Regression on Time Series Data - Part I

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Regression on Time Series Data- Part I

• Dynamics of Residuals

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Forecasting Using Regression

1. Watch for Spurious Regression
2. Check Stationarity of Residuals for Stability of
   the Relationship Co-integration
3. Learn Modeling Techniques for Reducing Residuals
   to WN
4. Aware of Translating The Forecasting Problem to
   That of Independent Variables

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Integrated Process

• Integrated Process I(1)
• (1 - L)Yt ARMA(p, q)t
• Key Model (Hypothesis) for Macroeconomic
  Variables
• Uncertainty of the Long Run Path

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Spurious Regression

• Two I(1) variables could exhibit significant
  correlation, without an underlying relationship.

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Spurious Regression Demonstration

• Two independent random walk series are
  generated
• Y1t Y1t-1 e1t e1t is WN(s1)
• Y2t Y2t-1 e2t e2t is WN(s2)
• Regression of Y1 on Y2 is computed.

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Key Reminders

• The regression must make economic sense
• Check the residual if stationary

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Regression Modeling-1 - AR (1) Error -

• e t r e (t-1) u t
• ut is WN(s)

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Regression Modeling 2 - Using the First
Difference -

• Use the first difference to reduce each series to
  stationary
• DYt a b DXt et
• Simple and practical, but may not be a best
  approach.

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Regression Strategy - 3

• The distributed lag regression model with lagged
  dependent variables (Text. Ch. 10.5)
• In a simplest form
• Yt a0 a1 Y(t-1) b1 X(t-1) et
• Does not require forecasting of the right hand
  side variables.

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Regression Modeling - 4

• Error Correction Model
• Yt a0 a1Y(t-1) b0Xt b1X(t-1)
  et
• It can be shown that the model is equivalent to
  (see the next page for the the definition of l
  and g and the derivation)
• DYt a0 b0 DXt - l(Y(t-1) g
  X(t-1)) et

Long run equilibrium relationship
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Regression Modeling 4 (cont.)
Write the model as follows
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Demand for Gasoline

• Regression 1
• Residuals might be non-stationary (t -1.80,
  p0.07)
• Coefficient of price is positive
• Forecasting PG
• Regression 2
• Coefficient of price is positive
• DW is too low -gtserial correlation of the
  residuals

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Demand for Gasoline cont.

• Regression 3
• DW is too low -gtserially correlated residuals
• Forecasting DPG
• Contaminated by influential observations
• Regression 4
• Low R-squared

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Demand for Gasoline cont.

• Regression 5
• AR(1) for the residual for generating WN error
• Possibly unequal variance?
• An important modeling approach
• Regression 6
• Using lagged Y for generating WN error
• Inferior to Regression 5
• Still a useful modeling approach

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Demand for Gasoline cont.

• Regression 7
• Needs the value of the independent variable for
  forecasting
• A theoretical problem the coefficient of G(-1)
  should be less than 1.0 for stationarity.

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AppendixAR(1) Error
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Cochrane-Orcutt Transformation

• For all Variables

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Simple Regression Case

• Model
• Transformation

Use
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Estimation of r

• 1. Use r 1
• 2. Use r1 of the residual of the
• standard regression.

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Implications of Using r 1

• 1) First Differences as Regression Variables
• DY t Y t - Y (t-1)
• DX t X t - X (t-1)
• 2) Regression without the Intercept
• DY t b1 DX t at