Comparison of the different levels of Divisia
Monetary Aggregates with respect to output.
Thesis for M.Sc. Economics
Specialization: Monetary Policy and BankingStudent: Eleni Papaioannou (10392351) Supervisor : Ed Westerhout
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Table of Contents
1. Introduction ... 3
2. Construction of Divisia Monetary Aggregates ... 4
3. Results from past literature ... 9
4. Explanation of Data ... 11
5. SVAR model ... 12
6. Granger Causality test ... 14
7. Impulse Response Function and Forecast Error Variance Decomposition ... 15
i. Impulse Response Function and Forecast Error Variance Decomposition 1967Q1 to 2013Q2... 16
ii. Impulse Response Function and Forecast Error Variance Decomposition 1967Q1 to 1980Q1 ... 22
iii. Impulse Response Function and Forecast Error Variance Decomposition 1980Q1 to 2013Q2 ... 27
8. Conclusion ... 32
9. References ... 34
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1. Introduction
If we pose a simple question as “what is the basic concept of economic science?”, most of the answers would be “the science that deals with money issues in the society.” But, what is actually the role of money in the economy? What are the effects of an increase in money supply? Is it possible to use money as a tool for the construction of monetary policy? These questions have been in the center of interest in the economic society for many decades. In most of the economic theoretical models, which are used for academic purposes, money is considered to be neutral or totally ignored. But in real life as well, Central Banks do not really use the word “money” in their announcements. Especially, after ‘80s and ever since the monetary policy is focused on the control of the short-run interest rate in order to stabilize the aggregate indicators of the economy.
The recent financial crisis has turned the conventional monetary policy into a useless tool of the Central Banks, especially since the policy rate has reached the zero lower bound. Central Banks, in their effort to stimulate the economy, have resorted to unconventional monetary policy tools. Also the interest about the impulse of money growth volatility on real economic activity revived (Serletis and Rahman (2009, 2013),).
There has been great discussion whether or not the monetary aggregates are useful as indicators in the construction of policy. Some economists support the idea of a better measurement of monetary aggregates. The transition from the use of simple-sum monetary
Abstract
Simple-sum monetary aggregates have received a lot of criticism and many economists support the idea of using Divisia monetary aggregates instead. The purpose of this paper is to check whether the different levels of Divisia monetary aggregates, which are constructed for the Center for Financial Stability about US economy from 1967 until 2013, perform better with respect to output in different time periods. The results of Impulse Response Functions (IRF) and Forecast Error Variance Decomposition (FEVD) by applying an SVAR model estimation are used to determine which level contains more information about the output. According to IRF and FEVD analysis it is concluded that while from 1967Q1 until 2013Q2 the broad levels of Divisia monetary aggregates outperform the narrow levels, when the data are separated in two periods, 1967Q1 to 1980Q1 and 1980Q1 to 2013Q2, the narrow levels exceed the broad levels.
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aggregates to Divisia monetary aggregates is considered to be successful, as the latter give a better prediction of the economic activity. Plenty of papers support that differences in the construction method helps them to outperform simple-sum monetary aggregates, not only in the case of U.S. but also for other countries (see for example; Thorton and Yue (1992), Ishida (1984), Starcca (2004), Dahalan et al. (2005) etc.).
Taking for granted that Divisia Monetary Aggregates outperform simple-sum monetary aggregates, the basic question of my thesis is “Which of the different levels of the new Divisia Monetary Aggregates, constructed for the Center for Financial Stability, are more informative about the changes of output, which in our case is the real GDP?”. Supplementary, a robustness test will be conducted for three different time periods in order to examine whether or not there is a change in the performance of these different levels. The data are from 1967Q1 to 2013Q2 for the case of U.S. economy.
The paper is set out as follows; first, a brief description of the method that is used for the construction of Divisia Monetary Aggregates will be presented, so that the differences from simple-sum monetary aggregates become clear. Thereafter, the econometric model that is used for the analysis and comparison and is a Structural VAR model will be discussed. The next section will contain evidence from past literature. After that, the results of Impulse Response Function and Forecast Error Variance Decomposition will be presented, and the paper will end with the conclusion that will sums up the most outstanding results.
