Abstract
An important aspect of financial decision-making may depend on the forecasting effectiveness of the composite index of leading economic indicators, LEI. The leading indicators can be used as an input to a transfer function model of real gross domestic product, GDP. The previous chapter employed four quarterly lags of the LEI series to estimate regression models of association between current rates of growth of real US GDP and the composite index of leading economic indicators. This chapter asks the question as to whether changes in forecasted economic indexes help forecast changes in real economic growth. The transfer function model forecasts are compared to several naïve models in terms of testing which model produces the most accurate forecast of real GDP. No-change forecasts of real GDP and random walk with drift models may be useful forecasting benchmarks (Granger & Newbold, 1977; Mincer & Zarnowitz, 1969). Economists have constructed leading economic indicator series to serve as a business barometer of the changing US economy since the time of Wesley C. Mitchell (1913). The purpose of this study is to examine the time series forecasts of composite economic indexes produced by The Conference Board (TCB) and test the hypothesis that the leading indicators are useful as an input to a time series model to forecast real output in the United States.
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Notes
- 1.
Section “ARMA Model Identification in Practice” can be omitted with little loss of continuity with readers more interested in the application of time series models.
- 2.
This section draws heavily from Box and Jenkins, Time Series Analysis, Chapters 2 and 3. The time series analysis of Box and Jenkins was a great intellectual advance. The 1970s and 1980s were its high water mark period. The authors continue to use Box and Jenkins modeling in the SCA system, see Liu (2006).
- 3.
- 4.
A stationary AR(p) process can be expressed as an infinite weighted sum of the previous shock variables:
$$ {\tilde{Z}}_t={\phi}^{-1}(B){\alpha}_t. $$In an invertible time series, the current shock variable may be expressed as an infinite weighted sum of the previous values of the series:
$$ {\theta}^{-1}(B){\tilde{Z}}_t={\alpha}_t. $$ - 5.
Box and Jenkins, Time Series Analysis. Chapter 6; C.W.J. Granger and Paul Newbold, Forecasting Economic Time Series. Second Edition (New York: Academic Press, 1986), pp. 109–110, 115–117, 206.
- 6.
Granger and Newbo1d, Forecasting Economic Time Series. pp. 185–186.
- 7.
Box and Jenkins, Time Series Analysis. pp. 173–179.
- 8.
G.E. Box and D.R. Cox, “An Analysis of Transformations,” Journal of the Royal Statistical Society, B 26 (1964), 211–243.
- 9.
G.M. Jenkins, “Practical Experience with Modeling and Forecasting Time Series,” Forecasting (Amsterdam: North-Holland Publishing Company, 1979).
- 10.
Jenkins, op. cit., pp. 135–138.
- 11.
Box and Jenkins, Time Series Analysis, pp. 305–308.
- 12.
Box and Jenkins, op. cit.
- 13.
A similar answer with R is reported in Guerard and Thomakos, Financial Forecasting in the Chinese Stock Market, Peking University Press, forthcoming.
- 14.
Automatic time series modeling has advocated since the early days of Box and Jenkins (1970). Reilly (1980), with the Autobox system, pioneered early automatic time series model implementation. Tsay (1988) identified outliers, level shifts, and variance change models that were implemented in PC-SCA. SCA was used in modeling time series in MZTT (1998). Guerard (1990) used both Autobox and SCA systems to model the monthly NYSE stock volume series, January 1965 to December 1987. SCA correctly identified the introduction of the DOT (trading) system as a level shift in March 1976 and the market crash of October 1987 as an additive outlier. Autobox identified October 1987 as a pulse variable.
- 15.
Doornik and Hendry (2013a, 2013b) remind the reader that the data generation process (DGP) is impossible to model, and the best solution that one can achieve is estimate the models to reflect the local DGP, through reduction, described above. The Automatic Gets algorithm reduces GUM to nest LGDP, the locally relevant variables. Congruency, in which the LGDP has the same shape and size as the GUM, or models reflect the local DGP. See Hendry and Krolzig (2005) for original Gets modeling.
- 16.
In the selection process, one tests the null hypothesis that the parameter in front of a variable is zero. The relevant t-statistic from a two-sided test is used.
- 17.
The use of saturation variables avoids the issue of forcing a unit root to capture the shifts, leading to an upward biased estimate of the lagged dependent variable coefficient. The authors are indebted to Jenny Castle for her comments on the application of saturation variables, which she observes addresses this very well.
