Natural Gas Prices and Industrial Sector Responses:

An Experimental Module for STIFS

Prepared for

ASA Committee on Energy Statistics Meeting

April 22, 2004

Washington, DC

By

Frederick L. Joutz, Associate Professor

Department of Economics

The George Washington University

Washington, Dc 20052

\(202\) 994-4899

\[e-mail <bmark@gwu.edu>\]

Dave Costello

Office of Energy Markets and End Use (EMEU)

Energy Information Administration

U.S. Department of Energy

(202-586-1468),

### \[e-mail <dave.costello@eia.doe.gov>\] 

### 

### Abstract

The Short-Term Integrated Forecasting System (STIFS) generates monthly
forecasts of energy demand, supply and prices using some forecast
information that is incorporated into STIFS that is generated by other
models that do not run in an integrated framework with STIFS. This
includes the macroeconomic forecasts and projections for certain energy
supply variables. There is no direct feedback between the macroeconomic
models projections and STIFS components. Members of the STIFS Team can
attempt to coordinate iterations between the two models. However, this
is not desirable for two main reasons. First, it suffers from
specification problems in the richness and complexity of the dynamic
interactions because the feedback is not directly estimated. Second, the
iteration process requires staff time and resources that are limited.

This project tests an experimental model for the interaction between
natural gas prices, natural gas consumption, and industrial sector
activity. Two strategies are followed. The first involves a simple VAR
framework capturing the time series dynamics testing for Granger
causality and examining impulse response functions and forecast error
variance decompositions. In the second approach, energy and economic
variables are analyzed in terms of integration, cointegration for a
long-run relationship between oil and natural gas prices. **\[This
latter part of the project is ongoing but is not completed here\]** The
general to specific modeling methodology is used to develop a data
coherent parsimonious representation. Issues related to parameter
constancy, encompassing, and forecasting are discussed. The forecasting
performance of the two strategies is compared and the potential gain
from using the experimental module is discussed.

**INTRODUCTION**

The Short-Term Integrated Forecasting System (STIFS) generates monthly
forecasts of energy demand, supply and prices using some forecast
information that is incorporated into STIFS that is generated by other
models that do not run in an integrated framework with STIFS. This
includes the macroeconomic forecasts and projections for certain energy
supply variables. There is no direct feedback between the macroeconomic
models projections and STIFS components. Members of the STIFS Team can
attempt to coordinate iterations between the two models. However, this
is not desirable for two main reasons. First, it suffers from
specification problems in the richness and complexity of the dynamic
interactions because the feedback is not directly estimated. Second, the
iteration process requires staff time and resources that are limited.

This project tests an experimental model for the interaction between
natural gas prices, natural gas consumption, and industrial sector
activity. Two strategies are followed. The first involves a simple VAR
framework capturing the time series dynamics testing for Granger
causality and examining impulse response functions and forecast error
variance decompositions. In the second approach, energy and economic
variables are analyzed in terms of integration, cointegration for a
long-run relationship between oil and natural gas prices. The general to
specific modeling methodology is used to develop a data coherent
parsimonious representation. Issues related to parameter constancy,
encompassing, and forecasting are discussed. The forecasting performance
of the two strategies is compared and the potential gain from using the
experimental module is discussed.

# Why an Industrial Response Module (IRM) for STIFS?

The Short-Term Integrated Forecasting System (STIFS) generates monthly
forecasts of energy demand, supply and prices using an array of
econometric relationships (estimated with monthly data) and numerous
accounting identities and other relationships that tie together highly
aggregated energy flows and stocks covering major energy sources and
economic sectors. STIFS constitutes a major integrated information
system in EIA, bringing together energy quantities and prices from
various sources within EIA (and from elsewhere) in a consistent, time
series format. The energy information is coupled with other economic and
non-economic information to form a modeling database from which
forecasting equations are estimated, saved and later used to produce
monthly projections, analytical reports, and frame policy discussions.

Some forecast information that is incorporated into STIFS is generated
by other models that do not run in an integrated framework with STIFS.
This information includes the macroeconomic forecasts and projections
for certain energy supply variables.

There is no current direct feedback between the macroeconomic model
projections and STIFS components. Members of the STIFS Team can attempt
to coordinate iterations between the two models. However, this is not
desirable for two main reasons. First, it suffers from specification
problems in terms of the richness and complexity of the dynamic
interactions because the feedback is not directly estimated. Second, the
iteration process requires staff time and resources that are limited.

The current EViews^®^ system used to prepare projections for EIA's
Short-Term Energy Outlook takes as given macroeconomic drivers from a
prepared solution of a large commercial macroeconomic model of the U.S.
economy. Neither the particular sensitivities of key industrial drivers
to energy price shocks nor the stochastic properties of these key
drivers are retrievable from the standard macroeconomic inputs provided
to STIFS. These pieces of information are desirable to increase the
usefulness of STIFS in the context of scenario or uncertainty analysis
without the labor-intensive and costly process of iterating the flow of
information into and out of the large macro model.

# Purpose for Developing and Testing a Pilot IRM

The development of the IRM will provide a first attempt to address the
feedback issue. The objective is to focus on a limited set of STIFS and
macroeconomic sectors.

Since 2000 there have been two major natural gas price shocks. The
manufacturing sector has been particularly affected by the high natural
gas price increases and volatility.

Six of the most energy consumptive industry groups - chemicals,
petroleum refining, primary metals, paper, food processing, and
nonmetallic minerals (glass, lime, cement, etc.) - lead the way in
natural gas consumption. Among those, the chemical industry is dominant.
According to the 1998 Manufacturing Energy Consumption Survey (MECS),
the chemical industry used 36 percent of all natural gas consumed by all
U.S. manufacturers in 1998 (and in fact, 12 percent of TOTAL U.S. gas
consumption for all purposes). Together, the top six industry groups
accounted for 84 percent of manufacturing use of natural gas, so most of
the aggregate economic effect of natural gas price trends would be
evident in those industries.

Beyond the amount of natural gas used, the share of natural gas in total
production cost is another key factor in assessing impacts. Natural gas
costs represent a large proportion of the cost of doing business for
some sectors \-- 40 percent for the nitrogen fertilizer industry in
1998. For chemicals as a whole, natural gas costs were 2 to 3 percent of
overall costs in that same year.

Higher and more volatile natural gas prices have led industry analysts
to use a term -- "demand destruction" to refer to the impacts. The term
can have several connotations and has different meanings depending on
the speaker/writer. The first refers to the loss in natural gas
consumption as firms switch or substitute for other fuel sources. The
ability to do so depends on end-use of natural gas, the capital stock
and the technology available to the first. Conservation is another form
of reducing cost per unit of output. This has the aspect of "permanence"
to it, because there is little incentive for firms to return to less
efficient processes. Third, firms' output will decline in response to
higher input costs as they become less competitive domestically and
internationally. The IRM is not expected to have the capability to
decompose demand destruction into these three elements.

The purpose of this task is to develop a methodology for providing
response functions for key industrial output variables in the STIFS
model to energy price shocks, particularly those related to natural gas,
and to tie the resulting mechanism into the underlying macroeconomic
forecast in such a way as to preserve the expected relationships between
industrial output and aggregate income. A key output of the task will be
new EViews objects and programs that can readily be integrated in the
STIFS EViews system.

*Methodology*

Natural gas price movements and volatility will be analyzed individually
and relative to crude oil prices (specifically refiners' acquisition
cost of crude oil). Currently, industrial production indexes are used as
the macroeconomic drivers in parts of the natural gas consumption and
price module(s).

The VAR will be used to evaluate the current STIFS equations for
industrial natural gas prices and the forecasting implications of the
feedback models.

# THE DATA

Table 1 provides the variable names, short descriptions and the sources
of the data used in the study.

Industrial Natural Gas Consumption

The variable name NGINX is the mnemonic in STIFS for total industrial
natural gas demand (including gas used in industrial cogeneration
plants) in the United States. The series excludes gas used in oil and
gas field operations and at natural gas processing plants. The data
series extends from January 1989 through the current period (Figure 1).

**Figure 1.**

![](media/image1.wmf){width="6.0in" height="3.1368055555555556in"}

The industrial production index, calculated as the weighted average of
the top 6 gas-consuming manufacturing industries (where the weights are
year-2000 value added weights used by the Federal Reserve Board), is
illustrated in Figure 2.

**Figure 2.**

![](media/image2.wmf){width="6.372916666666667in"
height="3.3444444444444446in"}

Figure 3 presents monthly industrial natural gas consumption plotted
along with the gas-intensive industrial output index.

**Figure 3.**

![](media/image3.wmf){width="6.704166666666667in"
height="3.623611111111111in"}

Industrial Natural Gas Prices

Hamilton (1996) proposed a "net oil price" measure to capture the impact
of oil price shocks based on previous experience. It is calculated as
the differential between the maximum realized price in the previous
twelve months and the current price. If the current price is higher, the
term is equal to zero. Following Mork (1989), Gardner and Joutz (1996)
and Balke, Brown, and Yucel (1999) asymmetric effects are incorporated
by examining the impact of downward price shocks. A second "net oil
price" measure is constructed using the differential when it is
negative, otherwise it is zero.

Figure 4 indicates the monthly track for average industrial natural gas
prices, in nominal and real dollar terms. The industrial price shown
here is based on gas utility deliveries for own account, which represent
a relatively small fraction of total industrial sales. However, we note
that this price series exhibits month-to-month changes over time that
are consistent with monthly changes in average wellhead prices.

**Figure 4.**

![](media/image4.wmf){width="6.375694444444444in"
height="3.4902777777777776in"}

Figure 5 shows actual, maximum and "net price" concepts described above.

**Figu**![](media/image5.wmf){width="6.625in"
height="3.0520833333333335in"}**re 5.**

Figure 6 illustrates the "net price" calculation applied to the residual
fuel price (delivered to the electric power sector).

**Figure 6.**

![](media/image6.wmf){width="6.626388888888889in"
height="3.2395833333333335in"}

A variable which indicates the relative adequacy of natural gas in
underground storage at any point in time (gasvar in STIFS) is the
difference between beginning-period working gas storage and the previous
5-year average for the same month (Figure 7).

![](media/image7.wmf){width="6.625in"
height="3.2708333333333335in"}**Figure 7.**

Another important factor related to natural gas demand and price in the
short term is weather, particularly heating degree-days. Deviations from
normal heating degree-days (where normal is the trend-adjusted long-term
average), expressed on a per-day basis, are presented in Figure 8.

**Figure 8.**

![](media/image8.wmf){width="6.874305555555556in"
height="3.623611111111111in"}

**MODELING ISSUES AND SPECIFICATION AND EMPIRICAL RESULTS**

The research follows the general-to-specific modeling approach advocated
by Hendry (1986, 2000, and 2001). The general-to-specific modeling
approach is a relatively recent strategy used in econometrics. It
attempts to characterize the properties of the sample data in simple
parametric relationships which remain reasonably constant over time,
account for the findings of previous models, and are interpretable in an
economic and financial sense. Rather than using econometrics to
illustrate theory, the goal is to \"discover\" which alternative
theoretical views are tenable and test them scientifically.

The approach begins with a general hypothesis about the relevant
explanatory variables and dynamic process (i.e. the lag structure of the
model). The general hypothesis is considered acceptable to all
adversaries. Then the model is narrowed down by testing for
simplifications or restrictions on the general model.

The first step involves examining the time series properties of the
individual data series. We look at patterns and trends in the data and
test for stationarity and the order of integration. Second, we form a
Vector Autoregressive Regression (VAR) system. This step involves
testing for the appropriate lag length of the system, including residual
diagnostic tests and tests for model/system stability. Issues of
causality, impulse responses, and forecast error decomposition can be
addressed. Third, we examine the system for potential cointegration
relationship(s). \[**This latter part of the project is ongoing but not
completed here\]** Data series which are integrated of the same order
may be combined to form economically meaningful series which are
integrated of lower order. Fourth, we interpret the cointegrating
relations and test for weak exogeneity. Based on these results a
conditional error correction model of the endogenous variables is
specified, further reduction tests are performed and economic hypotheses
tested.

