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Time Series Building

-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

1. Model Identification

2. Parameter Estimation- Methods of moments, maximum likelihood, least squares

3. Diagnostic checking- check for adequacy

-Residual analysis:

Residuals behave like a white noiseAutocorrelation function of the residuals re(k) not differ significantly from zero- check the first k residual autocorrelations together

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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 (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 patternAR(p) model

Sample PACF plot: -Cuts off after lag 2Use AR(2) model

4

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

42.0ˆ&27.0ˆ21

5

Acceptable fit

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

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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

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Residuals plot: changing variance- Violates the constant variance assumption

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

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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)

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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

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Forecast ARIMA process

How to obtain Best forecast

The mean square error for which the expected value of the squared forecast errors is minimized

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Best forecast in the mean square sense

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Forecast error

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

Prediction Intervals

Mean & variance

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100(1-a) percent prediction interval

2

ˆ aT zTy

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iiTiTTTT yyyETy ,/ˆ 1, 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

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ARIMA(1,1,1) process

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1. Infinite MA representation of the model

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weights

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Random shocks 1ˆ1 Tyye TTT

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2. Use difference equations

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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.

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SEASONAL PROCESS

ttt NSy St deterministic with periodicity s Nt stochastic, modeled ARMA

01 ts

stt SBSS

ts

ts

ts NBSByB 111

wt seasonally stationary process

ts

t BBwB 1Model Nt with ARMA

Assume after seasonal differencing ts

t yBw 1 becomes stationary

Seasonal ARMA model

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ss BBBwBBB *2*2

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ts

t NBw 1

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Example

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

ts

tDsds BByBBBB ** 11

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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

tt yBBw 1211

Sample ACF: significant value at lag 1Sample PACF : exponentially decaying values at the first 8 lagsUse non seasonal MA(1) model

Remaining seasonality: Sample ACF: significant value at lag 12Sample PACF : lags 12, 24,36 alternate in signUse seasonal MA(1) model

Example

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Coefficient estimates: MA(1) & seasonal MA(1) significantSample ACF & PACF plots: still some significant valuesBox-Pierce statistic: most of the autocorrelation is out

ARIMA(0,1,1)x(0,1,1)12

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