Lead-lag Relationship b/w Economic Data and Financial Market
Kingsly Wang
10/12/2019
Variable Explanatory Statistics: 1. Date: 2000/3 - 2019/9 2. SPXMoM: S&P 500 Index log first difference 3. ISM_chng_lag: US ISM (Purchasing Managers’ Index), first difference with a lag (the reason for a lag setting after differencing is to control for the announcement lag. For instance, in the current october month, ISM is not available until mid October, so the contemporaneous relationship may not be too helpful. This should justify for the lag because now we try to examine if the September reading will help invstors’ adjust their positions. A seperate test with ISM_chng without lag is done as well. 4. GDPMoM: Bloomberg has made it available a trailing monthly GDP which is useful for modeling purposes, but the validity of this estimate of quarterly GDP needs to be confirmed.
## Index SPXMoM ISM_chng_lag
## Min. :2000-03-31 Min. :-0.185636 Min. :-9.00000
## 1st Qu.:2005-02-14 1st Qu.:-0.017793 1st Qu.:-1.10000
## Median :2009-12-31 Median : 0.009320 Median : 0.00000
## Mean :2009-12-30 Mean : 0.003313 Mean :-0.03234
## 3rd Qu.:2014-11-15 3rd Qu.: 0.029306 3rd Qu.: 1.05000
## Max. :2019-09-30 Max. : 0.102307 Max. : 4.40000
## GDPMoM
## Min. :-0.016763
## 1st Qu.:-0.001481
## Median : 0.001807
## Mean : 0.001670
## 3rd Qu.: 0.005093
## Max. : 0.017093#VAR We then follow the operation by conducting a VAR model. First, we conduct information criterion test to help decide order of VAR. As it shows up, we decide to apply an order of two to enrich information interpretation.
## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 2 1 2
##
## $criteria
## 1 2 3 4
## AIC(n) -1.580136e+01 -1.587235e+01 -1.583424e+01 -1.578690e+01
## HQ(n) -1.571004e+01 -1.572624e+01 -1.563333e+01 -1.553120e+01
## SC(n) -1.557504e+01 -1.551024e+01 -1.533634e+01 -1.515321e+01
## FPE(n) 1.372669e-07 1.278683e-07 1.328538e-07 1.393283e-07
## 5 6 7 8
## AIC(n) -1.576438e+01 -1.575403e+01 -1.571123e+01 -1.569755e+01
## HQ(n) -1.545388e+01 -1.538874e+01 -1.529114e+01 -1.522267e+01
## SC(n) -1.499490e+01 -1.484876e+01 -1.467016e+01 -1.452070e+01
## FPE(n) 1.425550e-07 1.441141e-07 1.505253e-07 1.527408e-07##
## VAR Estimation Results:
## =========================
## Endogenous variables: SPXMoM, ISM_chng_lag, GDPMoM
## Deterministic variables: none
## Sample size: 233
## Log Likelihood: 859.101
## Roots of the characteristic polynomial:
## 0.4624 0.4116 0.4116 0.3946 0.3946 0.2098
## Call:
## VAR(y = VAR_model, p = 2, type = "none")
##
##
## Estimation results for equation SPXMoM:
## =======================================
## SPXMoM = SPXMoM.l1 + ISM_chng_lag.l1 + GDPMoM.l1 + SPXMoM.l2 + ISM_chng_lag.l2 + GDPMoM.l2
##
## Estimate Std. Error t value Pr(>|t|)
## SPXMoM.l1 0.0441107 0.0671874 0.657 0.5121
## ISM_chng_lag.l1 0.0027606 0.0015860 1.741 0.0831 .
## GDPMoM.l1 1.0574185 0.5153452 2.052 0.0413 *
## SPXMoM.l2 -0.0938324 0.0678955 -1.382 0.1683
## ISM_chng_lag.l2 0.0004907 0.0015654 0.314 0.7542
## GDPMoM.l2 0.8550768 0.5112902 1.672 0.0958 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.04179 on 227 degrees of freedom
## Multiple R-Squared: 0.04634, Adjusted R-squared: 0.02114
## F-statistic: 1.838 on 6 and 227 DF, p-value: 0.09263
##
##
##
## Covariance matrix of residuals:
## SPXMoM ISM_chng_lag GDPMoM
## SPXMoM 1.746e-03 0.0067022 4.593e-05
## ISM_chng_lag 6.702e-03 2.8378728 -2.156e-04
## GDPMoM 4.593e-05 -0.0002156 2.546e-05
##
## Correlation matrix of residuals:
## SPXMoM ISM_chng_lag GDPMoM
## SPXMoM 1.00000 0.09521 0.21785
## ISM_chng_lag 0.09521 1.00000 -0.02537
## GDPMoM 0.21785 -0.02537 1.00000As can be seen, both ISM_chng_lag.l1 and GDPMoM.l1 are significant, with GDP even more so. Further test of the model shows that the process passes a portmanteau test as the p-value is small to oconfirm there’s no serial correlation problem.
