1. Time Series Analysis Lecture 11
Materials for this lecture
Lecture 11 Lags.XLSX
Lecture 11 Stationarity.XLSX
Read Chapter 15 pages 30-37
Read Chapter 16 Section 15

2. Forecast a variable based only on its past values
Time Series Analysis
Forecast a variable based only on its past values
Called autoregressive models
We are going to focus on the application and less on the estimation calculations because they are just simply OLS
Simetar estimates TS models easily with a menu and provides forecasts of the time series model

3. Time Series Analysis
TS is a comprehensive forecasting methodology for univariate distributions
Best for variables unrelated to other variables in a structural model
Autoregressive process in the simplest form is a regression model such as:
Ŷt = f(Yt-1, Yt-2, Yt-3, Yt-4, … )

4. 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 using the Chain Rule

5. Two Time Series Analysis Assumptions
Assume series you are forecasting is really random
No trend, seasonal, or cyclical pattern remains in series
Series is stationary or “white noise”
To guarantee this assumption we must make the data stationary
Assume the variability is constant, i.e., the same for the future as for the past, in other words
σ2T+i = σ2t
Can test for constant variance by testing correlation over time
Crucial assumption because if σ2 changes overtime, then forecast will explode overtime

6. First Make the Data Series Stationary
Take differences of the data to make it stationary
First difference of raw data in 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

7. Make Data Series Stationary
Example the Difference table for a time series data set
D1t in period 1 is 0.41= – 71.06
D1t in period 2 is -1.41= – 71.47
D2t in period 1 is = – 0.41
D2t in period 2 is 1.66 = 0.25 – (– 1.41)
Etc.

8. Make Data Series Stationary
Example the Difference table for a time series data set
D1t in period 1 is 0.41= – 71.06
D1t in period 2 is -1.41= – 71.47
D2t in period 1 is = – 0.41
D2t in period 2 is 1.66 = 0.25 – (– 1.41)
Etc.

9. Dickie-Fuller Test for stationarity
First D-F test First is original data stationary?
Estimate regression Ŷ = 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; if more negative than -2.9, we declare the dependent series (Ŷ in this case) stationary at alpha 0.05 level (-2.90 is a 1 tailed t test critical value)
Thus a D-F statistic of -3.0 is more negative than so the original series is stationary and differencing is not necessary

10. Next Level of Testing for Stationarity
Second D-F Test for stationarity of the D1,t series, in other words
Here we are testing if the Y series will be stationary with one differencing?
Or we are asking if the D1,t series is stationary
Estimate regression for D1,t = a + b D2,t
t-statistic on slope b is the second D-F test statistic

11. Next Level of Testing for Stationarity
Third D-F Test for stationarity of the D2,t series
In other words will the Y series be stationary with two differences?
So we are testing if D2,t is stationary
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

12. Test for Stationarity How does D-F test work?
A series is stationary, if it oscillates about the 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.

13. Test for Stationarity
Estimated regression for: Ŷt = a + b D1,t
DF is You can see that by examining the t ratio on slope
See a trend in original data and no trend in the residuals
Intercept not zero so mean of Y is not constant

14. Test for Stationarity
Estimated regression for: D1,t = a + b D2,t
DF is , which is the t ratio on the slope parameter “b”
See the residuals oscillate about a mean of zero, no trend in either series
Intercept is or about zero, so the mean is more likely to be constant

15. Test for Stationarity DF is -24.967 the t ratio on the slope variable
Estimated regression for: D2,t = a + b D3,t
DF is the t ratio on the slope variable
Intercept is about zero

16. DF Stationarity Test in Simetar
Dickie Fuller (DF) function in Simetar
=DF ( Data Series, Trend, No Lags, No. Diff to Test)
Where: Data series is the location of the data,
Trend is True or False
No. Lags is optional, set to ZERO
No. Diff is the number of differences to test

17. Test for Stationarity
Number of differences that make the data series stationary is the one that has a DF test statistic more negative than -2.9
Here it is 1 difference no matter if trend included or not
Lecture 5

18. 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
To guarantee the assumption that the mean is constant
Dickie Fuller test in Simetar to determine the no. of differences needed to make series stationary
=DF(Data, Trend, , No. of Differences)
Test 0 to 6 differences with and without trend and zero lags using =DF()
Select the first difference with a DF test statistic more negative than -2.90

19. NEXT We Determine the Number of Lags
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, estimate the OLS model
D1,t = a + b1 D1,t-1 + b2 D1,t-2 + …
The only question that remains is how many lags (q) of D1,t will we need to forecast the series
To determine the number of lags we use several tests

20. Number of Lags for AR(p,q) Model
Partial autocorrelation coefficients used to estimate number of lags for Di,t in model.
If D1,t is stationary then:
Test for one lag use b1 from OLS regression model
D1,t = a + b1 D1,t-1
Test for two lags use b2 from OLS regression model
D1,t = a + b1 D1,t-1 + b2 D1,t-2
Test for three lags use b3 from OLS 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, i.e., the bold ones above
Use the t stat on the bi to determine contribution of the last lag to explaining D1,t
So this calls for running lots of regressions with different numbers of lags, right?

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

22. Number of Lags for Time Series Model
Bar chart of autocorrelation coefficients in Sample AUTOCORR() Table
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.

23. Autocorrelation Charts of Sample Autocorrelation Coefficients (SAC)
Dying down in a dampened exponential fashion – oscillation 1 -1
Lag k
Dying down quickly
Dying down extremely slowly 1 -1
Lag k
A SAC that cuts off
Dying down in a dampened exponential fashion – no oscillation

24. Number of Lags for Time Series Model
Schwarz Criteria (SIC) tests for the optimal number of lags in a TS model
Objective 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 test
=ARLAG(Data Series, Constant, No. of Differences)

25. 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

26. Summarize Stationarity/Lag Determination
Make the data series stationary by differencing the data
Use the Dickie-Fuller Test (df < -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

27. 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

28. 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