Matching and Propensity Score Matching

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Matching and Propensity Score Matching

• Lecturer Zhigang Li

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A Setting for Illustration

• Consider the following simple setting
• Each observation has three characteristics X, Y,
  and Z.
• Both X and Z could affect Y but we are only
  interested in the effect of X on Y.
• X and Z could be correlated.
• The function relating Z and Y is unknown.

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Solution 1 Parametric Approach

• Assume that the effect of Z on Y follows a
  specific function.
• Regress Y on X and on the function of Z.
• YaXf(Z)e
• The approach would work if the assumed function
  between Z and Y is a good approximation of the
  true relationship.
• Otherwise, part of the relationship between Z and
  Y may reflect on the coefficient of X.

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Solution 2 Matching

• Find observations with the same value of Z.
• Estimate the relationship between Y and X only
  for observations with the same Z. This may be
  done with the following regression
• YaXßiD(Zi)e
• What are we assuming with this regression?

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Some Implementation Problems with the Matching
Approach

• When the value of Z is continuous, it may not be
  easy to find the matching observations with
  exactly the same value of Z
• We may select Z with similar value
• When Z is not a single variable but includes many
  variables, it may be even harder to find matching
  observations because there are many possible
  combinations.
• Propensity score matching

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A Matching Approach(Berger and Waldfogel, 2005)

• Control Variables
• Set 1 Demographic controls (age, education,
  race, marital status, test score, family income),
  parity, child weight, child disabled and child
  female.
• Set 2 Above plus state level control variables
  (fraction of people with insurance, average
  income, party fraction, unemployment rate)
• Set 3 Above plus region dummies
• Set 4 Set 23 plus state dummies
• Propensity Score Matching

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Propensity Score Matching

• Step 1 Using all control variables to predict
  the probability for each mother to return to
  work within 12 weeks.
• Estimate P(return1x1, , xk)
• Predict P(returni) (this is the propensity score)
• Step 2 Compare mothers with similar propensity
  scores but some return to work early and some
  return to work late.

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