2. Construction of Divisia Monetary Aggregates
Consumers prefer to hold assets in equilibrium. These assets are means of exchange and provide them service by increasing their utility.
Innovation and liberation, that characterized financial market in the past decades, made difficult to accurately measure money. This situation made more obvious the separation between money and macroeconomic variables. Monetary aggregates were no more useful in the construction of monetary policy by Central Banks and made them choose short term interest rate as their key variable.
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A possible explanation of money’s failure to predict macroeconomic variables can be, as Barnett (1980) suggests, that in simple-sum monetary aggregates the different assets are assigned an equal and intertemporally constant weight. In addition, Fisher (1921) states that “index numbers that are computed with a simple arithmetic mean average should not be used
under any circumstances”.
A simple presentation of the construction of simple-sum and Divisia monetary aggregates follows, in order to make clear the differences between these two aggregate measures.
The simple-sum method of aggregation, stated as index , is equal to: ∑ ( ).
Equation (1) assumes that each monetary component is assigned a constant and equal unitary weight.
The simple-sum monetary aggregate can be interpreted as the discounted present value of current and expected expenditures on monetary services, plus the discounted present value of all present and expected future investments that yield from holding a good, which in this case is a monetary asset.
Moreover, equation (1) implies that the components of monetary aggregates must be perfect and dollar-for-dollar substitutes. This suggests that relative prices of the monetary components are constant and equal over time, which was rejected by many empirical results. Divisia monetary aggregates are based on microeconomic foundations of decision making by consumers and firms. By that they do not impose strong ex ante assumptions regarding the elasticities of substitution among the monetary assets.
Barnett (1980) constructed the Divisia Monetary Aggregates which were named after the Törnquist-Theil Divisia index by using the aggregation theory and index number theory. The basic equation for computing the CFS Divisia Monetary Aggregate, stated as , is:
∑
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With
∑ being the expenditure share of asset i during period t and
[ ] the average of expenditure shares from the two adjacent periods.
From equation (2) it is obvious that the growth rate of the index is a weighted average of the growth rates of its components. The weights are the share contributions of each component to the total value of the services of all components.
The price , of each component quantity of asset , is needed in order to compute equation (2) for aggregation over monetary assets. The appropriate price of a component durable good, according to the economic quantity aggregation theory, is the user cost. If we focus on the consumer’s decision model, the present value of the interest foregone by holding a good (monetary asset) is the consumer’s user cost of the good (monetary asset).
Barnett’s (1980) formulation of user cost when tax is included is:
( )( )
( ) ( ) : the true cost of living index
: the own current period holing yield on component
: the maximum available expected holing-period yield in the economy or benchmark rate : the marginal tax rate.
Following (3) the corresponding real user cost is ⁄ and depends on the interest foregone of the asset. The foregone interest depends on the interest of the asset and the higher expected rate of return on the benchmark rate. The benchmark rate should also meet some requirements. The benchmark rate ( ) is the own rate of return on an asset that will provide no monetary services in all periods, except of the last period. For simplicity, an asset that is used by consumers in order to transfer wealth from one period to another is used.
From equations (1) and (3) it is clear that a basic difference between these two equations is that simple-sum aggregates is based on a simple summation index while the divisia index requires the growth rate of the aggregate to be equal to the weighted average of the growth rates of the component quantities. Moreover, the computation of the Divisia Monetary Aggregates requires the price of the asset in each period.
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The construction of Divisia monetary aggregates contains the same assets as the simple sum. Below a short description of components of Divisia Monetary Aggregates is illustrated.
Divisia M1 Aggregate
It is the sum of seasonally adjusted levels of currency, traveler’s checks and non-interest-bearing deposits which are paired with a zero interest rate. Thus, it contains the most liquid monetary-asset components. Demand deposits and other-checkable-deposits (OCDs) are adjusted for retail sweeps, because otherwise M1 would be underestimated.
Divisia M2, MZM and M2M Aggregate
Besides the components of M1, they contain also savings deposits, money-market deposit accounts, small-denomination time deposits and retail money funds with their interest rates. Each of the divisia M2-variant contains different components. More specific and following the structure of simple-sum aggregates used by Fed:
→ M2M contains saving deposits, MMDA, and retail money-market funds but no small
time deposits.
