Univariate Time Series

1
Univariate Time Series

• Stationary and
• Nonstationary Series

2
Strictly Stationary Processes

• Issues relating to the stationarity of a time
  series are important because they help us
  understand the behavior and properties of a
  series.
• A series that is not stationary has some
  undesirable properties that make hypothesis
  testing using standard techniques incorrect.
• A series is strictly stationary if the
  distribution of its values remains the same as
  time progresses the probability that y falls
  within a particular interval is the same now as
  at any time in the past or future.

3
A Weakly Stationary Series

• If a series satisfies the following conditions
  for t1, 2, 3, , 8 it is said to be weakly or
  covariance stationary
• 1 means that the series has constant mean
• 2 means that the series has constant variance
• 3 means that the series has constant
  autocovariance

4
Autocovariance

• Autocovariance determines how y is related to its
  previous values. For a (strictly or weakly)
  stationary series they depend only on the
  difference between t1 and t2 so that the
  covariance between y1 and yt-1 is the same as the
  covariance between yt-10 and yt-11
• The moment
• Is the autocovariance function. When s0 the
  autocovariance at lag zero is obtained
  (covariance of yt and yt).

5
Autocorrelation

• Recall that the covariance is not that useful of
  a measure as it depends on the units of
  measurement of y.
• Autocorrelation covariances normalized by
  dividing by the variance
• The series t0 has the standard properties of
  correlation coefficients it is bounded by 1
• If we plot ts against s we get the
  autocorrelation function (acf) or correlogram.

6
Autocorrelation of tbill.ac tbill
7
Autocorrelation of grow.ac grow
8
A White Noise Process

• Definition a white noise process is one with on
  discernible structure. Formally
• A white noise process has constant mean and
  variance and zero autocovariances (except at lag
  zero)

9
Generating White Noise in STATA

• set obs 100
• gen obs_n
• tsset obs
• gen einvnorm(uniform())
• tsline e
• ac e

10
Testing Autocorrelations

• If the process that generates yt has a standard
  normal distribution then the sample
  autocorrelation coefficients are also distributed
  normally.
• This lets us construct confidence intervals for
  autocorrelations (see ac graphs above)
• It is also possible to test the joint hypothesis
  that all m of the tk correlations coefficients
  are simultaneously equal to zero using the Q
  statistic developed by Box and Pierce (1970)

11

• BP Q-statistic
• The correlation coefficient is squared so that
  the positive and negative coefficients do not
  cancel each other out. The Q-statistic is
  asymptotically distributed as ?2 with df equal to
  the number of squares in the sum (m).
• However, the Box-Pierce test has small sample
  properties leading to incorrect inferences in
  small samples.

12
Ljung-Box (1978) Q

• Ljung-Box Q statistic
• Asymptotically the (T2) and (T-k) terms cancel
  each other out so that the LB formulation is
  equivalent to the BP test

13
Q Statistics in STATA

• . corrgram grow, lag(20) /default is 40 lags/
• -1
  0 1 -1 0 1
• LAG AC PAC Q ProbgtQ
  Autocorrelation Partial Autocor
• --------------------------------------------------
  -----------------------------
• 1 0.8342 0.8346 146.84 0.0000
  ------ ------
• 2 0.5408 -0.5108 208.84 0.0000
  ---- ----
• 3 0.2059 -0.2212 217.88 0.0000
  - -
• 4 -0.0894 -0.0723 219.59 0.0000
  
• 5 -0.2101 0.3722 229.09 0.0000
  - --
• 6 -0.2225 -0.1546 239.79 0.0000
  - -
• 7 -0.1648 -0.0888 245.69 0.0000
  -
• 8 -0.0756 -0.0107 246.94 0.0000
  
• 9 -0.0098 0.1056 246.96 0.0000
  
• 10 0.0004 -0.2159 246.96 0.0000
  -
• 11 -0.0420 -0.1205 247.36 0.0000
  
• 12 -0.1234 -0.0698 250.75 0.0000
  
• 13 -0.1668 0.3070 256.98 0.0000
  - --
• 14 -0.1600 -0.1141 262.75 0.0000
  -

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• .wntestq grow, lag(20)
• Portmanteau test for white noise
• ---------------------------------------
• Portmanteau (Q) statistic 246.9643
• Prob gt chi2(10) 0.0000

15

• . corrgram e, lag(20) /simulation example/
• -1
  0 1 -1 0 1
• LAG AC PAC Q ProbgtQ
  Autocorrelation Partial Autocor
• --------------------------------------------------
  -----------------------------
• 1 0.0737 0.0746 .56021 0.4542
  
• 2 0.0512 0.0467 .83278 0.6594
  
• 3 -0.0023 -0.0091 .83331 0.8415
  
• 4 -0.0283 -0.0312 .91835 0.9219
  
• 5 0.0833 0.0954 1.664 0.8934
  
• 6 0.0347 0.0277 1.7949 0.9376
  
• 7 0.0050 -0.0143 1.7977 0.9702
  
• 8 -0.0959 -0.1036 2.8165 0.9453
  
• 9 -0.0205 0.0087 2.8635 0.9695
  
• 10 -0.0627 -0.0533 3.3091 0.9732
  
• 11 -0.1578 -0.1807 6.1612 0.8624
  - -
• 12 -0.1765 -0.1834 9.7712 0.6360
  - -
• 13 -0.1144 -0.0809 11.307 0.5851
  