A VAR with lag length (or order) p can be expressed as

![](media/image9.wmf)

Where Y is (n \* 1) the number of variables

> A~0~ is (n \* 1) the matrix of intercepts and deterministic variables
> (n \* 1) the intercept plus seasonal dummies and or trend variables
>
> A~i~ is (n \* n) matrix of coefficients at lag i
>
> e~t~ is (n \* 1) the vector of error terms with mean 0 and
> contemporaneous variance covariance matrix ![](media/image10.wmf)
>
> X is (m \* 1) the set of exogenous variables
>
> B(L) Lag operator for exogenous variables

The econometric modeling techniques will involve issues related to
stationarity, integration, cointegration, and conditional volatility.
Applied econometricians face the following issues when working with VAR
models:

-   The choice of variables (energy consumption and supply, energy
    prices, income, drilling and exploration costs, taxes, and
    demographic ) to answer the problem

-   The determination of the lag length to capture the data generating
    process (DGP).

-   The parameterization of the model. Clearly it will be over
    parameterized. Many coefficients will be effectively zero. For
    example, variable j has no explanatory power for variable i at lag
    k.

-   The issue of multicollinearity. This makes it difficult to determine
    the important regressors using individual t-tests. Thus, F-tests or
    Chi-Square tests can help to detect this problem.

-   Model and parameter stability issues are important tests for the
    robustness of the model, due to changes in market structure, demand,
    regulatory effects, and technological change.

-   Granger Causality is a way to test for overparameterization and
    understanding the dynamic effects.

-   The issues of stationarity, integration, and cointegration are
    important for model specification and design.

-   The degrees of exogeneity: weak, strong, and super, are linked to
    inference testing, forecasting, and policy analysis.

What have been the contributions of VARs to empirical research and
forecasting?

-   A simple means to represent the time series dynamics of a set of
    variables looking at issues like: short-run vs. long-run effects,
    tests of Granger "Causality", tests of exogeneity, and cointegration
    analysis

-   A method to study the dynamic response of endogenous variables to
    shocks using Impulse Response Functions and Structural VARs (See
    Gamber and Joutz 1993 and 1995)

-   A method to examine empirically testable implications from economic
    theory. For example, can any parameters from a structural model be
    identified? What restrictions are necessary for identification?

-   A simple approach to generate forecasts incorporating theoretical
    models.

-   A method for decomposing time series into economically meaningful
    components. What is the contribution of different variables to
    explaining the movement in the series of interest? Forecast error
    variance decompositions are a means for answering this question.

*Integration and Stationarity Issues*

Several features of the data discussed in the previous section need to
be considered. Energy and economic data are often characterized by
trends of some sort, seasonality, shifts, and structural breaks. The
industrial natural gas sector data contains all of these.

The seasonal factors appear to be deterministic as they are relatively
constant. Peaks and troughs do not appear to shift during the course of
a year. Trends and their specification are important for time series
analysis, specification, estimation, and inference. There do appear to
be possible changes, breaks and shifts, in the natural gas series. A
partial explanation is a result of definitional and measurement issues
in the 1980s. (This is not a focus of the current research.) The sample
period for the analysis was chosen to begin in January 1989 to avoid
these problems and it is appropriate from an industrial market or
institutional perspective.

Analysis of the autocorrelation functions and partial autocorrelation
functions suggested that there were not seasonal unit roots, but that
there might be simple zero order unit roots. The null hypothesis of a
unit root is tested using the Augmented Dickey Fuller Test with seasonal
dummy variables, represented by:

![](media/image11.wmf)

Table 2 presents the ADF results test for: nominal industrial natural
gas prices; the net natural gas price; industrial natural gas
consumption, and industrial production. Conditional on the lag length
selection to remove serial correlation from the estimated residuals, the
null hypothesis is that ![](media/image12.wmf) Rejection suggests that
the series are stationary. There are eight columns in the table. The
first gives the maximum lag length for the dependent variable in each
test equation. The column headed beta Y_1 presentes the implied lagged
value in level form, and the third column, headed sigma, is the standard
error of the ADF regression. The next four columns are used for
selecting the lag lengh for the regressions: t_DY_lag is the t-statistic
associated with the maximum lag in ADF, t-prob is the p-value for the
statistic, and F-prob is the significance of the F-test on lags dropped
to that point. It appears as though ten lags should be used in each
test. We report the I(1) tests which all suggest that all four series
are integrated of order one, that is they have a unit root. Testing for
I(2) processes was performed, but not reported here.

*VAR Model Specification*

Two systems with five alternative model specifications are considered.
The first system has three equations for the industrial natural gas
price, industrial natural gas consumption, and the industrial production
index of the six major consuming industries. The second system has four
equations; a net industrial natural gas price equation is added to the
first three equations. Five alternative exogenous variable groupings
were considered in each VAR system. The most general model included
twelve lags of the endogenous variables, eleven seasonal dummy variables
for January through November, three event dummy variable for
unseasonable weather, and five exogenous variables. The five models are:

1.  No seasonal and event dummies and exogenous variables

2.  No event dummies and exogenous variables

3.  No exogenous variables

4.  No event dummies

5.  Include seasonal and event dummies and exogenous variables

The fifth model is considered the most general one and the other four
impose restriction on it.

The general form of the vector autoregressive (VAR) system that captures
the extent of feedback between the industrial sectors and (industrial)
natural gas prices is:

![](media/image13.wmf)

where

-   ngicuus(p) -- nominal price of natural gas to industry deflated by
    the producer price index for non-energy and food products, \$/MCF
    (1982-84=0).

-   pgi_n(net) - Hamilton "net price" of industrial natural gas, the
    difference between the highest monthly price in the last three years
    and the current price.

-   nginx(c) - natural gas demand by industry including CHP, BCFD.

-   gbig6(y) - is the industrial production index for the six largest
    industrial sectors consuming 80% of that sector's natural gas.

A(L) is a matrix polynomial lag operator where each matrix is 4x4

![](media/image14.wmf)

D is a 4x12 matrix of a constant and coefficients for 11 monthly dummy
variables in each

equation with December as the constant.

C is a 4x3 matrix of coefficients for the three extreme events. This
matrix can be

expanded later to include more events.

B is 4x5 matrix for the vector, x, of exogenous variables.

![](media/image15.wmf){width="0.8055555555555556in"
height="0.3055555555555556in"}

The vector is the vector of white noise residual terms which may be
contemporaneously correlated. The symbols (p, net, c, and y) represent
the price, "net price", consumption, and industrial production
respectively.

*Initial Model Selection*

Table 3.a and Table 3.b present the results of the zero restrictions
used for both systems. There are five columns in each table with the
model number in the first column. Columns two through five contain the
Log Likelihood, AIC, SC, and HQ statistics respectively. Model 4 appears
to be the most parsimonious representation in both systems, except by
the HQ statistic in the four variable model. Below each table is another
form of reduction test in finite samples. A sequence of F-tests is
compared against alternative models. Again, model 4's specification
appears to be appropriate. The p-value associated with reducing from
model 5 to model 4, removing the event dummies, is 0.80 and 0.24 in the
three variable and four variable VARs, respectively.

*VAR Model Lag Length Selection*

A primary benefit of VAR models is that they can capture the dynamics of
the system. The seasonal nature of the data and sample size suggests
that twelve lags may be an appropriate starting point. The selection
criteria for the appropriate lag length of the unrestricted VAR model
follow the asymptotic $\chi^{2}$ test(s) suggested by Sims (1980). The
maximum possible lag length considered was twelve, the number of months
in a year. Each model included eleven dummy monthly dummy variables and
the constant. The chi-squared statistic used was:

![](media/image16.wmf){width="3.3291666666666666in"
height="0.40694444444444444in"}

where det $\Sigma_{r}$ and det $\Sigma_{u}$ are the determinants of the
restricted and unrestricted covariance matrices of residuals
respectively. The number of observations is T, and c is a correction for
the number of variables in the unrestricted model. The adjustment is
used to improve the small sample properties of the statistic. This is
calculated as the number of estimated coefficients in all equations. If
there are n variables (equations), p lags of each variable, and an
intercept, we have c = n ^2^ \* p+ n coefficients. In our comparison the
test statistic is distributed as ![](media/image17.wmf) with n\*n(u-r)
degrees of freedom where u is the maximum number of lags and r is the
restricted lag length. The difference between u and r represents the
number of zero restrictions imposed on the lags for each variable in the
system.

In finite samples, the likelihood ratio approach is not as appealing
since it relies on asymptotic theory. In addition, this test applies
only in the case where one model is a restricted version of the other.
The Bayesian Scwhartz Criterion (BSC), the Akaike Information Criterion
(AIC), and the Hannan-Quinn Criterion are used as alternative criterion.
They rely on information similar to the Chi-Squared tests and are
derived as follows:

$\begin{array}{r}
\text{AIC = }\text{log}(\text{Det}\hat{\Sigma})\text{ + 2*c*}T^{\text{-1}} \\
\text{BSC = }\text{log}(\text{Det}\hat{\Sigma})\text{ + c*}\text{log}(T) \ast T^{\text{-1}} \\
\text{HC = }\text{log}(\text{Det}\hat{\Sigma})\text{ + 2*c*}\text{log}(\text{log}(T)) \ast T^{\text{-1}}
\end{array}$

Intuitively, the log determinant will decline as the number of
regressors increases, just as in a single equation ordinary least
squares regression. It is similar to the residual sum of squares or
estimated variance. The second term on the right hand side acts as a
penalty for including additional regressors; it increases the statistic.
We calculate these statistics for each lag length and choose the lag
length based on the model(s) with the minimum value for the statistics.
The three tests do not always agree as to the optimal lag lengths. The
AIC is biased towards selecting more lags than is actually needed, but
this is not necessarily bad.

Tables 4, 5, and 6 (4-variable model) and 7,8 and 9 (3-variable model)
were used in the lag length selection process. Model 4 for both systems
was estimated with twelve lags of the dependent variables. In the four
variable system with 176 observations, 48 degrees of freedom are used up
for these parameter estimates alone. The model is more than likely over
parameterized. An alternative model with the first six lags of each
variable and lag twelve was considered. The estimation results for these
models are shown in Tables 10-13. Tables 4 and 7 present the Lagrange
multiplier test for serial correlation through lag 12 in both systems.
It appears as though there is no serial correlation in the residuals
beyond two to three lags. These results at important lags may be
dominated by the over-parameterization of other lags. Similar results
are obtained in Tables 5 and 8. The AIC, SC, and HQ statistics are
minimized at 3,1, and 2 lags, respectively, in both the four variable
and three variable VAR systems. Tables 6 and 9 report the Wald Lag
Exclusion tests in the VARS where lags seven through eleven have been
omitted. The rows in the table report the contribution of the lagged
terms at given lags in each equation. For example, in the four equation
system at lag 1 the lag values do not provide explanatory power for the
LNGICUUS equation, but do for the other three variables: LPGI_N, LNGINX,
and LGBIG6. The last column in each row is a joint test for explanatory
power of all the variables at a particular lag across all equations. In
the four equation VAR, zero restriction on the lagged values provide
explanatory power at lag six and twelve. The respective p-values for the
Wald tests are 0.068 and 0.001. In the three equation system, the same
values are 0.35 and 0.026 respectively.

*VAR Model Residual Diagnostics*

Residual diagnostic tests examine the estimates for normality and serial
correlation. Tables 14 and 15 contain test results for skewness,
kurtosis, and the Jarque-Bera test. In both systems, there does not
appear to be a problem with skewness. However, there does appear to be a
problem with excess kurtosis. The equations for industrial consumption
and production have outliers that give a sharp peak to the estimated
residual distribution. This leads to rejection of the Jarque-Bera test
for normality in these equations and the systems as a whole. We could
model the outliers using dummy variables, but have chosen not too.

*VAR Model Stability*

Natural gas markets have experienced periods of "excess" supply or
"gluts" and tightness, and increases in price volatility. Any or all of
these events could have contributed to a "structural" break in the data
generating process between natural gas prices, industrial consumption,
and industrial output.

The textbook approach to model constancy assumes that modeler knows the
date of a possible structural break in the sample. He/she fits the model
over the full sample and for the two "halves" of the sample. The full
sample implicitly imposes the same model structure throughout and can be
considered a restricted model. This is evaluated against the
unrestricted model comprised of the two "halves" using an F-test. We
take an agnostic view on the possibility of structural breaks over the
1989-2003 sample.

