A Taylor Rule with Monthly Data

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A Taylor Rule with Monthly Data

• A.G. Malliaris
• Mary .E. Malliaris
• Loyola University Chicago

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Fed Funds 1957-2005
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Unemployment Rate 1957-2005
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CPI-All Items 12 month logarithmic change rate
Jan 1957-Nov 2005
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CPI, All Items, 1957 - 2005
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Standard Approaches

• Random Walk
• rt a ßrt-1 e
• Taylor Model
• rt a ß1 (CPI-2) ß2 (Un-4) e
• Econometric Model
• rt a ß1rt-1 ß2(CPI-2) ß3(Un-4)
  e

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Neural Network Architecture
w1
F(sum inputsweights)node output
w2
w19
w3
F(sum inputsweights)output
w20
w21
Input, Hidden and Output Layers with sigmoid
function applied to weighted sum
w17
w16
w18
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Network Process

• The neural network adjusts the weights and
  recalculates the total error.
• This process continues to some specified ending
  point (amount of error, training time, or number
  of weight changes).
• The final network is the one with the lowest
  error from the sets of possible weights tried
  during the training process

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Variable Designations

• rt the Fed Funds rate at time t, the dependent
  variable
• CPIt-1 the Consumer Price Index at time t-1
• Adjusted CPIt-1 CPI minus 2 at time t-1
• Unt-1 the Unemployment Rate at time t-1
• Gapt-1 the Unemployment Rate minus 4 at time
  t-1

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Variables Per Model
rt-1 CPIt-1 Gapt-1
Random Walk X
Taylor X X
Econometric X X X
Neural Net X X X
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Data Sets
Data Set Training Validation Total
PreGreenspan Jan 58 to Jul 87 319 36 355
Greenspan Aug 87 to Nov 05 197 22 219
rt-1 0 to 5 219 24 243
rt-1 5.01 to 10 243 27 270
rt-1 over 10 55 6 61
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Random Walk
Intercept Coefficient of r at t-1
PreGreenspan 0.177 0.973
Greenspan 0.006 0.995
High 1.481 0.879
Medium 0.021 0.995
Low 0.022 0.995
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Taylor Equation

• Original Equation
• rt 2 1.5CPI .5Gap
• Calculated Equation

Intercept CPI Gap
PreGreenspan 2.334 0.789 0.296
Greenspan 1.797 1.477 -0.935
High 5.005 0.564 0.910
Medium 5.755 0.197 0.161
Low 2.837 0.496 -0.490
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Econometric Model
Intercept Fed Funds Adj. CPI Gap
PreGreenspan 0.291 0.965 0.019 -0.035
Greenspan 0.047 0.994 -0.007 -0.024
High 1.442 0.862 0.066 -0.027
Medium 0.007 1.002 -0.003 -0.019
Low 0.125 0.983 0.018 -0.022
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Neural NetworksSignificance of Variables
PreGreenspan Greenspan Low Medium High
Fed Funds Fed Funds Fed Funds Fed Funds CPI
UnRate CPI CPI CPI UnRate
CPI UnRate UnRate UnRate Fed Funds
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Mean Squared Error Comparisons on Validation Sets

Model / Data Set PreGreenspan Greenspan Low Medium High
Random Walk 0.676 0.034 0.122 0.271 0.574
Taylor 10.036 8.392 6.651 9.701 16.754
Taylor2 6.793 3.001 0.985 2.221 1.263
Econometric 0.657 0.030 0.124 0.262 0.613
Neural Network 1.121 0.129 0.104 0.269 0.372
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Summary

• Several approaches to modeling
• Econometric approach best when applied to
  pre-Greenspan and Greenspan
• Neural Network best when sample is divided to
  low, medium and high