Integrated Time Series and Cointegration

1
Integrated Time Series and Cointegration

• Course Applied Econometrics
• Lecturer Zhigang Li

2
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).

3
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

4
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

5
Cointegration and Error Correction Model

• If 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.

6
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.

7
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
• 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

8
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 critical values are the same as the
  Dickey-Fuller test in last slide.

9
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 changes
  when a time trend is included
• Significance Level 1 5 10
• Critical Value (w/o trend) -3.43 -2.86 -2.57
• Critical Value (Trend) -3.96 -3.41 -3.12

10
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.

11
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

12
Pre-test (D-F test) for Nonstationarity
13
(No Transcript)
14
(No Transcript)
15
Bank Lending and Property Prices in Hong Kong
(Gerlach and Peng, 2005)
16
(No Transcript)
17
(No Transcript)
18
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).

19
(No Transcript)
20
Short-term effects
21
Findings

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