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Forecasting with Judgment Forecasting with Judgment Simone Manganelli DG-Research European Central Bank (Frankfurt am Main, Germany) Disclaimer: The views expressed in this paper are our own and do not necessarily reflect the views of the


  1. Forecasting with Judgment Forecasting with Judgment Simone Manganelli DG-Research European Central Bank (Frankfurt am Main, Germany) Disclaimer: The views expressed in this paper are our own and do not necessarily reflect the views of the ECB or the Eurosystem

  2. Classical Theory of Forecasting Classical Theory of Forecasting • Forecasts serve to make decisions about the future. • Forecast errors impose costs on the decision-maker. • Agents want to minimize expected loss associated to forecast errors. • Classical forecast is the minimizer of the sample equivalent of the expected loss. • See, e.g., Haavelmo (1944), Granger and Newbold (1986), Granger and Machina (2005).

  3. Optimal Point Forecast Optimal Point Forecast Simplest case: • i.i.d. normally distributed observations with variance 1 • quadratic loss function T { y } Want to forecast y T +1 t t 1 = 2 min Q ( ) E [( y ) ] � � � � 0 T 1 + � * 2 E [ y ] 0 E [ y ] FOC: � � � = � = T 1 T 1 + + T ˆ 1 � T y � = � = T t t 1

  4. The Problem The Problem ˆ is the minimiser of: � T ˆ T 1 2 � min Q ( ) T [( y ) ] � = � � � � T t t 1 � T 2 1 2 2 � min { E [( y ) ] T [( y ) ] E [( y ) ]} � � � + � � � � � T 1 t T 1 + + t 1 = � ( � ) � � T 2 min { E [( y ) ] ( )} � � + � � T 1 T + �

  5. Outline Outline Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion

  6. Introducing Judgment Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Start from a Subjective Guess Example 2: Forecasting US GDP Start from a Subjective Guess Relationship with Bayesian Conclusion 2 min Q ( ) E [( y ) ] � � � � 0 T 1 + � Q ( ) E [ y ] 0 � � � � � = 0 T 1 + The sample equivalent of the FOC is: ~ ~ ~ ˆ T 1 ˆ Q ( ) T [ y ] y � � = � � � � � = � � T t T t 1 ~ where � is a subjective guess of the decision maker. NOTE : This is a random variable, which may be different from zero just because of statistical error.

  7. Introducing Judgment Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Then Do Hypothesis Testing Then Do Hypothesis Testing Conclusion ~ H : is the true mean 0 � ~ ˆ y T ~ N ( 0 , 1 / T ) � � • Choose a confidence level α • η α /2 is the critical value If null is rejected: → First derivatives statistically ≠ 0 → Can be confident to decrease the true objective function ˆ → Only up to the point where new H 0 cannot be rejected * � T

  8. Introducing Judgment Introducing Judgment Risk Analysis of New Estimator Graphical Illustration Graphical Illustration Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion Distribution of FOC under H 0 ~ ˆ � Q ( ) � T / 2 � 0 ~ ˆ ˆ ˆ � * Q � ( ) Q ( ) � � T T T

  9. Intuition Intuition Introducing Judgment Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion 2 2 E [( y ) ] ( y ) � � � � 1 2 ( y ) � � 1 100*(1- α )% confidence interval ˆ ˆ � � � � + � 1 / 2 1 / 2 � � ~ ˆ � y � = � 1 1 First derivatives not statistically different from zero

  10. Introducing Judgment Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion New estimator: ~ ˆ ˆ y if y � � � � � > � T / 2 T / 2 � � � ~ ~ � * ˆ if | y | � = � � � < � � T T / 2 � ~ � ˆ ˆ y if y + � � � < � � � T / 2 T / 2 � � �

  11. Introducing Judgment Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Interpretation Interpretation Conclusion • α is the probability of committing type I errors, i.e. of rejecting the null when it is true. • Choose low α when confident in subjective guess or if the cost of type I errors is high. • Classical paradigm sets α =1: – Always FOC equal to zero; – No room for subjective guess; – It commits type I errors with probability 1.

