Time Series Analysis

1
Time Series Analysis

• Materials for this lecture
• Lecture 6 Lags.XLS
• Lecture 6 Stationarity.XLS
• Read Chapter 15 pages 30-37
• Read Chapter 16 Section 15

2
Time Series Analysis

• Comprehensive forecasting methodology for
  univariate distributions
• Best for variables unrelated to other variables
  in a structural model
• Autoregressive process in past will prevail
• Yt f(Yt-1, Yt-2, Yt-3, Yt-4, )
• Assumptions are
• E(Mean of ê) 0.0
• Variance of êt constant

3
Time Series Analysis

• Steps for estimating a times series model are
• Graph the data series to see what patterns are
  present trend, cycle, seasonal, and random
• Test data for Stationarity with a Dickie Fuller
  Test
• If original series is not stationary then
  difference it until it is
• Number of Differences (p) to make a series
  stationary is determined using the D-F Test
• Use the stationary (differenced) data series to
  determine the number of Lags that best forecasts
  the historical period
• Given a stationary series use the Schwarz
  Criteria (SIC) or autocorrelation table to
  determine the best number of lags (q) to include
  when estimating the model
• Estimate the AR(p,q) Model and make a forecast

4
Time Series Analysis Assumptions

• Assume series you are forecasting is random
• No trend, seasonal, or cyclical pattern remains
  in series
• Series is stationary or white noise
• To guarantee this assumption we make the data
  stationary
• Assume the variability is the same for the future
  as for the past
• s2Ti s2t
• Can test for constant variance by testing
  correlation over time
• Crucial assumption because if s2 changes
  overtime, then forecast will explode overtime

5
Make Data Series Stationary

• Take differences of the data to make it
  stationary
• First difference all T obs. of Y to calculate
  D1,t
• D1,t Yt Yt-1
• Calculate first difference of D1,t for a Second
  Difference of Y or
• D2,t D1,t D1,t-1
• Calculate first difference of D2,t for a Third
  difference of Y or
• D3,t D2,t D2,t-1
• Stop differencing data when series is stationary

6
Make Data Series Stationary

• Example Difference table for a time series data
  set
• D1t in period 1 is 0.41 71.47-71.06
• D2t in period 1 is -1.82 -1.41 0.41
• Etc.

7
Test for Stationarity

• Dickie-Fuller Test for stationarity
• First D-F test -- is original data stationary?
• Estimate regression Y a b D1,t
• Calculate the t ratio for b to see if a
  significant slope exists
• D-F Test statistic is the t-statistic more
  negative than -2.9, we declare the dependent
  series (Y in this case) stationary at alpha 0.05
  level
• Thus D-F statistic of -3.0 is more negative than
  -2.90 so the original series is stationary and
  differencing is not necessary

8
Test for Stationarity

• Second D-F Test for stationarity of the D1,t
  series, in other words will the series be
  stationary with one differencing?
• Estimate regression for D1,t a b D2,t
• t statistic on slope b is the second D-F test
  statistic
• Third D-F Test for stationarity of the D2,t
  series, in other words will the series be
  stationary with two differences?
• Estimate regression for D2,t a b D3,t
• t statistic on slope b is the third D-F test
  statistic
• Continue with a fourth and fifth DF test if
  necessary

9
Test for Stationarity

• How does D-F test work?
• If a series is stationary, it oscillates about
  its own mean and has a slope of zero
• If a 1 period lag Yt-1 is used to predict Yt ,
  they are good predictors of each other because
  they have the same mean. The t-statistic on b
  will be significant for D1,t a
  b D2,t
• Lagging the data 1 period causes the two series
  to be inversely related so the slope is negative
  in the OLS equation. That explains why the t
  coefficient of -2.9 is negative. The -2.9
  represents a 5 1 tail test statistic.

