4.2.1 Reduced form modeling
Beaudry and Portier (2006) have raised the issue if in a bivariate system comprising a technology measure (TFP) and stock prices (SP) as an indicator of future expectations about the business cycle, surprises i.e. news are released in expectations (i.e. stock prices) or technology. Interestingly, the former case would suggest that technological change is to some extent ’foreseen’ or processed in stock prices. The quarterly data on US total factor productivity and stock prices spans the period 1947 to 2000 and can be drawn from the net (http://www.aeaweb.org/articles.php?doi=10.1257/aer.96.4.1293). To clarify the origin
of news, Beaudry and Portier (2006) also employ higher dimensional systems comprising consumption and/or hours worked which is beyond the scope of the ITSI approach as introduced in this work. For the bivariate system two alternative identification schemes are applied in Beaudry and Portier (2006) one of which relies on long term identifying restrictions that allow a non vanishing response of the news shock on total factor produc-tivity. Alternatively, Beaudry and Portier (2006) use a Cholesky decomposition excluding on impact dynamics operating from stock prices on total factor productivity. It turns out that these two identification schemes obtain highly correlated news shocks. The Cholesky type lower diagonal identification scheme used by Beaudry and Portier (2006) can be subjected to loss comparison with an alternative upper triangular decomposition scheme.
It is this particular aspect of the relationship between technology and stock prices that can be subjected to a loss assessment in the ITSI framework.
To investigate the empirical linkage of stock prices and factor productivity four al-ternative VAR specifications are used to extract reduced form residuals. According to standard model selection criteria (AIC, BIC, HQ) applied to level data with determin-istic trend, lag order 2 is broadly supported. Accordingly, the four systems analyzed are a VAR(2) with trend for level data, a VAR(2) for level data without trend and a VAR(2) and VAR(1) model for first differences of stock prices and total factor produc-tivity both excluding a deterministic trend. Reduced form residuals extracted from these model specifications are throughout rather similar and the empirical residual correlation in these bivariate systems is around ˆσ ≈ 0.16. VAR estimates are provided in Table 4.
Noting an only weak correlation of reduced form disturbances in the TFP/SP system and the relatively small time series dimension the discriminatory content of the ITSI loss statistics is likely limited on the one hand. On the other hand, documented statistics testing the presumption of joint reduced form normality are significant at any reasonable level such that ITSI loss statistics naturally apply for a comparison of distinct structural assumptions.
4.2.2 Structural analysis
ITSI diagnostics for the alternative systems comprising reduced form disturbances of TFP and stock prices in the US are documented in Table 5. Detecting ’dependence’ minimizing
matrices D0 by means of systematically rotating Dl or Du obtains loss statistics and con-temporaneous causation pattern that are close to the loss statistics attached to the lower triangular recursive structure. While for most bD0 matrices some asymmetric feedback relations are detected, it is worthwhile to point out that the lower left element of these matrices is throughout larger in absolute magnitude than the upper right element. Dis-tinguishing alternative loss statistics L•, • ∈ {l, s, u} almost all estimated VAR systems deliver smallest loss statistics for the lower triangular decomposition (Ll < Lu) implying that innovations in stock prices have no contemporaneous effect on reduced form innova-tions characterizing total factor productivity. Thus, total factor productivity is confirmed to bear the interpretation of a news process. Moreover, with regard to inferential results it turns out that elements of implied structural innovations ξt(u) lack independence with 5% significance throughout. Thus, while it is difficult to distinguish the dependence level of structural innovations ξt(l) and ξt(s), the empirical evidence is markedly at odds with the presumption that iid news enters the bivariate system through stock prices. Overall, however, the evidence in favor of symmetric instantaneous causation is weaker as it is for the most likely (i.e. the lower triangular) recursive scheme. Interestingly, Beaudry and Portier (2006) rely on this assumed recursion on the basis of a-priori reasoning. Combined with arguments in Beaudry and Poitier (2006) this also supports the view that the news shock impacts on TFP in the long run.
5 Conclusions
In this paper a loss functional is introduced that carries informational content to dis-criminate between competing structural relations in a non iid Gaussian framework. In-dependence targeted structural innovations (ITSI) can provide a ranking of alternative just identifying structural data representations. Thus, ITSI assist the analyst in deter-mining a data supported structural view at the economy, or put differently, highlight to which extent particular identifying restrictions are not supported by empirical processes.
