1. 2-1 Lecture 2: Estimating a Slope (Chapter 2.1–2.7)

2. 2-2 Recall 計量經濟學的構成 1) 經濟理論 經濟理論 2) 數理模型 數理模型 3) 統計理論 統計理論 4) 計量模型 計量模型 經濟理論 數理模型統計理論 計量模型 現象

3. 2-3 計量與經濟理論之差異 ? Economic theory: qualitative results— Demand Curves Slope Downward Econometrics: quantitative results— price elasticity of demand for milk = -.75

4. 2-4 計量與統計之差異 ? Statistics: “summarize the data faithfully”; “let the data speak for themselves.” Econometrics: “ what do we learn from economic theory AND the data at hand?”

5. 2-5 計量能做啥事 ? Estimation: What is the marginal propensity to consume of Taiwan? ( 結構分析 ) Hypothesis Testing: Do Korean college workers’ productivity higher than Taiwan?( 檢定 假說 ) Prediction & Forecasting: What will Personal Savings be in 2004 if GDP is $14,864? And will it grow in the near future (2008)?( 預期及預測 )

6. 2-6 Economists Ask: “What Changes What and How?” Higher Income, Higher Saving Higher Price, Lower Quantity Demanded Higher Interest Rate, Lower Investment

7. 2-7 Savings Versus Income Theory Would Assume an Exact Relationship, e.g., Y =  X 0 1000 2000 3000 4000 5000 6000 24000480007200096000

8. 2-8 Slope of the Line Is Key! Slope is the change in savings with respect to changes in income Slope is the derivative of savings with respect to income If we know the slope, we’ve quantified the relationship!

9. 2-9 Never So Neat: Savings Versus Income

10. 2-10 Long-run Consumption Function 特點 向上斜 經原點 猜猜斜率 ?

11. 2-11 Underlying Mean + Random Part ( 憑直覺 ) 四大猜法 four intuitively appealing ways to estimate 

12. 2-12 估計策略 1.Min Σ (Y – Y ) 2.Min Σ ∣ Y – Y ∣ 3.Min Σ (Y – Y ) 優劣點 Y 為配適值 2

13. 2-13 “Best Guess 1” Mean of Ratios: X Y 共有 n 個

14. 2-14 Figure 2.4 Estimating the Slope of a Line with Two Data Points

15. 2-15 “Best Guess 2” Ratio of Means:

16. 2-16 Figure 2.5 Estimating the Slope of a Line:  g 2

17. 2-17 “Best Guess 3” Mean of Changes in Y over Changes in X : X y

18. 2-18 “Best Guess 4” Ordinary Least Squares: (minimizes  squared residuals in sample)

19. 2-19 Four Ways to Estimate 

20. 2-20 Underlying Mean + Random Part Are lines through the origin likely phenomena?

21. 2-21 Regression’s Greatest Hits!!! An Econometric Top 40

22. 2-22 Two Classical Favorites!! Friedman’s Permanent Income hypothesis: Capital Asset Pricing Model (CAPM) :

23. 2-23 A Golden Oldie !! Engel on the Demand for Rye:

24. 2-24 Four Guesses How to Choose?

25. 2-25 What Criteria Did We Discuss? Pick The One That's Right Make Mean Error Close to Zero Minimize Mean Absolute Error Minimize Mean Square Error

26. 2-26 What Criteria Did We Discuss? (cont.) Pick The One That's Right… – In every sample, a different estimator may be “right.” – Can only decide which is right if we ALREADY KNOW the right answer— which is a trivial case.

27. 2-27 What Criteria Did We Discuss? (cont.) Make Mean Error Close to Zero …seek unbiased guesses – If E(  g-  ) = 0,  g is right on average – If BIAS = 0,  g is an unbiased estimator of 

28. 2-28 Checking Understanding Question: Which estimator does better under the “minimize mean error” condition?  g -  is always a positive number less than 2 (our guesses are always a little high), or  g -  is always +10 or -10 (50/50 chance)

29. 2-29 Checking Understanding (cont.) If our guess is wrong by +10 for half the observations, and by -10 for the other half, then E(  g-  ) = 0! – The second estimator is unbiased! Mistakes in opposite directions cancel out. The first estimator is always closer to being right, but it does worse on this criterion.

30. 2-30 What Criteria Did We Discuss? Minimize Mean Absolute Error… – Mistakes don’t cancel out. – Implicitly treats cost of a mistake as being proportional to the mistake’s size. – Absolute values don’t go well with differentiation.

