Autocorrelation in Time Series data

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Autocorrelation in Time Series data

• KNN Ch. 12 (pp. 481-501)

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Quantitative Forecasting
Quantitative
Forecasting
Causal
Time Series
Models
Models
Exponential
Trend
Moving
Regression
Smoothing
Average
Models
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Time Series vs. Cross Sectional Data

• Time series data is a sequence of observations
• collected from a process
• with equally spaced periods of time.
• Contrary to restrictions placed on
  cross-sectional data, the major purpose of
  forecasting with time series is to extrapolate
  beyond the range of the explanatory variables.

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Autoregressive Forecasting
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Regression Model with AR(1) error

• The errors ut are independent and normally
  distributed N(0, s2)
• The autoregressive parameter r has r lt 1

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Multiple Regression Model with AR(1) error

• The previous simple regression model can be
  expanded to accommodate multiple predictors

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Autoregressive expansion

• The autocorrelation parameter r is the
  correlation coefficient between adjacent error
  terms
• Expanding the definition of et,

Random error component
Autoregressive component
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Autoregressive expansion

• The correlation coefficient diminishes over
  time, since r lt 1
• This is why an ACF plot exhibits a diminishing
  correlation pattern for AR(1) models

ACF
PACF
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Remedial measures for AR errors in regression
models

• Cochrane Orcutt procedure
• Hildreth Lu procedure
• First differences procedure
• All estimates are close to each other, the last
  procedure is the simplest

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First Differences procedure
(regression through the origin)
Back transformations
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The Blaisdell Company Example (Blaisdell.xls)
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The Blaisdell Company Example
(regression through the origin)
Back transformations
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Forecasting

• Forecasts obtained with autoregressive error
  regression models are conditional on the past
  observations
• Using recursive relations, two or three-step
  ahead forecasts can be obtained, but prediction
  intervals will expand very fast