Empirical Modeling Process 1 - Identify Problem / Question 2 - Conceptualize model 3 - Collect data 4 - Examine and Summarize data 5 - Estimate model.

Empirical Modeling Process

1 - Identify Problem / Question2 - Conceptualize model3 - Collect data4 - Examine and Summarize data5 - Estimate model6 - Examine model performance7 - Revise model as needed

Demand Curve

Price

Quantity

Price:

Deflated Constant Dollar Price

Quantity:

Per Capita Consumption

Mathematical Conceptual Model:

Model 1TP = b0+ b1TQ + e

Where:TP= real quarterly retail turkey

price (cents/lb.)TQ=quarterly per capita turkey

consumption (lbs./capita)b0, b1 are parameters to be estimatede is an error term

Turkey Demand Curve

TP

TQ

b0

b1

TP = b0+ b1TQ

Objective:

To quantify determinants of quarterly retail price of Turkey over 1980-2008

Estimate the demand curve



Nominal Quarterly Retail Turkey Prices, 1980-2008

80.0

85.0

90.0

95.0

100.0

105.0

110.0

115.0

120.0

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Pri

ce (

cen

ts/l

b.)

Quarterly Per Capita Turkey Consumption, 1980-2008

1.0

2.0

3.0

4.0

5.0

6.0

7.0

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Co

nsu

mp

tio

n (

lbs.

/cap

ita)

Quarterly Nominal Retail Turkey Price vs. Per Capita Consumption, 1980-2008

80.0

85.0

90.0

95.0

100.0

105.0

110.0

115.0

120.0

125.0

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Pri

ce (

cen

ts/lb

.)

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Co

nsu

mp

t. (

lbs.

/cap

ita)

Turkey Price

Turkey Consumption

Real Quarterly Retail Turkey Prices, 1980-2008

100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Pri

ce (

cen

ts/lb

., 20

06.3

do

llars

) .

Quarterly Real Retail Turkey Price vs. Per Capita Consumption, 1980-2008

80.0

100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Pri

ce (

cen

ts/lb

., 20

08.3

$)

1.0

2.0

3.0

4.0

5.0

6.0

7.0

Co

nsu

mp

t. (

lbs.

/cap

ita)

Turkey Price

Turkey Consumption

Real Turkey Price vs. Turkey Consumption Scatter Plot, Quarterly Data, 1980-2008

100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

260.0

1.0 2.0 3.0 4.0 5.0 6.0 7.0

Consumption (lbs./capita)

Pri

ce (

cen

ts/lb

.)



Summary Statistics of Quarterly Real Turkey Price and Per Capita Turkey Consumption, 1980-2008

Turkey Turkey Summary Statistics Price Quantity

(cents/lb.) (lbs./capita)Average 157.1 3.9Minimum 109.6 1.6Maximum 238.5 6.5Median 144.1 3.9Standard Deviation 35.8 1.2Coefficient of Variation 22.8% 30.3%

Model 1TP = b0 + b1*TQ

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.58R Square 0.33Adjusted R Square 0.33Standard Error 29.35Observations 116

ANOVAdf SS MS F Significance F

Regression 1 49082.453 49082.453 56.963 1.18055E-11Residual 114 98227.962 861.649Total 115 147310.415

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 227.48 9.71 23.42 0.00 208.24 246.72TQ -17.81 2.36 -7.55 0.00 -22.49 -13.14

Goodness of Fit Measures

SUMMARY OUTPUT


Residual Summary



Parameter Estimates

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 227.48 9.71 23.42 0.00 208.24 246.72TQ -17.81 2.36 -7.55 0.00 -22.49 -13.14

Model 1 Regression Estimates:

TP = 227.48 – 17.81 TQ (0.00) (0.00) p-values

Adj. R-Sq.=0.33 RMSE=29.35 cents/lb.

Observations=116

Turkey Demand Curve

TP

TQ

227.48

-17.81

TP = 227.48 – 17.81 TQ


100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

260.0

1.0 2.0 3.0 4.0 5.0 6.0 7.0


Pri

ce (

cen

ts/lb

.)