31. 2-31 What Criteria Did We Discuss? (cont.) Minimize Mean Square Error… – Implicitly treats cost of mistakes as disproportionately large for larger mistakes. – Squared expressions are mathematically tractable.

32. 2-32 What Criteria Did We Discuss? (cont.) Pick The One That’s Right… – only works trivially Make Mean Error Close to Zero… – seek unbiased guesses Minimize Mean Absolute Error… – mathematically tough Minimize Mean Square Error… – more tractable mathematically

33. 2-33 Criteria Focus Across Samples Make Mean Error Close to Zero Minimize Mean Absolute Error Minimize Mean Square Error What do the distributions of the estimators look like?

34. 2-34 Try the Four in Many Samples Pros will use estimators repeatedly— what track record will they have? Idea: Let’s have the computer create many, many data sets. We apply all our estimators to each data set.

35. 2-35 Try the Four in Many Samples (cont.) We use our estimates on many datasets that we created ourselves. We know the true value of  because we picked it! We can compare estimators. We run “horseraces.”

36. 2-36 Try the Four in Many Samples (cont.) Pros will use estimators repeatedly— what track record will they have? Which horse runs best on many tracks? Don’t design tracks that guarantee failure. What properties do we need our computer-generated datasets to have to avoid automatic failure for one of our estimators?

37. 2-37 Building a Fair Racetrack Under what conditions will each estimator fail?

38. 2-38 To Preclude Automatic Failure...

39. 2-39 Why Does Viewing Many Samples Work Well? We are interested in means: mean error, mean absolute error, mean squared error. Drawing many (m) independent samples lets us estimate means with variance  e 2 /m, where  e 2 is the variance of that mean’s error. If m is large, our estimates will be quite precise.

40. 2-40 How to Build a Race Track... n = ? – How big is each sample?  = ? – What slope are we estimating? Set X 1, X 2, …, X n – Do it once, or for each sample? Draw  1,  2,...,  n – Must draw randomly each sample.

41. 2-41 What to Assume About the  i ? What do the  i represent? What should the  i equal on average? What variance do we want for the  i ?

42. 2-42 Checking Understanding n = ? – How big is each sample?  = ? – What slope are we estimating? Set X 1, X 2, …, X n – Do it once, or for each sample? Draw  1,  2, …,  n – Must draw randomly each sample. Form Y 1, Y 2, …, Y n –Y i =  X i +  i We create 10,000 datasets with X and Y. For each dataset, what do we want to do?

43. 2-43 Checking Understanding (cont.) We create 10,000 datasets with X and Y For each dataset, we use all four of our estimators to estimate  g 1,  g 2,  g 3, and  g 4 We save the mean error, mean absolute error, and mean squared error for each estimator

44. 2-44 What Have We Assumed? We are creating our own data. We get to specify the underlying “Data Generating Process” relating Y to X. What is our Data Generating Process (DGP)?

45. 2-45 What Is Our Data Generating Process? E(  i ) = 0 Var(  i ) =  2 Cov(  i,  k ) = 0 i ≠ k X 1, X 2, …, X n are fixed across samples GAUSS–MARKOV ASSUMPTIONS

46. 2-46 What Will We Get? We will get precise estimates of: 1.Mean Error of each estimator 2.Mean Absolute Error of each estimator 3.Mean Squared Error of each estimator 4.Distribution of each estimator By running different racetracks (DGPs), we check the robustness of our results.

47. 2-47 Review We want an estimator to form a “best guess” of the slope of a line through the origin. Y i =  X i +  i We want an estimator that works well across many different samples: low average error, low average absolute error, low squared errors…

48. 2-48 Review (cont.) We have brainstormed 4 “best guesses”:

49. 2-49 Review (cont.) We will compare these estimators in “horseraces” across thousands of computer-generated datasets We get to specify the underlying relationship between Y and X We know the “right answer” that the estimators are trying to guess We can see how each estimator does

50. 2-50 Review (cont.) We choose all the rules for how our data are created. The underlying rules are the “Data Generating Process” (DGP) We choose to use the Gauss– Markov Rules.

51. 2-51 What Is Our Data Generating Process? E(  i ) = 0 Var(  i ) =  2 Cov (  i,  k ) = 0 i ≠ k X 1, X 2, …, X n are fixed across samples GAUSS–MARKOV ASSUMPTIONS