→ MZM contains all components of M2M and the institutional money-market funds → M2 contains small–denomination time-deposit level and rates but does not include institutional money-market funds
→ M2-ALL contains M2 and the institutional money-market funds
Divisia M3 Aggregate
Contains all the components of M2-ALL and the level of large time deposits and overnight and term repurchase agreements.
Divisia M4M and M4 Aggregates
They are substituting the simple-sum L aggregate that has been discontinued by the Federal-Reserve. The M4 aggregates include apart from M3, also the levels and rates of return on the primary, negotiable, money-market securities, including commercial paper, large-denomination time deposits, overnight repurchase agreements (repos) and short term treasury bills (T-bills).Divisia M4 includes T-bills rates to round out the most general Divisia aggregate within this project while M4- excludes it for use in applications that require separation of monetary from fiscal policy effects. Divisia M4 is the broadest aggregate and is only provided by the CFS’s program AMFM.
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Table 1
Construction of Divisia Monetary Aggregates
Assets Divisia M1 Divisia M2M Divisia MZM Divisia M2 Divisia M3 Divisia M4- Divisia M4 Currency
*
*
*
*
*
*
*
Travel Checks*
*
*
*
*
*
*
Demand Deposits*
*
*
*
*
*
*
OCD Commercial*
*
*
*
*
*
*
OCD Thrift*
*
*
*
*
*
*
Saving Deposits Commercial*
*
*
*
*
*
Saving Deposits Thrift*
*
*
*
*
*
Retail Money Market Funds*
*
*
*
*
*
Small Time Deposits Commercial*
*
*
*
Small time Deposit Thrift*
*
*
*
Institutional Money Market Funds*
*
*
*
Large Time Deposits*
*
*
Overnight and Term Repos*
*
*
Commercial Paper*
*
T-Bills*
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More details about the construction and the specific data that were used for the computation are presented on Barnett et.al (2012) and all the data are available by the Center for Financial Stability upon request. The purpose of this paper is to compare the different levels of Divisia Monetary Aggregates’ ability to explain the fluctuations of GDP in US economy and not to compute them. For that reason, data provided by the Central for Financial Stability (CFS) will be used.
3. Results from past literature
According to the aforementioned, the question about the relation between money and real economic activity has been again a matter of great interest among the economic community after the 1980s. There is a great amount of literature on this topic, using different economic tools to give an answer. After Barnett’s (1980) combination of Divisia Index theory with aggregation theory, many economists compared Divisia Monetary Aggregates with simple-sum Monetary Aggregates. They were interested on whether or not this superior way of money measurement makes any difference in the explanation of the link between money volatility and real economic activity.
In Belognia’s (1996) paper, the author tries to replicate results from five previous studies: Stock and Watson (1989), Kydland and Prescott (1990), Cover (1992), Friedman and Kuttner (1992, 1993), and Rotemberg (1993), that used simple-sum monetary aggregates. In four out of these five cases the initial results were rejected simply by using Divisia Monetary Aggregates instead of sum monetary aggregates. He concludes that the use of simple-sum aggregates sabotaged the connection between money and macroeconomic variables. The superiority of Divisia Monetary Aggregates relative to the simple-sum is also proved in the paper of Darrat et al. (2004) with respect to four breaking points that represent four major events in the early 1980s. These events are the FED’s swift of operating target from the Federal funds to nonborrowed reserves (October 1979), the passage of the Depository Institutions Deregulation and Monetary Control Act (DIDMCA) on March 1980, the introduction of NOW accounts (January 1981) and finally the change in the operating target of FED from monetary aggregates to borrowed reserves (October 1982). Their results are in favor of a break between macroeconomic measures and money when data after the 1980s were included. The results are opposite when instead of simple-sum monetary aggregates,
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Divisia monetary aggregates are used. Their findings suggest that Divisia measures are more appropriate to use in order to examine the relation among money and macroenomic variables. Chrystal and Mac Donald (1994) also provide results in favor of the Divisia Monetary Aggregates, at least, in the case of US economy. They follow two approaches in their study is about seven economies (U.S., U.K., Germany, Australia, Switzerland, Canada and Japan). The first one is, to compare the results when simple-sum or Divisia monetary aggregates are used in the context of St.Louis Equations, with the nominal GDP serving as the dependent variable. The other approach used is the time-series methodology, in order to examine the short-run and long-run causal links of money with real economic activity. They conclude, for the case of U.S., that while the simple-sum M1 level dominates the Divisia M1, the results are reversed after the M2 level in the case of nominal GDP. Also, consistent with the findings of Darrat et al. (2004), they claim that the advantages of using the Divisia Monetary Aggregates are more obvious after the 1980s. Before 1980s there is not great difference between the two different monetary aggregates. This suggests that innovations and liberation on the financial sector have made the use of simple-sum aggregates have less predictive power.