Diagnostic Testing of Regression

1. Predicted vs. Actual Graph

2. Graphical analysis of residuals

Predicted TP = 227.48 – 17.81 TQResidual = TP – Predicted TP

Predicted Values & Residuals (Errors)

Model 1Actual Predicted Model 1

TP TP ErrorYear.Qtr (cents/lb) (cents/lb) (cents/lb)

80.1 237.8 195.55 42.27218.5 192.66 25.89223.6 181.75 41.80238.5 157.38 81.10

81.1 238.1 198.67 39.40233.2 193.55 39.60234.5 183.13 51.33210.8 146.37 64.43

82.1 207.5 196.49 11.02201.7 191.21 10.53207.5 181.50 25.98201.3 152.52 48.73

83.1 200.0 191.16 8.85199.0 188.89 10.14194.7 183.65 11.01191.2 149.87 41.37

84.1 196.3 193.34 3.01199.7 189.04 10.64206.8 181.36 25.40207.1 149.61 57.53

Model 1: Actual and Predicted Real Quarterly Retail Turkey Prices, 1980-2008

100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Rea

l Pri

ce (

cen

ts/lb

.)

Actual

Predicted

Model 1: Residuals (Actual minus Predicted) for Predicting

Real Quarterly Retail Turkey Prices, 1980-2008

-60.00

-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

100.00

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Err

or

(cen

ts/lb

.)

Performance Summary Model 1

•Adj. R-square only 0.33•RMSE = 29.35 cents/lb (std dev TP=35.8)•Sign on coefficient is as expected•TQ is statistically significant• Not predicting well• We suspect we have left out some relevant important factors

(omitted relevant variable problem)



Mathematical Conceptual Model

Model 2TP = b0 + b1TQ + b2BfQ + b3PkQ

+ b4ChQ + b5INC + e

Where:TP and TQ are as defined in model 1,BfQ, PkQ, ChQ quarterly beef, pork and chicken consumption (lbs./capita)INC = real disposable income ($/capita)



+ b4ChQ + b5INC + e

Sign Expectations:b1

b2

b3

b4

b5


100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

260.0

1.0 2.0 3.0 4.0 5.0 6.0 7.0


Pri

ce (

cen

ts/lb

.)

Real Turkey Price vs. Beef Consumption Scatter Plot, Quarterly Data, 1980-2008

100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

260.0

14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0

Beef Consumption (lbs./capita)

Tu

rkey

Pri

ce (

cen

ts/lb

.)

Real Turkey Price vs. Pork Consumption Scatter Plot, Quarterly Data, 1980-2008

100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

260.0

11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5

Pork Consumption (lbs./capita)

Tu

rkey

Pri

ce (

cen

ts/lb

.)

Real Turkey Price vs. Chicken Consumption Scatter Plot, Quarterly Data, 1980-2008

100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

260.0

11.0 13.0 15.0 17.0 19.0 21.0 23.0 25.0

Pork Consumption (lbs./capita)

Tu

rkey

Pri

ce (

cen

ts/lb

.)

Real Turkey Price vs. Real Income Scatter Plot, Quarterly Data, 1980-2008

100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

260.0

20,000 22,000 24,000 26,000 28,000 30,000 32,000 34,000 36,000

Income ($/capita)

Tu

rkey

Pri

ce (

cen

ts/lb

.)

Summary Statistics of Data Used to Explain Quarterly Real Turkey Price, 1980-2008

Turkey Turkey Beef Pork Chicken RealSummary Statistics Price Quantity Quantity Quantity Quantity Income

(cents/lb.) (lbs./capita) (lbs./capita) (lbs./capita) (lbs./capita) ($/capita)Average 157.1 3.9 17.5 12.8 16.7 28848.5Minimum 109.6 1.6 15.0 11.3 11.5 22387.2Maximum 238.5 6.5 20.8 14.8 22.4 34927.6Median 144.1 3.9 17.0 12.7 17.0 28230.4Standard Deviation 35.8 1.2 1.4 0.7 3.2 3489.4Coefficient of Variation 22.8% 30.3% 8.0% 5.7% 19.0% 12.1%

Correlation MatrixTP TQ BfQ PkQ ChQ Inc

TP 1TQ -0.577 1BfQ 0.868 -0.598 1PkQ 0.186 0.315 0.022 1ChQ -0.933 0.427 -0.763 -0.230 1Inc -0.911 0.476 -0.751 -0.200 0.957 1