The ITSI concept fully relies on the notion of data being independent and identically dis-tributed over the time dimension such that simple resampling and Monte Carlo techniques are applicable to resolve inferential issues with regard to competing a-priori settings of
structural data relations. A Monte Carlo study underpins that ITSI may reasonbly used to contrast rival schemes of recursive causality even if sample sizes are small to moder-ate. Applied to empirical systems of break even inflation rates, ITSI diagnostics indicate that the Bank of Canada is more dependent on US monetary policy in comparison with the Bank of England. For the US system comprising total factor productivity and stock prices suggesting that news arrives through TFP innovations (Beaudry and Poitier 2006) is found to be in line with sample information.
For the purpose of simplicity and computational tractability the outline of ITSI in this paper has addressed the bivariate case exclusively. The generalization towards higher order systems is straightforward and feasible in principle. The determination of ITSI in higher dimensional systems is an interesting direction of future research. Moreover, in the framework of multivariate GARCH models, it has become a common practice to use the eigenvalue decomposition for the extraction of vector model innovations. Such innovations are typically found to exhibit remaining leptokurtosis as it has been the case in this study for the systems of univariate GARCH innovations extracted from breakeven inflation rates. In consequence, ITSI may also support an analyst to decide upon the most suitable, data driven decomposition of time varying covariance matrices in the multivariate GARCH framework. Opening a further direction of future research one might notice from the empirical analysis of systems of breakeven inflation rates - and in particular of the UK/FR subsystem - that the assumption of time invariant structural relations might lack support empirically. In this respect it is of interest in how far the notion of independent structural innovation could be helpful to uncover time variation in contemporaneous causal relations.
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Dl vs. Ds Dl vs. Du
Table 1: Simulation results: The table shows empirical frequencies for events indicated in the top row, i.e., frequencies of minimum loss measures obtained for Ll (columns 4 and 9), size estimates for nominal significance levels 5% and 10% (columns 5 and 6) and power estimates with respect to these nominal levels (columns 7,8,10,11).
co. αˆ0 αˆ1 βˆ1 LM1 LM5 corrs (ˆσ)
Table 2: GARCH(1,1) parameter estimates and model diagnostics for changes of breakeven inflation rates in Canada (CA), France (FR) the UK and the US, denoted εt. For a particular variance process the conditional variance σt2 = E[ε2t|Ft−1] character-izing the time series εt is, σt2 = α0 + α1ε2t−1+ β1σt−12 . Values in parentheses are either t-ratios (for parameter estimates) or p−values (for the LM-statistic testing against con-ditional heteroskedasticity in GARCH implied standardized residuals). The right hand side panel shows unconditional correlations for the standardized GARCH(1,1) residuals.
D• Db L Db L Db L
CAUS 1st subs 2nd subs
Du 0.998 0.441 176.9∗∗ 0.999 0.517 110.3∗∗ 0.998 0.361 99.34∗
0 0.899 0 0.843 0 0.949
Dl 0.896 0 252.9∗∗∗ 0.851 0 158.7∗∗∗ 0.933 0 133.7∗∗∗
0.440 1.002 0.523 0.989 0.355 1.015
Ds 0.972 0.226 212.7∗∗∗ 0.961 0.270 141.7∗∗∗ 0.981 0.182 113.2∗∗∗
0.226 0.976 0.270 0.951 0.182 0.999
D0 0.764 0.643 172.02∗ 0.679 0.733 100.50 0.877 0.476 93.75
-0.243 0.972 -0.267 0.952 -0.136 1.006
FRUS 1st subs 2nd subs
Du 1.001 0.213 192.8∗∗∗ 1.028 0.200 106.1∗∗ 0.972 0.228 106.0∗∗
