Integrated Time Series and Cointegration

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Integrated Time Series and Cointegration

• Course Applied Econometrics
• Lecturer Zhigang Li

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Integrated Series and Cointegration

• A series xt is integrated of order d (we call it
  I(d) process) if the series becomes stationary
  after differencing d times.
• Two series x and y are cointegrated if
• Both series are of the same order d
• A linear combination of the two series is
  integrated to the order b (bltd).

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Unit Root Process

• Unit Root Process
• ytyt-1ut (ut is a weakly dependent process)
• A random walk (ut is i.i.d. with mean zero) is a
  special case of the unit root process.
• No matter how far in the future we look and how
  much information we have for the past, our best
  prediction of future is todays value.
• The expected value of a random walk does not
  depend on t
• The variance of a random walk increases as a
  linear function of time (nonstationary).
• High persistency Corr(yt, yth)t/(th)1/2

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Spurious Regression

• Spurious regression X and Y are not related at
  all but regressing Y on X shows a significant
  statistical correlation between them. This could
  happen for
• Omitted variable Z that drives both X and Y
• Trending X and Y
• Even if series Xt and Yt are not trending, the
  regression between them may be spurious if X and
  Y are independent and are both I(1).
• In this case, the error term of the regression is
  an I(1) process, thus strongly dependent,
  violating consistency assumptions.
• Solutions
• Include omitted variables
• First difference
• Cointegration

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Cointegration and Error Correction Model

• If a ß exists such that yt-ßxt is an I(0)
  process, then y and x are cointegrated.
• ytaßxte
• A cointegration model between X and Y can be
  equally rewritten as
• ?yta??xtd(yt-1-ßxt-1)u
• While the cointegration model emphasizes the
  long-run equilibrium relationship between y and
  x, the error correction model characterizes the
  short-run adjustment processes towards the
  equilibrium relationship.

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The Engle-Granger Procedure

• If the series X and Y are integrated to the same
  order d, cointegration between X and Y can be
  tested through the following two-stage procedure
• Cointegrating Regression Regress Y on X (and
  other control variables) by OLS
• The residuals from the regression are tested for
  the order of integration. If the residuals are
  integrated to lower order, then X and Y are
  cointegrated.

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Visual Inspection for Nonstationarity (Unit Root)

• One may investigate whether a series is
  stationary by visual inspection of the graph of
  the sample autocorrelations against time interval
  t, known as correlogram.
• A series is likely to be nonstationary if the
  correlograms decay slowly.

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Rigorous Unit Root Test I(Dickey-Fuller Test)

• To test whether ?1 in yta?yt-1e, rewrite it
  as ?yta(?-1)yt-1et
• H0 ?-10 H1 ?-1lt0
• The model is assumed to be dynamically complete
  (martingale E(etyt-1, yt-2,,y0)0)
• Because the series yt is I(1) under H0, usual
  t-test critical values need to be adjusted
  (following Dickey-Fuller) as follows
• Significance Level 1 5 10
• Critical Value -3.43 -2.86 -2.57

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Rigorous Unit Root Test II(Augmented
Dickey-Fuller Test)

• ?yta(?-1)yt-1 ?yt-1?yt-2?yt-pet
• H0 ?-10 H1 ?-1lt0
• Enough lagged dependent variables are added so
  that the model is dynamically complete.
• The lag length is often dictated by the frequency
  of the data. For annual data, one or two lags
  usually suffice. For monthly data, twelve lags
  might be needed.
• The t statistics on the lagged changes have
  approximate t distributions, so standard tests
  might be used to determine lag lengths.
• The critical values are the same as the
  Dickey-Fuller test in last slide.

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Rigorous Unit Root Test III(Dickey-Fuller Test
with Time Trend)

• ?ytadt(?-1)yt-1et
• H0 ?-10 H1 ?-1lt0
• A trend-stationary process can be mistaken for a
  unit root process if the time trend is not
  controlled for.
• Critical values of the Dickey-Fuller test chanes
  when a time trend is included
• Significance Level 1 5 10
• Critical Value -3.43 -2.86 -2.57
• Critical Value (Trend) -3.96 -3.41 -3.12

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Limitations of Cointegration Analysis

• Pre-test procedures (unit root test of individual
  variables) are often inconclusive.
• There may be substantial small-sample bias.
• Structural breaks in the time series can cause
  difficulties in unit root test and cointegration
  analysis.

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Aggregated Consumption and the Demand for Imports
(Clarida, 1992)

• Empirical model
• m Import of nonduarable goods
• h Domestic nondurable goods consumption
• p Relative import prices
• v Stationary disturbance

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Pre-test (D-F test) for Nonstationarity
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Endogeneity Issue

• Reverse causality
• The endogeneity problem is addressed with the
  Phillips-Loretan procedure

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Bank Lending and Property Prices in Hong Kong
(Gerlach and Peng, 2005)

• Theory
• Bank lending can affect property prices by
  affecting the demand for property (through
  mortgage) and supply for property (real estates
  firms).
• Property prices can affect bank lending by
  affecting banks capital position and thus
  lending capacity.

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What is so special about Hong Kong?

• Property prices in Hong Kong are more volatile
  than most other countries.
• Asset price swings in Hong Kong is less affected
  by monetary policy due to the currency board
  regime.
• Banking sector remains sound despite the collapse
  in property prices since 1998.

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Data

• Property prices
• The residential property price index published by
  the Rating and Valuation Department.
• Bank lending
• Total lending for use in Hong Kong

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Methodology

• Estimating the cointegration relationship between
  bank lending, GDP, and Property price index.
• Estimating the error-correction model of bank
  lending and property prices, respectively, to
  investigate the short-run causal relationship
  (using predetermined variables as IVs).

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Short-term effects
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Findings

• Bank lending, GDP, and Property price index are
  all I(1) and are cointegrated.
• Property prices affect bank lending but not the
  reverse.

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Interest Rate and Inventory(Maccini et al. 2004)

• Inventory investment decision should be affected
  by interest rate.
• Interest rate reflects the opportunity costs of
  holding inventory.

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A Puzzle

• Real interest rates seems to have little impact
  on inventory investment. The following empirical
  strategies have been tried but failed
• Decision-rule Estimation
• Nta0a1Nt-1a2Nt-2a3rt a4Xt et
• Euler Equation Estimation Expected net present
  value of investments today and tomorrow are the
  same.

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Explanation

• Mean interest rate is highly persistent.
  Expecting this, investors do not respond to
  short-run interest rate changes but only to
  long-run interest rate changes. This suggests a
  cointegration regression.
• D-F tests suggests that all variables in the
  regressions are I(1) processes.
• Error-correction type of model is estimated using
  the Johansen-Juselius test of cointegration.
• The effect of interest rate on inventory is
  significant in the long-run adjustment term of
  the error correction model.