Model 2TP = b0 + b1*TQ + b2*BfQ + b3*PkQ + b4*ChQ + b5*Inc

SUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 105.295 31.643 3.328 0.001 42.586 168.003TQ -4.955 0.985 -5.029 0.000 -6.907 -3.002BfQ 7.656 1.004 7.625 0.000 5.666 9.646PkQ 4.410 1.296 3.403 0.001 1.842 6.978ChQ -6.261 0.897 -6.978 0.000 -8.039 -4.483Inc 0.000 0.001 -0.493 0.623 -0.002 0.001

Model 2: Actual and Predicted Real Quarterly Retail Turkey Prices, 1980-2008

100.0

120.0

140.0

160.0

180.0

200.0

220.0

240.0

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Rea

l Pri

ce (

cen

ts/lb

.)

Actual

Predicted

Model 2: Residuals (Actual minus Predicted) for Predicting Real Quarterly Retail Turkey Prices, 1980-2008

-60

-40

-20

0

20

40

60

80

100

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Err

or

(cen

ts/lb

.)

Model 1 vs. Model 2: Residuals (Actual minus Predicted) for Predicting Real Quarterly Retail Turkey Prices, 1980-2008

-60

-40

-20

0

20

40

60

80

100

80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 00.1 02.1 04.1 06.1 08.1

Date (Year.Quarter)

Err

or

(cen

ts/lb

.)

Model 1

Model 2

Model 2TP = b0 + b1*TQ + b2*BfQ + b3*PkQ + b4*ChQ + b5*Inc

SUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 105.295 31.643 3.328 0.001 42.586 168.003TQ -4.955 0.985 -5.029 0.000 -6.907 -3.002BfQ 7.656 1.004 7.625 0.000 5.666 9.646PkQ 4.410 1.296 3.403 0.001 1.842 6.978ChQ -6.261 0.897 -6.978 0.000 -8.039 -4.483Inc 0.000 0.001 -0.493 0.623 -0.002 0.001

Performance Summary of Model 2

•Adj. R-square 0.94 better than twice 1•RMSE = 8.68 cents/lb less than 1/3 of 1•BfQ & PkQ unexpected signs

& statistically significant

• Appears to be seasonality not accounted for in other regressors


Model 3 Seasonality AdjustmentTP = b0 + b1TQ + b2BfQ + b3PkQ+ b4ChQ + b5INC + b6Q1Dum + b7Q2Dum + b8Q3Dum + e

Where:Q1Dum = 1 in qtr 1 and 0 otherwiseQ2Dum = 1 in qtr 2 and 0 otherwiseQ3Dum = 1 in qtr 3 and 0 otherwise

Dummy variables in spreadsheet

Year Qtr Q1Dum Q2Dum Q3Dum Q4Dum1980 1 1 0 0 0

2 0 1 0 03 0 0 1 04 0 0 0 1

1981 1 1 0 0 02 0 1 0 03 0 0 1 04 0 0 0 1

etc....

Drop one dummy variable column when estimate regression

Model 3TP = b0 + b1*TQ + b2*BfQ + b3*PkQ + b4*ChQ + b5*Inc + b6*Qtr1 Dummy + b7*Qtr2 Dummy + b8*Qtr3 Dummy

SUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 207.993 47.559 4.373 0.000 113.713 302.272TQ -10.790 2.284 -4.724 0.000 -15.319 -6.262BfQ 4.910 1.462 3.358 0.001 2.011 7.808PkQ 2.821 1.540 1.832 0.070 -0.232 5.874ChQ -6.183 1.040 -5.947 0.000 -8.244 -4.122Inc -0.00051 0.001 -0.578 0.565 -0.002 0.001Qtr 1 Dummy -16.583 6.144 -2.699 0.008 -28.762 -4.404Qtr 2 Dummy -14.330 5.525 -2.594 0.011 -25.284 -3.377Qtr 3 Dummy -8.143 4.510 -1.806 0.074 -17.084 0.797

