate to be statistically different from zero when, in fact, it is zero • Probability of a Type I Error • 1 minus level of significance = level of confidence 4 -23

Performing a t-Test • t-ratio is computed as • Use t-table to choose critical t-value with n – k degrees of freedom for the chosen level of significance • n = number of observations • k = number of parameters estimated 4 -24

Student t Distributions The fewer the degrees of freedom, the flatter is the distribution. 4 -25

Degrees of Freedom • Number of observations you have less the minimum number of observations needed to fit the curve • 2 observations are needed to fit a straight line • 3 observations are needed to fit a plane in 3 -D space 4 -26

Performing a t-Test • If the absolute value of t-ratio is greater than the critical t, the parameter estimate is statistically significant at the given level of significance 4 -27

df = n – k n=7 k=2 df = 5 4 -28

Statistical Output - Excel Y=11573. 0 + 4. 97191 X t* = 2. 571 4 -29

Using p-Values • Treat as statistically significant only those parameter estimates with p-values smaller than the maximum acceptable significance level • p-value gives exact level of significance • Also the probability of finding significance when none exists 4 -30

Statistical Output - Excel Y=11573. 0 + 4. 97191 X t* = 2. 571 4 -31

Coefficient of Determination • R 2 measures the percentage of total variation in the dependent variable (Y) that is explained by the regression equation • Ranges from 0 to 1 • High R 2 indicates Y and X are highly correlated 4 -32

High and Low Correlation 4 -33

Coefficient of Determination (R 2) Y SSE (unexplained) SST SSR (explained) _ Y X R 2 – ratio of explained to total variation 4 -34

Statistical Output - Excel Y=11573. 0 + 4. 97191 X t* = 2. 571 4 -35

F-Test • Used to test for significance of overall regression equation • Measures goodness of fit • F value (ratio of explained to unexplained sum of squares) • Compare F-statistic to critical F-value from Ftable • Two degrees of freedom, k – 1 & n – k • Level of significance • If F-statistic exceeds the critical F, the regression equation overall is statistically significant 4 -36

Example: n-k = 5 k-1 = 1 4 -37

Statistical Output - Excel 4 -38

F-Test • If F-statistic exceeds the critical F, the regression equation overall is statistically significant at the specified level of significance 4 -39

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Multiple Regression • Uses more than one explanatory variable • Coefficient for each explanatory variable measures the change in the dependent variable associated with a one-unit change in that explanatory variable, all else constant 4 -41

Quadratic Regression Models • Use when curve fitting scatter plot is shaped or ∩-shaped • Y = a + b. X + c. X 2 U- • For linear transformation compute new variable Z = X 2 • Estimate Y = a + b. X + c. Z • ΔY/ΔX = b + 2 c. X • At min or max X= -b/2 c • c is positive (negative) if there is a minimum (maximum) 4 -42

Quadratic Regression 4 -43

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Log-Linear Regression Models • Use when relation takes the form: Y = a. Xb. Zc Percentage change in Y • b= Percentage change in X Percentage change in Y • c= Percentage change in Z • Transform by taking natural logarithms: • b and c are elasticities 4 -45

Log Linear Regression 4 -46

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