UNIVARIATE ARIMA MODELS

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UNIVARIATE ARIMA MODELS

• AUTOREGRESSIVE INTEGRATED MOVING AVERAGE MODELS
• BOX JENKINS MODEL

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• A technique that tries to model the underlying
  process (stochastic) of the time series
• Forecasting by concentrating only on the past
  patterns of the time series
• Find a model that accurately represents the past
  and future patterns of a time series

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• Yt Pattern et
• The pattern can be random, seasonal, trend,
  cyclical or a combination of patterns

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• MODEL
• Similar to a machine that takes the observed
  time series and turns them into forecasts and
  white noise errors
• Actual series ? Machine (Black Box)? Accurate
  Forecast and White Noise residuals
• AIM OF ARIMA DESIGN THE RIGHT MACHINE (IDENTIFY
  THE RIGHT PATTERN)

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• CONFIRMATION THAT CORRECT PATTERN HAS BEEN
  IDENTIFIED REQUIRES WHITE NOISE RESIDUALS
• Meaning
• ERRORS normally and independently distributed
  (NID)
• Errors have no pattern ? cannot be predicted
  using past values
• Errors have zero mean

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• Steps in ARIMA modeling
• Model Identification
• Use graphs, statistics, ACF, PACF, etc. to
  identify pattern and model components
• 2. Parameter Estimation
• Determine model coefficients through software
  applications

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• Model Diagnostics
• Use graphs, statistics, ACF, PACF of residuals to
  determine if the model is valid.
• If valid, then use the model. If not, repeat 1,
  2 and 3 again
• 4. Forecast

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• Breaking Down ARIMA
• Model 1
• AUTOREGRESSIVE MODEL AR(p)
• The time series is predicted using its own
  previous values previous values
• Yt ß1Yt-1 ß2Yt-2 ..ßpYt-p et

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• AR(p) Autoregressive of order p
• i.e. using p past periods to forecast
• AR(1)
• Yt ß1Yt-1 et
• How do we determine p?
• Check ACF and PACF

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• ACF should decline rapidly to insignificant
  values (tend to decrease toward zero)
• The number of statistically significant spikes in
  PACF is your p (order of autoregression)

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• MOVING AVERAGE MODEL MA(q)
• Model that predict time series based on past
  forecast errors.
• Yt et d1et-1 d 2Yt-2 .. dqYt-q et
• Similar to weighted average model
• q order of MA

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• How do you determine q?
• If PACF gradually declines to zero, then the
  number of significant spikes in ACF q

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• AUTOREGRESSIVE MOVING AVERAGE MODELS ARMA(p,q)
• Predicting time series using past values of the
  series and past values of the forecast errors
• Mixed model
• Yt ß1Yt-1 ß2Yt-2 ..ßpYt-p d1et-1
• d2Yt-2 .. dqYt-q et

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• To define an ARMA model, Check ACF and PACF
• Both ACF and PACF should gradually fall to Zero
• Count the number of AR and MA terms significantly
  different from zero
• For AR count PACF, for MA count ACF

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• STATIONARITY
• Autocorrelations pattern dominate non-stationary
  series.
• So to determine the correct model and pattern,
  wed have to stationarize the data
• One of the ways to achieve stationarity is
  differencing

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• Sometimes wed have to find 2nd differences to
  stationarize
• Number of differences order of Integration d
• When we use differencing to achieve stationarity
  the resulting model ARIMA(p,d,q)
• Seasonal differencing ARIMA(p,d,q) (p,d,q)

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DIAGNOSTIC CHECK OF MODEL

• To check if model is adequate (valid), check if
  the errors generated by the model are white noise
  by
• 1. Create ACF for the residuals and check if
  they have any significant spikes. If white
  noise? no significant spikes

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• Check the Ljung-Box Q-statistics.
• This is a chi-square test on the residuals with
  m-p-q degrees of freedom
• m number of time lags to be tested
• Calculated Q-statistics is compared to chi-square
  value from tables.
• If calculated Q is less than table value ? errors
  are white noise

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• If ACF plot of residual or Q-statistics shows
  that the errors are not white noise ? model must
  be redefined.
• If two models yield white noise errors, pick one
  with the lowest AIC or BIC
• AIC Akaike Information Criterion
• BIC Bayesian Information Criterion

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• AIC n ln(SSE) 2k
• K of parameters that are fitted in the model
• SSE sum of the squared errors
• n number of observations in series
• BIC n ln(SSE) k ln(n)

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• You cannot compare the AIC or BIC of one series
  with another series
• You cannot compare AIC to BIC