1. Time Series Building 1. Model Identification
Simple time series plot : preliminary assessment tool for stationarity
Non stationarity: consider time series plot of first (dth) differences
Unit root test of Dickey & Fuller : check for the need of differencing
Sample ACF & PACF plots of the original time series or dth differences
20-25 sample autocorrelations sufficient
2. Parameter Estimation
- Methods of moments, maximum likelihood, least squares
3. Diagnostic checking- check for adequacy
Residual analysis:
Residuals behave like a white noise
Autocorrelation function of the residuals re(k) not differ significantly from zero - check the first k residual autocorrelations together

2. Example
May exist some autocorrelation between the number of applications in the current week & the number of loan applications in the previous weeks

3. Weekly data:
Short runs
Autocorrelation
A slight drop in the mean for the 2nd year ( weeks) :
Safe to assume stationarity
Sample ACF plot:
Cuts off after lag 2 (or 3)
MA(2) or MA(3) model
- Exponential decay pattern
AR(p) model
Sample PACF plot:
Cuts off after lag 2
Use AR(2) model

4. - Parameter estimates:
Significant
- Box-Pierce test: no autocorrelation
-sample ACF & PACF: confirm also

5. Acceptable fit
Fitted values: smooth out the highs and lows in the data

6. Example Signs of non-stationarity: changing mean & variance
Sample ACF: slow decreasing
Sample PACF: significant value at lag 1, close to 1
Significant sample PACF value at lag 1: AR(1) model

7. Residuals plot: changing variance
- Violates the constant variance assumption

8. - AR(1) model coefficient:
Significant
- Box-Pierce test: no autocorrelation
-sample ACF & PACF: confirm also

9. Plot of the first difference:
-level of the first difference remains the same
Sample ACF & PACF plots: first difference white noise
RANDOM WALK MODEL, ARIMA(0,1,0)

10. Decide between the 2 models AR(1) & ARIMA(0,1,0)
Use specific criteria
Use subjective matter /process knowledge
Do we expect a financial index as the Dow Jones Index to be around a fixed mean, as implied by AR(1)?
ARIMA(0,1,0) takes into account the inherent non stationarity of the process
However
A random walk model means the price changes are random and cannot be predicted
Not reliable and effective forecasting model
Random walks models for financial data

11. Forecast ARIMA process
How to obtain Best forecast
The mean square error for which the expected value of the squared forecast errors is minimized
Best forecast in the mean square sense
Forecast error

12. 100(1-a) percent prediction interval
Mean & variance
Variance of the forecast errors: bigger with increasing forecast lead times
Prediction Intervals
100(1-a) percent prediction interval

13. Need to know the magnitude of random shocks in the past
2 problems
Infinite many terms in the past, in practice have finite number of data
Need to know the magnitude of random shocks in the past
Estimate past random shocks through one-step forecasts
ARIMA model

14. forecast
ARIMA(1,1,1) process
1. Infinite MA representation of the model
forecast
weights
Random shocks

15. 2. Use difference equations

16. Seasonality in a time series is a regular pattern of changes that repeats over S time periods, where S defines the number of time periods until the pattern repeats again.

17. SEASONAL PROCESS Seasonal ARMA model
St deterministic with periodicity s
Nt stochastic, modeled ARMA
wt seasonally stationary process
Model Nt with ARMA
becomes stationary
Assume after seasonal differencing
Seasonal ARMA model

18. ARIMA model (p,d,q)x(P,D,Q) with period s
Example

19. Example Monthly seasonality, s=12:
ACF values at lags 12, 24,36 are significant & slowly decreasing
PACF values: lag 12 significant value close to 1
Non-stationarity: slowly decreasing ACF
Remedy of non-stationarity: take first differences & seasonal differencing
Eliminates seasonality
stationary
Sample ACF: significant value at lag 1
Sample PACF : exponentially decaying values at the first 8 lags
Use non seasonal MA(1) model
Remaining seasonality:
Sample ACF: significant value at lag 12
Sample PACF : lags 12, 24,36 alternate in sign
Use seasonal MA(1) model

20. ARIMA(0,1,1)x(0,1,1)12
Coefficient estimates: MA(1) & seasonal MA(1) significant
Sample ACF & PACF plots: still some significant values
Box-Pierce statistic: most of the autocorrelation is out

21. ARIMA(0,1,1)x(0,1,1)12 : reasonable fit