Copyright © 2006 Pearson Addison-Wesley. All rights reserved. Lecture 2: Econometrics (Chapter 2.1–2.7)

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


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?”


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?


Economists Ask: “What Changes What and How?”

• Higher Income, Higher Saving

• Higher Price, Lower Quantity Demanded

• Higher Interest Rate, Lower Investment


Savings Versus Income

• Theory Would Assume an Exact Relationship, e.g., Y =X

0

1000

2000

3000

4000

5000

6000

24000 48000 72000 96000


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!


Never So Neat: Savings Versus Income

0

1

2

3

4

5

6

0 20 40 60 80 100 120

saving


Yi Xi i

Underlying Mean + Random Part

• We devised four intuitively appealing ways to estimate


“Best Guess 1”

Mean of Ratios:

g

1

1

n

Yi

Xi


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


“Best Guess 2”

Ratio of Means:

2i

i

Yg

X


Figure 2.5 Estimating the Slope of a Line: g2


“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


“Best Guess 4”

Ordinary Least Squares:

(minimizes squared residuals in sample)

g4

XiY

iX

i2


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


Underlying Mean + Random Part

• Are lines through the origin likely phenomena?

Yi Xi i


Regression’s Greatest Hits!!!

• An Econometric Top 40


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


A Golden Oldie !!

• Engel on the Demand for Rye:

E(%change in quantity) elasticity·(%change in price)


Four Guesses

• How to Choose?


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


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.



• 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


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)


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


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 - |)



• 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 ]



• 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


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?


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.


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.”


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?


Building a Fair Racetrack


g11

n

Yi

Xi

g3 1

n 1

Yi Y

i 1

Xi X

i 1


g2

Yi

Xi g

4

YiX

iX

i2

Under what conditions will each estimator fail?


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


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.


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


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 ?


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


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


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)?


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


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.


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…


Review (cont.)

• We have brainstormed 4 “best guesses”:


g11

n

Yi

Xi

g3 1

n 1

Yi Y

i 1

Xi X

i 1


g2

Yi

Xi g

4

YiX

iX

i2


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


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.


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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Copyright © 2006 Pearson Addison-Wesley. All rights reserved. Lecture 2: Econometrics (Chapter 2.1–2.7)