Introduction to Econometrics

1
Introduction to Econometrics

• Lecture week 11
• Time Series Econometrics
• Stationary versus nonstationary series

2
Stationary versus nonstationary series

• Spurious regression a regression of one time
  series variable on another yielding a very high
  R2 although the two have no meaningful economic
  or inherent relationship. This often occurs
  precisely because both series exhibit a common
  strong trend.
• Spurious regression a regression of one time
  series variable on another which gives a series
  of residuals which violate the classical linear
  regression assumptions. This could again be
  attributed to the presence of trend in one or two
  of the variables.
• Exception the latter may not hold true if a
  linear combination of the time series variables
  in question (i.e. the residual) is free from
  trend.
• In general any series with a trend is considered
  as a nonstationary series.
• Consequence of spurious regression the usual t
  and F tests will be invalid.
• Objective of the lecture the concept of
  stationarity, how to test for it, what
  implications of nonstationarity are, and
  conditions under which nonstationarity proves not
  to be problematic.

3
Requirements for a time series variable to be
stationary.

• For a time series variable to be considered as a
  stationary series, the following conditions must
  be satisfied
• Mean E(Yt) ?
• Variance Var (Yt) E(Yt - ?)2 ?2
• Covariance ?k E(Yt - ?)(Ytk - ?) 0

4
Examples of SAS Nonstationary TIME SERIES Data
5
Graphs of Logarithms of SAs Total Agricultural
Production Index (LAGPRO) and Combined Producer
Price Index of Agricultural Production (LAGPRICE)
6
Tests for Stationarity correlogram unit root

• Correlogram test plot the sample autocorrelation
  function (ACF) at successive lags against the
  length of the lag.
• Unit root this is a formal test for stationarity
  of a variable. Test for unit root is conducted
  using either Dickey-Fuller (DF) or Augmented
  Dickey-Fuller (ADF) procedures.
• DF test ?LAGPROt ?1 ?2t ? LAGPROt-1
  ut
• ADF test

7
Correlogram test for stationarity of logarithms
of index of agricultural production (on levels)
and combined index of producer prices (on levels)
8
Graph of the logarithm of Ag Prodn on levels
(LAGPRO) and after trend is removed (D(LAGPRO))
9
Graph of the logarithms of the combined producer
price index on levels (LAGPRPRICE) and after
trend is removed (D(LAGPRPRICE))
10
Correlogram test for stationarity of logarithms
of index of combined index of producer prices
before and after trend is removed
11
CONCLUSION ON THE STATIONARITY OF PRODUCTION AND
PRICE DATA USING CORRELOGRAM TEST

• Production is stationary but price is not in
  levels.
• Price was differenced once and the test was
  carried out again to check if trend was
  completely removed. According to the result, the
  no autocorrelation null was not rejected.
• Therefore, production is integrated of order zero
  (I(0)) because no significant trend was detected.
  But price is integrated of order two (I(1))
  because it was differenced once to make it
  stationary.

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Unit Root Test A formal test for the
stationarity of a variable

• Random walk has a coefficient (?) which is unity
  and its variance increases or decreases over
  time.
• Yt ?Yt-1 ut

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Unit Root Test A formal test for the
stationarityof a variable (cont)

• Yt ?Yt-1 ut ........................
  .........(1)
• Ut is N (0, ?2)
• Subtract Yt-1 from each side of equation 1
• Yt- Yt-1 (1- ?) Yt-1 ut
• ?Yt ? Yt-1 ut ..........................
  .........(2)
• Where ?Yt Yt- Yt-1 , ? (1- ?)
• If ? 0, ? 1 therefore equation 2 becomes
• ?Yt ut .......(3)
• Equation 3 means that first difference of a
  random walk series is
• stationary b/c from slide 3 we know that the mean
  and variance of a
• stationary series are time invariant and the
  series is not autocorrelated.
• A random walk is I(1).
• Note stationarity of Yt is checked by running
  equation 2 and testing for
• the significance of ?. If not significant, ? 1
  and Yt is random walk or
• nonstationary.
• Dickey and Fuller tabulated the appropriate ?
  (tau) statistics

14
ADF APPROACH TO TEST FOR THE PRESENCE OF A UNIT
ROOT IN THE LOGARITHM OF SAS AGRICULTURAL
PRODUCTION
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ADF APPROACH TO TEST FOR THE PRESENCE OF A UNIT
ROOT IN THE LOGARITHM OF SAS AGRICULTURAL
PRODUCER PRICE ON LEVELS
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ADF APPROACH TO TESTING THE PRESENCE OF A UNIT
ROOT IN THE LOGARITHM OF SAS AGRICULTURAL
PRODUCER PRICE ON FIRST DIFFERENCE
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ADF APPROACH TO TESTING THE PRESENCE OF A UNIT
ROOT IN THE LOGARITHM OF SAS AGRICULTURAL
PRODUCER PRICE ON SECOND DIFFERENCE
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COMPARISON OF AGRICULTURAL PRODUCER PRICE ON
FIRST SECOND DIFFERENCE
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CONCLUSION ON THE STATIONARITY OF PRODUCTION AND
PRICE DATA USING ADF TEST

• Production is stationary i.e. I(0).
• Price is I(2) not I(1)
• The ADF test is a robust test.

20
Nonstationary processes trend stationary process
(TSP) and difference stationary process (DSP)

• The distinction is useful to make good forecast
• TSP the trend is deterministic (predictable)

21
Nonstationary processes trend stationary process
(TSP) and difference stationary process (DSP)
Cont..

• DSP the trend is non deterministic or stochastic
  (unpredictable).
• Most time series data are characterized by
  deterministic trend.

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Techniques to distinguish TSP from DSP

• The ADF test is a more convenient method.
• If ?2 ? 0 and d ?0 ? LAGPROt is TSP (detrended
  LAGPRO follows a stationary AR process)
• If ?2 0 and d 0 ? LAGPROt is DSP (LAGPRO
  follows a nonstationary RW process)
• When we test for the presence of a unit root in a
  series, we are testing against the alternative of
  trend stationarity