VAR_SPX %>% irf (response = "SPXMoM") %>% plot()
serial.test(VAR_SPX, lags.pt = 16, type = "PT.asymptotic")##
## Portmanteau Test (asymptotic)
##
## data: Residuals of VAR object VAR_SPX
## Chi-squared = 195.75, df = 126, p-value = 6.835e-05serial.test(VAR_SPX, lags.pt = 16, type = "BG")##
## Breusch-Godfrey LM test
##
## data: Residuals of VAR object VAR_SPX
## Chi-squared = 71.151, df = 45, p-value = 0.007759#The null hypothesis is: H_0: B_1 = … = B_h = 0The residual of the VAR process also contains no clustered volatility structure, as it passes the arch test.
arch1 <- arch.test(VAR_SPX, lags.multi = 5)
arch1##
## ARCH (multivariate)
##
## data: Residuals of VAR object VAR_SPX
## Chi-squared = 328.46, df = 180, p-value = 9.256e-11plot(arch1, names = "resid")## Warning in plot.varcheck(arch1, names = "resid"):
## Invalid residual name(s) supplied, using residuals of first variable.
Finally, here are some metrics for normality test
norm <- normality.test(VAR_SPX)
norm$jb.mul## $JB
##
## JB-Test (multivariate)
##
## data: Residuals of VAR object VAR_SPX
## Chi-squared = 21.713, df = 6, p-value = 0.001365
##
##
## $Skewness
##
## Skewness only (multivariate)
##
## data: Residuals of VAR object VAR_SPX
## Chi-squared = 12.136, df = 3, p-value = 0.006933
##
##
## $Kurtosis
##
## Kurtosis only (multivariate)
##
## data: Residuals of VAR object VAR_SPX
## Chi-squared = 9.5777, df = 3, p-value = 0.02252plot(norm)
#Appendix Below you can find the result for using ISM_chng without lag. From the result of significance of L2 in ISM_chng, we can posit that it’s useful to use the lag of the first difference term of ISM.
VARselect(VAR_model2, lag.max = 8, type = "both")## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 1 1 2
##
## $criteria
## 1 2 3 4
## AIC(n) -1.586515e+01 -1.589846e+01 -1.583756e+01 -1.583411e+01
## HQ(n) -1.577412e+01 -1.575282e+01 -1.563730e+01 -1.557923e+01
## SC(n) -1.563953e+01 -1.553748e+01 -1.534121e+01 -1.520239e+01
## FPE(n) 1.287845e-07 1.245727e-07 1.324132e-07 1.329026e-07
## 5 6 7 8
## AIC(n) -1.580044e+01 -1.577673e+01 -1.573324e+01 -1.571306e+01
## HQ(n) -1.549094e+01 -1.541262e+01 -1.531451e+01 -1.523971e+01
## SC(n) -1.503335e+01 -1.487427e+01 -1.469541e+01 -1.453986e+01
## FPE(n) 1.375045e-07 1.408775e-07 1.472439e-07 1.503843e-07VAR_SPX <- VAR(VAR_model2, p = 2, type="none")
summary(VAR_SPX, equation = "SPXMoM")##
## VAR Estimation Results:
## =========================
## Endogenous variables: SPXMoM, ISM_chng, GDPMoM
## Deterministic variables: none
## Sample size: 234
## Log Likelihood: 861.749
## Roots of the characteristic polynomial:
## 0.4896 0.4195 0.4195 0.3972 0.3972 0.185
## Call:
## VAR(y = VAR_model2, p = 2, type = "none")
##
##
## Estimation results for equation SPXMoM:
## =======================================
## SPXMoM = SPXMoM.l1 + ISM_chng.l1 + GDPMoM.l1 + SPXMoM.l2 + ISM_chng.l2 + GDPMoM.l2
##
## Estimate Std. Error t value Pr(>|t|)
## SPXMoM.l1 0.018609 0.067163 0.277 0.7820
## ISM_chng.l1 0.002458 0.001623 1.514 0.1313
## GDPMoM.l1 0.998778 0.510207 1.958 0.0515 .
## SPXMoM.l2 -0.112154 0.069041 -1.624 0.1057
## ISM_chng.l2 0.002938 0.001579 1.861 0.0641 .
## GDPMoM.l2 0.745735 0.510614 1.460 0.1455
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.04159 on 228 degrees of freedom
## Multiple R-Squared: 0.05308, Adjusted R-squared: 0.02816
## F-statistic: 2.13 on 6 and 228 DF, p-value: 0.05088
##
##
##
## Covariance matrix of residuals:
## SPXMoM ISM_chng GDPMoM
## SPXMoM 1.730e-03 0.0149043 4.405e-05
## ISM_chng 1.490e-02 2.9372174 9.353e-04
## GDPMoM 4.405e-05 0.0009353 2.602e-05
##
## Correlation matrix of residuals:
## SPXMoM ISM_chng GDPMoM
## SPXMoM 1.0000 0.2091 0.2076
## ISM_chng 0.2091 1.0000 0.1070
## GDPMoM 0.2076 0.1070 1.0000