Global Analysts
Eirik Skeid, Anders Graham,
Bradley Moore, Matthew Scott Tor Seim, Steven Comstock
Project Purpose
Investigate the correlation between USA’s unemployment and inflation rates.
Construct a model which can be used for approximate inflation forecasting.
Outline
Characterize the data Test for Unit Roots Pre-Whitten the time series Investigate Causality Bivariate Model Construction Remodel Forecast and Evaluate Results
DataTraces
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INFLATION
Inflation Calculation based on:
CPI-CPI(-12) / CPI(-12).
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12
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UNEM
Unemployment Trace
Histograms
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-0.025 0.000 0.025 0.050 0.075 0.100 0.125 0.150
Series : INFLATIONSample 1949:01 2007:03Observations 699
Mean 0.037713Median 0.030880Max imum 0.147560Minimum -0.028690Std. Dev. 0.030048Skewness 1.283760Kurtos is 4.836957
Jarque-Bera 290.2763Probability 0.000000
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Series : UNEMSample 1949:01 2007:04Observations 700
Mean 5.635000Median 5.600000Max imum 10.80000Minimum 2.500000Std. Dev. 1.500707Skewness 0.567713Kurtos is 3.516460
Jarque-Bera 45.38114Probability 0.000000
Inflation:•Multi-peaked•Positively Skewed•Slightly Kurtotic•Not Normal
Unemployment:•Multi-peaked•Not Normal
CorrelogramsInflation UnemploymentBoth appear to be random walks and require unit root test.
Unit Roots Tests
ADF Test Statistic -2.793594 1% Critical Value*
-3.4424
5% Critical Value
-2.8661
10% Critical Value
-2.5692
Inflation Unemployment
ADF Test Statistic -1.916942 1% Critical Value*
-3.4422
5% Critical Value
-2.8660
10% Critical Value
-2.5692
Both are evolutionary.
Differencing
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DINFLATION
Inflation Unemployment
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DUNEM
Differenced Histograms
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Series : DINFLATIONSample 1949:02 2007:03Observations 698
Mean 0.002168Median -0.002000Max imum 2.153000Minimum -2.039000Std. Dev. 0.424951Skewness 0.124286Kurtos is 7.772432
Jarque-Bera 664.2021Probability 0.000000
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Series : DUNEMSample 1949:02 2007:04Observations 699
Mean 0.000286Median 0.000000Max imum 1.300000Minimum -1.500000Std. Dev. 0.214115Skewness 0.340677Kurtos is 9.419683
Jarque-Bera 1213.830Probability 0.000000
Inflation:•Kurtotic•Not Normal
Unemployment:•Kurtotic•Not normal
Differenced Inflation Correlogram
Looks like a seasonal ARMA(2,2).
The reason for the spike at 12 is because of the definition of inflation CPI-CPI(-12) / CPI(-12).
It does not look over differenced.
Unemployment Correlogram
Granger Causality Test
Pairwise Granger Causality Tests Date: 05/31/07 Time: 19:51 Sample: 1949:01 2007:04 Lags: 12
Null Hypothesis: Obs F-Statistic Probability
DUNEM does not Granger Cause DINFLATION 686 4.17866 2.4E-06 DINFLATION does not Granger Cause DUNEM 0.47418 0.93005
Results show one way causality, unemployment affects inflation
Cross Correlations
VAR Impulse Response
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Response of DINFLATION to DINFLATION
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Response of DINFLATION to DUNEM
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Response of DUNEM to DINFLATION
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Response of DUNEM to DUNEM
Response to One S.D. Innovations ± 2 S.E.
VAR Impulse Response
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Response of DINFLATION to DINFLATION
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Response of DINFLATION to DUNEM
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Response of DUNEM to DINFLATION
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Response of DUNEM to DUNEM
Response to One S.D. Innovations ± 2 S.E.
Inflation Model
Dependent Variable: DINF Method: ML - ARCH Date: 06/01/07 Time: 10:45 Sample(adjusted): 1951:10 2007:03 Included observations: 666 after adjusting endpoints Convergence achieved after 23 iterations Backcast: 1950:10 1951:09
Coefficient Std. Error z-Statistic Prob.
DUNEM(-8) -0.095073 0.038123 -2.493818 0.0126 AR(1) 0.208462 0.042508 4.904030 0.0000 AR(2) 0.187172 0.039713 4.713126 0.0000 AR(7) 0.123481 0.037743 3.271624 0.0011 AR(9) 0.085019 0.039976 2.126754 0.0334 AR(10) 0.118897 0.038506 3.087717 0.0020 AR(12) -0.113026 0.038087 -2.967558 0.0030 AR(15) 0.110151 0.036492 3.018500 0.0025 AR(24) -0.146959 0.032104 -4.577601 0.0000 MA(11) 0.104528 0.019561 5.343553 0.0000 MA(12) -0.782336 0.019995 -39.12726 0.0000
Variance Equation
C 0.003712 0.001110 3.343397 0.0008 ARCH(1) 0.151702 0.028644 5.296180 0.0000
GARCH(1) 0.802292 0.034546 23.22355 0.0000
R-squared 0.530402 Mean dependent var -0.006288 Adjusted R-squared 0.521039 S.D. dependent var 0.375694 S.E. of regression 0.260006 Akaike info criterion 0.073361 Sum squared resid 44.07740 Schwarz criterion 0.167983 Log likelihood -10.42924 F-statistic 56.64776 Durbin-Watson stat 1.938664 Prob(F-statistic) 0.000000
Inverted AR Roots .95 -.08i .95+.08i .86+.35i .86 -.35i .74+.58i .74 -.58i .60+.70i .60 -.70i .34 -.86i .34+.86i .15 -.89i .15+.89i -.13+.92i -.13 -.92i -.34+.85i -.34 -.85i -.54 -.72i -.54+.72i -.75+.56i -.75 -.56i -.85+.34i -.85 -.34i -.91 -.15i -.91+.15i
Inverted MA Roots .97 .84+.48i .84 -.48i .50 -.84i .50+.84i .01 -.98i .01+.98i -.49 -.86i -.49+.86i -.85+.50i -.85 -.50i -.99
ResidualsCorrelograms of the residuals and the squared residuals.
