Econ 3790: Business and Economics Statistics

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Econ 3790: Business and Economics Statistics. Instructor: Yogesh Uppal yuppal@ysu.edu. Sampling Distribution of b 1. Expected value of b 1 : E(b 1 ) = b 1 Variance of b 1 : Var(b 1 ) = σ 2 /SS x. Estimate of σ 2. The mean square error (MSE) provides the estimate of σ 2. - PowerPoint PPT Presentation

Transcript of Econ 3790: Business and Economics Statistics

  • Econ 3790: Business and Economics StatisticsInstructor: Yogesh Uppalyuppal@ysu.edu

  • Sampling Distribution of b1Expected value of b1:

    E(b1) =b1

    Variance of b1:Var(b1) = 2/SSx

  • Estimate of 2The mean square error (MSE) provides the estimate of 2.

    s 2 = MSE = SSE/(n - 2)where:

  • Sample variance of b1Estimate of variance of b1:

    Standard error of b1:

    s is called the standard error of the estimate.

  • Interval Estimate of b1:(1-a)100% confidence interval for b1 is:

    Where ta/2 is the value from t distribution with (n-2) degrees of freedom such that probability in the upper tail is a/2.

  • Example: Reed Auto Saless2 = MSE = SSE/(n - 2) = 8.2/3 =2.73

    95% confidence interval for b1:

    We can say we 95% confidence that b1 will lie between 1.87 and 7.13.

  • Testing for Significance: t TestHypotheses

    Test Statistic

    Where b1 is the slope estimate and SE(b1) is the standard error of b1.

  • Rejection RuleTesting for Significance: t Testwhere: t is based on a t distributionwith n - 2 degrees of freedomReject H0 if p-value < a or t < -tor t > t

  • 1. Determine the hypotheses.2. Specify the level of significance.3. Select the test statistic.a = .054. State the rejection rule.Reject H0 if p-value < .05or t 3.182 or t 3.182Testing for Significance: t Test

  • Testing for Significance: t Test5. Compute the value of the test statistic.6. Determine whether to reject H0.t = 5.42 > ta/2 = 3.182. We can reject H0.

  • Some Cautions about theInterpretation of Significance Tests Just because we are able to reject H0: b1 = 0 and demonstrate statistical significance does not enableus to conclude that there is a linear relationshipbetween x and y. Rejecting H0: b1 = 0 and concluding that therelationship between x and y is significant does not enable us to conclude that a cause-and-effectrelationship is present between x and y.

  • Multiple Regression ModelThe equation that describes how the dependent variable y is related to the independent variables x1, x2, . . . xp and an error term is called the multiple regression model.y = b0 + b1x1 + b2x2 + . . . + bpxp + ewhere:b0, b1, b2, . . . , bp are the parameters, ande is a random variable called the error term

  • A simple random sample is used to compute sample statistics b0, b1, b2, . . . , bp that are used as the point estimators of the parameters b0, b1, b2, . . . , bp.Estimated Multiple Regression EquationThe estimated multiple regression equation is:

  • Interpreting the Coefficients In multiple regression analysis, we interpret each regression coefficient as follows: bi represents an estimate of the change in y corresponding to a 1-unit increase in xi when all other independent variables are held constant.

  • Example: Car Sales Suppose we believe that number of cars sold (y) isnot only related to the number of ads (x1), but also to the minimum down payment required at the (x2). The regression model can be given by:Multiple Regression Modelwhere y = number of cars sold x1 = number of ads x2 = minimum down payment required (000)y = 0 + 1x1 + 2x2 +

  • Estimated Regression Equationy = 14.4 + 3.7 x1 + 0.251 x2Interpretation? Estimated values of y?Error?Prediction?

  • Multiple Coefficient of DeterminationRelationship Among SST, SSR, SSEwhere: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to errorSST = SSR + SSE

  • Multiple Coefficient of DeterminationR2 = 84.63/89.2 = .949Adjusted Multiple Coefficient of DeterminationStandard Error of EstimateR2 = SSR/SST

  • Testing for Significance: t TestHypothesesRejection RuleTest StatisticsReject H0 if p-value < a orif t < -tor t > t where t is based on a t distributionwith n - p - 1 degrees of freedom.

  • Example: Testing for significance of coefficientsHypothesesRejection RuleFor = .05 and d.f. = ?, t.025 = Test Statistics

  • Testing for Significance of Regression: F TestHypothesesRejection RuleTest Statistics H0: 1 = 2 = . . . = p = 0 Ha: One or more of the parameters is not equal to zero.F = MSR/MSEReject H0 if p-value < a or if F > F,where F is based on an F distributionwith p d.f. in the numerator andn - p - 1 d.f. in the denominator.

