Post on 16-Jan-2016
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A Taylor Rule with Monthly Data
A.G. Malliaris
Mary .E. Malliaris
Loyola University Chicago
Fed Funds 1957-2005
02468
101214161820
Unemployment Rate 1957-2005
0.0
2.0
4.0
6.0
8.0
10.0
12.0
J an-57 J an-60 J an-63 J an-66 J an-69 J an-72 J an-75 J an-78 J an-81 J an-84 J an-87 J an-90 J an-93 J an-96 J an-99 J an-02 J an-05
CPI-All Items 12 month logarithmic change rate Jan 1957-Nov 2005
0
2
4
6
8
10
12
14
16
CPI, All Items, 1957 - 2005
0.0
50.0
100.0
150.0
200.0
250.0
J an-57 J an-61 J an-65 J an-69 J an-73 J an-77 J an-81 J an-85 J an-89 J an-93 J an-97 J an-01 J an-05
Date
Standard Approaches
• Random Walkrt = α + βrt-1 + ε
• Taylor Model
rt = α + β1 (CPI-2) + β2 (Un-4) + ε
• Econometric Model
rt = α + β1rt-1 + β2(CPI-2) + β3(Un-4) + ε
Neural Network Architecture
Input, Hidden and Output Layers with sigmoid function applied to weighted sum
w1
w2
w3
w16w17
w18
w19
w20
w21
F(sum inputs*weights)=node output
F(sum inputs*weights)=output
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
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
Variables Per Model
rt-1 CPIt-1 Gapt-1
Random Walk X
Taylor X X
Econometric X X X
Neural Net X X X
Fed Funds t-1 vs. Fed Funds tsorted by Fed Funds t-1
0
5
10
15
20
25
0.6
3
1.5
3
2.3
3
2.9
6
3.2
9
3.7
3
4.0
4
4.6
3
4.9
5
5.3
5.5
1
5.8
6.1
4
6.7
7.6
8.2
9
8.9
9
9.9
1
11
.4
18
.9
Fed Funds t-1
CPI t-1 vs Fed Funds tsorted by CPI t-1
0
5
10
15
20
25
0.35
1.09
1.35
1.59
1.81
2.21
2.56
2.74
2.94
3.14
3.39
3.62
3.89
4.38
4.78
5.48
6.23
7.61
10.2
13.4
CPI t-1
Gap t-1 vs Fed Funds tsorted by Gap t-1
0
5
10
15
20
25
-0.6
-0.2 0
0.5
0.9
1.1
1.3
1.4
1.5
1.6
1.7
1.9 2
2.3
2.8
3.1
3.3
3.6
4.3
6.4
Gap t-1
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
Fed Funds Validation Set for PreGreenspan and Greenspan Data Sets
0
5
10
15
20
25
Fed Funds Validation Set for Low, Medium and High Data Sets
0
5
10
15
20
Apr-5
8
Apr-6
3
Apr-6
8
Apr-7
3
Apr-7
8
Apr-8
3
Apr-8
8
Apr-9
3
Apr-9
8
Apr-0
3
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
Taylor Equation• Original Equation
rt = 2 + 1.5*CPI + .5*Gap
• 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
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
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
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
Mean Squared Error Comparisons on Validation Sets
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