1. Diploma in Statistics Introduction to Regression Lecture 4.11 Introduction to Regression Lecture 4.2 Indicator variables for estimating seasonal effects in time series –another application, Meter Sales analysis Correlated explanatory variables

2. Diploma in Statistics Introduction to Regression Lecture 4.12 Housing Completions case study

3. Diploma in Statistics Introduction to Regression Lecture 4.13 Table 1.7 Completions and Quarterly Indicators

4. Diploma in Statistics Introduction to Regression Lecture 4.14 Model formulation Completions =  1  Q 1 +  2  Q 2 +  3  Q 3 +  4  Q4 +   Time + . Quarter 1:Completions =  1 +   Time Quarter 2:Completions =  2 +   Time, Homework 4.1.1:Write down the prediction formulas for future third and fourth quarters.

5. Diploma in Statistics Introduction to Regression Lecture 4.15 Prediction formula Predicted completions = 3,248  Q1 + 3901  Q2 + 4174  Q3 + 5031  Q4 + 250  Time  500 Exercise:Write down separate prediction formulas for each of the four quarters. Make predictions for each quarter of 2001 and of 2002.

6. Diploma in Statistics Introduction to Regression Lecture 4.16 A sales forecasting problem Southern Oil Products vegetable oil producer raw material supply is seasonal, variety of sources / countries problems in second quarter of last year prompts business review forecasts required for –budgetting and staff planning –quantifying extent of last year's problem

7. Diploma in Statistics Introduction to Regression Lecture 4.17 Table 9.1Quarterly production of vegetable oil, in numbers of 50 litre drums, for a six year period

8. Diploma in Statistics Introduction to Regression Lecture 4.18 Initial data analysis

9. Diploma in Statistics Introduction to Regression Lecture 4.19 A simple linear model for trend (Years 1-5) Regression Analysis: Production versus Time The regression equation is Production = 1177 + 36.2 Time Exercise: Estimate quarterly/annual growth in production Predictor Coef SE Coef T P Constant 1176.61 53.61 21.95 0.000 Time 36.223 4.475 8.09 0.000 Exercise: Calculate a confidence interval for quarterly/annual growth in production S = 115.405 Exercise: Comment on prediction

10. Diploma in Statistics Introduction to Regression Lecture 4.110 Quarterly indicator variables Special variables Q1, Q2, Q3 and Q4 called indicator variables may be added to the simple regression model to produce a multiple regression model incorporating the seasonal effects. Each quarterly indicator takes the value 1 in the relevant quarter and 0 otherwise. Note that, in each row, only one of the quarterly indicators takes the value 1, while the other three take the value 0. Thus, for each time (row), the indicator with value 1 indicates the corresponding quarter, 1, 2, 3 or 4

11. Diploma in Statistics Introduction to Regression Lecture 4.111 Table 9.2 Quarterly production of vegetable oil, in numbers of 50 litre drums, with Time and quarterly indicators, for a five year period

12. Diploma in Statistics Introduction to Regression Lecture 4.112 Multiple regression model Production =  1  Q 1 +  2  Q 2 +  3  Q 3 +  4  Q 4 +   Time + .

13. Diploma in Statistics Introduction to Regression Lecture 4.113 Regression Analysis Production vs Q1, Q2, Q3, Q4, Time The regression equation is Production = 1030 Q1 +1292 Q2 + 1210 Q3 + 1279 Q4 + 33.7 Time Predictor Coef SE Coef T P Noconstant Q1 1029.87 23.41 43.99 0.000 Q2 1292.35 24.45 52.85 0.000 Q3 1210.42 25.55 47.37 0.000 Q4 1278.70 26.71 47.88 0.000 Time 33.725 1.619 20.83 0.000 S = 40.9654

14. Diploma in Statistics Introduction to Regression Lecture 4.114 Exercise Predict the first quarter production levels for Year 6 and Year 7 Comment on prediction error with respect to (i) its previous vale (ii) recent production levels Next: Diagnostic analysis

15. Diploma in Statistics Introduction to Regression Lecture 4.115 Exercise Following calculation of a revised regression, make a table of initial and revised coefficient estimates and residual standard deviations. Compare. Which would you choose? Why?

