Time Series Econometrics

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Time Series Econometrics

• Ch7 Some Basic Concepts

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Ch7 Basics

• 1. Stochastic Processes
• 2. Stationarity Processes
• 3. Purely Random processes
• 4. Nonstationary Processes
• 5. Random Walk Models
• 6. Unit Root Tests
• 7. Cointegration and Cointegration Tests
• 8. Error Correction Mechanism
• 9. Granger Causality Test

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Background

1. Regression analysis based on time series data
   implicitly assumes that the underlying time
   series are stationary. The classical t test, F
   tests, etc. are based on this assumption.
2. In practice most economic time series are
   nonstationary. (spurious/ nonsensical regression)

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Spurious Regression

• Regression of one time series variable on one or
  more time series variables often can give
  nonsensical or spurious results.
• Spurious regression often shows a significant
  relationship between variables, but in fact, this
  kind of relationship does not exist.
• This phenomenon is known as spurious
  regression.
• An is a good rule of thumb to
  suspect that the estimated regression is
  spurious.

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1. Stochastic Processes

• A random or stochastic process is a collection of
  random variables ordered in time.
• The term stochastic comes from the Greek word
  stochos, which means a target or bulls eye.

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2. Stationary Stochastic Processes

• A stochastic process is said to be stationary if
  its mean and variance are constant over time and
  the value of the covariance between the two time
  periods depends only on the distance or gap or
  lag between the two time periods and not the
  actual time at which the covariance is computed.
  That is, they are time invariant.
• In the time series literature, such a stochastic
  process is known as a weakly stationary. A
  stationary time series will tend to return to its
  mean ( called mean reversion) and fluctuate
  around this mean.

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Stationary Stochastic Processes, continued

• Properties of stationarity
• Let Yt be a stochastic time series
• Mean
• Variance
• Covariance

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Stationary Stochastic Processes, continued

• A time series is strictly stationary if all the
  moments of its probability distribution and not
  just the first two (i.e., mean and variance) are
  invariant over time. If, however, the stationary
  process is normal, the weakly stationary
  stochastic process is also strictly stationary,
  for the normal stochastic process is fully
  specified by its two moments, the mean and
  variance.
• Nonstationary time series if a time series is
  not stationary in the sense just defined, it is
  called a nonstationary time series. In other
  words, a nonstationary time series will have a
  time-varying mean or a time-varying variance or
  both.

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Stationary Stochastic Processes, continued

• White Noise a special case of stationary
  stochastic process.
• We call a stochastic process purely random or
  white noise if it has zero mean, constant
  variance and is serially uncorrelated.
• IID identically and independently
  distributed

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3. Nonstationary stochastic process

• Random walk model (RWM) a classical example of
  nonstationary time series
• The term random walk is often compared with a
  drunkards walk. Leaving a bar, the drunkard
  moves a random distance ut at time t, and
  continuing to walk indefinitely, will eventually
  drift farther and farther away from the bar. The
  same is said about stock prices. Todays stock
  price is equal to yesterdays stock price plus a
  random shock.

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4. Unit Root Process

• If, , it becomes a RWM (without drift).
  If is in fact 1, we face what is known as the
  unit root problem, that is, a situation of
  nonstationary we already know that in this case
  the variance of Yt is not stationary. The name
  unit root is due to the fact that .
• Thus the terms nonstationarity, random walk, and
  unit root can be treated as synonymous.

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5. Spurious Regression Again

• If Y and X have unit roots then all the usual
  regression results might be misleading and
  incorrect.
• One way to guard against it is to find out if the
  time series are cointegrated.
• An is a good rule of thumb to
  suspect that the estimated regression is
  spurious.

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6.Tests of Nonstationarity

• How do we find out whether a given time series is
  stationary?
• At the informal level, weak stationarity can be
  tested by the correlogram of a given time series.
  
• At the formal level, stationarity can be checked
  by finding out if the time series contains a unit
  root.

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Tests of Nonstationarity the correlogram test

• Autocorrelation function (ACF)
• The correlogram is a graph of autocorrelation at
  various lags for a given time series
• For stationary time series, the corelogram tapers
  off quickly, whereas for nonstationary time
  series it dies off gradually.
• Q statistic
• testing the joint hypothesis that all the
  up to certain lags are simultaneously equal
  to zero.

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Tests of Nonstationarity the Unit Root Test

• The Unit Root Test
• If , then , that is we have a
  unit root, meaning the time series under
  consideration is nonstationary.

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Tests of Nonstationarity Dickey-Fuller (DF) Test

• The DF test is estimated in three different
  forms. It was assumed that the error term was
  uncorrelated.