In a different study, Donald L.Schunk (2001) attempts to compare the true out-of-sample forecasting ability of Divisia and simple-sum monetary aggregates in terms of real GDP and U.S. GDP deflator, using data from 1960Q1 until 1997Q4. His results are in favor of the superiority of the Divisia Monetary Aggregates compared to their equivalent simple-sum. Moreover, it is clear from the results that in the case of real GDP, the broad levels of Divisia Monetary Aggregates perform better. Simple-sum monetary aggregates, on the other hand, outperform Divisia monetary aggregates in predicting prices. More specifically the narrowest levels of aggregates in both cases result to smaller root mean square errors and mean error of price forecasts.
Gogas et al. (2013) attempted to investigate the relation between the Divisia and simple-sum monetary aggregates on predicting the real GDP, using data for the U.S. economy from 1967Q1 until 2011Q4, that were provided both by the FED and from the CFS databases. They state that Divisia Monetary Aggregates are better than simple-sum monetary aggregates in terms of standard forecast evaluation statistics. In their case, the results also suggest that the best forecasts for real GDP are provided when either CFS or MSI Divisia M1 are used.
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The most accurate forecast for the next quarter’s real GDP is obtained by the CFS Divisia MZM, the MSI Divisia M1 and the simple-sum M2M aggregates while the most accurate out-of-sample forecast of real GDP is given by the CFS Divisia M1 level.
4. Explanation of Data
The data that will be used in this case are quarterly United States data. The sources of these data are the Center for Financial Stability (CFS) and the Federal Reserve Bank of St. Louis (FED) over the period from 1967Q1 until 2013Q2.
More specifically, the series that are used are the CFS Divisia monetary aggregates DM1, DM2, DMZM, DM2M, DM3, DM4 and DM4M computed for the Advances in Monetary and Financial Measurement (AMFM) program. The CFS Divisia Monetary Aggregates are preferred over the MSI, which are provided by the FED, mainly because they tend to provide more accurate results (see Gogas et al. (2013)). Moreover, the FED no longer provides the broad Divisia Monetary Aggregates, making CFS the only provider.
In the construction of SVAR model, that is used to estimate the Impulse Response Functions and the Forecast Error Variance Decomposition, the following variables are also included: (a) the seasonally adjusted Consumer Price Index for All Urban Consumers: All Items in place for the Prices ( ), that measures the changes in the price of a basket of goods and services for consumers in urban areas only. It is an often statistic tool to identify inflation or deflation.
(b) the Effective Federal Funds Rate (R) which is the weighted average of the interest rate that the bank borrows funds by the lending bank and is provided by the FED. This measurement is closely monitored since it is an indicator for the credit market conditions in the banking system and for the monetary policy.
(c) the Real Gross Domestic Product, that is the inflation adjusted value of the goods and services produced by labor in the United States and ( )
(d) a Commodity Price Index which in this case is the Oil Prices ( )
These four data series are provided by the Federal Reserve Economic Database (FRED) of the Federal Bank of St. Louis.
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5. SVAR model
A vector autoregression model (SVAR) is used for the analysis, not only because it has been widely used in previous research on this topic, but also because it provides statistic results that are pretty straight forward to interpret. In VAR models, the endogenous variables are explained only by their own history, without taking into account deterministic regressors, while the structural VAR (SVAR) models allow the modeling of contemporaneous interdependence between the left-hand side variables.