Model constancy of the VAR system is evaluated using recursive
estimation techniques. Suppose the original model has T observations.
The technique begins by estimating the model over the first s\<T
observations in the sample and then fitting the model using s+1, s+2,
\..., up to T observations. At this point there are a number of tests
for evaluating model (and parameter) constancy. They are often best
presented in graphical form, because of the large number of statistics
which are calculated.

A familiar statistical presentation is the 1-step ahead residuals plus
the standard error bound used to search for outliers. The 1-step
residuals are given by
${\hat{e}}_{t}\text{ = }y_{t}\text{ - }x_{t^{'}}{\hat{\beta}}_{t}$ and
plotted with the current estimate of +/-2${\hat{\sigma}}_{t}$ on either
side of zero. When ${\hat{e}}_{t}$ is outside the band it can be
interpreted as an outlier. Standardized innovations are another way to
illustrate the presence of outliers.

We report two types of recursive Chow tests. The first is the 1-step
Chow test. This looks at the sequence of one period ahead predictions
from the recursive estimation for period s to T. The tests are
F(1,t-k-1) under the null hypothesis of parameter constancy. The
statistic is calculated as:

The test assumes that the dependent variables,$y_{t},$ are approximately
normally distributed.

The *N-down test or* *Break-Point Chow test* plots the test statistic
over the sample scaled by either the 5% or 1% critical value and can be
interpreted as a forecast stability test. The model is estimated for the
first s observations. A forecast is constructed from period s+1 through
T and the F-test is calculated. The null hypothesis is that the
estimated model and forecast will explain or predict the full sample the
same as model for the full sample. The number of periods in the
forecasts goes down from T-s+1 to 1 as the model is estimated and
forecasts are performed recursively. Recursive estimation permits
construction of Chow tests over the full sample and lets the data do the
talking. The

*Break-Point Chow test* is calculated as:

The results of the 1-step ahead Chow tests and the Break-point Chow test
are presented graphically in Figures 9 and 10 respectively. At each
observation the normalized test statistic calculated as the ratio of the
statistic to the appropriate critical value at 1%. The sequence of tests
is plotted. When the normalized value exceeds unity, this indicates a
rejection of the null hypothesis of no structural break.

The estimation sample begins in January 1989 and the model is fit
through December 1994 before the recursive analysis begins. The graphs
for the four-equation VAR system are provided; there is little
difference with the three equation VAR system. (They are available upon
request.) By construction the equations for the industrial natural gas
price and the "net" natural gas price are going to be unstable. In the
figures, the test statistics are normalized by the critical values.
Thus, when the lines exceed unity it represents a rejection of the null
hypothesis of insignificant errors or model stability. A single graph is
shown for each equation and the fifth graph is for the system as a
whole.

**Figure 9. 1 Step-Ahead Chow Test Results**

![](media/image18.wmf){width="5.997222222222222in" height="3.75in"}
**(4-Variable model)**

**Figure 10. Recursive N-Step Down Break Test Results**

**(4-Variable model)**

![](media/image19.wmf){width="6.0in" height="4.0in"}

The 1-step Chow graphs show only a few rejections for the price and net
price equations out of 104 tests. These occur in 1997, 1999, 2000, and
2003.

The N-step down tests have a similar but stronger pattern. The equation
estimates for the industrial natural gas price and the "net" natural gas
price show rejections up until the last few months. The models appear
unstable, because of the tremendous instability in the natural gas price
series in the last year but also in 2000. This is driven by the price
movements and heights which had previously not been realized. If the
recursive tests were performed on shorter samples, the models would have
appeared (more) stable. There are no rejections for the natural gas
consumption equation and industrial production equation. The graph for
the system as a whole shows the same rejection pattern as the price
equations.

*Impulse Responses*

A VAR representation, even in finite form, has an infinite VMA, vector
moving average representation. This important feature is used to trace
out the impact of a shock to the jth variable to the ith variable in the
system h periods ahead or the impulse response. The accumulated impulses
are calculated to construct long-run multipliers.

The VAR as estimated here is a reduced form or primitive system. The
contemporaneous innovation terms, ![](media/image20.wmf), are
correlated. Thus, the shocks are a combination of terms and have common
component(s) which cannot be associated with a specific variable.
Knowing the autoregressive coefficient matrices and the variance
covariance matrix of the estimated innovations is not sufficient to
identify the system. Calculation and interpretation of the impulse
responses would be subject to the ordering of the equations and not
unique.

The Choleski decomposition offers one possible identification approach
to transforming the innovations so that they are uncorrelated. This
imposes a "structure" on the system, which comes from economic theory
and industry knowledge. The uncorrelated or structural shocks,
![](media/image21.wmf) can be derived from a transformation G to the
innovations so that they are uncorrelated.

![](media/image22.wmf){width="5.384722222222222in"
height="3.161111111111111in"}

The transformed innovations,![](media/image23.wmf), retain the mean zero
property, but the variance-covariance matrix is diagonal. The
specification in the four variable VAR system is based on a general
assumption that prices are more flexible than real measures like
consumption and industrial production. The net price measure captures
the impact of the current price shock to previous maximum prices.
Innovations to consumption and industrial activity do not have a
contemporaneous impact on either the net price or price level. This
feeds into the shock to the current price level. The "structural" net
price shock has a contemporaneous impact,![](media/image24.wmf), on the
current level shock. There is no contemporaneous impact of innovations
to the current price level from current consumption, and industrial
activity. The net price and price level innovations do impact
consumption and economic activity. The specification in the three
variable VAR system uses the same ordering except that the net price
measure and innovations are not in the model.

> ![](media/image25.wmf){width="4.5in" height="2.9402777777777778in"}

Figure 11 and 12 show the cumulative impulse responses with two standard
error bounds. The Standard errors for the impulse responses are
calculated using 1000 repetitions in a Monte Carlo procedure. There are
three rows and four columns for the four variable VAR system and three
rows and three columns for the three variable VAR system. Impulse
responses for the net oil price measure are not computed. The rows show
the responses for a particular variable through twelve periods from the
"structural" shocks of each variable. The columns provide the impulse
responses from a particular variable on the variables in the system.

The "net" oil price measure has significant effects on the industrial
price series and industrial production. The price response (in Figure ZZ
first row and first column)is negative and consistent with the notion
that there has been a shock to the previous maximum experienced so
prices should begin to moderate. The revelation of a shock to the "net"
price has a negative impact on industrial production. This could be
interpreted as a signal to the economy of a forthcoming recession. The
impact of price shocks on the industrial price does have permanent and
significant impacts on industrial gas price level.

![](media/image26.wmf){width="5.992361111111111in"
height="4.030555555555556in"}**Figure 11. Accumulated Response to
Cholesky One s.d. Innovations**

**(4-Variable model)**

**Figure 12. Accumulated Response to Cholesky One s.d. Innovations**

![](media/image27.wmf){width="5.995138888888889in"
height="3.96875in"}**(3-Variable model)**

(This result occurs for all three own innovation shocks and is
consistent with the non-stationary properties of the data.) In addition,
this shock to appears to reduce industrial consumption and industrial
activity. The former occurs right away, while the later takes 6-12
months to occur and is not as statistically significant. There may be a
minor 1-2 month positive impact of industrial natural gas consumption
shocks on the industrial price level. The impact of this innovation on
industrial activity is indistinguishable from zero. Industrial
production innovations do not appear to have an impact on industrial
natural gas prices. There is a positive and permanent impact on
industrial natural gas consumption which is consistent with theory. An
explanation for this is that prices are strongly determined by other
market factors like petroleum prices, and non-market factors such as
weather.

The impulse responses in the three variable VAR model are similar to the
four variable model. The main difference is in the strength and
significance of the industrial price shocks. Consumption does not appear
to decline by as much, but appears to be more significant. The impact on
industrial activity is less significant.

*Forecast Error Variance Decompositions*

Tables 16 and 17 show the forecast error variance decompositions for the
4-variable system and the three variable system. In the four variable
system the orthogonalized innovations are ordered as the net-price
shock, the current natural gas price level shock, industrial natural gas
consumption, and the industrial production index. The three variable
system is the same except there is no net-price shock. The decomposition
for the net price variable is not presented as this is a created
variable.

Ninety-five percent of the forecast error variance for the natural gas
price level is explained by itself or the net price measure. Initially,
the net price explains slightly over half, but falls to forty-three
percent after one year. The own price shocks share of the variance moves
in a mirror image to the net price. Consumption shocks explain
ninety-five percent of the forecast error in the first month and decline
to about two-thirds by twelve months. The natural gas price shock and
production index explain sixteen percent and thirteen percent by the end
of a year respectively. Industrial production shocks also explain about
ninety five percent of their forecast error variance initially, but
decline to only eight-three percent after a year. The contributions of
the price level, net price, and natural gas consumption shocks are
eight, five, and four percent respectively by twelve months ahead.
However these three estimates are imprecise.

The decomposition of the forecast error variance in the three variable
system for the natural gas price is similar to the four variable system.
Over ninety-five percent is explained by the price measure over the
twelve month period. Short-run shocks to consumption of natural gas and
industrial production do not have a strong influence for forecast errors
for prices. Industrial consumption own shocks are major factors in its
own forecast errors in the first three months. After a year industrial
production and natural gas price level shocks contribute about twelve
percent and nine percent respectively to the forecast error variance for
consumption. Industrial production forecast shocks explain over ninety
percent of the forecast error variance through twelve months. The
difference between the four variable system and the three variable
system, despite the imprecision is that including the net price measure
in the model appears to inject price effects on industrial activity.

*Granger "Causality" Testing*

Table 18 presents the Granger Causality tests for the 4-variable model.
Each of the 4 variables appears to have explanatory power for one or
more of the other variables at the 10% level in the system. It is not
clear how to interpret the causal relationship from the real variables
to the net price. Lagged values of the natural gas price have very high
power for explaining the net price, but do not appear to preceed
consumption or industrial consumption. The net price term does help to
explain industrial production albeit at only 7%. Both lagged values of
industrial consumption and the production index appear to explain both
price measures, but not each other. There may some imprecision in the
later result. The p-value for industrial production on consumption is
13% and neither price measure appears to "cause" consumption. However,
all three variables do at the 5% level.

Table 19 presents the Granger Causality tests for the 3-variable model.
Each of the 3 variables appears to have explanatory power for one of the
other variables at the 10% level in the system. Lagged values of the
natural gas price have marginal power explaining the industrial
consumption at just above the 5% level. Both lagged values of industrial
consumption and the production index appear to explain the industrial
natural gas price measure at 6% and less than 1% respectively. But they
do not appear to help explain each other. The p-value for industrial
production on consumption is 20%. Jointly the lagged price measure and
industrial production index have a p-value of just under 8%. Industrial
production does not appear to "cause" the natural gas price level or
industrial consumption.

*Model Simulation*

The four variable model was simulated over the period January 1998
through August 2003. The dynamic stochastic solution was solved for in
Eviews 4.1. The values for the exogenous variables were used. Figure X
presents graphs of the "forecast" and actual levels for each series.
Figure XX presents graphs of the percent deviation of the "forecasted"
values from the actuals. Discontinuities in the net natural gas price
graph reflect ratios where the denominator was zero, the natural gas
price was at a new high.

> **Figure 13. Forecast Simulation Levels**

![](media/image28.wmf){width="6.0in" height="6.063194444444444in"}

**Figure 14. Forecast Simulation Percent Deviation from Actual**

![](media/image29.wmf){width="6.0in" height="5.939583333333333in"}

**CONCLUSIONS**

**QUESTIONS FOR THE COMMITTEE**

Does the VAR approach to modeling relationships between the variables
(industrial natural gas price, industrial natural gas consumption,
industrial production) appear fruitful?

Would you recommend another approach?

What do you think are the strengths and weaknesses of the approach?

How can the current model be improved?

Should VAR models for other sectors and fuels be considered? Which ones?

What do you think of the "net" price variable?

Are there other "net" price variables that you think should be
considered? Are

their better types of shock variables?

Is modeling the natural gas consumption of the BIG 6 industrial sectors

appropriate?

They are currently weighted by value-added in 2000. Are there other
weighting

schemes which should be considered like natural gas consumption weights?

