Bài giảng Managerial Economics - Chapter 04: Basic Estimation Techniques

Chapter 4: Basic Estimation TechniquesMcGraw-Hill/IrwinCopyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.Basic EstimationParametersThe coefficients in an equation that determine the exact mathematical relation among the variables Parameter estimationThe process of finding estimates of the numerical values of the parameters of an equationRegression AnalysisRegression analysisA statistical technique for estimating the parameters of an equation and testing for statistical significance 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:Simple Linear Regression Simple linear regression model relates dependent variable Y to one independent (or explanatory) variable XRandom EffectFirm expects $10,000 in sales from each agency plus an additional $5 in sales from each additional $1 of advertising.Simple Linear RegressionParameter estimates are obtained by choosing values of a & b that minimize the sum of squared residualsThe residual is the difference between the actual and fitted values of Y: Yi – ŶiThe sample regression line is an estimate of the true regression lineSample DataTime series data – values taken by a variable over timeCross sectional data – values for multiple occurrences of a variable at a point in timeSample Regression Line (Figure 4.2)A08,0002,00010,0004,0006,00010,00020,00030,00040,00050,00060,00070,000Advertising expenditures (dollars)Sales (dollars)S•••••••Sample regression line Ŝi = 11,573 + 4.9719AŜi = 46,376eiSi = 60,000Population regression line – true regression lineSample regression line – estimate of the true regression lineThe Method of Least SquaresStatistical Output - ExcelY=11573.0 + 4.97191XThree Kinds of CorrelationUnbiased EstimatorsThe distribution of values the estimates might take is centered around the true value of the parameter The estimates â & do not generally equal the true values of a & b â & are random variables computed using data from a random sampleUnbiased EstimatorsAn estimator is unbiased if its average value (or expected value) is equal to the true value of the parameterExample of an Unbiased EstimateYou 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.Statistical SignificanceMust 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 zeroTest for statistical significance using t-tests or p-valuesRelative Frequency Distribution* (Figure 4.3)*Also called a probability density function (pdf)0821046113579Errors Around Regression LineX1X2XY f(e)Confidence intervalAn 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. Areas under the PDF measure probabilitiesStatistical SignificanceRelative Frequency Distribution* (Figure 4.3)-53-35-111-4-2024*Also called a probability density function (pdf)Relative frequency of estimated b when true b is zeroProbability of Type I error- finding parameter significant when it is notEstimated bTrue bTest for Statistical SignificanceTo test for statistical significance we need a statistic for measuring deviations from the mean valueThe standard error of the estimate provides that measuret-value measures how many standard errors we are from the meanThe t-test indicates whether the slope parameter is statistically significantStatistical SignificanceStatistical Output - ExcelY=11573.0 + 4.97191XPerforming a t-TestFirst determine the level of significanceProbability of finding a parameter estimate to be statistically different from zero when, in fact, it is zeroProbability of a Type I Error1 minus level of significance = level of confidencePerforming a t-TestUse 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 t-ratio is computed asStudent t DistributionsThe fewer the degrees of freedom, the flatter is the distribution.Degrees of FreedomNumber of observations you have less the minimum number of observations needed to fit the curve2 observations are needed to fit a straight line3 observations are needed to fit a plane in 3-D spacePerforming a t-TestIf the absolute value of t-ratio is greater than the critical t, the parameter estimate is statistically significant at the given level of significancedf = n – kn = 7k = 2df = 5Statistical Output - ExcelY=11573.0 + 4.97191Xt* = 2.571Using p-ValuesTreat as statistically significant only those parameter estimates with p-values smaller than the maximum acceptable significance levelp-value gives exact level of significanceAlso the probability of finding significance when none existsStatistical Output - ExcelY=11573.0 + 4.97191Xt* = 2.571Coefficient of DeterminationR2 measures the percentage of total variation in the dependent variable (Y) that is explained by the regression equationRanges from 0 to 1High R2 indicates Y and X are highly correlatedHigh and Low CorrelationCoefficient of Determination (R2)YXYSSTSSE (unexplained)SSR (explained) _R2 – ratio of explained to total variationStatistical Output - ExcelY=11573.0 + 4.97191Xt* = 2.571F-TestUsed to test for significance of overall regression equationMeasures goodness of fitF value (ratio of explained to unexplained sum of squares)Compare F-statistic to critical F-value from F-tableTwo degrees of freedom, k – 1 & n – kLevel of significanceIf F-statistic exceeds the critical F, the regression equation overall is statistically significantExample:n-k = 5k-1 = 1Statistical Output - ExcelF-TestIf F-statistic exceeds the critical F, the regression equation overall is statistically significant at the specified level of significanceMultiple RegressionUses more than one explanatory variableCoefficient for each explanatory variable measures the change in the dependent variable associated with a one-unit change in that explanatory variable, all else constantQuadratic Regression ModelsUse when curve fitting scatter plot is U-shaped or ∩-shaped Y = a + bX + cX2For linear transformation compute new variable Z = X2Estimate Y = a + bX + cZΔY/ΔX = b + 2cXAt min or max X= -b/2c c is positive (negative) if there is a minimum (maximum)Quadratic RegressionLog-Linear Regression Models Use when relation takes the form: Y = aXbZc Percentage change in YPercentage change in X b = Percentage change in YPercentage change in Z c = b and c are elasticities Transform by taking natural logarithms:Log Linear Regression