Empirical Modeling Process 1 - Identify Problem / Question 2 - Conceptualize model 3 - Collect data 4 - Examine and Summarize data 5 - Estimate model.
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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
Empirical Modeling Process
1 - Identify Problem / Question2 - Conceptualize model3 - Collect data4 - Examine and Summarize data5 - Estimate model6 - Examine model performance7 - Revise model as needed
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
.)
Empirical Modeling Process
1 - Identify Problem / Question2 - Conceptualize model3 - Collect data4 - Examine and Summarize data5 - Estimate model6 - Examine model performance7 - Revise model as needed
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
Regression StatisticsMultiple R 0.58R Square 0.33Adjusted R Square 0.33Standard Error 29.35Observations 116
Residual Summary
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
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
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
.)
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)
Empirical Modeling Process
1 - Identify Problem / Question2 - Conceptualize model3 - Collect data4 - Examine and Summarize data5 - Estimate model6 - Examine model performance7 - Revise model as needed
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)
Mathematical Conceptual Model
Model 2TP = b0 + b1TQ + b2BfQ + b3PkQ
+ b4ChQ + b5INC + e
Sign Expectations:b1
b2
b3
b4
b5
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
.)
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
Regression StatisticsMultiple R 0.971R Square 0.944Adjusted R Square 0.941Standard Error 8.684Observations 116
ANOVAdf SS MS F Significance F
Regression 5 139014.884 27802.977 368.672 5.59838E-67Residual 110 8295.531 75.414Total 115 147310.415
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
Regression StatisticsMultiple R 0.971R Square 0.944Adjusted R Square 0.941Standard Error 8.684Observations 116
ANOVAdf SS MS F Significance F
Regression 5 139014.884 27802.977 368.672 5.59838E-67Residual 110 8295.531 75.414Total 115 147310.415
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
Mathematical Conceptual Model
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
Regression StatisticsMultiple R 0.974R Square 0.948Adjusted R Square 0.944Standard Error 8.463Observations 116
ANOVAdf SS MS F Significance F
Regression 8 139645.990 17455.749 243.693 5.07518E-65Residual 107 7664.425 71.630Total 115 147310.415
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
Mathematical Conceptual Model
Model 4TP = b0 + b1TQ + b2BfQ + b3PkQ
+ 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
Regression StatisticsMultiple R 0.971R Square 0.942Adjusted R Square 0.939Standard Error 8.857Observations 116
ANOVAdf SS MS F Significance F
Regression 7 138838.516 19834.074 252.845 6.4152E-64Residual 108 8471.899 78.444Total 115 147310.415
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…..
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