**REFERENCES**

Table 1 Variable Definitions

+----------+--------------------------------------------------+-------+
| **Name** | **Description and Units**                        | **Sou |
|          |                                                  | rce** |
+----------+--------------------------------------------------+-------+
| **N      | nominal price of natural gas to industry,        |       |
| GICUUS** | \$/MCF.                                          |       |
+----------+--------------------------------------------------+-------+
| **NG     | real price of natural gas to industry deflated   |       |
| ICUUSR** | by the producer price index for non-energy and   |       |
|          | food products, \$/MCF, WPIINUS (1982-84=0).      |       |
+----------+--------------------------------------------------+-------+
| *        | Net Industrial Natural Gas Price Measure, the    |       |
| *PGI_N** | difference between the highest monthly price in  |       |
|          | the last three years and the current price.      |       |
+----------+--------------------------------------------------+-------+
| *        | Natural gas demand: industrial sector (Incl      |       |
| *NGINX** | CHP), BCFD, NSA                                  |       |
+----------+--------------------------------------------------+-------+
| *        | Weighted Industrial Production Index for six     | Fe    |
| *GBIG6** | largest natural gas consuming 2-digit sectors,   | deral |
|          | consuming 80% of that sector's natural gas.      | Re    |
|          |                                                  | serve |
|          | Weights by value-added in 2000                   | Board |
|          |                                                  |       |
|          | 0.0994\*g331+0.2760\*g311+0.117                  |       |
|          | 6\*g322+0.3544\*g325+0.0838\*g327+0.06886\*g324, |       |
|          | NSA, 1997=100                                    |       |
+----------+--------------------------------------------------+-------+
| **NG     | the lagged relative price of, ngspuus, to the    |       |
| SPRFTC** | lagged rftcuus, converted to MCF. The spot       |       |
|          | price, (in DMMB) is divided by 1.03 and the fuel |       |
|          | price (in \$/G) is divided by 6.287.             |       |
+----------+--------------------------------------------------+-------+
| **R      | price of retail residual fuel price, \$/MMBTU    |       |
| FTCEUS** |                                                  |       |
+----------+--------------------------------------------------+-------+
| **N      | spot natural gas, \$/TCF                         |       |
| GSPUUS** |                                                  |       |
+----------+--------------------------------------------------+-------+
| **R      | the price of residual fuel oil to electric       |       |
| FEUDUS** | utilities, (\$/MMBTU)                            |       |
+----------+--------------------------------------------------+-------+
| x(4)     | the deviation of heating degree days from        |       |
|          | normal, (ZGHDPUS-ZGHNPUS). They are population   |       |
|          | weighted and divided by the number of days in    |       |
|          | the month, ZSAJQUS.                              |       |
+----------+--------------------------------------------------+-------+
| **Z      | Monthly population weighted degree days          |       |
| WHDPUS** |                                                  |       |
+----------+--------------------------------------------------+-------+
| **Z      | Monthly population weighted normal heating       |       |
| WHNPUS** | degree-days. Normal = trend-adjusted 1971-2000   |       |
|          | average.                                         |       |
+----------+--------------------------------------------------+-------+
| x(5)     | Interaction term between cold weather monthly    |       |
|          | dummy variables and deviations of heating degree |       |
|          | days from normal. The monthly variables are      |       |
|          | October through April.                           |       |
+----------+--------------------------------------------------+-------+
| **Z      | The number of days per month                     |       |
| SAJQUS** |                                                  |       |
+----------+--------------------------------------------------+-------+
| **       | deviation of natural gas from normal storage     |       |
| GASVAR** | levels (BCFD)                                    |       |
+----------+--------------------------------------------------+-------+
| **       | Dummy variable for Hurricane Andrew which caused |       |
| D_HAND** | production shutdowns in September and October    |       |
|          | 1992.                                            |       |
+----------+--------------------------------------------------+-------+
| **       | Dummy variable for February 1994 as a result of  |       |
| D_9402** | cold weather, ice storms, and transmission       |       |
|          | problems.                                        |       |
+----------+--------------------------------------------------+-------+
| **       | Dummy variable for February 1996 for             |       |
| D_9602** | expectations of cold weather and low inventories |       |
|          | leading to panic in the spot price market.       |       |
+----------+--------------------------------------------------+-------+

**Table 2 Augmented Dickey-Fuller Tests**

sample is 1987 (2) - 2003 (8)

LNGICUUSR: ADF tests (T=199, Constant+Trend+Seasonals; 5%=-3.43
1%=-4.01)

D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob

12 -2.480 0.91413 0.06819 -0.9607 0.3380 -5.250

11 -2.736 0.90729 0.06817 0.1887 0.8506 -5.254 0.3380

[10 -2.764 0.90860 0.06799 3.782 0.0002 -5.264 0.6201]{.underline}

9 -2.086 0.92947 0.07051 -1.048 0.2961 -5.196 0.0022

8 -2.317 0.92295 0.07053 -0.02822 0.9775 -5.200 0.0034

7 -2.373 0.92278 0.07033 -0.8551 0.3936 -5.210 0.0076

6 -2.613 0.91692 0.07028 1.351 0.1784 -5.216 0.0112

5 -2.375 0.92603 0.07044 -0.4690 0.6396 -5.215 0.0105

4 -2.626 0.92184 0.07029 1.361 0.1753 -5.224 0.0171

3 -2.383 0.93058 0.07045 1.115 0.2663 -5.224 0.0153

2 -2.204 0.93703 0.07050 -1.096 0.2743 -5.227 0.0166

1 -2.541 0.92949 0.07054 1.134 0.2582 -5.231 0.0180

0 -2.351 0.93620 0.07059 -5.234 0.0189

LPGI_N: ADF tests (T=199, Constant+Trend+Seasonals; 5%=-3.43 1%=-4.01)

D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob

12 -3.616\* 0.77512 0.05189 1.725 0.0863 -5.796

11 -3.269 0.80216 0.05219 0.9763 0.3303 -5.789 0.0863

[10 -3.126 0.81642 0.05218 2.116 0.0358 -5.793 0.1425]{.underline}

9 -2.703 0.84361 0.05269 -0.2356 0.8140 -5.778 0.0404

8 -2.863 0.84016 0.05255 1.280 0.2023 -5.788 0.0790

7 -2.624 0.85776 0.05264 -0.6851 0.4942 -5.789 0.0753

6 -2.919 0.84780 0.05257 1.188 0.2364 -5.796 0.1055

5 -2.681 0.86680 0.05263 -0.06527 0.9480 -5.798 0.1040

4 -2.883 0.86569 0.05248 1.584 0.1148 -5.808 0.1542

3 -2.504 0.88867 0.05270 0.2134 0.8312 -5.805 0.1089

2 -2.586 0.89180 0.05256 0.2234 0.8235 -5.815 0.1521

1 -2.691 0.89516 0.05243 2.767 0.0062 -5.824 0.2030

0 -1.768 0.93482 0.05336 -5.794 0.0391

LNGINX: ADF tests (T=199, Constant+Trend+Seasonals; 5%=-3.43 1%=-4.01)

D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob

12 -0.07121 0.99809 0.02888 0.4462 0.6560 -6.968

11 -0.01693 0.99955 0.02881 0.2902 0.7720 -6.977 0.6560

[10 0.01415 1.0004 0.02874 -2.701 0.0076 -6.986 0.8682]{.underline}

9 -0.2360 0.99371 0.02925 -0.01246 0.9901 -6.956 0.0610

8 -0.2378 0.99369 0.02917 -3.713 0.0003 -6.966 0.1167

7 -0.5016 0.98627 0.03020 -1.013 0.3124 -6.901 0.0010

6 -0.5709 0.98441 0.03020 -2.634 0.0092 -6.905 0.0015

5 -0.7732 0.97861 0.03069 -0.4872 0.6267 -6.877 0.0002

4 -0.8209 0.97742 0.03063 -2.528 0.0123 -6.886 0.0004

3 -1.072 0.97025 0.03108 -1.430 0.1546 -6.861 0.0001

2 -1.243 0.96564 0.03117 -2.699 0.0076 -6.860 0.0001

1 -1.698 0.95296 0.03169 -3.323 0.0011 -6.831 0.0000

0 -2.345 0.93461 0.03254 -6.783 0.0000

LGBIG6: ADF tests (T=199, Constant+Trend+Seasonals; 5%=-3.43 1%=-4.01)

D-lag t-adf beta Y_1 sigma t-DY_lag t-prob AIC F-prob

12 -2.114 0.94987 0.006324 0.4178 0.6766 -10.01

11 -2.078 0.95156 0.006309 -0.5501 0.5830 -10.01 0.6766

[10 -2.219 0.94923 0.006296 2.812 0.0055 -10.02 0.7886]{.underline}

9 -1.802 0.95841 0.006418 0.9372 0.3500 -9.989 0.0431

8 -1.671 0.96197 0.006416 -0.9338 0.3517 -9.994 0.0603

7 -1.869 0.95816 0.006414 0.4676 0.6407 -9.999 0.0776

6 -1.823 0.95974 0.006400 0.2439 0.8076 -10.01 0.1181

5 -1.812 0.96038 0.006383 0.1993 0.8423 -10.02 0.1750

4 -1.807 0.96100 0.006366 0.1105 0.9121 -10.03 0.2437

3 -1.817 0.96137 0.006349 2.845 0.0049 -10.04 0.3225

2 -1.439 0.96908 0.006471 -0.2632 0.7927 -10.00 0.0536

1 -1.496 0.96827 0.006454 -3.594 0.0004 -10.01 0.0771

0 -2.039 0.95596 0.006659 -9.956 0.0025

beta Y_1 is the implied lagged value in level form, sigma is the
standard error of the ADF regression, t_DY_lag is the t-statistic
associated with the maximum lag in ADF, t-prob is the p-value for the
statistic, and F-prob is the significance of the F-test on lags dropped
to that point. \* is a rejection of the null for a unit root.

**Table 3.a Specification Selection from Alternative VAR Models with Net
Price**

**Sample 1989.01 -- 2003.08**

**Model Reduction via F-test**

MODEL(5) \--\> MODEL(4): F(32,444)= 1.1753 \[0.2382\]

MODEL(5) \--\> MODEL(3): F(40,456)= 2.2661 \[0.0000\]\*\*

MODEL(5) \--\> MODEL(2): F(52,466)= 1.9536 \[0.0002\]\*\*

MODEL(4) \--\> MODEL(1): F(96,477)= 3.7888 \[0.0000\]\*\*

MODEL(4) \--\> MODEL(3): F(8,256) = 6.8220 \[0.0000\]\*\*

MODEL(4) \--\> MODEL(2): F(20,425)= 3.1852 \[0.0000\]\*\*

MODEL(4) \--\> MODEL(1): F(64,503)= 5.0520 \[0.0000\]\*\*

MODEL(3) \--\> MODEL(2): F(12,344)= 0.83868 \[0.6104\]

MODEL(3) \--\> MODEL(1): F(56,507)= 4.4594 \[0.0000\]\*\*

MODEL(2) \--\> MODEL(1): F(44,510)= 5.4807 \[0.0000\]\*\*

The five models are:

1)  No seasonal and event dummies and exogenous variables

2)  No event dummies and exogenous variables

3)  No exogenous variables

4)  No event dummies

5)  Include seasonal and event dummies and exogenous variables

The fifth model is considered the most general one and the other four
impose restriction on it.

**Table 3.b Specification Selection from Alternative VAR Models without
Net Price**

**Sample 1989.01 -- 2003.08**

**Model Reduction via F-test**

MODEL( 5) \--\> MODEL( 4): F(12,349)= 0.64854 \[0.8001\]

MODEL( 5) \--\> MODEL( 3): F(20,438)= 3.6005 \[0.0000\]\*\*

MODEL( 5) \--\> MODEL( 2): F(32,488)= 2.5246 \[0.0000\]\*\*

MODEL( 5) \--\> MODEL(1): F(76,522)= 4.6889 \[0.0000\]\*\*

MODEL( 4) \--\> MODEL( 3): F(8,270) = 8.4265 \[0.0000\]\*\*

MODEL( 4) \--\> MODEL( 2): F(20,448)= 3.6998 \[0.0000\]\*\*

MODEL( 4) \--\> MODEL(1): F(64,530)= 5.4962 \[0.0000\]\*\*

MODEL( 3) \--\> MODEL( 2): F(12,362)= 0.69928 \[0.7524\]

MODEL( 3) \--\> MODEL(1): F(56,535)= 4.6585 \[0.0000\]\*\*

MODEL( 2) \--\> MODEL(1): F(44,537)= 5.7906 \[0.0000\]\*\*

The five models are:

1)  No seasonal and event dummies and exogenous variables

2)  No event dummies and exogenous variables

3)  No exogenous variables

4)  No event dummies

5)  Include seasonal and event dummies and exogenous variables

The fifth model is considered the most general one and the other four
impose restriction on it.