Performance Summary of model 3

•Adj. R-square 0.94 comparable to 2•RMSE = 8.46 cents/lb bit less than 2•Signs on coeff. ok except BfQ & PkQ

See what happens when we drop BfQ



+ b4ChQ + b5INC + b6Q1Dum + b7Q2Dum + b8Q3Dum + e

or

TP = b0 + b1TQ + b2PkQ + b3ChQ+b4INC + b5Q1Dum + b6Q2Dum + b7Q3Dum + e

Model 4TP = b0 + b1*TQ + b2*PkQ + b3*ChQ + b4*Inc + b5*Qtr1 Dummy + b6*Qtr2 Dummy + b7*Qtr3 Dummy

SUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 337.344 29.181 11.560 0.000 279.503 395.186TQ -16.097 1.726 -9.326 0.000 -19.518 -12.676PkQ 2.245 1.601 1.402 0.164 -0.929 5.419ChQ -7.410 1.019 -7.274 0.000 -9.429 -5.390Inc 0.000 0.001 -0.131 0.896 -0.002 0.002Qtr 1 Dummy -29.098 5.110 -5.694 0.000 -39.228 -18.969Qtr 2 Dummy -21.879 5.282 -4.142 0.000 -32.348 -11.410Qtr 3 Dummy -11.915 4.571 -2.607 0.010 -20.975 -2.855

Performance Summary of model 4

•Adj. R-square 0.94 , about as high as any other•RMSE = 8.85 cents/lb bit more than 3•Signs on coeff. o.k. except pork •Pork marginally statistically significant•Income not close to statistically significant•All rest are significant

What do you think????

Testing Between Models

We use an F-test to statisticallycompare two NESTED models.

Nested models are those in whichone is a subset of the other andboth models have same dependentvariable and were estimated oversame time period.

Nested vs Nonnested Models

Model 1TP = f (TQ)Model 2TP = f (TQ, BfQ, PkQ, ChQ, INC)Model 3TP = f (TQ, BfQ, PkQ, ChQ, INC, Q1D, Q2D, Q3D)Model 4TP = f (TQ, PkQ, ChQ, INC, Q1D, Q2D, Q3D)

Nested Model pairs:1 is nested in 2; 2 is nested in 3;4 is nested in 3; 1 is nested in 4, etc.

Testing Between Models 2 and 3

Model 2TP = f (TQ, BfQ, PkQ, ChQ, INC)Model 3TP = f (TQ, BfQ, PkQ, ChQ, INC, Q1D, Q2D, Q3D)

Null Hypothesis: Alternative Hypothesis:Ho: b6=b7=b8=0 Not Ho

Use an F-test to test this hypothesis.

F - Test:

(SSRr- SSRf) / (q) (SSRf / DFf)

Where:SSRr is the sum of squared residuals from

reduced or smaller modelSSRf is the sum of squared residuals from

full or bigger modelq is the number of restrictionsDFf is the degrees of freedom full model

F =

F - Test Criteria:

(SSRr - SSRf) / (q) (SSRf / DFf)

Compare to critical F-table valueof F (q, DFf, alpha) or F(v1, v2, 0.05).

If F > F(q, DFf, 0.05) then reject null andconclude that full model is better thanreduced model

F =

Testing Between Models 2 and 3:

Null Hypothesis: Alternative HypothesisHo: b6=b7=b8=0 Not Ho

F = (8,295.5 – 7,664.4) / 3 (7,664.4 / 107 )

= 2.94

F (3, 107, 0.05) = ~2.70

Conclusion: Reject Ho, 95% certain model 3is better, i.e., seasonal variables significant

Summary of Regression Performance

•R-square •RMSE •Signs on parameters correct?•Statistical Significance of Parameters?•Comparison between models•Economic logic consistency

•Economic importance of estimates

Summary of Regression Challenges

• Conceptual model wrong • Fail to include important variable(s)• Patterns in residuals• Omitted Relevant Variables• Spurious relationships?• other problems…..

Empirical Modeling Process 1 - Identify Problem / Question 2 - Conceptualize model 3 - Collect data 4 - Examine and Summarize data 5 - Estimate model.

Documents

data5 estimate model6

turkey consumption lbs

problem question2

b0 b1tqobjective

b0 b1tq ewhere