The structural VAR is based on the fact that it “uses economic theory to short out the
contemporaneous links among the variable” as Sims (1986) wrote. To do so, it is necessary
to identify the links between the variables that are included in the model. The basic difference from VAR models is that the identifications in the case of SVARs are not opposed to the estimated coefficients, but in the errors of the model. The errors of the model are assumed to be linear combinations of exogenous shocks in the system. By that, instrumental variables could be estimated by using instrumental variables regression, allowing the estimation of contemporaneous links.
The definition of SVAR model is the following:
( )
In equation (4) , with :1,2,…, , are the structural coefficient matrices and the structural errors are basicly white noise disturbances.
Multiplying with the inverse matrix of equation (1) can also be written as :
( )
In equation (2) is the reduced form of the residuals, , and the covariance
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The structural model in this case has
{ }
with output ( ), prices ( ), oil-prices ( ), money ( ) and nominal interest rate ( ).
[ ] and [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] These are the coefficients of interest, equal to the estimations that are produced by the model, according to the restrictions that are used in this case, by using recursive identification for the matrices.
What separates the SVAR models from the traditional dynamic simultaneous equation approach is that in SVAR we assume that the structural innovations are orthogonal, meaning that the shocks are uncorrelated. Bernanke (1986) states that matrix can have arbitrary off-diagonal elements. This suggests that stochastic parts of equations are allowed to be contemporaneously correlated in an arbitrary way.
In the case of SVAR models we want to examine the structural shocks that can be translated as monetary shocks. Since we cannot observe the shocks directly we should first make some assumptions in order to interpret the identification of shocks’ results appropriately. The process of identification through assumptions has always been under question in the literature. Posing different identification schemes can lead to significantly different results in our analysis, so theory and sensitive analysis are crucial for the model. In his paper, Sims (1980) states that “... for forecasting and policy analysis, structural identification is not
ordinarily needed and that false restrictions may not hurt, may even help a model to function in these capacities”. Furthermore, he argues that in big macroeconomic models, very large
mistakes on the identification procedure are more likely to be “detected and eliminated”. Following the previous analysis, he assumes that estimating a large-scale macroeconomic
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model without opposing any restrictions, coming from previous knowledge, should be possible.
The suggested procedure when a structural VAR is estimated, is to present both results of Granger causality tests, an impulse response analysis and the forecast error variance decomposition. The estimated coefficients of the model are usually not presented due to the fact that they are less informative compared to the previous mentioned statistics, mainly due to the existence of complicated dynamics that characterize VAR models. Since we are interested in observing the response of one variable after an unexpected shock in money supply is occurred, we will proceed in the analysis of the Impulse Response Functions (IRF) and the Forecast Error Variance Decomposition (FEVD) of our model.
A five-variables structural VAR is constructed. It consists of the log of real GDP ( ), the log of the price level (P), the log of a commodity price index, here the oil-prices (O) will be used, the log of a monetary aggregate ( ) and finally the interest rate (R). The order I used for the estimation of the SVAR model is the following [ ]. This specific order stems from the assumption that before deciding to set the monetary base, FED takes into account the price level ( ) and the aggregate output ( ). Differnet order of the variables were tested based on suggestions from the literature and the theory behind tha construction of the identification restrictions needed in the SVAR models in order to find the order that could be closer to reality and also in line with the economic theory. Though, the initially mentioned order is the one that gave the best fits.
6. Granger Causality Test
The basic idea of Granger Causality test is that a time series has a causal influence on another time series, if by including the second in the estimation procedure of the first, the predictions we get are better. This implies that the forecast error has a smaller variance. In our case we are interested in examining whether or not there is a causal relation between the money, expressed by the monetary aggregates, and the output.
Granger Causality test is a simple statistical test of whether the introduction of one variable will have any forecasting power over the variable of interest, or not.
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Table 2
Granger Causality Test P-values SVAR with Y 1967Q1 to 2013Q2 1967Q1 to 1980Q1 1980Q1 to 2013Q2 DM1 0.98 0.19 0.56 DM2 0.92 0.39 0.68 DM2M 0.91 0.15 0.48 DMZM 0.79 0.15 0.4 DM3 0.22 0.49 0.58 DM4 0.79 0.63 0.9 DM4M 0.16 0.35 0.61
On table 2, the p-values that correspond to the F-statistics are presented and are computed by testing if the relevant coefficients are zero or not for the different levels of monetary aggregates in different time periods.