**Table 4 VAR Residual Serial Correlation LM Tests -- 4-variable Model**

================================

VAR Residual Serial Correlation LM Tests

H0: no serial correlation at lag order h

Date: 03/17/04 Time: 11:20

Sample: 1989:01 2003:08

Included observations: 176

================================

Lags LM-Stat Prob

================================

1 31.39807 0.0120

2 27.79299 0.0335

3 15.76848 0.4692

4 22.91516 0.1160

5 20.40675 0.2025

6 15.84146 0.4641

7 19.22812 0.2570

8 19.63372 0.2372

9 27.39964 0.0372

10 20.91918 0.1816

11 23.10515 0.1110

12 14.56703 0.5565

13 15.96538 0.4554

================================

Probs from chi-square with 16 df.

================================

================================

VAR Residual Serial Correlation LM Tests

H0: no serial correlation at lag order h

Date: 03/17/04 Time: 11:20

Sample: 1989:01 2003:08

Included observations: 176

================================

Lags LM-Stat Prob

================================

1 31.39807 0.0120

2 27.79299 0.0335

3 15.76848 0.4692

4 22.91516 0.1160

5 20.40675 0.2025

6 15.84146 0.4641

================================

Probs from chi-square with 16 df.

================================

**Table 5 VAR Lag Order Selection Criteria -- 4-variable Model**

=======================================================================

VAR Lag Order Selection Criteria

Endogenous variables: LNGICUUS LPGI_N LNGINX LGBIG6

Exogenous variables: D_SDUM GASVAR
(LNGSPUUS(-1)/1.03)/(LRFTCUUS(-1)/6.287) LRFEUDUS(-1)(

Date: 03/17/04 Time: 11:20

Sample: 1989:01 2003:08

Included observations: 176

=======================================================================

Lag LogL LR FPE AIC SC HQ

=======================================================================

0 414.9680 NA 2.18E-07 -3.988273 -2.835370 -3.520661

1 638.322 2168.673 2.41E-13 -17.70821 -16.26708\* -17.12369

2 1666.619 48.87676 2.10E-13 -17.84795 -16.11859 -17.14653\*

3 1683.195 27.87764 2.09E-13\* -17.85449\* -15.83691 -17.03617

4 1697.975 24.18582 2.13E-13 -17.84063 -15.53482 -16.90540

5 1710.113 19.31002 2.25E-13 -17.79674 -15.20271 -16.74461

6 1726.091 24.69390 2.27E-13 -17.79649 -14.91424 -16.62746

7 1737.091 16.49861 2.42E-13 -17.73967 -14.56918 -16.45373

8 1744.581 10.89518 2.71E-13 -17.64297 -14.18426 -16.24013

9 1764.430 27.96874 2.63E-13 -17.68670 -13.93977 -16.16696

10 1780.312 21.65814 2.69E-13 -17.68537 -13.65021 -16.04873

11 1800.372 26.44253 2.62E-13 -17.73150 -13.40812 -15.97796

12 1826.468 33.21300\* 2.40E-13 -17.84623 -13.23462 -15.97578

13 1839.125 15.53289 2.57E-13 -17.80823 -12.90840 -15.82088

=======================================================================

\* indicates lag order selected by the criterion

LR: sequential modified LR test statistic (each test at 5% level)

FPE: Final prediction error

AIC: Akaike information criterion

SC: Schwarz information criterion

HQ: Hannan-Quinn information criterion

**Table 6 VAR Lag Exclusion Wald Tests -- 4-variable Model**

======================================================================

VAR Lag Exclusion Wald Tests

Date: 03/17/04 Time: 11:20

Sample: 1989:01 2003:08

Included observations: 176

======================================================================

Chi-squared test statistics for lag exclusion:

Numbers in \[ \] are p-values

=======================================================================

LNGICUUS LPGI_N LNGINX LGBIG6 Joint

=======================================================================

Lag 1 5.270330 64.65935 35.49432 74.77634 285.7763

\[ 0.260667\] \[ 3.04E-13\] \[ 3.68E-07\] \[ 2.22E-15\] \[ 0.000000\]

Lag 2 2.047120 1.865048 7.109933 10.93084 23.17889

\[ 0.727092\] \[ 0.760562\] \[ 0.130192\] \[ 0.027352\] \[ 0.109035\]

Lag 3 3.947669 2.316255 10.08036 2.489416 16.30239

\[ 0.413134\] \[ 0.677811\] \[ 0.039096\] \[ 0.646532\] \[ 0.432063\]

Lag 4 1.472194 0.852193 1.591576 14.38032 20.00502

\[ 0.831555\] \[ 0.931317\] \[ 0.810305\] \[ 0.006175\] \[ 0.219995\]

Lag 5 4.794720 6.198192 0.926794 5.059021 18.62108

\[ 0.309016\] \[ 0.184828\] \[ 0.920686\] \[ 0.281295\] \[ 0.288802\]

Lag 6 5.153824 7.345143 0.642838 4.319195 25.08342

\[ 0.271875\] \[ 0.118735\] \[ 0.958187\] \[ 0.364528\] \[ 0.068368\]

Lag 12 14.99228 16.55351 15.58120 1.883287 38.32953

\[ 0.004717\] \[ 0.002360\] \[ 0.003636\] \[ 0.757215\] \[ 0.001358\]

=======================================================================

df 4 4 4 4 16

=======================================================================

**Table 7 VAR Residual Serial Correlation LM Tests -- 3-variable Model**

================================

VAR Residual Serial Correlation LM Tests

H0: no serial correlation at lag order h

Date: 03/17/04 Time: 11:33

Sample: 1989:01 2003:08

Included observations: 176

================================

Lags LM-Stat Prob

================================

1 25.58759 0.0024

2 15.32495 0.0824

3 8.403164 0.4941

4 3.360731 0.9483

5 7.150594 0.6214

6 3.291020 0.9516

7 6.122042 0.7276

8 8.944652 0.4424

9 10.34945 0.3229

10 7.257987 0.6103

11 5.552807 0.7837

12 7.863126 0.5480

13 19.32661 0.0226

================================

Probs from chi-square with 9 df.

================================

================================

VAR Residual Serial Correlation LM Tests

H0: no serial correlation at lag order h

Date: 03/17/04 Time: 11:33

Sample: 1989:01 2003:08

Included observations: 176

================================

Lags LM-Stat Prob

================================

1 25.58759 0.0024

2 15.32495 0.0824

3 8.403164 0.4941

4 3.360731 0.9483

5 7.150594 0.6214

6 3.291020 0.9516

================================

Probs from chi-square with 9 df.

================================

**Table 8 VAR Lag Order Selection Criteria** **-- 3-variable Model**

=======================================================================

VAR Lag Order Selection Criteria

Endogenous variables: LNGICUUS LNGINX LGBIG6

Exogenous variables: D_SDUM GASVAR
(LNGSPUUS(-1)/1.03)/(LRFTCUUS(-1)/6.287) LRFEUDUS(-1)(

Date: 03/17/04 Time: 11:33

Sample: 1989:01 2003:08

Included observations: 176

=======================================================================

Lag LogL LR FPE AIC SC HQ

=======================================================================

0 336.7440 NA 7.55E-06 -3.281182 -2.416505 -2.930473

1 1299.724 1718.045 1.48E-10 -14.12187 -13.09506\* -13.70540

2 1317.669 31.40273 1.34E-10 -14.22351 -13.03458 -13.74129\*

3 1330.250 21.58727 1.29E-10\* -14.26420\* -12.91314 -13.71622

4 1336.728 10.89522 1.33E-10 -14.23554 -12.72236 -13.62180

5 1340.208 5.733706 1.42E-10 -14.17281 -12.49750 -13.49331

6 1345.924 9.224497 1.48E-10 -14.13550 -12.29806 -13.39025

7 1348.999 4.857553 1.59E-10 -14.06818 -12.06861 -13.25716

8 1350.599 2.471871 1.74E-10 -13.98408 -11.82238 -13.10731

9 1359.210 13.01442 1.76E-10 -13.97966 -11.65584 -13.03713

10 1362.882 5.424074 1.88E-10 -13.91911 -11.43316 -12.91082

11 1373.092 14.73586 1.87E-10 -13.93287 -11.28479 -12.85882

12 1385.133 16.96660\* 1.82E-10 -13.96742 -11.15722 -12.82762

13 1391.689 9.014742 1.89E-10 -13.93965 -10.96732 -12.73409

=======================================================================

\* indicates lag order selected by the criterion

LR: sequential modified LR test statistic (each test at 5% level)

FPE: Final prediction error

AIC: Akaike information criterion

SC: Schwarz information criterion

HQ: Hannan-Quinn information criterion

**Table 9. VAR Lag Exclusion Wald Tests -- 3-variable Model**

============================================================

VAR Lag Exclusion Wald Tests

Date: 03/17/04 Time: 11:33

Sample: 1989:01 2003:08

Included observations: 176

============================================================

Chi-squared test statistics for lag exclusion:

Numbers in \[ \] are p-values

============================================================

LNGICUUS LNGINX LGBIG6 Joint

============================================================

Lag 1 7.378234 30.97330 73.43093 116.8882

\[ 0.060771\] \[ 8.61E-07\] \[ 7.77E-16\] \[ 0.000000\]

Lag 2 3.384349 2.672137 9.408560 14.50934

\[ 0.336074\] \[ 0.444983\] \[ 0.024324\] \[ 0.105326\]

Lag 3 2.191392 9.177107 2.136779 11.98391

\[ 0.533646\] \[ 0.027027\] \[ 0.544508\] \[ 0.214218\]

Lag 4 2.172937 1.925789 6.618795 11.54303

\[ 0.537299\] \[ 0.587951\] \[ 0.085093\] \[ 0.240313\]

Lag 5 4.190698 0.710040 0.335169 5.441438

\[ 0.241595\] \[ 0.870840\] \[ 0.953284\] \[ 0.794256\]

Lag 6 6.618718 1.101272 2.243517 9.905260

\[ 0.085096\] \[ 0.776767\] \[ 0.523428\] \[ 0.358209\]

Lag 12 6.415887 13.97250 0.618776 18.89453

\[ 0.093039\] \[ 0.002943\] \[ 0.892121\] \[ 0.026105\]

============================================================

df 3 3 3 9

============================================================

**Table 10 VAR Model 4 Estimates -- 4-variable Model**

================================================================

Vector Autoregression Estimates

Date: 03/17/04 Time: 11:19

Sample: 1989:01 2003:08

Included observations: 176

Standard errors in ( ) & t-statistics in \[ \]

================================================================

LNGICUUS LPGI_N LNGINX LGBIG6

================================================================

LNGICUUS(-1) 0.287897 0.744002 -0.129315 -0.000272

(0.13736) (0.12256) (0.06148) (0.01477)

\[ 2.09596\] \[ 6.07027\] \[-2.10331\] \[-0.01838\]

LNGICUUS(-2) -0.144766 -0.097603 0.190289 -0.014991

(0.17842) (0.15920) (0.07986) (0.01919)

\[-0.81138\] \[-0.61307\] \[ 2.38276\] \[-0.78134\]

LNGICUUS(-3) 0.354016 -0.229617 -0.138990 0.000209

(0.18121) (0.16170) (0.08111) (0.01949)

\[ 1.95359\] \[-1.42005\] \[-1.71357\] \[ 0.01074\]

LNGICUUS(-4) 0.019742 0.068920 0.061742 0.039021

(0.18319) (0.16347) (0.08200) (0.01970)

\[ 0.10776\] \[ 0.42162\] \[ 0.75297\] \[ 1.98086\]