  12. Introducing Judgment Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Problems with Pretest Estimators Problems with Pretest Estimators Conclusion Test the following null hypothesis, for given confidence level α : ~ ˆ H : � = � 0 T Then: • If do not reject, keep the subjective guess • If reject, take the maximum likelihood estimator This is wrong!

  13. Introducing Judgment Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion 2 2 E [( y ) ] ( y ) � � � � 1 2 ( y ) � � 1 (1- α )% confidence interval ˆ ˆ � � � � + � 1 / 2 1 / 2 � � ~ ˆ � y � = � 1 1 First derivatives not statistically different from zero

  14. Introducing Judgment Risk Analysis of New Estimator Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion

  15. Introducing Judgment Risk Analysis of New Estimator Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion 0 Let y t ~ i . i . d . N ( , 1 ) � Estimate the mean, given a single observation y 1 (Magnus 02) Compare the risk properties of two estimators: 1) Subjective classical estimator (coincides with Magnus 02) ~ y if y � � � � � > � 1 / 2 1 / 2 � � � ~ ~ � * if | y | � = � � � < � � 1 1 / 2 � ~ � y if y + � � � < � � � � 1 / 2 1 / 2 � � 2) Pretest estimator ~ ~ if | y | � � � � < � � ˆ 1 / 2 P � � = ~ � y if | y | � � � > � � 1 1 / 2 �

  16. Introducing Judgment Risk Associated to f f ( ( y y ) ) Risk Associated to Risk Analysis of New Estimator Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion ~ 0 2 E [( f ( y ) ) ] , 0 � � � = 0 � 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 OLS (_=1) Subjective Guess (_=0) Pretest (_=0.10) Subjective Classical (_=0.10)

  17. Introducing Judgment Risk Analysis of New Estimator Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Monte Carlo Simulation Monte Carlo Simulation Conclusion • Random draws from standard normal • Sample sizes = 5, 20, 60, 120, 240, 1000 • Quadratic loss function • Two estimators: classical, new ( α =0.10) • Evaluated expected loss with MC simulation • Repeat 5000 times and average

  18. Introducing Judgment Risk Analysis of New Estimator Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion 0.7 0.6 0.5 Mean 0 0.4 0.05 0.3 0.1 0.5 0.2 1 0.1 0 5 20 60 120 240 1000

  19. Introducing Judgment Risk Analysis of New Estimator Risk Analysis of New Estimator Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Implications Implications Conclusion • Good guess (gut feelings) are as important as good econometric models. • Organization of forecasting process: – Subjective guess based on maximum likelihood estimates can never be rejected by construction; – Clear separation b/w: Who provides guess, based on judgment; Who tests the guess, based on econometric models. • Shared responsibility for the quality of the forecasts: – High confidence in bad judgment results in bad forecast.

  20. Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Example 1: Asset Allocation Example 2: Forecasting US GDP Relationship with Bayesian Conclusion

  21. Introducing Judgment Risk Analysis of New Estimator Example 1: Asset Allocation Example 1: Asset Allocation Generalization Example 2: Forecasting US GDP Generalization Relationship with Bayesian Conclusion d ˆ 0 T U ( ) N ( 0 , ) � T � � � � ~ ˆ * ( ) ( 1 ) [ 0 , 1 ] � � � � � + � � � � � T T * 0 Under the null H : ( ) � � = � 0 T ˆ ˆ ˆ * ' * 1 * ˆ z ( ( )) T U ( ( )) � U ( ( )) � � � � � � � � � � � � T T T T T T � � ˆ * ˆ if z ( ( 0 )) 0 � � � � � = T T , k � ˆ ˆ * * ˆ if z ( ( 0 )) arg max U ( ( )) � > � � � = � � T T , k T T � [ 0 , 1 ] � � * ˆ s.t. z ( ( )) � � = � T T , k �

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