Forecasting Meandering No Trend Series: Random Walk

1
Forecasting Meandering (No Trend) Series Random
Walk Auto-regressive Models

• Topics
• Characteristics of Random Walk
• Forecasting DJIA with Random Walk
• Characteristics of Auto-Regressive
• Distinguishing Between Random Walk and
  Auto-Regressive
• Forecasting with Auto-Regressive Model

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Random Walk Model

• Typically stock price time series data are NOT
  random. But its differences - that is the
  changes from one period to the next - are random
• Use random walk model to forecast this type of
  time series

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Random Walk Model Forecast Equation

• K-Step-Ahead Forecast
• Where the mean differences in the series
  up until time t

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Random Walk Model Standard Error

• For K-Step-Ahead Forecast
• Where the Standard dev. of differences in
  the series up until time t

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Problem Scenario

• A financial analyst would like to forecast the
  closing value of the DJIA on March 5th in a
  given year based on daily closing values for the
  previous year Feb. Feb.
• Is the random walk model appropriate?

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DJIA Time Series Plot Check for Randomness
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Runs Test on DJIA Time Series Reject Randomness
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Correlogram for DJIA Time Series Reject
Randomness
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Time Series Plot of Differences Check for
Randomness
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Runs Test DJIA Differences Randomness Doubtful
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Correlogram for Differences Cannot reject
Randomness
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Forecast for 5th Trading Day
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Forecast for 5th Trading Day

• 95 probability interval for forecast of closing
  index on 5th trading day 10,836 2157
  10,522 to 11,150

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Autoregressive Model Description

• Forecast value depends on lagged values of the
  series, Yt
• Can have lag1, lag2, lag3lagk
• Ft a bYt-1 cYt-2 dYt-3 e
• Regression based model

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Autoregressive Models When Applicable

• High autocorrelation between Yt and first one to
  three lagged series
• Autocorrelation quickly dies away as k increases
  (unlike situations suitable for random walk
  model)
• No noticeable trend or seasonality

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Problem Scenario

• The Consumer Confidence Index (CCI) measures
  peoples feelings about general business
  conditions
• Is an autoregressive model reasonable to forecast
  future values of the CCI?
• Is a random walk model applicable?

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Time Series Plot Check for Meandering Pattern
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Correlogram Check for high Autocorrelation of
few lags
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Time Series Plot of Differences Check for
Randomness
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Runs Test of Randomness for Differences
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Assessment of Appropriate Forecast Models

• Meandering Pattern of Time Series suggests either
  Autoregressive or Random Walk
• Reject Random Walk because
• Correlogram dies away quickly
• However, Series Plot of Differences appear random
  and Runs Test p-value high

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Regression Model with One lagged Variable
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Regression Model with Two lagged Variables
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Assessing Forecasts of Model with Two lagged
Variables
Random Residual Plot
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Assessing Forecasts of Model with Two lagged
Variables

• MAPE 11.8 (reasonable?)
• CCI forecast for next year Use
• Ft a bYt-1 cYt-2
• F35 50.4 0.866110.9 0.406107.8 102.6
• Compare with actual Y35 when it becomes available