LNGICUUS(-5) -0.162058 -0.095696 -0.005212 -0.043685

(0.18308) (0.16336) (0.08195) (0.01969)

\[-0.88518\] \[-0.58579\] \[-0.06360\] \[-2.21899\]

LNGICUUS(-6) 0.011517 0.229775 -0.024301 0.020543

(0.13179) (0.11760) (0.05899) (0.01417)

\[ 0.08738\] \[ 1.95388\] \[-0.41195\] \[ 1.44953\]

LNGICUUS(-12) -0.098526 0.032786 0.010449 0.003024

(0.07152) (0.06382) (0.03201) (0.00769)

\[-1.37755\] \[ 0.51373\] \[ 0.32639\] \[ 0.39317\]

LPGI_N(-1) 0.105696 0.940494 -0.173931 0.003039

(0.13759) (0.12277) (0.06158) (0.01480)

\[ 0.76820\] \[ 7.66060\] \[-2.82425\] \[ 0.20542\]

LPGI_N(-2) -0.096189 -0.108337 0.176760 -0.023913

(0.19698) (0.17577) (0.08817) (0.02118)

\[-0.48832\] \[-0.61637\] \[ 2.00479\] \[-1.12893\]

LPGI_N(-3) 0.331026 -0.179966 -0.037264 -0.010962

(0.19961) (0.17811) (0.08935) (0.02146)

\[ 1.65835\] \[-1.01040\] \[-0.41708\] \[-0.51071\]

LPGI_N(-4) -0.108473 0.142557 0.032840 0.060972

(0.20323) (0.18134) (0.09096) (0.02185)

\[-0.53375\] \[ 0.78613\] \[ 0.36102\] \[ 2.79005\]

LPGI_N(-5) -0.073425 -0.196914 0.002894 -0.042782

(0.20478) (0.18272) (0.09166) (0.02202)

Table 10 VAR Model 4 Estimates -- 4-variable Model (cont.)

\[-0.35856\] \[-1.07767\] \[ 0.03158\] \[-1.94285\]

LPGI_N(-6) -0.004763 0.109936 -0.007020 0.006597

(0.14173) (0.12647) (0.06344) (0.01524)

\[-0.03360\] \[ 0.86929\] \[-0.11066\] \[ 0.43287\]

LPGI_N(-12) 0.073573 -0.045359 -0.016278 0.003592

(0.03573) (0.03188) (0.01599) (0.00384)

\[ 2.05937\] \[-1.42286\] \[-1.01795\] \[ 0.93488\]

LNGINX(-1) 0.201483 -0.108195 0.420108 0.006009

(0.19562) (0.17456) (0.08756) (0.02104)

\[ 1.02995\] \[-0.61983\] \[ 4.79786\] \[ 0.28565\]

LNGINX(-2) -0.195245 0.182893 0.046376 -0.003170

(0.21783) (0.19437) (0.09750) (0.02342)

\[-0.89634\] \[ 0.94097\] \[ 0.47565\] \[-0.13534\]

LNGINX(-3) -0.015049 -0.077879 0.104828 0.024400

(0.22274) (0.19875) (0.09970) (0.02395)

\[-0.06756\] \[-0.39185\] \[ 1.05146\] \[ 1.01874\]

LNGINX(-4) 0.002234 0.019978 0.024278 -0.023239

(0.21823) (0.19473) (0.09768) (0.02347)

\[ 0.01024\] \[ 0.10260\] \[ 0.24854\] \[-0.99030\]

LNGINX(-5) -0.042128 -0.119468 0.087621 -0.014810

(0.21692) (0.19356) (0.09709) (0.02333)

\[-0.19421\] \[-0.61723\] \[ 0.90245\] \[-0.63491\]

LNGINX(-6) 0.101346 -0.072345 -0.004898 0.020340

(0.19020) (0.16972) (0.08513) (0.02045)

\[ 0.53284\] \[-0.42627\] \[-0.05753\] \[ 0.99448\]

LNGINX(-12) -0.419928 0.496882 0.200590 0.007475

(0.15115) (0.13488) (0.06766) (0.01625)

\[-2.77814\] \[ 3.68400\] \[ 2.96480\] \[ 0.45991\]

LGBIG6(-1) -0.202540 0.074533 -0.285934 0.702550

(0.76504) (0.68264) (0.34243) (0.08227)

\[-0.26475\] \[ 0.10918\] \[-0.83501\] \[ 8.54001\]

LGBIG6(-2) 0.792735 -0.645274 0.568402 0.284152

(0.92929) (0.82921) (0.41595) (0.09993)

\[ 0.85306\] \[-0.77818\] \[ 1.36651\] \[ 2.84356\]

LGBIG6(-3) 0.051056 -0.348207 0.383378 0.066576

(0.92426) (0.82472) (0.41370) (0.09939)

\[ 0.05524\] \[-0.42221\] \[ 0.92671\] \[ 0.66986\]

LGBIG6(-4) -0.360190 0.041935 -0.353259 -0.172181

(0.91125) (0.81311) (0.40788) (0.09799)

\[-0.39527\] \[ 0.05157\] \[-0.86609\] \[-1.75715\]

LGBIG6(-5) -1.638171 1.574120 -0.003206 0.004452

Table 10 VAR Model 4 Estimates -- 4-variable Model (cont.)

(0.89376) (0.79750) (0.40005) (0.09611)

\[-1.83290\] \[ 1.97381\] \[-0.00802\] \[ 0.04632\]

LGBIG6(-6) 1.741810 -0.939555 0.206487 0.057740

(0.78645) (0.70175) (0.35202) (0.08457)

\[ 2.21477\] \[-1.33886\] \[ 0.58658\] \[ 0.68276\]

LGBIG6(-12) -0.036252 -0.064098 -0.423171 0.040753

(0.40704) (0.36320) (0.18219) (0.04377)

\[-0.08906\] \[-0.17648\] \[-2.32266\] \[ 0.93108\]

D_JAN 0.126452 -0.103603 0.010359 0.015432

(0.04283) (0.03821) (0.01917) (0.00461)

\[ 2.95273\] \[-2.71119\] \[ 0.54040\] \[ 3.35097\]

D_FEB 0.069101 -0.091028 0.032387 0.023024

(0.04740) (0.04230) (0.02122) (0.00510)

\[ 1.45778\] \[-2.15211\] \[ 1.52645\] \[ 4.51698\]

D_MAR -0.005711 -0.020717 -0.049405 0.023733

(0.05322) (0.04749) (0.02382) (0.00572)

\[-0.10730\] \[-0.43623\] \[-2.07388\] \[ 4.14684\]

D_APR -0.077436 0.040126 -0.023078 0.027852

(0.05550) (0.04953) (0.02484) (0.00597)

\[-1.39512\] \[ 0.81019\] \[-0.92892\] \[ 4.66652\]

D_MAY -0.136495 0.110981 -0.073138 0.025592

(0.05858) (0.05227) (0.02622) (0.00630)

\[-2.33005\] \[ 2.12317\] \[-2.78932\] \[ 4.06264\]

D_JUN -0.114032 0.102069 -0.029495 0.044057

(0.06446) (0.05751) (0.02885) (0.00693)

\[-1.76917\] \[ 1.77469\] \[-1.02234\] \[ 6.35648\]

D_JUL -0.145142 0.135995 -0.041745 0.015624

(0.07049) (0.06290) (0.03155) (0.00758)

\[-2.05908\] \[ 2.16218\] \[-1.32311\] \[ 2.06124\]

D_AUG -0.147739 0.123219 -0.000750 0.039345

(0.06053) (0.05401) (0.02709) (0.00651)

\[-2.44083\] \[ 2.28142\] \[-0.02768\] \[ 6.04489\]

D_SEP -0.094972 0.093653 0.013841 0.034050

(0.06295) (0.05617) (0.02818) (0.00677)

\[-1.50862\] \[ 1.66722\] \[ 0.49119\] \[ 5.02991\]

D_OCT -0.084950 0.072979 0.021403 0.024897

(0.05364) (0.04787) (0.02401) (0.00577)

\[-1.58363\] \[ 1.52467\] \[ 0.89141\] \[ 4.31617\]

D_NOV 0.045905 -0.052810 0.027995 -0.000861

(0.04654) (0.04152) (0.02083) (0.00500)

\[ 0.98643\] \[-1.27177\] \[ 1.34399\] \[-0.17198\]

**Table 10 VAR Model 4 Estimates -- 4-variable Model (cont.)**

GASVAR -2.57E-05 1.43E-05 -1.12E-05 -3.62E-06

(2.3E-05) (2.0E-05) (1.0E-05) (2.5E-06)

\[-1.12144\] \[ 0.69975\] \[-1.09279\] \[-1.46700\]

(LNGSPUUS(-1)/1.0 0.229781UUS-0.1971637) -0.015937 0.000652

(0.02638) (0.02353) (0.01181) (0.00284)

\[ 8.71204\] \[-8.37757\] \[-1.34991\] \[ 0.22979\]

LRFEUDUS(-1) 0.122034 -0.080275 0.021330 -0.005302

(0.03244) (0.02894) (0.01452) (0.00349)

\[ 3.76241\] \[-2.77365\] \[ 1.46923\] \[-1.52024\]

(ZGHDPUS-ZGHNPUS) 0.007998 -0.005333 0.001057 0.001170

(0.01247) (0.01112) (0.00558) (0.00134)

\[ 0.64154\] \[-0.47943\] \[ 0.18936\] \[ 0.87281\]

((ZGHDPUS-ZGHNPUS-0.004989)\*( 0.005551OV+ 0.002602AN+-0.002115AR+D_APR)

(0.01290) (0.01151) (0.00577) (0.00139)

\[-0.38670\] \[ 0.48220\] \[ 0.45065\] \[-1.52446\]

================================================================

R-squared 0.963781 0.942207 0.944071 0.993703

Adj. R-squared 0.951982 0.923381 0.925851 0.991652

Sum sq. resids 0.427287 0.340208 0.085606 0.004941

S.E. equation 0.056895 0.050767 0.025466 0.006118

F-statistic 81.68578 50.04688 51.81672 484.4450

Log likelihood 280.0957 300.1508 421.5733 672.5700

Akaike AIC -2.682906 -2.910804 -4.290606 -7.142841

Schwarz SC -1.890285 -2.118183 -3.497985 -6.350220

Mean dependent 1.198498 0.191695 3.049033 4.564978

S.D. dependent 0.259640 0.183407 0.093522 0.066961

================================================================

Determinant Residual Covaria 8.59E-14

Log Likelihood (d.f. adjuste 1648.564

Akaike Information Criteria -16.73368

Schwarz Criteria -13.56320

================================================================

**Table 11. Residual Covariance Matrix and Residual Correlation Matrix
-- 4-variable Model**

Residual Covariance Matrix

============================================================

LNGICUUS LPGI_N LNGINX LGBIG6

============================================================

LNGICUUS 0.003237 -0.002093 -0.000298 1.19E-05

LPGI_N -0.002093 0.002577 8.77E-05 -4.58E-05

LNGINX -0.000298 8.77E-05 0.000649 2.37E-05

LGBIG6 1.19E-05 -4.58E-05 2.37E-05 3.74E-05

============================================================

Residual Correlation Matrix

============================================================

LNGICUUS LPGI_N LNGINX LGBIG6

============================================================

LNGICUUS 1.000000 -0.724639 -0.205955 0.034305

LPGI_N -0.724639 1.000000 0.067869 -0.147444

LNGINX -0.205955 0.067869 1.000000 0.152363

LGBIG6 0.034305 -0.147444 0.152363 1.000000

============================================================

**Table 12 VAR Model 4 Estimates -- 3-variable Model**

====================================================

Vector Autoregression Estimates

Date: 03/17/04 Time: 11:33

Sample: 1989:01 2003:08

Included observations: 176

Standard errors in ( ) & t-statistics in \[ \]

====================================================

LNGICUUS LNGINX LGBIG6

====================================================

LNGICUUS(-1) 0.273367 -0.014864 -0.002008

(0.10617) (0.04735) (0.01148)

\[ 2.57476\] \[-0.31392\] \[-0.17489\]