16. Diploma in Statistics Introduction to Regression Lecture 4.116 Exercise Confirm and quantify the extent of the problem in Year 6, Q2. Homework 4.2.2 Confirm and quantify the extent of the recovery in Year 6, Q3.

17. Diploma in Statistics Introduction to Regression Lecture 4.117 Multiple regression model, alternative formulation Production =  1  Q 1 +  2  Q 2 +  3  Q 3 +  Time  Time + .

18. Diploma in Statistics Introduction to Regression Lecture 4.118 Alternative regression * Q4 is highly correlated with other X variables * Q4 has been removed from the equation. The regression equation is Production = 1279 - 249 Q1 + 13.7 Q2 - 68.3 Q3 + 33.7 Time Predictor Coef SE Coef T P Constant 1278.70 26.71 47.88 0.000 Q1 -248.82 26.36 -9.44 0.000 Q2 13.65 26.11 0.52 0.609 Q3 -68.27 25.96 -2.63 0.019 Time 33.725 1.619 20.83 0.000 S = 40.9654

19. Diploma in Statistics Introduction to Regression Lecture 4.119 Homework 4.2.3 List correspondences between the output from the original regression and the output from the alternative regression. Confirm that the coefficients of Q1, Q2 and Q3 in the original are the corresponding coefficients in the alternative with the Q4 coefficient added.

20. Diploma in Statistics Introduction to Regression Lecture 4.120 Introduction to Regression Lecture 4.2 Indicator variables for estimating seasonal effects in time series –another application, Meter Sales analysis Correlated explanatory variables

21. Diploma in Statistics Introduction to Regression Lecture 4.121 Another application, meter sales analysis Recall the analysis of Meter sales, discussed in Lab 1 Feedback.doc.

22. Diploma in Statistics Introduction to Regression Lecture 4.122 Another application, Meter Sales analysis Meter Sales jumped when nominal Phone Charge increased. Model these jumps by adding "indicators" defined to be 0 for all years prior to the jump and 1 for all years during and after the jump. Thus, the first jump occurred during 1952, so the corresponding indicator will be 0 from 1949 to 1952 and 1 from 1953 to 1983. Multiplying this explanatory variable by regression coefficient  adds 0 to predicted Meter Sales from 1949 to 1952 and adds  from 1953 to 1983.

23. Diploma in Statistics Introduction to Regression Lecture 4.123 Another application, Meter Sales analysis Regression Analysis: Meter Sales versus GNP, RLP,... Predictor Coef SE Coef T P Constant 38.87 51.62 0.75 0.461 GNP 0.15905 0.04881 3.26 0.004 RLP -73.53 14.76 -4.98 0.000 RPC -14.36 65.76 -0.22 0.830 Jump1953 13.385 9.688 1.38 0.184 Jump1956 23.110 8.140 2.84 0.011 Jump1964 41.93 15.17 2.76 0.013 Jump1970 16.39 10.19 1.61 0.125 S = 5.42373

24. Diploma in Statistics Introduction to Regression Lecture 4.124 Another application, Meter Sales analysis Note that the t-value for RPC is negligible so that RPC may be omitted. The variation explained by RPC is captured by the four indicator variables. Also, the s value is lower than before, suggesting that the variation in Meter Sales is better explained by the indicators than by RPC alone. N.B.Additional uses for indicators may be found in Extra Notes, Indicators.doc

25. Diploma in Statistics Introduction to Regression Lecture 4.125 Introduction to Regression Lecture 4.2 Indicator variables for estimating seasonal effects in time series –another application, Meter Sales analysis Correlated explanatory variables

26. Diploma in Statistics Introduction to Regression Lecture 4.126 Correlated explanatory variables Ref:Extra Notes Homework 4.2.4 Calculate the simple linear regressions of Jobtime on each of T_Ops and Units. Confirm the corresponding t-values. Calculate the simple linear regression of Jobtime on Ops per Unit. Comment of the negative correlation of Jobtime with Ops per Unit in the light of the corresponding t-value. Confirm the calculation of the R 2 values.

27. Diploma in Statistics Introduction to Regression Lecture 4.127 Reading SA §§ 9.1- 9.2 Hamilton, pp 82-84, 133-136