Review I: How to detect a relationship statistically

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Review I How to detect a relationship
statistically?

• Econ6001 Applied Econometrics
• Instructor Zhigang Li

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Variable

• Variable A measure that may take different
  values
• over time or
• across observations (e.g. people, firms, regions,
  etc.)

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Relationship and Causality
X
Y
Y
X
X causes Y
X
Y
Z
Y causes X
Spurious Relationship between X and Y
X
Y
Simultaneous Causation
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How to Represent the Relationship between X and Y

• Theory f(X,Y)0
• A linear relationship X-2Y-20
• A non-linear relationship log(X)-Y0
• Reality f(X,Y,u)0, where u is an unobserved
  variable.
• E.g. Y2-4X u
• How to represent the relationship between X and Y
  in this case?

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Statistical Relationship between X and Y

• Correlation (coefficient)
• COVXYE(X-E(X))(Y-E(Y))E(XY)-E(X)E(Y)
• CORXYCOVXY/sd(X)sd(Y)
• If Y2-4Xe, then COVXY-4var(X)-4sd(X)sd(X)
• Regression coefficient
• RYXCOVXY/ var(X)
• If Y2-4Xe, then RYX-4
• Correlation is unit-free and it ranges between 1
  and 1, measuring how tightly X and Y are related.
• Regression coefficient is not unit-free. It
  measures the average change of Y when X changes
  by one unit on average.

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Note

• Correlation and regression coefficient are
  statistical measures of the linear relationship
  between X and Y.
• Correlation and regression coefficient always
  have the same sign.
• Correlation and regression coefficient do not
  tell the causal relation between X and Y

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How to Measure RYX Empirically?

• Suppose that economic theory suggests that
  Ya-bXe and we want to know how much b is. How
  can we measure it?
• If we know all values of Y and X for the
  population, we can calculate RYX by its
  definition. Note that RYX is a number.
• Otherwise, we can randomly sample the population
  for Y and X (getting y1, y2, , yn and x1, x2, ,
  xn), then we can calculate the regression
  coefficient for this sample. Call it rYX. Note
  that rYX may vary for different samples. So rYX
  is a variable, while Ryx is a number.

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Statistical Inference

• The rYX from a particular sample is called an
  estimate (or a proxy) of RYX.
• How much do we know about RYX if we know rYX?
• If rYX follows an unbiased normal distribution
  with variance s2, then the 95 two-sided
  confidence interval of RYX is
• rYX 1.96s, rYX 1.96s

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Hypothesis Testing

• Theory often predicts a relationship to be
  positive or negative. This prediction (or
  hypothesis) can be formally checked (or tested)
  in the following two equal ways.
• t test A simple t statistic is a measure of the
  statistical distance between an estimate and
  zero. The larger the t statistic, the larger the
  distance, and the more likely is the estimate
  different from zero.
• P-value A p-value is roughly the possibility for
  you to make mistake if you conclude that the
  relationship is there. So the smaller the
  p-value, the more confident you should be that
  the relationship exists.