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Appendices
Time Series Analysis Appendix 1: Table 19 (Continued) with Identified Outliers
The outlier analysis estimation, using a critical two standard deviation criteria for outliers, is: | ||||||||
PARAMETER LABEL | VARIABLE NAME | NUM./DENOM. | FACTOR | ORDER | CONSTRAINT | VALUE | STD ERROR | T-VALUE |
1 | DUE | MA | 1 | 1 | NONE | .7714 | .1054 | 7.32 |
2 | DUE | D-AR | 1 | 1 | NONE | .8752 | .0823 | 10.64 |
SUMMARY OF OUTLIER DETECTION AND ADJUSTMENT | |||
TIME | ESTIMATE | T-VALUE | TYPE |
8 | 0.332 | 3.12 | TC |
11 | −0.532 | −4.96 | TC |
14 | 0.766 | 5.86 | IO |
17 | 0.325 | 2.51 | IO |
21 | 0.577 | 4.46 | IO |
23 | 0.426 | 3.34 | AO |
31 | −0.424 | −3.28 | IO |
34 | −0.307 | −2.92 | TC |
43 | 0.339 | 2.66 | AO |
46 | 0.327 | 2.56 | AO |
53 | −0.314 | −2.43 | IO |
66 | −0.307 | −2.41 | AO |
70 | −0.324 | −2.54 | AO |
74 | −0.429 | −3.32 | IO |
132 | 0.418 | 3.99 | TC |
142 | 0.352 | 2.76 | AO |
166 | −0.266 | −2.53 | TC |
188 | 0.369 | 2.88 | AO |
190 | 0.619 | 5.64 | TC |
192 | 0.595 | 4.47 | AO |
194 | 0.299 | 2.79 | TC |
197 | −0.361 | −3.39 | TC |
209 | 0.245 | 2.34 | TC |
216 | −0.298 | −2.31 | IO |
227 | −0.373 | −2.89 | IO |
234 | 0.363 | 2.85 | AO |
247 | 0.312 | 2.42 | IO |
255 | 0.605 | 5.75 | TC |
259 | −0.298 | −2.84 | TC |
264 | 0.346 | 2.68 | IO |
268 | 0.322 | 2.49 | IO |
270 | −0.354 | −2.77 | AO |
273 | 0.329 | 3.12 | TC |
284 | 0.316 | 3.00 | TC |
288 | −0.506 | −3.91 | IO |
294 | −0.674 | −5.22 | IO |
296 | −0.340 | −3.24 | TC |
306 | 0.373 | 2.93 | AO |
319 | −0.308 | −2.38 | IO |
325 | 0.547 | 4.29 | AO |
378 | 0.316 | 3.01 | TC |
405 | −0.309 | −2.39 | IO |
TOTAL NUMBER OF OBSERVATIONS.............. | 419 |
EFFECTIVE NUMBER OF OBSERVATIONS............ | 418 |
RESIDUAL STANDARD ERROR (WITHOUT OUTLIER ADJUSTMENT).. | 0.186459E + 00 |
RESIDUAL STANDARD ERROR (WITH OUTLIER ADJUSTMENT)... | 0.129131E + 00 |
Time Series Analysis Appendix 2: Table 22 (Continued)
LAGS 13 THROUGH 18 | |||||
. − | −. | .. | .. | .. | .. |
. + | +. | . + | .. | .. | .. |
LAGS 19 THROUGH 24 | |||||
.. | .. | .. | .. | .. | .. |
.. | .. | .. | .. | .. | .. |
========== STEPWISE AUTOREGRESSION SUMMARY ========== | ||||||||||
I | RESIDUAL | I | EIGENVAL. | I | CHI-SQ | I | I | SIGNIFICANCE | ||
LAG | I | VARIANCES | I | OF SIGMA | I | TEST | I | AIC | I | OF PARTIAL AR COEFF. |
---- | + | ---------- | + | ---------- | + | --------- | + | ---------- | + | ---------------------- |
1 | I | .320E − 01 | I | .297E − 01 | I | 224.58 | I | −6.067 | I | − |
I | .743E − 01 | I | .766E − 01 | I | I | I | + | |||
NOTE: CHI-SQUARED CRITICAL VALUES WITH 4 DEGREES OF FREEDOM ARE | ||||||||||
5%: 9.5 1%: 13.3 | ||||||||||
NOTE: THE PARTIAL AUTOREGRESSION COEFFICIENT MATRIX FOR LAG L IS THE | ||||||||||
ESTIMATED PHI(L) FROM THE FIT WHERE THE MAXIMUM LAG USED IS L | ||||||||||
(I.E., THE LAST COEFFICIENT MATRIX). THE ELEMENTS ARE | ||||||||||
STANDARDIZED BY DIVIDING EACH BY ITS STANDARD ERROR | ||||||||||
-- | ||||||||||
MTSMODEL ARMATF11. SERIES ARE DUE,DLEI. @ | ||||||||||
