Econ 2148, fall 2017 Statistical decision theory

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Econ 2148, fall 2017 Statistical decision theory

Transcript Of Econ 2148, fall 2017 Statistical decision theory

Statistical Decision Theory
Econ 2148, fall 2017 Statistical decision theory
Maximilian Kasy
Department of Economics, Harvard University
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Statistical Decision Theory
Takeaways for this part of class
1. A general framework to think about what makes a “good” estimator, test, etc.
2. How the foundations of statistics relate to those of microeconomic theory.
3. In what sense the set of Bayesian estimators contains most “reasonable” estimators.
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Statistical Decision Theory
Examples of decision problems
Decide whether or not the hypothesis of no racial discrimination in job interviews is true Provide a forecast of the unemployment rate next month Provide an estimate of the returns to schooling Pick a portfolio of assets to invest in Decide whether to reduce class sizes for poor students Recommend a level for the top income tax rate
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Statistical Decision Theory
Agenda
Basic definitions Optimality criteria Relationships between optimality criteria Analogies to microeconomics Two justifications of the Bayesian approach
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Statistical Decision Theory Basic definitions
Components of a general statistical decision problem
Observed data X A statistical decision a
A state of the world θ A loss function L(a, θ ) (the negative of utility) A statistical model f (X |θ ) A decision function a = δ (X )
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Statistical Decision Theory Basic definitions
How they relate
underlying state of the world θ ⇒ distribution of the observation X . decision maker: observes X ⇒ picks a decision a her goal: pick a decision that minimizes loss L(a, θ ) (θ unknown state of the world) X is useful ⇔ reveals some information about θ ⇔ f (X |θ ) does depend on θ .
problem of statistical decision theory:
find decision functions δ which “make loss small.”
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Statistical Decision Theory Basic definitions
Graphical illustration
Figure: A general decision problem

observed data X

decision function a=δ(X)

decision a

statistical model
X~f(x,θ)

state of the world θ

loss L(a,θ)

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Statistical Decision Theory Basic definitions
Examples
investing in a portfolio of assets: X : past asset prices a: amount of each asset to hold θ : joint distribution of past and future asset prices L: minus expected utility of future income
decide whether or not to reduce class size: X : data from project STAR experiment a: class size θ : distribution of student outcomes for different class sizes L: average of suitably scaled student outcomes, net of cost
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Statistical Decision Theory Basic definitions
Practice problem For each of the examples on slide 2, what are
the data X , the possible actions a,
the relevant states of the world θ , and
reasonable choices of loss function L?
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Statistical Decision Theory Basic definitions
Loss functions in estimation
goal: find an a
which is close to some function µ of θ . for instance: µ(θ ) = E[X ]
loss is larger if the difference between our estimate and the true value is larger Some possible loss functions: 1. squared error loss,
L(a, θ ) = (a − µ(θ ))2
2. absolute error loss,
L(a, θ ) = |a − µ(θ )|
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Decision TheoryDecisionUtilityAssetsPortfolio