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Introduction to RegressionLecture 4.2

- Indicator variables for estimating seasonal effects in time series
- another application, Meter Sales analysis
- Correlated explanatory variables

Diploma in Statistics

Introduction to Regression

Model formulation

Completions = a1Q1 + a2Q2 + a3Q3 + a4Q4

+ b Time

+ e.

Quarter 1: Completions = a1 + b Time

Quarter 2: Completions = a2 + b Time,

Homework 4.1.1: Write down the prediction formulas for future third and fourth quarters.

Diploma in Statistics

Introduction to Regression

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.

Diploma in Statistics

Introduction to Regression

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

Diploma in Statistics

Introduction to Regression

Table 9.1 Quarterly production of vegetable oil, in numbers of 50 litre drums, for a six year period

Diploma in Statistics

Introduction to Regression

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

Diploma in Statistics

Introduction to Regression

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

Diploma in Statistics

Introduction to Regression

Table 9.2 Quarterly production of vegetable oil, in numbers of 50 litre drums, with Time and quarterly indicators, for a five year period

Diploma in Statistics

Introduction to Regression

Multiple regression model

Production = a1Q1 + a2Q2 + a3Q3 + a4Q4

+ b Time

+ e.

Diploma in Statistics

Introduction to Regression

Regression AnalysisProduction 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

Diploma in Statistics

Introduction to Regression

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

Diploma in Statistics

Introduction to Regression

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?

Diploma in Statistics

Introduction to Regression

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.

Diploma in Statistics

Introduction to Regression

Multiple regression model,alternative formulation

Production = a + b1Q1 + b2Q2 + b3Q3

+ bTime Time

+ e.

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Introduction to Regression

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

Diploma in Statistics

Introduction to Regression

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.

Diploma in Statistics

Introduction to Regression

Introduction to RegressionLecture 4.2

- Indicator variables for estimating seasonal effects in time series
- another application, Meter Sales analysis
- Correlated explanatory variables

Diploma in Statistics

Introduction to Regression

Another application, meter sales analysis

Recall the analysis of Meter sales, discussed in

Lab 1 Feedback.doc.

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Introduction to Regression

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 b adds 0 to predicted Meter Sales from 1949 to 1952 and adds b from 1953 to 1983.

Diploma in Statistics

Introduction to Regression

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

Diploma in Statistics

Introduction to Regression

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

Diploma in Statistics

Introduction to Regression

Introduction to RegressionLecture 4.2

- Indicator variables for estimating seasonal effects in time series
- another application, Meter Sales analysis
- Correlated explanatory variables

Diploma in Statistics

Introduction to Regression

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 R2 values.

Diploma in Statistics

Introduction to Regression

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