10
Test for Stationarity

• Estimated regression for Y a b D1,t
• DF is -1.868 the t ratio
• Intercept not zero so mean of Y is not constant

11
Test for Stationarity

• Estimated regression for D1,t a b D2,t
• DF is -12.948 the t ratio
• Intercept is 0.121 or about zero, so the mean
  is more likely to be constant

12
Test for Stationarity

• Estimated regression for D2,t a b D3,t
• DF is -24.967 the t ratio
• Intercept is about zero

13
Test for Stationarity

• Dickie Fuller test in Simetar
• DF ( Data Series, Trend, No Lags, No. Diff )
• Where Data series is the location of the data,
• Trend is True or False
• No. Lags is optional if data are to be
  lagged
• No. Diff is the number of differences to
  test

14
Test for Stationarity

• The number of differences that make the data
  series stationary is the one that has a DF test
  stat more negative than -2.9
• Here it is 1 difference no matter if trend
  included or not

Lecture 5
15
Summarize Stationarity

• Yt is the original data series
• Di,t is the ith difference of the Yt series
• We difference the data to make it stationary
• This guarantees the assumption that mean is
  constant
• Dickie Fuller test is used to determine the Di,t
  difference needed to make series stationary
• DF(Data, Trend, , No. of Differences)
• Test 0 to 6 differences with and without trend
  and zero lags
• Select the difference with a DF test stat. More
  negative than -2.90

16
Test for Stationarity

• Once we have determined the number of differences
  we know what form the auto regressive (AR) model
  will be fit
• AR ( No. Differences, No. Lags) or AR(p,q)
• If the series is stationary with 1 difference we
  will fit the model
• D1,t a b1 D1,t-1 b2 D1,t-2
• The only question that remains is how many lags
  (q) of Di,t will we need to forecast the series
• To determine the number of lags we use several
  tests

17
Number of Lags for TS Model

• Partial autocorrelation coefficients used to
  estimate number of lags of Di,t to include in
  model. Assume D1,t is stationary then
• To test for one lag use b1 from regression model
• D1,t a b1 D1,t-1
• To test for two lags use b2 from regression model
• D1,t a b1 D1,t-2 b2 D1,t-2
• To test for three lags use b3 from regression
  model
• D1,t a b1 D1,t-1 b2 D1,t-2 b3 D1,t-3
• After each regression we only use the beta (bi)
  for the longest number of lags.
• Use the t stat on the bi to determine
  contribution of the lags to explain D1t
• So this calls for running lots of regressions
  with different numbers of lags, right?

18
Number of Lags for TS Model

• Find the optimal number of lags the easy way
• Use Simetar to build a SAC table
• AUTOCORR(Data Series, No. Lags, No. Diff)
• Pick best no. of lags based on the last lag with
  a statistically significant t value

19
Number of Lags for TS Model

• Bar chart of autocorrelation coefficients in
  AUTOCORR(Data Series, No. Lags, No. Diff)
• The explanatory power of the distant lags is not
  large enough to warrant including in the model,
  based on their t stats, so do not include them.

20
Autocorrelation Charts of Sample Autocorrelation
Coefficients (SAC)
21
Note Partial vs. Sample Autocorrelation

• Partial autocorrelation coefficients (PAC) show
  the contribution of adding one more lag.
• It takes into consideration the impacts of lower
  order lags
• A beta for the 3rd lag shows the contribution of
  3rd lag after having lags 1-2 in place
• D1,t a b1 D1,t-1 b2 D1,t-2 b3 D1,t-3
• Sample autocorrelation coefficients (SAC) show
  contribution of adding a particular lag.
• A SAC for 3 lags shows the contribution of just
  the 3rd lag.
• Thus the SAC does not equal the PAC

22
Number of Lags for Time Series Model

• Some authors suggest using SAC to determine the
  number of differences to achieve stationarity
• If the SAC cuts off or dies down rapidly it is an
  indicator that the series is stationary
• If the SAC dies down very slowly, the series is
  not stationary
• This is a good check of the DF test, but we will
  rely on the DF test for stationarity

23
Number of Lags for Time Series Model

• Schwarz Criteria (SIC) tests for the number of
  lags in a TS model
• Goal is to find the number of lags which
  minimizes the SIC
• In Simetar use the ARLAG() function which returns
  the optimal number of lags based on SIC
• ARLAG(Data Series, No. of Diff, No. of Lags)

24
Schwarz Criteria and ARLAG Tables
ARLAG indicates no. of lags to minimize SC 1
lag, given the no. of differences.
Pick the no. of lags to Minimize the SC 1 lag
Autocorr finds no. of lags where AC drops 1 lag
25
Summarize Stationarity/Lag Determination

• Make the data series stationary by differencing
  the data
• Use the Dickie-Fuller Test (df lt -2.90) to find
  Di series that is stationary
• Use the DF() function in Simetar
• Determine the number of Lags for the Di series
• Use the sample autocorrelation coefficients for
  alternative lag models
• AUTOCORR() function in Simetar
• Minimize the Schwarz Criteria
• ARLAG() or ARSCHWARZ() functions in Simetar