MODEL IS (1-PHI*B)SERIES=C+(1-TH1*B)NOISE. | ||||||||||
SUMMARY FOR MULTIVARIATE ARMA MODEL – ARMATF11 | ||||||||||
VARIABLE DIFFERENCING | ||||||||||
DUE | ||||||||||
DLEI | ||||||||||
PARAMETER | FACTOR | ORDER | CONSTRAINT | |
1 | C | CONSTANT | 0 | CC |
2 | PHI | REG AR | 1 | CPHI |
3 | TH1 | REG MA | 1 | CTH1 |
-- | ||||
CAUSALTEST MODEL ARMATF11. OUTPUT PRINT(CORR). alpha.01 | ||||
SUMMARY OF THE TIME SERIES | ||||
SERIES | NAME | MEAN | STD DEV | DIFFERENCE ORDER(S) |
1 | DUE | 0.0012 | 0.1940 | |
2 | DLEI | 0.1081 | 0.3502 | |
ERROR COVARIANCE MATRIX | ||
1 | 2 | |
1 | .037737 | |
2 | −.026752 | .122671 |
ITERATIONS TERMINATED DUE TO: CHANGE IN (−2*LOG LIKELIHOOD)/NOBE.LE. 0.100E − 03 TOTAL NUMBER OF ITERATIONS IS 12 MODEL SUMMARY WITH MAXIMUM LIKELIHOOD PARAMETER ESTIMATES | ||
CONSTANT VECTOR (STD ERROR) | |
0.007 | (0.003) |
−0.011 | (0.007) |
PHI MATRICES ESTIMATES OF PHI(1) MATRIX AND SIGNIFICANCE | ||
.782 | −.064 | + − |
.511 | 1.095 | + + |
STANDARD ERRORS | |
.055 | .022 |
.129 | .052 |
THETA MATRICES ESTIMATES OF THETA(1) MATRIX AND SIGNIFICANCE | ||
.805 | .062 | +. |
.294 | .732 | . + |
STANDARD ERRORS | |
.063 | .032 |
.151 | .068 |
ERROR COVARIANCE MATRIX | ||
1 | 2 | |
1 | .029350 | |
2 | −.006695 | .065507 |
================================================= SUMMARY OF FINAL PARAMETER ESTIMATES AND THEIR STANDARD ERRORS ================================================= | |||
PARAMETER NUMBER | PARAMETER DESCRIPTION | FINAL ESTIMATE | ESTIMATED STD. ERROR |
1 | CONSTANT(1) | 0.007181 | 0.003101 |
2 | CONSTANT(2) | −0.010655 | 0.007280 |
3 | AUTOREGRESSIVE (1, 1, 1) | 0.781951 | 0.055383 |
4 | AUTOREGRESSIVE (1, 1, 2) | −0.064217 | 0.022034 |
5 | AUTOREGRESSIVE (1, 2, 1) | 0.510876 | 0.129085 |
6 | AUTOREGRESSIVE (1, 2, 2) | 1.095485 | 0.051517 |
7 | MOVING AVERAGE (1, 1, 1) | 0.804953 | 0.063443 |
8 | MOVING AVERAGE (1, 1, 2) | 0.061788 | 0.031849 |
9 | MOVING AVERAGE (1, 2, 1) | 0.293963 | 0.150721 |
10 | MOVING AVERAGE (1, 2, 2) | 0.732353 | 0.068170 |
CORRELATION MATRIX OF THE PARAMETERS | ||||||||||
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
1 | 1.00 | |||||||||
2 | −.83 | 1.00 | ||||||||
3 | −.71 | .61 | 1.00 | |||||||
4 | −.79 | .62 | .89 | 1.00 | ||||||
5 | .55 | −.72 | −.81 | −.68 | 1.00 | |||||
6 | .62 | −.79 | −.77 | −.79 | .90 | 1.00 | ||||
7 | −.67 | .52 | .89 | .84 | −.69 | −.65 | 1.00 | |||
8 | −.58 | .44 | .54 | .73 | −.49 | −.56 | .56 | 1.00 | ||
9 | .52 | −.68 | −.75 | −.65 | .93 | .86 | −.73 | −.49 | 1.00 | |
10 | .52 | −.65 | −.60 | −.66 | .67 | .82 | −.53 | −.62 | .68 | 1.00 |
THE RESIDUAL COVARIANCE MATRIX IS SET TO FULL MATRIX ALL ELEMENTS IN THE MATRIX PARAMETERS ARE ALLOWED TO BE ESTIMATED −2*(LOG LIKELIHOOD AT FINAL ESTIMATES UNDER H5) IS −0.17857400E + 04 |
THE RESIDUAL COVARIANCE MATRIX IS SET TO DIAGONAL MATRIX ALL ELEMENTS IN THE MATRIX PARAMETERS ARE ALLOWED TO BE ESTIMATED −2*(LOG LIKELIHOOD AT FINAL ESTIMATES UNDER H5*) IS −0.17751251E + 04 |
