Lecture 2: Estimating a Slope

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

2. Recall 計量經濟學的構成 • 經濟理論 • 數理模型 • 統計理論 • 計量模型 現象 經濟理論 數理模型 統計理論 計量模型

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. 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. 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 2004if GDP is $14,864? And will it grow in the near future (2008)?(預期及預測)

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

7. 6000 5000 4000 3000 2000 1000 0 24000 48000 72000 96000 Savings Versus Income • Theory Would Assume an Exact Relationship, e.g., Y =bX

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. Never So Neat: Savings Versus Income

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

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

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

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

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

15. “Best Guess 2” Ratio of Means:

16. Figure 2.5 Estimating the Slope of a Line: bg2

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

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

19. Four Ways to Estimate b

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

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

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

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

24. Four Guesses • How to Choose?

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. 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. What Criteria Did We Discuss? (cont.) • Make Mean Error Close to Zero …seek unbiased guesses • IfE(bg-b) = 0, bgis right on average • If BIAS = 0, bg is an unbiased estimator of b

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

29. Checking Understanding (cont.) • If our guess is wrong by +10 for half the observations, and by -10 for the other half, then E(bg-b) = 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. 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. 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. 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. 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. 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. Try the Four in Many Samples (cont.) • We use our estimates on many datasets that we created ourselves. • We know the true value of b because we picked it! • We can compare estimators. • We run “horseraces.”

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

38. To Preclude Automatic Failure...

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 e2/m, where e2 is the variance of that mean’s error. • If m is large, our estimates will be quite precise.

40. How to Build a Race Track... • n= ? • How big is each sample? • b = ? • What slope are we estimating? • Set X1, X2, … , Xn • Do it once, or for each sample? • Draw e1, e2, ... , en • Must draw randomly each sample.

41. What to Assume About the ei ? • What do the eirepresent? • What should the eiequal on average? • What variance do we want for the ei?

42. n= ? How big is each sample? b= ? What slope are we estimating? Set X1 , X2 , … , Xn Do it once, or for each sample? Draw e1 , e2 , … , en Must draw randomly each sample. Form Y1 , Y2 , … , Yn Yi =bXi+ ei We create 10,000 datasets with X and Y. For each dataset, what do we want to do? Checking Understanding

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 bg1, bg2, bg3,and bg4 • We save the mean error, mean absolute error, and mean squared error for each estimator

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. What Is Our Data Generating Process? • E(ei ) = 0 • Var(ei ) = 2 • Cov(ei ,ek ) = 0 i ≠ k • X1 , X2 , … , Xn are fixed across samples GAUSS–MARKOV ASSUMPTIONS

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

47. Review • We want an estimator to form a “best guess” of the slope of a line through the origin. • Yi = bXi +ei • We want an estimator that works well across many different samples: low average error, low average absolute error, low squared errors…

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

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. 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.