In all cases of money as we can see from the table, we fail to reject the null hypothesis of no Granger Causality from money to output. It is interesting that the p-values for the same levels of aggregates are different, depending on time-period which is under question. Except of DM3 level, the p-values of Granger-causality tests are lower in the case of 1967Q1 until 1980Q1 compared to the other periods. The results for Granger causality p-values for the rest of the variables of the model are also presented in the appendix.
7. Impulse Response Functions and Forecast Error Variance Decomposition
The Impulse Response Function (IRF) analysis is useful for observing the time needed for a system to be affected by an unexpected exogenous change, a shock. In other words, IRF is
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the description of the reaction as a function of time. The IRFs are the time profiles of the effect of a hypothetical shock in the economy at time t and are compared with a baseline profile at time , taking into account the information provided by the past values of the variables of the system. This approach has been largely used in economic analysis when macroeconomic theory is tested. In order to construct the impulse response function a one-time shock of one standard deviation increase in monetary aggregate is introduced.
We also have to present the results of the Forecast Error Variance Decomposition (FEVD) in order to be able to draw conclusions about the previous analysis. As Stock and Watson (2001) state “forecast decomposition is like a partial R2 for the forecast error, by forecast
horizon”. The forecast error decomposition presents the percentage of variance of the error
made while forecasting a variable, subject to a specific shock at a given horizon. Following the impulse response analysis, we assume ten lags for the SVAR model regression and the forecast horizon is ten quarters (two years and six months). As it is suggested from economic theory, changes on output after a shock in the level of money in economy should have an instant effect.
Staying focused on the main question, which of the Divisia Monetary Aggregates performed better in explaining changes and variations of output in the US and whether or not these results are different when the monetary policy regime is different, the results in different time periods will be presented and described. At the last part the results of the comparisons will be presented. In the graphs of Impulse Response Function the area between the dotted lines represent the Confidence Interval which is 95%. Also the vertical axis stands for the percentage of the changes in the output while the horizontal the forecast horizon.
i. Impulse Response Function and Forecast Error Variance Decomposition from 1967Q1 to 2013Q2
The impulse response analysis of a one-time shock in the monetary aggregate DM1 (Figure 1) gives some surprising results. Based on the economic theory, what would be expected is that after the shock the output should start rising immediately. On the contrary, the effect of this change is visible almost after the 2ndquarter. Initially, there is a small decline on the level of output and there are small fluctuations for the next 7 quarters, which finally leads to a slight increase at the beginning of 10th period.
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On the other hand, there is a great difference when in the estimation of the SVAR model, the DM2 monetary aggregate is used (Figure 2). The response of output in a one standard deviation of money is instantaneously and leads to a sharp increase on its level until the 3rd quarter, after which it remains stable with very small fluctuations.
From the two narrower Divisia Monetary aggregates, DM2 dominates over DM1, presenting a more realistic movement of the output after a change in money. The differences between DM2 and DM2M (Figure 3), DMZM (Figure 4) are very few, due to the fact that they are consisted of almost the same assets. So when the all data are included, meaning that the time period is from 1967Q1 until 2013Q2, they seem to have identical reactions to a one-time shock of money. Figure 1 -.006 -.004 -.002 .000 .002 .004 .006 1 2 3 4 5 6 7 8 9 10 Response of Y to DM1 Innovation Figure 2 -.002 -.001 .000 .001 .002 .003 .004 .005 .006 .007 1 2 3 4 5 6 7 8 9 10 Response of Y to DM2 Innovation
18 Figure 3 -.004 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DM2M Innovation Figure 4 -.004 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DMZM Innovation Figure 5 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DM3 Innovation
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Considering the broader levels of Divisia Monetary Aggregates their IRFs will also be discussed.
In the case of output’s IRF (Figure 5) when DM3 is used, as we would expect the effect is positive during the first 3 quarters, after which it is stabilized leading to a higher level of output.
Following that, the response of output to an innovation in DM4 (Figure 6), is similar with the only exception being that the increase is of smaller scale. A more stable reaction is described by the IRF when the DM4M (Figure 7) level is used. An initial increase takes place, until the third quarter, followed by an almost stable output. From the eighth quarter until the end of the forecast horizon, there is a very slow decrease of the output.