THE RESIDUAL COVARIANCE MATRIX IS SET TO FULL MATRIX THE (2,1)TH ELEMENTS IN THE MATRIX PARAMETERS ARE SET TO ZERO −2*(LOG LIKELIHOOD AT FINAL ESTIMATES UNDER H4) IS −0.17693824E + 04 |
THE RESIDUAL COVARIANCE MATRIX IS SET TO DIAGONAL MATRIX THE (2,1)TH ELEMENTS IN THE MATRIX PARAMETERS ARE SET TO ZERO −2*(LOG LIKELIHOOD AT FINAL ESTIMATES UNDER H4*) IS −0.17625596E + 04 |
THE RESIDUAL COVARIANCE MATRIX IS SET TO FULL MATRIX THE (1,2)TH ELEMENTS IN THE MATRIX PARAMETERS ARE SET TO ZERO −2*(LOG LIKELIHOOD AT FINAL ESTIMATES UNDER H3) IS −0.17192010E + 04 |
THE RESIDUAL COVARIANCE MATRIX IS SET TO DIAGONAL MATRIX THE (1,2)TH ELEMENTS IN THE MATRIX PARAMETERS ARE SET TO ZERO −2*(LOG LIKELIHOOD AT FINAL ESTIMATES UNDER H3*) IS −0.17085436E + 04 |
THE RESIDUAL COVARIANCE MATRIX IS SET TO FULL MATRIX THE (2,1)TH ELEMENTS IN THE MATRIX PARAMETERS ARE SET TO ZERO THE (1,2)TH ELEMENTS IN THE MATRIX PARAMETERS ARE SET TO ZERO −2*(LOG LIKELIHOOD AT FINAL ESTIMATES UNDER H2) IS −0.17073081E + 04 |
THE RESIDUAL COVARIANCE MATRIX IS SET TO DIAGONAL MATRIX THE (2,1)TH ELEMENTS IN THE MATRIX PARAMETERS ARE SET TO ZERO THE (1,2)TH ELEMENTS IN THE MATRIX PARAMETERS ARE SET TO ZERO −2*(LOG LIKELIHOOD AT FINAL ESTIMATES UNDER H1) IS −0.16916883E + 04 |
CAUSALITY TEST BETWEEN VARIABLES DUE AND DLEI |
P, PP, Q, QQ, NP, NAR, NSAR, NMA, NSMA: |
1 0 1 0 0 4 0 4 0 |
−2*(LOG LIKELIHOOD) UNDER H5,H5*,H4,H4*,H3,H3*,H2,H1 ARE: |
1 −0.17857400E + 04 |
2 −0.17751251E + 04 |
3 −0.17693824E + 04 |
4 −0.17625596E + 04 |
5 −0.17192010E + 04 |
6 −0.17085436E + 04 |
7 −0.17073081E + 04 |
8 −0.16916883E + 04 |
−2*(LOG LIKELIHOOD) UNDER H1,H2,H3,H3*,H4,H4*,H5,H5* ARE: |
−0.16916884E + 04 |
−0.17073081E + 04 |
−0.17192010E + 04 |
−0.17085436E + 04 |
−0.17693824E + 04 |
−0.17625597E + 04 |
−0.17857400E + 04 |
−0.17751251E + 04 |
LR1 TO LR10: 66.53894042968750 16.35754394531250 11.89294433593750 62.07434082031250 15.61975097656250 78.43188476562500 10.65747070312500 6.822753906250000 10.61486816406250 94.05163574218750 |
DDF1 TO DDF10: 2.000000000000000 2.000000000000000 2.000000000000000 2.000000000000000 1.000000000000000 4.000000000000000 1.000000000000000 1.000000000000000 1.000000000000000 5.000000000000000 |
CHI1 TO CHI10: 9.210340416679674 9.210340416679674 9.210340416679674 9.210340416679674 6.634896640840623 13.27670418742523 6.634896640840623 6.634896640840623 6.634896640840623 15.08627252353956 |
T1 TO T10: 57.3286018 7.14720345 2.68260384 52.8640022 8.98485470 65.1551819 4.02257395 0.187857270 3.97997141 78.9653625 |
IT1 TO IT10: 1 1 1 1 1 1 1 1 1 1 |
I1 TO I6: 3 3 3 3 3 3 |
RESULT BASED ON THE BACKWARD PROCEDURE (Y:DUE, X: DLEI) |
DUE <=> DLEI (Y AND X HAVE FEEDBACK RELATION) |
RESULT BASED ON THE FORWARD PROCEDURE (Y:DUE, X: DLEI) |
DUE <=> DLEI (Y AND X HAVE FEEDBACK RELATION) |
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Guerard, J.B., Saxena, A., Gultekin, M. (2021). Time Series Modeling and the Forecasting Effectiveness of the US Leading Economic Indicators. In: Quantitative Corporate Finance. Springer, Cham. https://doi.org/10.1007/978-3-030-43547-9_13
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