LNGICUUS(-2) -0.116986 0.044866 0.000356

(0.09354) (0.04171) (0.01011)

\[-1.25067\] \[ 1.07556\] \[ 0.03523\]

LNGICUUS(-3) 0.127767 -0.103248 0.004244

(0.09498) (0.04236) (0.01027)

\[ 1.34526\] \[-2.43766\] \[ 0.41325\]

LNGICUUS(-4) 0.131427 0.047361 -0.005870

(0.09714) (0.04332) (0.01050)

\[ 1.35298\] \[ 1.09327\] \[-0.55888\]

LNGICUUS(-5) -0.109748 -0.006394 -0.005719

(0.09936) (0.04431) (0.01074)

\[-1.10455\] \[-0.14430\] \[-0.53234\]

LNGICUUS(-6) 0.080073 -0.035292 0.008059

(0.07667) (0.03419) (0.00829)

\[ 1.04444\] \[-1.03225\] \[ 0.97210\]

LNGICUUS(-12) -0.022562 0.029290 -0.000105

(0.03984) (0.01777) (0.00431)

\[-0.56636\] \[ 1.64864\] \[-0.02428\]

LNGINX(-1) 0.245431 0.459438 0.013588

(0.19075) (0.08507) (0.02063)

\[ 1.28667\] \[ 5.40091\] \[ 0.65877\]

LNGINX(-2) -0.235751 0.023319 0.000890

(0.21851) (0.09745) (0.02363)

\[-1.07890\] \[ 0.23930\] \[ 0.03768\]

LNGINX(-3) -0.076772 0.115495 0.016812

(0.22355) (0.09969) (0.02417)

\[-0.34342\] \[ 1.15849\] \[ 0.69550\]

LNGINX(-4) 0.004582 0.034346 -0.024428

(0.21926) (0.09778) (0.02371)

\[ 0.02090\] \[ 0.35125\] \[-1.03032\]

LNGINX(-5) -0.013000 0.075493 -0.007529

(0.21935) (0.09782) (0.02372)

Table 12 VAR Model 4 Estimates -- 3-variable Model (cont.)

\[-0.05926\] \[ 0.77173\] \[-0.31742\]

LNGINX(-6) 0.084283 0.001369 0.017236

(0.19245) (0.08582) (0.02081)

\[ 0.43795\] \[ 0.01595\] \[ 0.82824\]

LNGINX(-12) -0.344889 0.187180 -0.002210

(0.13974) (0.06232) (0.01511)

\[-2.46811\] \[ 3.00364\] \[-0.14627\]

LGBIG6(-1) -0.374625 -0.153760 0.676567

(0.73859) (0.32938) (0.07987)

\[-0.50722\] \[-0.46681\] \[ 8.47133\]

LGBIG6(-2) 1.024924 0.468911 0.296977

(0.89884) (0.40085) (0.09719)

\[ 1.14028\] \[ 1.16980\] \[ 3.05552\]

LGBIG6(-3) -0.207756 0.341637 0.119909

(0.91966) (0.41013) (0.09944)

\[-0.22591\] \[ 0.83300\] \[ 1.20579\]

LGBIG6(-4) -0.562040 -0.360435 -0.221223

(0.90461) (0.40342) (0.09782)

\[-0.62130\] \[-0.89344\] \[-2.26157\]

LGBIG6(-5) -1.501972 0.070735 0.002698

(0.90115) (0.40188) (0.09744)

\[-1.66672\] \[ 0.17601\] \[ 0.02769\]

LGBIG6(-6) 1.848446 0.034942 0.078845

(0.78583) (0.35045) (0.08497)

\[ 2.35222\] \[ 0.09971\] \[ 0.92788\]

LGBIG6(-12) 0.086181 -0.319208 0.032678

(0.38705) (0.17261) (0.04185)

\[ 0.22266\] \[-1.84931\] \[ 0.78078\]

D_JAN 0.127624 0.003192 0.016252

(0.04244) (0.01893) (0.00459)

\[ 3.00698\] \[ 0.16866\] \[ 3.54120\]

D_FEB 0.060128 0.028736 0.025855

(0.04657) (0.02077) (0.00504)

\[ 1.29114\] \[ 1.38368\] \[ 5.13437\]

D_MAR -0.005397 -0.057179 0.024360

(0.05359) (0.02390) (0.00580)

\[-0.10071\] \[-2.39240\] \[ 4.20363\]

D_APR -0.070200 -0.029669 0.029076

(0.05575) (0.02486) (0.00603)

\[-1.25911\] \[-1.19327\] \[ 4.82297\]

D_MAY -0.119669 -0.085314 0.026356

Table 12 VAR Model 4 Estimates -- 3-variable Model (cont.)

(0.05767) (0.02572) (0.00624)

\[-2.07512\] \[-3.31728\] \[ 4.22652\]

D_JUN -0.097035 -0.041265 0.044590

(0.06375) (0.02843) (0.00689)

\[-1.52216\] \[-1.45150\] \[ 6.46864\]

D_JUL -0.121911 -0.054568 0.016892

(0.06936) (0.03093) (0.00750)

\[-1.75754\] \[-1.76402\] \[ 2.25215\]

D_AUG -0.129213 -0.012117 0.040805

(0.05983) (0.02668) (0.00647)

\[-2.15956\] \[-0.45411\] \[ 6.30696\]

D_SEP -0.068003 -0.002869 0.034453

(0.06152) (0.02743) (0.00665)

\[-1.10544\] \[-0.10458\] \[ 5.17932\]

D_OCT -0.062770 0.006301 0.026705

(0.05189) (0.02314) (0.00561)

\[-1.20958\] \[ 0.27226\] \[ 4.75907\]

D_NOV 0.054702 0.017055 -0.000537

(0.04604) (0.02053) (0.00498)

\[ 1.18810\] \[ 0.83063\] \[-0.10787\]

GASVAR -2.51E-05 -9.49E-06 -2.60E-06

(2.2E-05) (9.7E-06) (2.3E-06)

\[-1.15970\] \[-0.98193\] \[-1.10744\]

(LNGSPUUS(-1)/1.0 0.218352UUS-0.0149167) 0.002096

(0.02516) (0.01122) (0.00272)

\[ 8.67942\] \[-1.32954\] \[ 0.77041\]

LRFEUDUS(-1) 0.127404 0.020053 -0.006051

(0.03274) (0.01460) (0.00354)

\[ 3.89144\] \[ 1.37345\] \[-1.70913\]

(ZGHDPUS-ZGHNPUS) 0.010049 0.000481 0.001003

(0.01248) (0.00557) (0.00135)

\[ 0.80526\] \[ 0.08635\] \[ 0.74364\]

((ZGHDPUS-ZGHNPUS-0.006277)\*( 0.002839OV+-0.002004AN+D_FEB+D_M

(0.01289) (0.00575) (0.00139)

\[-0.48676\] \[ 0.49369\] \[-1.43691\]

====================================================

R-squared 0.960590 0.939589 0.993072

Adj. R-squared 0.950383 0.923943 0.991277

Sum sq. resids 0.464932 0.092466 0.005436

S.E. equation 0.057835 0.025792 0.006254

F-statistic 94.11177 60.05272 553.4372

Log likelihood 272.6655 414.7899 664.1596

Akaike AIC -2.678017 -4.293067 -7.126813

Schwarz SC -2.011495 -3.626545 -6.460291

Table 12 VAR Model 4 Estimates -- 3-variable Model (cont.)

Mean dependent 1.198498 3.049033 4.564978

S.D. dependent 0.259640 0.093522 0.066961

====================================================

Determinant Residual Covaria 8.16E-11

Log Likelihood (d.f. adjuste 1294.964

Akaike Information Criteria -13.45413

Schwarz Criteria -11.45457

**Table 13 Residual Covariance Matrix and Residual Correlation Matrix --
3-variable Model**

Residual Covariance Matrix

================================================

LNGICUUS LNGINX LGBIG6

================================================

LNGICUUS 0.003345 -0.000312 9.30E-06

LNGINX -0.000312 0.000665 2.06E-05

LGBIG6 9.30E-06 2.06E-05 3.91E-05

================================================

Residual Correlation Matrix

================================================

LNGICUUS LNGINX LGBIG6

================================================

LNGICUUS 1.000000 -0.209370 0.025716

LNGINX -0.209370 1.000000 0.127941

LGBIG6 0.025716 0.127941 1.000000

================================================

**Table 14 VAR Residual Normality Tests -- 4-variable Model**

============================================================

VAR Residual Normality Tests

Orthogonalization: Cholesky (Lutkepohl)

H0: residuals are multivariate normal

Date: 03/17/04 Time: 11:20

Sample: 1989:01 2003:08

Included observations: 176

============================================================

Component Skewness Chi-sq df Prob.

============================================================

1 0.136160 0.543830 1 0.4608

2 -0.090889 0.242316 1 0.6225

3 0.107840 0.341129 1 0.5592

4 0.044076 0.056986 1 0.8113

============================================================

Joint 1.184261 4 0.8807

============================================================

Component Kurtosis Chi-sq df Prob.

============================================================

1 2.706974 0.629673 1 0.4275

2 2.627047 1.020024 1 0.3125

3 1.846095 9.764309 1 0.0018

4 1.848915 9.716634 1 0.0018

============================================================

Joint 21.13064 4 0.0003

============================================================

Component Jarque-Bera df Prob.

================================================

1 1.173503 2 0.5561

2 1.262340 2 0.5320

3 10.10544 2 0.0064

4 9.773620 2 0.0075

================================================

Joint 22.31490 8 0.0044

================================================

**Table 15 VAR Residual Normality Tests -- 3-variable Model**

============================================================

VAR Residual Normality Tests

Orthogonalization: Cholesky (Lutkepohl)

H0: residuals are multivariate normal

Date: 03/17/04 Time: 11:33

Sample: 1989:01 2003:08

Included observations: 176

============================================================

Component Skewness Chi-sq df Prob.

============================================================

1 0.122927 0.443254 1 0.5056

2 0.093147 0.254509 1 0.6139

3 -0.001845 9.99E-05 1 0.9920

============================================================

Joint 0.697863 3 0.8737

============================================================

Component Kurtosis Chi-sq df Prob.

============================================================

1 2.972848 0.005406 1 0.9414

2 2.038016 6.786367 1 0.0092

3 1.981236 7.611122 1 0.0058

============================================================

Joint 14.40290 3 0.0024

============================================================

Component Jarque-Bera df Prob.

================================================

1 0.448661 2 0.7991

2 7.040876 2 0.0296

3 7.611222 2 0.0222

================================================

Joint 15.10076 6 0.0195

================================================

**Table 16 Forecast Error Variance Decomposition-- 4-variable Model**

Accumulated Response of LNGICUUS to Cholesky (d.f. adjusted) One S.D.
Innovations

======

Variance Decomposition

=============================================================

=============================================================

Variance Decomposition of LNGICUUS:

Period S.E. LNGICUUS LPGI_N LNGINX LGBIG6

=============================================================

1 0.056895 47.48981 52.51019 0.000000 0.000000

(5.15201) (5.15201) (0.00000) (0.00000)

2 0.058330 48.27076 51.00714 0.679370 0.042732

(5.39924) (5.41679) (1.63272) (0.82944)

3 0.058432 48.10515 50.83175 0.715138 0.347967

(5.37982) (5.34985) (1.82084) (1.36956)

4 0.060178 50.45118 48.07494 0.707163 0.766714

(5.53839) (5.43560) (1.99055) (1.80815)

5 0.062305 53.66571 44.90191 0.665675 0.766699

(5.88288) (5.59733) (2.08205) (1.88103)

6 0.063220 52.72743 43.99587 0.659477 2.617216

(5.87343) (5.59345) (2.19998) (2.45390)

7 0.063353 52.62825 43.87896 0.709927 2.782861

(5.82457) (5.52575) (2.24148) (2.52092)

8 0.063908 53.28807 43.27494 0.698409 2.738579

(5.93138) (5.56973) (2.29607) (2.53317)

9 0.064239 53.62137 42.84445 0.799314 2.734862

(5.98532) (5.65561) (2.34534) (2.57913)

10 0.064384 53.42614 42.98891 0.795739 2.789218

(6.03269) (5.67330) (2.38778) (2.64054)