Forecasting of Inflation
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INFFINFFL
INFFUINF
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INF07UINF07L
INF07INF
Tracking Plot Forecast
Model Improvement
Removing imputed structure in CPI calculation.
Old Equation: CPI-CPI(-12) / CPI(-12)
New Equation: CPI-CPI(-1) / CPI(-1)
Data Identification
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INFLATION2
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Series: INFLATION2Sample 1949:02 2007:04Observations 699
Mean 3.709728Median 3.582090Maximum 21.67043Minimum -10.08403Std. Dev. 4.293659Skewness 0.410974Kurtosis 4.386076
Jarque-Bera 75.63192Probability 0.000000
Trace Histogram
Unit Root Test for CPI Data
ADF Test Statistic -3.431205 1% Critical Value*
-3.4424
5% Critical Value
-2.8661
10% Critical Value
-2.5692
Differenced CPI Data
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DIN
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Series: DINSample 1949:03 2007:04Observations 698
Mean 0.025495Median -0.005849Maximum 18.95550Minimum -19.00968Std. Dev. 4.279287Skewness -0.162033Kurtosis 4.915232
Jarque-Bera 109.7353Probability 0.000000
Stationary, yet not normal.
Granger causality test
Pairwise Granger Causality Tests Date: 06/01/07 Time: 11:46 Sample: 1949:04 2007:04 Lags: 12
Null Hypothesis: Obs F-Statistic Probability
DUNEM does not Granger Cause DIN 685 2.78029 0.00104 DIN does not Granger Cause DUNEM 0.45937 0.93785
Unemployment seems to cause inflation but not the opposite
Estimation Output for Best Model
Dependent Variable: DIN Method: ML - ARCH Date: 06/01/07 Time: 10:20 Sample(adjusted): 1952:05 2006:04 Included observations: 648 after adjusting endpoints Convergence achieved after 41 iterations Backcast: 1952:02 1952:04
Coefficient Std. Error z-Statistic Prob.
C 0.000943 0.014862 0.063476 0.9494 DUNEM(-3) -0.909459 0.181043 -5.023453 0.0000
AR(11) 0.081456 0.038564 2.112251 0.0347 AR(12) 0.191082 0.040177 4.756017 0.0000 AR(34) -0.134611 0.032547 -4.135958 0.0000 AR(36) 0.199666 0.031593 6.319992 0.0000 MA(1) -0.802523 0.031746 -25.27981 0.0000 MA(3) -0.108571 0.030626 -3.545056 0.0004
Variance Equation
C 0.578962 0.249201 2.323276 0.0202 ARCH(1) 0.125292 0.022222 5.638220 0.0000 ARCH(2) 0.074787 0.027533 2.716271 0.0066
GARCH(1) -0.111933 0.033232 -3.368218 0.0008 GARCH(2) 0.857783 0.031214 27.48052 0.0000
R-squared 0.431494 Mean dependent var 0.008715 Adjusted R-squared 0.420750 S.D. dependent var 3.957843 S.E. of regression 3.012254 Akaike info criterion 4.961863 Sum squared resid 5761.783 Schwarz criterion 5.051617 Log likelihood -1594.644 F-statistic 40.16349 Durbin-Watson stat 1.966808 Prob(F-statistic) 0.000000
Inverted AR Roots .95 .92+.14i .92 -.14i .88+.31i .88 -.31i .84+.47i .84 -.47i .75+.59i .75 -.59i .62+.73i .62 -.73i .49 -.84i .49+.84i .35+.90i .35 -.90i .17 -.95i .17+.95i -.00 -.98i -.00+.98i -.16 -.95i -.16+.95i -.34 -.90i -.34+.90i -.50+.84i -.50 -.84i -.62 -.73i -.62+.73i -.74 -.60i -.74+.60i -.84 -.46i -.84+.46i -.89+.31i -.89 -.31i -.92 -.15i -.92+.15i -.94
Inverted MA Roots .93 -.06+.34i -.06 -.34i
Forecasting
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D IN FTEST ± 2 S.E.
Forecast: DINFTESTActual: DINSample: 2006:05 2007:04Include observations: 12
Root Mean Squared Error 4.460440Mean Absolute Error 3.774718Mean Abs. Percent Error 160.3528Theil Inequality Coefficient 0.647216 Bias Proportion 0.001770 Variance Proportion 0.035402 Covariance Proportion 0.962828
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Forec as t of Var ianc e
Recolored Forecast
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INFEINFLATIONF
INFLATIONF+2*SEF2INFLATIONF-2*SEF2
Date Inflation Forecast
2007:05:00 6.41 2007:06:00 4.25 2007:07:00 2.60 2007:08:00 1.59 2007:09:00 1.03 2007:10:00 2.66 2007:11:00 1.69 2007:12:00 0.81
Conclusion The best model uses the CPI based on monthly
changes instead of annual. The coefficient on dunem is negative. So an
increase in unemployment rate will result in a decrease in inflation, all else held constant.
This is correct according to macroeconomic theory. Inflation and unemployment rate are inversely related probably with a small lag, in our case 3 months.
The inclusion of additional correlated variables could increase forecasting accuracy