  • The years of experience, score on the aptitudetest, and corresponding annual salary ($1000s) for a sample of 20 programmers is shown on the nextslide.Example 2: Programmer Salary SurveyMultiple Regression Model A software firm collected data for a sampleof 20 computer programmers. A suggestionwas made that regression analysis couldbe used to determine if salary was relatedto the years of experience and the scoreon the firms programmer aptitude test.

  • 471581001669210568463378100868286847580839188737581748779947089244323.734.335.83822.223.130333826.636.231.6293430.133.928.230Exper.ScoreScoreExper.SalarySalaryMultiple Regression Model

  • Suppose we believe that salary (y) isrelated to the years of experience (x1) and the score onthe programmer aptitude test (x2) by the following regression model:Multiple Regression Modelwhere y = annual salary ($1000) x1 = years of experience x2 = score on programmer aptitude testy = 0 + 1x1 + 2x2 +

  • Solving for b0, b1 and b2:

    Sheet1

    ABCDEFGHIJ

    1ProgrammerExperienceTest ScoreSalary

    2147824.0

    32710043.0

    4318623.7

    5458234.3

    6588635.8

    76108438.0

    8707522.2

    9818023.1

    10968330.0

    111069133.0

    121198838.0

    131227326.6

    1413107536.2

    151458131.6

    161567429.0

    171688734.0

    181747930.1

    191869433.9

    201937028.2

    212038930.0

    22

    23

    24SUMMARY OUTPUT

    25

    26Regression Statistics

    27Multiple R0.9133340588

    28R Square0.8341791029

    29Adjusted R Square0.814670762

    30Standard Error2.4187620762

    31Observations20

    32

    33ANOVA

    34dfSSMSFSignificance F

    35Regression2500.3285303144250.164265157242.76012552080.0000002328

    36Residual1799.45696968565.8504099815

    ABCDEFGHI

    38

    39Coeffic.Std. Err.t StatP-valueLo. 95%Up. 95%Lo. 95.0%Up. 95.0%

    40Intercept3.17393626986.15606682880.51557859230.6127886957-9.814247562716.1621201022-9.814247562716.1621201022

    41Experience1.40390248510.1985669127.07017332840.00000188060.98496233591.82284263440.98496233591.8228426344

    42Test Score0.25088544780.0773541273.24333629880.00478001840.08768227780.41408861780.08768227780.4140886178

    43

    Sheet1

    0

    0

    0

    0

    0

    TV Ads

    Residuals

    TV Ads Residual Plot

    Sheet2

    Sheet3

  • Anova Table

  • Estimated Regression EquationSALARY = 3.174 + 1.404(EXPER) + 0.251(SCORE) b1 = 1.404 implies that salary is expected to increase by $1,404 for each additional year of experience (when the variable score on programmer attitude test is held constant).b2 = 0.251 implies that salary is expected to increase by $251 for each additional point scored on the programmer aptitude test (when the variable years of experience is heldconstant).

  • PredictionSuppose Bob had an experience of 4 years and had a score of 78 on the aptitude test. What would you estimate (or expect) his score to be? = 3.174 + 1.404*(4) + 0.251(78) = 28.358Bobs estimated salary is $28,358.

  • ErrorBobs actual salary is $24000. How much error we made in estimating his salary based on his experience and score?

    So, we shall overestimate Bobs salary.

  • Multiple Coefficient of DeterminationRelationship Among SST, SSR, SSEwhere: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to errorSST = SSR + SSE

  • Multiple Coefficient of DeterminationR2 = 500.3285/599.7855 = .83418R2 = SSR/SSTAdjusted Multiple Coefficient of Determination

  • Testing for Significance: t TestHypothesesRejection RuleTest StatisticsReject H0 if p-value < a orif t < -tor t > t where t is based on a t distributionwith n - p - 1 degrees of freedom.

  • ExampleHypothesesRejection RuleFor = .05 and d.f. = 17, t.025 = 2.11Reject H0 if p-value < .05 or if t > 2.11Test StatisticsSince t=7.07 > t0.025 =2.11, we reject H0.

  • Testing for Significance of Regression: F TestHypothesesRejection RuleTest Statistics H0: 1 = 2 = . . . = p = 0 Ha: One or more of the parameters is not equal to zero.F = MSR/MSEReject H0 if p-value < a or if F > F,where F is based on an F distributionwith p d.f. in the numerator andn - p - 1 d.f. in the denominator.

  • ExampleHypotheses H0: 1 = 2 = 0 Ha: One or both of the parameters is not equal to zero.Rejection RuleFor = .05 and d.f. = 2, 17; F.05 = 3.59Reject H0 if p-value < .05 or F > 3.59Test StatisticsF = MSR/MSE = 250.17/5.86 = 42.8F = 42.8 > F0.05 = 3.59, so we can reject H0.