0 0.979 0 0.968 0 0.989
Dl 0.978 0 211.7∗∗∗ 1.007 0 115.1∗∗ 0.947 0 111.7∗∗
0.213 1.002 0.208 0.989 0.218 1.015
Ds 0.995 0.107 197.3∗∗∗ 1.023 0.103 111.1∗∗ 0.965 0.112 106.6∗∗
0.107 0.996 0.103 0.983 0.112 1.009
D0 0.930 0.302 187.26∗∗ 1.009 0.280 104.96∗ 0.900 0.293 97.20
-0.107 1.018 -0.076 0.965 -0.106 1.033
UKUS 1st subs 2nd subs
Du 1.001 0.159 188.1∗∗∗ 0.991 0.220 96.20 1.011 0.097 110.4∗∗
0 0.989 0 0.964 0 1.010
Dl 0.988 0 180.5∗∗ 0.966 0 107.0∗∗ 1.006 0 100.4
0.159 1.002 0.220 0.989 0.097 1.015
Ds 0.997 0.080 176.7∗∗ 0.984 0.111 100.9∗ 1.010 0.049 102.5∗
0.080 0.998 0.111 0.983 0.049 1.014
D0 0.985 0.078 176.5∗ 0.931 0.337 93.94 1.010 -0.031 100.25
0.080 1.011 -0.122 0.981 0.128 1.007
UKFR 1st subs 2nd subs
Du 1.001 0.293 216.5∗∗∗ 0.991 0.408 110.2∗∗ 1.011 0.175 98.87
0 0.957 0 0.944 0 0.956
Dl 0.957 0 181.8∗∗ 0.909 0 103.6∗ 0.994 0 86.73
0.293 1.001 0.393 1.028 0.182 0.972
Ds 0.990 0.148 185.9∗∗∗ 0.969 0.205 104.6∗∗ 1.007 0.090 82.92
0.148 0.990 0.205 1.008 0.090 0.967
D0 0.998 0.070 176.93∗ 0.990 -0.027 103.44 0.991 0.078 82.83
0.225 0.975 0.434 0.932 0.106 0.983
Table 3: ITSI estimates and diagnostics for bivariate systems of breakeven inflation. Ma-trices Du, (Dl), Ds, and D0 refer to upper (lower) triangular schemes, symmetric impact relations and impact relations resulting in minimum dependence of structural relations.
Full samples comprises 1956 daily observations. 1st and 2nd subs refer to almost equal sized subsamples covering 1000 and 956 observations, respectively. Significant bootstrap based diagnostics against the independence assumption are indicated with ∗∗∗ (1% signif-icance), ∗∗ (5% significance) and ∗ (10% significance). For further notes see Table 2.
TFP SP TFP SP ∆ TFP ∆ SP ∆ TFP ∆ SP
Table 4: VAR parameter estimates for bivariate systems of US total factor productivity and stock prices (Beaudry and Portier 2006). Model selection criteria (AIC, BIC, HQ, not documented) are in favor of a VAR order two if the model is specified in levels and contains a linear trend). VAR parameter estimates are documented with t-ratios in parentheses. LM is the Lagrange Multiplier test (p-value in parentheses) for multivariate serial correlation up to lag order 10. JB is the Jarque-Bera statistic on joint normality of both reduced form residual processes. Under the null hypothesis of normality JB is χ2 distributed with 4 degrees of freedom, p−values are not provided. Dependent variables are listed in the top row, where ∆ is short for the first difference operator. The left hand side column lists the conditioning variables from which TFP and SP are either in levels or in first differences. Moreover, c and t signify a constant and a trend, respectively, entering the VAR. Estimation and diagnostic results are obtained from Eviews 6.0.
D• D · 1000b L• D · 1000b L•
levels, with trend levels, no trend Du 9.241 8.293 83.29∗∗∗ 9.248 8.770 85.31∗∗∗
0 51.50 0 52.95
Dl 9.123 0 21.12 9.123 0 22.81
1.469 52.16 1.511 53.67
Ds 9.156 1.250 22.50 9.157 1.291 21.29
1.250 52.14 1.291 53.66
D0 9.095 0.716 21.10 9.157 1.291 21.29
-2.628 52.11 1.291 53.66
1st diff, VAR(2) 1st diff, VAR(1) Du 9.344 9.532 92.10∗∗∗ 9.395 8.774 86.76∗∗∗
0 52.51 0 53.08
Dl 9.194 0 23.57 9.270 0 23.86
1.669 53.37 1.532 53.80
Ds 9.235 1.423 23.91 9.304 1.307 24.43
1.423 53.35 1.307 53.78
D0 9.081 -1.438 22.81 9.155 -1.450 22.62
9.997 52.45 9.929 52.90
Table 5: ITSI estimates and diagnostics for the US system comprising reduced form errors of TFP and stock prices. For further notes see Table 4 and Table 3
.
US
CA
UK
FR
Figure 1: Liquidity adjusted breakeven inflation rates for Canada and the US (left hand side) and France and the UK (right hand side).