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Dickey-Fuller Test, continued

• Suppose Yt can be described by the following
  equation
• Using OLS to run the unrestricted regression
• Using OLS to run the restricted regression
• 
  or
• Using the sums of squared residuals in the
  restricted and unrestricted regressions to
  calculate F statistic, then compare F value with
  the critical value.

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Tests of Nonstationarity the Augmented Dickey-
Fuller (ADF) Test

• ADF test is developed if the are correlated.
• Under the circumstance, the unit root test
  is run the same way as before.

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7. Cointegration

• Regressing one random walk against another can
  lead to spurious results.
• Differencing variables before using them in a
  regression may result in a loss of long-run
  information.
• Cointegration means that despite two or more time
  series follow random walks, a linear combination
  of them can be stationary. If this is the case,
  we say that these time series are co-integrated.
• The AEG,augmented Engle-Granger test,and other
  tests can be used to find out whether two or more
  time series are cointegrated.
• Cointegration of two or more time series suggests
  that there is a long-run, or equilibrium,
  relationship between them.

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Testing for Cointegration ( ADF CRDW Tests)

• Testing whether there is a co-integrated
  relationship between two time series.
• Step 1 using the ADF test to confirm that
  variables in the regression are random walks.
• Step 2 using OLS to estimate the regression
  equation
• , then tests
  whether the residuals from the above
  co-integrating regression are stationary.
• 1. Perform Augmented Dickey-Fuller unit
  root test on the residual series to see if the
  residual is stationary.
• 2. Look at the Durbin-Watson statistic
  from the above regression
• , compare DW value
  with the corresponding critical vale at proper
  confidence level to decide whether reject or
  accept the null hypothesis.

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8. Error Correction Mechanism

• Granger representation theorem if two variables
  Y and X are cointergated, then the relationship
  between the two can be expressed as ECM, the
  error correction mechanism.
• where is the error obtained from the
  regression model with Y and X (i.e.
  ) and is the error in the ECM
  model.
• The ECM says that depends on - an
  intuitively sensible point (i.e. changes in X
  cause Y to change).
• In addition, depends on . This
  latter aspects is unique to the ECM and gives it
  its name.

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Error Correction Mechanism, continued

• Remember that can be thought of an an
  equilibrium error. If it is non-zero, then the
  model is out of equilibrium. Consider the case
  where and is positive. The
  latter implies that is too high to be in
  equilibrium (i.e. is above its equilibrium
  level of ). Since the
  term will be negative and so will
  be negative. In other words, if is above
  its equilibrium level, then it will start falling
  in the next period and the equilibrium error will
  be corrected in the model hence the term
  error correct model. In the case where
  the opposite will hold (i.e. is below its
  equilibrium level, hence which causes to be
  positive, triggering Y to rise in period t).

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Error Correction Mechanism, continued

• A distinctive feature of the model is that the
  ECM has both long run and short run properties
  built into it.
• We use this error term, to tie the
  short-run behaviour of Y to its long-run
  value.The ECM developed by Engle and Granger is a
  means of reconciling the short-run behaviour of
  an economic variable with its long-run behaviour.

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9. Causality in Economics the Granger Causality
Test

• The test involves estimating the following pair
  of regressions

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Granger Causality Test, continued

• 1. Unidirectional causality from X to Y is
  indicated if the estimated coefficients on the
  lagged X in (1) are statistically different from
  zero as a group and the set of estimated
  coefficients on the lagged Y in (2) is not
  statistically different from zero
• 2. Conversely, unidirectional causality from Y to
  X exists if the set of lagged X coefficients in
  (1) is not statistically different from zero and
  the set of the lagged Y coefficients in (2) is
  statistically different from zero

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Granger Causality Test, continued

• 3. Feedback, or bilateral causality, is suggested
  when the sets of X and Y coefficients are
  statistically different from zero in both
  regression
• 4. Finally, independence is suggested when the
  sets of X and Y coefficients are not
  statistically significant in both the regression.

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Granger Causality Test, continued

• 1. Regress current Y on all lagged Y terms, but
  not include the lagged X variables, obtain the
  restricted residual sum of squares, RSSR.
• 2. Run the regression including the lagged X
  terms, obtain the unrestricted residual sum of
  squares, RSSu
• 3. The null hypothesis is , that
  is, lagged X terms do not belong in the
  regression.

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Granger Causality Test, continued

• 4. To test this hypothesis, we apply the F test
• Which follows the F distribution with m and
  (n-k) df. m is equal to the number of lagged X
  terms and k is the number of parameters estimated
  in the unrestricted regression.
• 5. If the computed F value exceeds the critical F
  value at the chose level of significance, we
  reject the null hypothesis, in which case the
  lagged X terms belong in the regression. This is
  another way of saying that X Granger-causes Y.
• 6. Steps 1 to 5 can be repeated to test model 2,
  that is, if Y Granger-causes X.