Econ 3790: Business and Economics Statistics

Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal yuppal@ysu.edu

Sampling Distribution of b1 • Expected value of b1: E(b1) =b1 • Variance of b1: Var(b1) = σ2/SSx

Estimate of σ2 • The mean square error (MSE) provides the estimate of σ2. s2 = MSE = SSE/(n - 2) where:

Sample variance of b1 • Estimate 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 Sales s2 = 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 Test • Hypotheses • Test Statistic • Where b1 is the slope estimate and SE(b1) is the standard error of b1.

Testing for Significance: t Test • Rejection Rule Reject H0 if p-value <a or t< -tor t>t where: tis based on a t distribution with n - 2 degrees of freedom

Testing for Significance: t Test 1. Determine the hypotheses. a = .05 2. Specify the level of significance. 3. Select the test statistic. 4. State the rejection rule. Reject H0 if p-value < .05 or t ≤ 3.182 or t ≥ 3.182

Testing for Significance: t Test 5. 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 • Rejecting H0: b1 = 0 and concluding that the • relationship between x and y is significant does not enable us to conclude that a cause-and-effect • relationship is present between x and y. • Just because we are able to reject H0: b1 = 0 and • demonstrate statistical significance does not enable • us to conclude that there is a linear relationship • between x and y.

Multiple Regression Model The equation that describes how the dependent variable y is related to the independent variables x1, x2, . . . xp and an error term is called the multipleregression model. y = b0 + b1x1 + b2x2 +. . . + bpxp + e where: b0, b1, b2, . . . , bp are the parameters, and e is a random variable called the error term

^ y = b0 + b1x1 + b2x2 + . . . + bpxp Estimated Multiple Regression Equation 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. The 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.

Multiple Regression Model Example: Car Sales Suppose we believe that number of cars sold (y) is not 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: y = 0 + 1x1 + 2x2 +  where y = number of cars sold x1 = number of ads x2 = minimum down payment required (‘000)

Estimated Regression Equation y = 14.4 + 3.7 x1 + 0.251 x2 • Interpretation? • Estimated values of y? • Error? • Prediction?

Multiple Coefficient of Determination • Relationship Among SST, SSR, SSE SST = SSR + SSE where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error

Multiple Coefficient of Determination R2 = SSR/SST R2 = 84.63/89.2 = .949 Adjusted Multiple Coefficient of Determination Standard Error of Estimate

Testing for Significance: t Test Hypotheses Test Statistics Rejection Rule Reject H0 if p-value <a or if t< -tor t>twhere t is based on a t distribution with n - p - 1 degrees of freedom.

Example: Testing for significance of coefficients Hypotheses • For  = .05 and d.f. = ?, t.025 = Rejection Rule Test Statistics

Testing for Significance of Regression: F Test H0: 1 = 2 = . . . = p = 0 Ha: One or more of the parameters is not equal to zero. Hypotheses F = MSR/MSE Test Statistics Rejection Rule Reject H0 if p-value <a or if F > F, where F is based on an F distribution with p d.f. in the numerator and n - p - 1 d.f. in the denominator.

Multiple Regression Model • Example 2: Programmer Salary Survey A software firm collected data for a sample of 20 computer programmers. A suggestion was made that regression analysis could be used to determine if salary was related to the years of experience and the score on the firm’s programmer aptitude test. The years of experience, score on the aptitude test, and corresponding annual salary ($1000s) for a sample of 20 programmers is shown on the next slide.

Multiple Regression Model Exper. Score Salary Exper. Score Salary 4 7 1 5 8 10 0 1 6 6 78 100 86 82 86 84 75 80 83 91 9 2 10 5 6 8 4 6 3 3 88 73 75 81 74 87 79 94 70 89 38 26.6 36.2 31.6 29 34 30.1 33.9 28.2 30 24 43 23.7 34.3 35.8 38 22.2 23.1 30 33

Multiple Regression Model Suppose we believe that salary (y) is related to the years of experience (x1) and the score on the programmer aptitude test (x2) by the following regression model: y = 0 + 1x1 + 2x2 +  where y = annual salary ($1000) x1 = years of experience x2 = score on programmer aptitude test

Solving for b0, b1 and b2:

Anova Table

Estimated Regression Equation SALARY = 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 held constant).

Prediction • Suppose 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) = 28.358 • Bob’s estimated salary is $28,358.

Error • Bob’s actual salary is $24000. How much error we made in estimating his salary based on his experience and score? • So, we shall overestimate Bob’s salary.

Multiple Coefficient of Determination • Relationship Among SST, SSR, SSE SST = SSR + SSE where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error

Multiple Coefficient of Determination R2 = SSR/SST R2 = 500.3285/599.7855 = .83418 Adjusted Multiple Coefficient of Determination

Testing for Significance: t Test Hypotheses Test Statistics Rejection Rule Reject H0 if p-value <a or if t< -tor t>twhere t is based on a t distribution with n - p - 1 degrees of freedom.

Example Hypotheses • For  = .05 and d.f. = 17, t.025 = 2.11 • Reject H0 if p-value < .05 or if t> 2.11 Rejection Rule Test Statistics • Since t=7.07 > t0.025 =2.11, we reject H0.

Testing for Significance of Regression: F Test H0: 1 = 2 = . . . = p = 0 Ha: One or more of the parameters is not equal to zero. Hypotheses F = MSR/MSE Test Statistics Rejection Rule Reject H0 if p-value <a or if F > F, where F is based on an F distribution with p d.f. in the numerator and n - p - 1 d.f. in the denominator.

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

Econ 3790: Business and Economics Statistics