Regression with Time Series Data

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Regression with Time Series Data

• Judge et al Chapter 15 and 16

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Distributed Lag
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Polynomial distributed lag
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Estimating a polynomial distributed lag
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Geometric Lag
Impact Multiplier change in yt when xt changes
by one unit
If change in xt is sustained for another period
Long-run multiplier
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The Koyck Transformation
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Autoregressive distributed lag
ARDL(1,1)
ARDL(p,q)
Represents an infinite distributed lag with
weights
Approximates an infinite lag of any shape when p
and q are large.
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Stationarity

• The usual properties of the least squares
  estimator in a regression using time series data
  depend on the assumption that the variables
  involved are stationary stochastic processes.
• A series is stationary if its mean and variance
  are constant over time, and the covariance
  between two values depends only on the length of
  time separating the two values

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Stationary Processes
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Non-stationary processes
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Non-stationary processes with drift
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Summary of time series processes

• AR(1)
• Random walk
• Random walk with drift
• Deterministic trend

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Trends

• Stochastic trend
• Random walk
• Series has a unit root
• Series is integrated I(1)
• Can be made stationary only by first differencing
• Deterministic trend
• Series can be made stationary either by first
  differencing or by subtracting a deterministic
  trend.

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Real data
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Spurious correlation
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Spurious regression
Variable DF B Value Std Error T ratio Approx prob
Intercept 1 14.204040 0.5429 26.162 0.0001
RW2 1 -0.526263 0.00963 -54.667 0.0001
R2 0.7495 D-W 0.0305
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Checking/testing for stationarity

• Correlogram
• Shows partial correlation observations at
  increasing intervals.
• If stationary these die away.
• Box-Pierce
• Ljung-Box
• Unit root tests

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Unit root test
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Dickey Fuller Tests

• Allow for a number of possible models
• Drift
• Deterministic trend
• Account for serial correlation

Drift
Drift against deterministic trend
Adjusting for serial correlation (ADF)
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Critical values
Table 16.4 Critical Values for the Dickey-Fuller Test Table 16.4 Critical Values for the Dickey-Fuller Test Table 16.4 Critical Values for the Dickey-Fuller Test Table 16.4 Critical Values for the Dickey-Fuller Test
Model 1 5 10
?2.56 ?1.94 ?1.62
?3.43 ?2.86 ?2.57
?3.96 ?3.41 ?3.13
Standard critical values ?2.33 ?1.65 ?1.28
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Example of a Dickey Fuller Test
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Cointegration

• In general non-stationary variables should not be
  used in regression.
• In general a linear combination of I(1) series,
  eg is I(1).
• If et is I(0) xt and yt are cointegrated and the
  regression is not spurious
• et can be interpreted as the error in a long-run
  equilibrium.

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Example of a cointegration test
Model 1 5 10
?3.90 ?3.34 ?3.04