• 14 -0.0179 -0.0031 11.345 0.6588
  

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Stationarity and unit root testing

• Focus on tests for weak stationarity. Recall
  that weak stationarity requires constant mean,
  constant variance and constant autocovariance for
  each lag
• Why is stationarity important?
• The stationarity of a series can influence its
  behavior. For example, we use the word shock
  to denote an unexpected change in the value of a
  variable (or error). For a stationary series a
  shock will gradually die away. That is, the
  effect of a shock during time t will have a
  smaller effect in time t1, a smaller effect in
  time t2, etc.
• A shock to a non-stationary series will be
  infinite

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• The use of non-stationary data can lead to
  spurious regressions. If two stationary series
  are generated as independent random series then
  when we regress one on the other we expect the
  t-statistic on the slope to not be different from
  zero and the value of R2 to be zero.
• This may seem obvious but if two series are
  trending over time then a regression of one on
  another could have a high R2 and a t-statistic
  that is different from zero even if the two
  series are totally unrelated

18
Two Types of Non-Stationarity

• Random Walk with a Drift
• Trend-stationary process (stationary around a
  trend)

19

• Note that the random walk model can be
  generalized to the case where yt is an explosive
  process.
• Where ?gt1. In practice this case is often
  ignored because there are few series where this
  condition is satisfied. The two more relevant
  cases are ?1 and ?lt1.

20
Random Walk

• yt yt-1 ut
• yt .95 yt-1 ut

21
Random Walk

• yt .5 yt-1 ut
• yt .1 yt-1 ut

22
Explosive Case

• yt 1.1 yt-1 ut

23
Trend-Stationary Case

• Recall
• We could run a regression on this model and
  obtain the residuals which would have the linear
  trend removed.

24

• set obs 100
• gen t_n
• gen einvnorm(uniform())
• tsset t
• gen y2te
• tsline y
• reg y t
• predict r, resid
• tsline r

25
Differencing

• If a non-stationary series, yt must be
  differenced d times before it becomes stationary
  then it is said to be integrated of order d.
• This is written ytI(d)
• If ytI(d) then ?dytI(0). This means that
  applying the difference operator (?) d times
  leads to an I(0) process a process with no unit
  roots.

26
Testing for Unit Roots

• One approach would be to examine the
  autocorrelation function of the series of
  interest. This would be incorrect, however,
  because the ACF for a unit root process (a random
  walk) will look like it decays slowly to zero.
  Thus it may be mistaken for a highly persistent
  but persistent process.
• Thus ACFs cannot be used to test for unit roots.

27
Dickey Fuller Test

• Basic idea is to test ?1 in
• Against the one-sided alternative ?lt1 . The
  hypotheses of interest are
• H0 series contains a unit root
• HA series is stationary.
• In practice we estimate
• so that a test of ?1 is equivalent to a test of
  ?0

28
Dickey Fuller Tests in Stata

• . dfuller grow
• Dickey-Fuller test for unit root
  Number of obs 207
• ----------
  Interpolated Dickey-Fuller ---------
• Test 1 Critical
  5 Critical 10 Critical
• Statistic Value
  Value Value
• --------------------------------------------------
  ----------------------------
• Z(t) -4.296 -3.474
  -2.883 -2.573
• --------------------------------------------------
  ----------------------------
• MacKinnon approximate p-value for Z(t) 0.0005
• . dfuller tbill
• Dickey-Fuller test for unit root
  Number of obs 207
• ----------
  Interpolated Dickey-Fuller ---------
• Test 1 Critical
  5 Critical 10 Critical
• Statistic Value
  Value Value

29
Phillips-Perron Tests

• Just like Dickey-Fuller tests but they include an
  automatic correction to the DF procedure to allow
  for autocorrelated residuals. The tests often
  give similar results.
• . pperron tbill
• Phillips-Perron test for unit root
  Number of obs 207
• 
  Newey-West lags 4
• ----------
  Interpolated Dickey-Fuller ---------
• Test 1 Critical
  5 Critical 10 Critical
• Statistic Value
  Value Value
• --------------------------------------------------
  ----------------------------
• Z(rho) -8.741 -20.157
  -13.914 -11.143
• Z(t) -2.227 -3.474
  -2.883 -2.573
• --------------------------------------------------
  ----------------------------
• MacKinnon approximate p-value for Z(t) 0.1965

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Criticism of DF and PP Tests

• The most important criticism is that their power
  is low if a process is stationary but with a root
  close to the non-stationary boundary (for example
  ?.95).
• Alternative is the KPSS test

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kpss is a user written do file

• .kpss grow
• KPSS test for grow
• Maxlag 14 chosen by Schwert criterion
• Autocovariances weighted by Bartlett kernel
• Critical values for H0 grow is trend stationary
• 10 0.119 5 0.146 2.5 0.176 1 0.216
• Lag order Test statistic
• 0 .113
• 1 .0619
• 2 .046
• 3 .0393
• 4 .0367
• 5 .036
• 6 .0363

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Next Topics

• Using time series in regression
• Instrumental variables
• Vector Autoregression
• Causality