Econ 3790: Business and Economics Statistics. Instructor: Yogesh Uppal

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1 Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal

2 Sampling Distribution of b 1 Expected value of b 1 : Variance of b 1 : E(b 1 ) = 1 Var(b 1 ) = σ 2 /SS x

3 Estimate of σ 2 The mean square error (MSE) provides the estimate of σ 2. s 2 = MSE = SSE/(n 2) where: SSE ( y i y ˆ i ) 2

4 Sample variance of b1 Estimate of variance of b 1 : Var( b1) s 2 SS MSE x SS x Standard error of b 1 : SE( b1) s 2 SS x MSE SS x s SS x s is called the standard error of the estimate.

5 Interval Estimate of 1 : (1-)100% confidence interval for 1 is: b t b 1 / 2 SE( ) 1 Where t /2 is the value from t distribution with (n-2) degrees of freedom such that probability in the upper tail is /2.

6 Example: Reed Auto Sales s 2 = MSE = SSE/(n - 2) = 8.2/3 = SE( b ) s SS x % confidence interval for 1 : We can say we 95% confidence that 1 will lie between 1.87 and 7.13.

7 Testing for Significance: t Test Hypotheses H H : a: 1 0 Test Statistic t b1 0 SE( b ) 1 Where b 1 is the slope estimate and SE(b 1 ) is the standard error of b 1.

8 Testing for Significance: t Test Rejection Rule Reject H 0 if p-value < or t < -t or t > t where: t is based on a t distribution with n - 2 degrees of freedom

9 Testing for Significance: t Test 1. Determine the hypotheses. H H 2. Specify the level of significance. : a: 1 0 = Select the test statistic. t b1 SE( b 1 ) 4. State the rejection rule. Reject H 0 if p-value <.05 or t or t 3.182

10 Testing for Significance: t Test 5. Compute the value of the test statistic. t b SE( b 1 ) Determine whether to reject H 0. t = 5.42 > t /2 = We can reject H 0.

11 Some Cautions about the Interpretation of Significance Tests Rejecting H 0 : 1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and and-effect relationship is present between x and y. Just because we are able to reject H 0 : 1 = 0 and demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y.

12 Multiple Regression Model The equation that describes how the dependent variable y is related to the independent variables x 1, x 2,... x p and an error term y = is 0 called + 1 x 1 + the 2 x 2 multiple p xregression p + model. where: 0, 1, 2,..., p are the parameters,, and is a random variable called the error term

13 Estimated Multiple Regression Equation A simple random sample is used to compute sample statistics b 0, b 1, b 2,..., b p that are used as the point estimators of the parameters 0, 1, 2,..., p. The estimated multiple regression equation is: ^y = b 0 + b 1 x 1 + b 2 x b p x p

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

15 Multiple Regression Model Example: Car Sales Suppose we believe that number of cars sold (y)( ) is not only related to the number of ads (x( 1 ), but also to the minimum down payment required at the (x( 2 ). The regression model can be given by: y = x x 2 + where y = number of cars sold x 1 = number of ads x 2 = minimum down payment required ( 000)(

16 Estimated Regression Equation y = *x1* 25* x2 Interpretation? Estimated values of y? Error? Prediction?

17 Multiple Coefficient of Determination Relationship Among SST, SSR, SSE SST = SSR + SSE where: ( y y ) i 2 ( y ˆ y ) i SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error 2 ( y y ˆ ) i i 2

18 Multiple Coefficient of Determination R 2 = 84.63/89.2 =.949 Adjusted Multiple Coefficient of Determination R R 2 = SSR/SST n 1 1 ( 1 R ) n p a Standard Error of Estimate s MSE SSE n p 1

19 Testing for Significance: t Test Hypotheses Test Statistics Rejection Rule H : 0 0 i H : 0 a i b t SE i ( b i ) Reject H 0 if p-value < or if t < -t or t > t where t is based on a t distribution with n - p - 1 degrees of freedom.

20 Example: Testing for significance of coefficients Hypotheses H H 0 a : i : i 0 0 Rejection Rule For =.05 and d.f. =?, t.025 = Test Statistics t b SE i ( b i )

21 Testing for Significance of Regression: F Test Hypotheses Test Statistics H 0 : 1 = 2 =... = p = 0 H a : One or more of the parameters is not equal to zero. F = MSR/MSE Rejection Rule Reject H 0 if p-value < 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.

22 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 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.

23 Exper Exper. Score Score Score Score Exper Exper. Salary Salary Salary Salary Multiple Regression Model Multiple Regression Model

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

25 Solving for 0, 1 and 2 : A B C Coeffic. Std. Err. 40 Intercept Experience Test Score

26 Anova Table Source of Variation Sum of Squares Degrees of Freedom Mean Square F-statistic Regression Error.... Total

27 Estimated Regression Equation SALARY = (EXPER) (SCORE) b 1 = 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 = 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).

28 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? ŷ = *(4) (78) = Bob s s estimated salary is $28,358.

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

30 Multiple Coefficient of Determination Relationship Among SST, SSR, SSE SST = SSR + SSE where: ( y y ) i 2 ( y ˆ y ) i SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error 2 ( y y ˆ ) i i 2

31 Multiple Coefficient of Determination R 2 = SSR/SST R 2 = / = Adjusted Multiple Coefficient of Determination R a R n 1 1 ( 1 R ) n p a ( )

32 Testing for Significance: t Test Hypotheses Test Statistics Rejection Rule H : 0 0 i H : 0 a i b t SE i ( b i ) Reject H 0 if p-value < or if t < -t or t > t where t is based on a t distribution with n - p - 1 degrees of freedom.

33 Example Hypotheses H 0 H a : 1 : Rejection Rule For =.05 and d.f. = 17, t.025 = 2.11 Reject H 0 if p-value <.05 or if t > 2.11 b Test Statistics t SE( b 1 ) Since t=7.07 > t =2.11, we reject H 0.

34 Testing for Significance of Regression: F Test Hypotheses Test Statistics H 0 : 1 = 2 =... = p = 0 H a : One or more of the parameters is not equal to zero. F = MSR/MSE Rejection Rule Reject H 0 if p-value < 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.

35 Example Hypotheses H 0 : 1 = 2 = 0 H a : One or both of the parameters is not equal to zero. Rejection Rule Test Statistics For =.05 and d.f. = 2, 17; F.05 = 3.59 Reject H 0 if p-value <.05 or F > 3.59 F = MSR/MSE = /5.86 = 42.8 F = 42.8 > F 0.05 = 3.59, so we can reject H 0.