Copyright © 2006 Pearson Addison-Wesley. All rights reserved. Lecture 2: Econometrics (Chapter...
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Transcript of Copyright © 2006 Pearson Addison-Wesley. All rights reserved. Lecture 2: Econometrics (Chapter...
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
Lecture 2: Econometrics
(Chapter 2.1–2.7)
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 2-2
How Does Econometrics Differ From Economic Theory?
• Economic theory: qualitative results— Demand Curves Slope Downward
• Econometrics: quantitative results— price elasticity of demand for milk = -.75
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How Does Econometrics Differ From Statistics?
• Statistics: “summarize the data faithfully”; “let the data speak for themselves.”
• Econometrics: “ what do we learn from economic theory AND the data at hand?”
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What’s Metrics For?
• Estimation: What is the marginal propensity to consume?
• Hypothesis Testing: Do unions raise workers’ wages?
• Forecasting: What will Personal Savings be in 2001 if GDP is $9.2 trillion?
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Economists Ask: “What Changes What and How?”
• Higher Income, Higher Saving
• Higher Price, Lower Quantity Demanded
• Higher Interest Rate, Lower Investment
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Savings Versus Income
• Theory Would Assume an Exact Relationship, e.g., Y =X
0
1000
2000
3000
4000
5000
6000
24000 48000 72000 96000
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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!
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Never So Neat: Savings Versus Income
0
1
2
3
4
5
6
0 20 40 60 80 100 120
saving
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Yi Xi i
Underlying Mean + Random Part
• We devised four intuitively appealing ways to estimate
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“Best Guess 1”
Mean of Ratios:
g
1
1
n
Yi
Xi
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Figure 2.4 Estimating the Slope of a Line with Two Data Points
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“Best Guess 2”
Ratio of Means:
2i
i
Yg
X
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Figure 2.5 Estimating the Slope of a Line: g2
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“Best Guess 3”
Mean of Changes in Y over Changes in X:
g
3
1
n 1
Yi Y
i 1
Xi X
i 1
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“Best Guess 4”
Ordinary Least Squares:
(minimizes squared residuals in sample)
g4
XiY
iX
i2
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Four Ways to Estimate
1) Mean of Ratios: 3) Mean of Ratio of Changes:
g1
1
n
Yi
Xi
g3
1
n 1
Yi Y
i 1
Xi X
i 1
2) Ratio of Means: 4) Ordinary Least Squares:
g2
Yi
Xi g
4
YiX
iX
i2
Yi Xi i
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 2-17
Underlying Mean + Random Part
• Are lines through the origin likely phenomena?
Yi Xi i
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Regression’s Greatest Hits!!!
• An Econometric Top 40
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Two Classical Favorites!!
• Friedman’s Permanent Income hypothesis:
• Capital Asset Pricing Model (CAPM) :
Consumption ·(Permanent Income)i
(Asset j’s Return Above a Riskless Rate) ·(Market’s Return Above a Riskless Rate) i
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A Golden Oldie !!
• Engel on the Demand for Rye:
E(%change in quantity) elasticity·(%change in price)
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Four Guesses
• How to Choose?
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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
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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.
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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
Mean Error E(g - )
BIAS of g in estimating
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Checking Understanding
• Question: Which estimator does better under the “minimize mean error” condition?
1. g- is always a positive number less than 2 (our guesses are always a little high), or
2. g- is always +10 or -10 (50/50 chance)
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 2-26
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.
Mean Error E(g - )
BIAS of g in estimating
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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.
Mean Absolute Error E(| g - |)
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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.
Mean Square Error E[(g - )2 ]
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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
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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?
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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.
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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.”
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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?
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Building a Fair Racetrack
1) Mean of Ratios: 3) Mean of Ratio of Changes:
g11
n
Yi
Xi
g3 1
n 1
Yi Y
i 1
Xi X
i 1
2) Ratio of Means: 4) Ordinary Least Squares:
g2
Yi
Xi g
4
YiX
iX
i2
Under what conditions will each estimator fail?
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To Preclude Automatic Failure...
1) g1
1
n
Yi
Xi
3) g3
1
n 1
Yi Y
i 1
Xi X
i 1
No X
i0 No successive X 's equal
2) g2
Yi
Xi 4) g
4
YiX
iX
i2
Xi0 Some X
i0
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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.
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How to Build a Race Track...
• n = ? – How big is each sample?
• = ? – What slope are we estimating?
• Set X1 , X2 , … , Xn
– Do it once, or for each sample?
• Draw 1 , 2 , ... , n – Must draw randomly each sample.
Yi Xi i i 1, 2, , n
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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 ?
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Checking Understanding
• n = ? – How big is each sample?
• = ? – What slope are we estimating?
• Set X1 , X2 , … , Xn
– Do it once, or for each sample?
• Draw 1 , 2 , … , n
– Must draw randomly each sample.
• Form Y1 , Y2 , … , Yn
– Yi = Xi + i
• We create 10,000 datasets with X and Y.
• For each dataset, what do we want to do?
Yi Xi i i 1, 2, , n
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Checking Understanding (cont.)
• We create 10,000 datasets with X and Y
• For each dataset, we use all four of our estimators to estimate g1 , g2 , g3 , and g4
• We save the mean error, mean absolute error, and mean squared error for each estimator
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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)?
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What Is Our Data Generating Process?
• E(i ) = 0
• Var(i ) = 2
• Cov(i ,k ) = 0 i ≠ k
• X1 , X2 , … , Xn are fixed across samples
GAUSS–MARKOV ASSUMPTIONS
Yi Xi i i 1, 2, , n
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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.
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Review
• We want an estimator to form a “best guess” of the slope of a line through the origin.
• Yi = Xi +i
• We want an estimator that works well across many different samples: low average error, low average absolute error, low squared errors…
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 2-45
Review (cont.)
• We have brainstormed 4 “best guesses”:
1) Mean of Ratios: 3) Mean of Ratio of Changes:
g11
n
Yi
Xi
g3 1
n 1
Yi Y
i 1
Xi X
i 1
2) Ratio of Means: 4) Ordinary Least Squares:
g2
Yi
Xi g
4
YiX
iX
i2
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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
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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.
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What Is Our Data Generating Process?
• E(i ) = 0
• Var(i ) = 2
• Cov (i ,k ) = 0 i ≠ k
• X1 , X2 , … , Xn are fixed across samples
GAUSS–MARKOV ASSUMPTIONS
Yi Xi i i 1, 2, , n