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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

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

- Weekly data:
- Short runs
- Autocorrelation
- A slight drop in the mean for the 2nd year (53-104 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

Significant

- Box-Pierce test: no autocorrelation

-sample ACF & PACF: confirm also

Fitted values: smooth out the highs and lows in the data

- 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

Residuals plot: changing variance

- Violates the constant variance assumption

Significant

- Box-Pierce test: no autocorrelation

-sample ACF & PACF: confirm also

-level of the first difference remains the same

Sample ACF & PACF plots: first difference white noise

RANDOM WALK MODEL, ARIMA(0,1,0)

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

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

Variance of the forecast errors: bigger with increasing forecast lead times

Prediction Intervals

100(1-a) percent prediction interval

2problems

- 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

ARIMA(1,1,1) process

1. Infinite MA representation of the model

forecast

weights

Random shocks

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.

SEASONAL PROCESS that repeats over S time periods, where S defines the number of time periods until the pattern repeats again.

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

ARIMA model ( that repeats over S time periods, where S defines the number of time periods until the pattern repeats again.p,d,q)x(P,D,Q) with period s

Example

Example that repeats over S time periods, where S defines the number of time periods until the pattern repeats again.

- 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

ARIMA(0,1,1)x(0,1,1) that repeats over S time periods, where S defines the number of time periods until the pattern repeats again.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

ARIMA(0,1,1)x(0,1,1) that repeats over S time periods, where S defines the number of time periods until the pattern repeats again.12 : reasonable fit

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