11 0.064569 53.46140 42.81164 0.796779 2.930182

(6.15356) (5.75426) (2.41768) (2.76735)

12 0.064690 53.52450 42.65829 0.836984 2.980223

(6.25453) (5.84753) (2.45288) (2.89672)

=============================================================

Variance Decomposition of LNGINX:

Period S.E. LNGICUUS LPGI_N LNGINX LGBIG6

=============================================================

1 0.025466 5.175461 0.460625 94.36391 0.000000

(3.18390) (1.31564) (3.46207) (0.00000)

2 0.028535 10.70294 1.143457 87.79773 0.355879

(5.32970) (2.02391) (5.36546) (1.08097)

3 0.029318 10.92888 1.743545 86.71040 0.617176

(5.62424) (2.40905) (5.86538) (1.35871)

4 0.030862 13.65918 2.273187 81.82410 2.243526

(6.11734) (2.44565) (6.48772) (2.44245)

5 0.031406 13.95292 2.195142 80.80724 3.044698

(6.39356) (2.50484) (6.79742) (3.05949)

6 0.032325 13.96075 2.144964 79.73188 4.162403

(6.59987) (2.48022) (7.15951) (3.76976)

7 0.033356 14.44442 2.019650 77.84041 5.695524

(6.85629) (2.43977) (7.54841) (4.35195)

8 0.034111 15.37175 2.213262 75.61729 6.797702

(7.39319) (2.52762) (7.98120) (4.94405)

9 0.034933 15.48773 2.111017 73.18139 9.219864

**Table 16 Forecast Error Variance Decomposition -- 4-variable Model
(cont.)**

(7.76537) (2.48450) (8.49868) (5.88049)

10 0.035502 15.55951 2.048614 71.76576 10.62612

(8.15700) (2.47953) (8.94351) (6.60581)

11 0.036048 15.80479 1.999634 70.31397 11.88161

(8.58035) (2.46069) (9.40412) (7.33417)

12 0.036711 16.09753 1.956545 68.50036 13.44557

(9.08310) (2.42459) (9.87381) (8.12237)

=============================================================

Variance Decomposition of LGBIG6:

Period S.E. LNGICUUS LPGI_N LNGINX LGBIG6

=============================================================

1 0.006118 1.108014 2.173979 2.030553 94.68745

(1.71028) (2.24977) (2.25780) (3.60181)

2 0.007481 1.184114 1.828749 2.393418 94.59372

(2.21720) (2.46628) (2.87555) (4.29693)

3 0.008962 2.220173 2.897503 2.378810 92.50351

(3.06784) (3.28216) (3.12830) (5.29093)

4 0.010598 4.542468 4.959113 3.478528 87.01989

(4.22593) (4.34734) (4.04407) (6.97011)

5 0.011401 4.822637 4.457173 3.441369 87.27882

(4.80965) (4.25132) (4.44076) (7.47867)

6 0.012184 5.661782 4.334024 3.099243 86.90495

(5.55601) (4.33589) (4.45552) (8.10825)

7 0.012972 5.838266 4.632218 3.290663 86.23885

(5.85853) (4.62128) (4.77594) (8.72004)

8 0.013555 6.177866 4.749761 3.383178 85.68919

(6.33086) (4.80101) (5.04431) (9.33102)

9 0.014105 6.568133 4.835848 3.617084 84.97893

(6.82983) (4.92153) (5.34463) (9.98340)

10 0.014687 7.309755 4.789435 3.850270 84.05054

(7.50367) (4.93052) (5.59941) (10.6891)

11 0.015158 7.612971 4.841800 3.969519 83.57571

(8.05004) (4.99085) (5.80123) (11.2908)

12 0.015583 8.039772 4.845988 4.130304 82.98394

(8.69460) (5.00323) (6.02420) (11.9464)

=============================================================

Cholesky Ordering: LPGI_N LNGICUUS LNGINX LGBIG6

Standard Errors: Monte Carlo (1000 repetitions)

**Table 17 Forecast Error Variance Decomposition-- 3-variable Model**

Accumulated Response to Cholesky (d.f. adjusted) One S.D. Innovations

======

Variance Decomposition

==================================================

==================================================

Variance Decomposition of LNGICUUS:

Period S.E. LNGICUUS LNGINX LGBIG6

==================================================

1 0.057835 100.0000 0.000000 0.000000

(0.00000) (0.00000) (0.00000)

2 0.059939 98.89096 0.959201 0.149834

(1.92624) (1.74818) (0.94286)

3 0.060115 98.44672 0.981278 0.571999

(2.50856) (1.96157) (1.57239)

4 0.060661 97.80122 1.344585 0.854192

(2.98330) (2.34670) (1.95148)

5 0.061480 97.82485 1.329310 0.845842

(3.10672) (2.40529) (2.11530)

6 0.062273 95.44628 1.321417 3.232302

(3.87552) (2.45135) (3.16048)

7 0.062384 95.27811 1.322299 3.399594

(3.95924) (2.53727) (3.24262)

8 0.062467 95.26395 1.318795 3.417252

(4.04580) (2.61615) (3.28767)

9 0.062484 95.22490 1.335799 3.439302

(4.11157) (2.66051) (3.34203)

10 0.062532 95.08378 1.364020 3.552202

(4.21032) (2.68854) (3.41873)

11 0.062625 94.90318 1.393777 3.703041

(4.31869) (2.71634) (3.50705)

12 0.062627 94.89772 1.396827 3.705449

(4.38186) (2.74526) (3.55448)

==================================================

Variance Decomposition of LNGINX:

Period S.E. LNGICUUS LNGINX LGBIG6

==================================================

1 0.025792 4.383600 95.61640 0.000000

(3.16991) (3.16991) (0.00000)

2 0.028438 5.006378 94.88149 0.112136

(4.17252) (4.23495) (0.86158)

3 0.029138 4.843088 94.64404 0.512871

(3.97496) (4.24076) (1.49580)

4 0.030704 7.689274 90.12447 2.186253

(4.52740) (4.99034) (2.59601)

5 0.031324 7.884667 89.24637 2.868962

(4.64338) (5.27084) (3.22541)

6 0.032172 7.533281 88.56162 3.905099

(4.53742) (5.59537) (4.08024)

7 0.033148 8.345063 86.54397 5.110964

(4.70635) (6.06885) (4.72220)

8 0.033775 9.307385 84.65260 6.040019

(5.05025) (6.57656) (5.33749)

9 0.034452 9.198521 82.64655 8.154925

**Table 17 Forecast Error Variance Decomposition-- 3-variable Model
(cont.)**

(5.02710) (7.13382) (6.24376)

10 0.034958 9.251518 81.48261 9.265873

(5.09686) (7.58295) (6.97348)

11 0.035432 9.397141 80.22703 10.37583

(5.20393) (8.04760) (7.68441)

12 0.036003 9.322856 78.57578 12.10136

(5.23730) (8.58796) (8.47594)

==================================================

Variance Decomposition of LGBIG6:

Period S.E. LNGICUUS LNGINX LGBIG6

==================================================

1 0.006254 0.066132 1.859055 98.07481

(0.95770) (2.13692) (2.29780)

2 0.007584 0.056286 2.734236 97.20948

(1.53408) (2.90059) (3.27697)

3 0.008965 0.045653 3.306685 96.64766

(1.60239) (3.66721) (3.98312)

4 0.010468 0.047263 4.762711 95.19003

(1.59353) (4.66940) (4.90352)

5 0.011268 0.099779 4.929649 94.97057

(1.80678) (5.18544) (5.46912)

6 0.012004 0.247418 4.742382 95.01020

(1.90277) (5.41635) (5.74599)

7 0.012749 0.229202 5.107096 94.66370

(1.87020) (5.77243) (6.06028)

8 0.013289 0.229839 5.405538 94.36462

(1.92823) (6.15904) (6.44865)

9 0.013830 0.244212 5.711616 94.04417

(1.96403) (6.50416) (6.79659)

10 0.014388 0.257035 6.085096 93.65787

(1.98858) (6.85510) (7.13775)

11 0.014865 0.260440 6.333732 93.40583

(2.04566) (7.17184) (7.46440)

12 0.015314 0.271367 6.595151 93.13348

(2.09680) (7.49410) (7.79960)

==================================================

Cholesky Ordering: LNGICUUS LNGINX LGBIG6

Standard Errors: Monte Carlo (1000 repetitions)

================================================

+-----------+-----------+-----------+----------+----------+----------+
| **Table   |           |           |          |          |          |
| 18:       |           |           |          |          |          |
| Granger   |           |           |          |          |          |
| "C        |           |           |          |          |          |
| ausality" |           |           |          |          |          |
| Tests: 4  |           |           |          |          |          |
| Variable  |           |           |          |          |          |
| Model**   |           |           |          |          |          |
|           |           |           |          |          |          |
| Sample    |           |           |          |          |          |
| 1989.01   |           |           |          |          |          |
| 2003.08   |           |           |          |          |          |
| (176      |           |           |          |          |          |
| obse      |           |           |          |          |          |
| rvations) |           |           |          |          |          |
+-----------+-----------+-----------+----------+----------+----------+
|           | **        |           |          |          |          |
|           | Dependent |           |          |          |          |
|           | V         |           |          |          |          |
|           | ariable** |           |          |          |          |
+-----------+-----------+-----------+----------+----------+----------+
| **        |           | LNGICUUS  | Net      | LNGINX   | LGBIG6   |
| Exclusion |           |           | Price    |          |          |
| Restr     |           |           |          |          |          |
| ictions** |           |           |          |          |          |
|           |           |           |          |          |          |
| **p       |           |           |          |          |          |
| -values** |           |           |          |          |          |
+-----------+-----------+-----------+----------+----------+----------+
|           | LNGICUUS  | \--       | 0.00     | 0.347    | 0.324    |
+-----------+-----------+-----------+----------+----------+----------+
|           | Net Price | 0.113     | \--      | 0.158    | 0.067    |
+-----------+-----------+-----------+----------+----------+----------+
|           | LNGINX    | 0.049     | 0.013    | \--      | 0.728    |
+-----------+-----------+-----------+----------+----------+----------+
|           | LGBIG6    | 0.002     | 0.005    | 0.130    | \--      |
+-----------+-----------+-----------+----------+----------+----------+
|           | All       | 0.000     | 0.00     | 0.044    | 0.5934   |
+-----------+-----------+-----------+----------+----------+----------+

+-----------+-----------+-----------+----------+----------+----------+
| **Table   |           |           |          |          |          |
| 19:       |           |           |          |          |          |
| Granger   |           |           |          |          |          |
| "C        |           |           |          |          |          |
| ausality" |           |           |          |          |          |
| Tests: 3  |           |           |          |          |          |
| -         |           |           |          |          |          |
| -Variable |           |           |          |          |          |
| Model**   |           |           |          |          |          |
|           |           |           |          |          |          |
| Sample    |           |           |          |          |          |
| 1989.01   |           |           |          |          |          |
| 2003.08   |           |           |          |          |          |
| (176      |           |           |          |          |          |
| obse      |           |           |          |          |          |
| rvations) |           |           |          |          |          |
+-----------+-----------+-----------+----------+----------+----------+
|           | **        |           |          |          |          |
|           | Dependent |           |          |          |          |
|           | V         |           |          |          |          |
|           | ariable** |           |          |          |          |
+-----------+-----------+-----------+----------+----------+----------+
| **        |           | LNGICUUS  | LNGINX   | LGBIG6   |          |
| Exclusion |           |           |          |          |          |
| Restr     |           |           |          |          |          |
| ictions** |           |           |          |          |          |
|           |           |           |          |          |          |
| **p       |           |           |          |          |          |
| -values** |           |           |          |          |          |
+-----------+-----------+-----------+----------+----------+----------+
|           | LNGICUUS  | \--       | 0.054    | 0.973    |          |
+-----------+-----------+-----------+----------+----------+----------+
|           | LNGINX    | 0.063     | \--      | 0.844    |          |
+-----------+-----------+-----------+----------+----------+----------+
|           | LGBIG6    | 0.002     | 0.201    | \--      |          |
+-----------+-----------+-----------+----------+----------+----------+
|           | All       | 0.000     | 0.078    | 0.979    |          |
+-----------+-----------+-----------+----------+----------+----------+