Empirical Modeling Process 1 - Identify Problem / Question 2 - Conceptualize model 3 - Collect data 4 - Examine and Summarize data 5 - Estimate model 6 - Examine model performance 7 - Revise model as needed

Demand Curve Price Quantity

Price: Deflated Constant Dollar Price Quantity: Per Capita Consumption

Mathematical Conceptual Model: Model 1 TP = b 0 + b 1 TQ + e Where: TP= real quarterly retail turkey price (cents/lb.) TQ=quarterly per capita turkey consumption (lbs./capita) b 0, b 1 are parameters to be estimated e is an error term

Turkey Demand Curve TP TQ b0b0 b1b1 TP = b 0 + b 1 TQ

Objective: To quantify determinants of quarterly retail price of Turkey over Estimate the demand curve

Empirical Modeling Process 1 - Identify Problem / Question 2 - Conceptualize model 3 - Collect data 4 - Examine and Summarize data 5 - Estimate model 6 - Examine model performance 7 - Revise model as needed

Empirical Modeling Process 1 - Identify Problem / Question 2 - Conceptualize model 3 - Collect data 4 - Examine and Summarize data 5 - Estimate model 6 - Examine model performance 7 - Revise model as needed

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

Goodness of Fit Measures

Residual Summary

Parameter Estimates

Model 1 Regression Estimates: TP = – TQ (0.00) (0.00) p-values Adj. R-Sq.=0.33 RMSE=29.35 cents/lb. Observations=116

Turkey Demand Curve TP TQ TP = – TQ

Diagnostic Testing of Regression 1. Predicted vs. Actual Graph 2. Graphical analysis of residuals

Predicted TP = – TQ Residual = TP – Predicted TP Predicted Values & Residuals (Errors)

Performance Summary Model 1 Adj. R-square only 0.33 RMSE = 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 / Question 2 - Conceptualize model 3 - Collect data 4 - Examine and Summarize data 5 - Estimate model 6 - Examine model performance 7 - Revise model as needed

Mathematical Conceptual Model Model 2 TP = b 0 + b 1 TQ + b 2 BfQ + b 3 PkQ + b 4 ChQ + b 5 INC + 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 2 TP = b 0 + b 1 TQ + b 2 BfQ + b 3 PkQ + b 4 ChQ + b 5 INC + e Sign Expectations: b 1 b 2 b 3 b 4 b 5

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

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 Adjustment TP = b 0 + b 1 TQ + b 2 BfQ + b 3 PkQ + b 4 ChQ + b 5 INC + b 6 Q1Dum + b 7 Q2Dum + b 8 Q3Dum + e Where: Q1Dum = 1 in qtr 1 and 0 otherwise Q2Dum = 1 in qtr 2 and 0 otherwise Q3Dum = 1 in qtr 3 and 0 otherwise

Dummy variables in spreadsheet YearQtr Q1Dum Q2Dum Q3Dum Q4Dum etc.... Drop one dummy variable column when estimate regression

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 4 TP = b 0 + b 1 TQ + b 2 BfQ + b 3 PkQ + b 4 ChQ + b 5 INC + b 6 Q1Dum + b 7 Q2Dum + b 8 Q3Dum + e or TP = b 0 + b 1 TQ + b 2 PkQ + b 3 ChQ +b 4 INC + b 5 Q1Dum + b 6 Q2Dum + b 7 Q3Dum + e

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 statistically compare two NESTED models. Nested models are those in which one is a subset of the other and both models have same dependent variable and were estimated over same time period.

Nested vs Nonnested Models Model 1 TP = f (TQ) Model 2 TP = f (TQ, BfQ, PkQ, ChQ, INC) Model 3 TP = f (TQ, BfQ, PkQ, ChQ, INC, Q1D, Q2D, Q3D) Model 4 TP = 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 2 TP = f (TQ, BfQ, PkQ, ChQ, INC) Model 3 TP = f (TQ, BfQ, PkQ, ChQ, INC, Q1D, Q2D, Q3D) Null Hypothesis:Alternative Hypothesis: H o : b 6 =b 7 =b 8 =0Not H o Use an F-test to test this hypothesis.

F - Test: (SSRr- SSRf ) / (q) (SSRf / DFf) Where: SSR r is the sum of squared residuals from reduced or smaller model SSR f is the sum of squared residuals from full or bigger model q is the number of restrictions DF f is the degrees of freedom full model F =

F - Test Criteria: (SSRr - SSRf ) / (q) (SSR f / DF f ) Compare to critical F-table value of F (q, DF f, alpha) or F(v 1, v 2, 0.05). If F > F(q, DF f, 0.05) then reject null and conclude that full model is better than reduced model F =

Testing Between Models 2 and 3: Null Hypothesis:Alternative Hypothesis H o : b 6 =b 7 =b 8 =0Not H o F = (8,295.5 – 7,664.4) / 3 (7,664.4 / 107 ) =2.94 F (3, 107, 0.05) = ~2.70 Conclusion: Reject H o, 95% certain model 3 is 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…..