How to Identify Causal Relationship: An Introduction

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How to Identify Causal Relationship An
Introduction

• Course Applied Econometrics
• Lecturer Zhigang Li

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The Setting

• Finding the linear causal impact of X1 on Y (e.g.
  X1 is education level and Y is wage).
• Is there an effect (positive or negative)?
• How big is the effect?
• Data is well behaved (e.g. no problem with
  sampling, outliers, measurement errors,
  multicollinearity, Heteroskedasticity)
• Endogeneity may be present

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Where is the Endogeneity from?

• Endogeneity Some independent variables are
  correlated with the error term.
• Yß0ß1X1ß2X2u
• Some variables that can affect Y and are
  correlated with X1 are omitted (therefore hiding
  in u), e.g. ability
• Other independent variables (e.g. X2) are
  correlated with u and with X1 at the same time.
  For example, X2 is the appearance of peoplee.
• Y can affect X1 (so-called reverse-causality)

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Solution 1 Include more control variables

• Control for confounding variables
• Yß0ß1X1ß2X2ß3X3ßkXku
• Why including control variables?
• If X1 is uncorrelated with the control variables,
  excluding them does not affect the estimate of ß1
  but may lower the precision (i.e. the variance)
  of the estimate of ß1. In this case, we want to
  include more control variables. (Hint p.96 or
  101)
• If X1 is correlated with some control variables,
  then excluding them will bias the estimate of ß1.
  The effect on the precision of estimates is
  unclear. (AK p. 11-12)

Causing Variable
Control Variables
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Problems with the Control-Variable Approach

• There are always some variables you do not
  observe.
• Adding too many control variables may take away
  too many degrees of freedom (or the information
  content of data), making estimates less precise.
• When X1 is measured with some error, adding
  control variables correlated with X1 may actually
  increase the bias. (AK p. 14)
• When control variables are themselves endogenous
  (i.e. affected by Y), then adding them may not
  improve the estimate of ß1. (AK p.14-15)

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Solution 2 Panel Data and the Fixed-Effects
Approach

• Panel data typically include the observations of
  agent i (e.g. individuals, firms, cities, etc.)
  at different time periods t.
• Yitß0ß1X1itßkXkitßiuit
• The fixed-effects approach (ßi) uses repeated
  observations on each agent to control for the
  effects of time-constant characteristics, which
  may be related to the Xs, e.g. ability.
• Using dummies for different agents
• Taking a first-difference before regressions

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Pitfalls of the Fixed-effect Approach

• Standard fixed-effect estimators can only be used
  to estimate the effects of time-varying
  regressors
• Bias from measurement error is usually
  aggravated.
• The assumption that unobserved variables do not
  change over time may not be valid.

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Solution 3 Natural (Quasi-) Experiment

• In an ideal experiment we have a treatment group
  and a control group. The two groups are the same
  except that only the treatment group is affected
  by some factor of interest while the control
  group is not affected by the factor (both groups
  can still be affected by other confounding
  factors).
• A natural experiment is like an experiment. The
  difference is that in the latter case we have
  good control on which group can be affected by
  which factors. In a natural experiment, we can
  not control the factors, but we try to find some
  natural settings in which the nature has
  generated two similar groups, one of which is
  affected by the factors of interest and the other
  group is not.

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Solution 4 Regression Discontinuity

• Consider a regression Yß0ß1X1ßkXku, in
  which X1 could be endogenous.
• A RD design is a situation in which some agents
  have (1) similar control variable X2 but have (2)
  dramatically different X1 due to (3) a functional
  relationship between X1 and X2 .
• If Y also differs dramatically for those agents
  at the break point, it suggests a causal effect
  from X1 to Y.

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Solution 5 Instrumental Variables (IV)

• Consider a regression Yß0ß1X1ßkXku, in
  which X1 is endogenous but control variables are
  not.
• An IV is a variable (call it Z) that is
  correlated with X1 but uncorrelated with u.
• A Two-Stage OLS (2SLS)
• First regress X1 on Z and predict X1 with Z.
• In the second stage replace X1 with the predicted
  (or fitted) value of X1 in the original
  regression.

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Problems NE vs. RD vs. IV

• All are demanding in the sense that they need
  strict conditions to be satisfied.
• The three approaches complement each other and
  some of them may be more suitable for different
  issues.
• Studies using the IV approach are more commonly
  seen than those using NE and RD. But perfect IVs
  are rarely available.
• Empirical findings without NE, RD, or IV are
  typically much less convincing (but using NE, RD,
  or IV does not guarantee that your findings will
  be accepted).

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Natural Experiment vs. The Matching Approach

• The matching approach also compare a treatment
  group and a control group.
• Consider a regression Yf(X1,,Xk,u) in which X1
  is exogenous (X1 is typically a dummy variable)
  after controlling for X2 through Xk.
• The control group is constructed by selecting
  from non-treated observations those with similar
  X2 through Xk as the treatment group.
• The matching approach is not an NE approach since
  X1 is assumed exogenous.
• The matching approach imposes no functional form
  assumptions.