Figure 6 -.004 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DM4 Innovation Figure 7 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DM4M Innovation
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In Table 3, the results for the Forecast Error Variance Decomposition of output (Y) are presented, with respect to the different levels of monetary aggregates. The lag length, as it has already been mentioned, is equal to ten and the forecast horizon is equal to ten quarters. In the following tables not all observations will be presented, since the differences from quarter to quarter are not necessary to draw conclusions for the comparison. There will be presented the results for every 4 quarters starting from the second quarter and the comparison will be based on the tenth quarter.
As it can be seen, innovation on money, when noted as DM1, has a very small explanation power over the variance of output (GDP) during all the quarters and in the 10th quarter is equal to 0,64%. On the other hand, the interest rate (R), while on the first quarters has a very small impact (2nd quarter equal to 0.05%), it increases as the forecast horizon increases ending close to 30%.
By including the DM2 aggregate the results are better, since the percentage of the error variance of output is higher and especially at the tenth quarter that is equal to 4,7%. In the case of DM2M, still the results are by far better compared to the DM1, but still there is a difference with DM2, which is better. On that case, compared to the previous situation the interest rate is still the one that explains better the variations of the output.
From the narrow Divisia Monetary Aggregates the one that explains the highest percentage of the output variations in the long-run is the DMZM. At the tenth quarter the percentage of the variation of the output that is explained by changes in money is almost 5%. Though for the first quarters the DM2 is the best.
Based on the previous results it can be concluded, that among the narrow measures of money, DMZM is superlative relative to the other narrow level of monetary aggregates.
On the broader levels of monetary aggregates, it can be easily inferred that relative to the narrow levels, they explain a higher percentage of the variations of output. This is the case, probably because they contain more information in the process of their construction. Among the broad aggregates, the DM4M explains more of the variations of the real output, since it has the highest percentages along the entire forecast horizon, ending to a 13, 05% at the tenth quarter. In the case of the broad monetary aggregates, the interest rate is the one with the highest percentages of the variations of the output.
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Table 3
1967q1 – 2013q2
Forecast Error Variance Decompostion of Y Forecast Horizon Y P O M R DM1 2 99.80 0.14 0.00 0.01 0.05 6 87.55 1.28 0.05 0.56 10.57 10 67.29 2.08 0.24 0.64 29.75 DM2 2 99.36 0.01 0.01 0.46 0.16 6 82.57 6.43 0.50 3.12 7.39 10 59.02 13.64 1.88 4.70 20.75 2 99.42 0.04 0.00 0.38 0.16 DM2M 6 85.46 4.26 0.06 2.65 7.57 10 64.25 11.25 0.29 3.57 20.64 DMZM 2 99.41 0.04 0.02 0.26 0.26 6 85.16 4.35 0.04 2.81 7.65 10 62.34 11.77 0.16 4.98 20.74 DM3 2 98.62 0.004 0.01 1.10 0.27 6 77.37 4.06 1.74 9.49 7.34 10 54.98 6.33 6.72 10.83 21.14 DM4 2 99.00 0.05 0.31 0.65 0.27 6 82.92 3.42 1.37 4.63 7.65 10 61.07 5.58 5.81 5.66 21.88 DM4M 2 98.16 0.02 0.05 1.52 0.25 6 76.63 2.86 2.40 11.09 7.03 10 53.79 3.95 8.42 13.05 20.79
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We can now conclude that when the model is estimated with data from 1967Q1 to 2013Q4 the broader levels of Divisia Monetary Aggregates are superlative to narrow levels. Among the narrow levels of aggregation the DMZM level is better and among the broader levels the DM4M.
ii. Impulse Response Function and Forecast Error Variance Decomposition from 1967Q1 to 1980Q1
The results are based on the SVAR model, with the same structure and identifications as presented on the fourth chapter. In this case though, the data are for the period during which the main tool that central banks and monetary authorities were using were the monetary aggregates. This period is from 1967Q1 until the 1980Q1. The only change that had to be applied is the length of the lags, which now is equal to six. That was necessary since the number of the observations now is less than when all the data are included. It would have been impossible to estimate such a big model with lag length equal to ten. It must be clear that during that time period there was small interest on shifting the interest rate in order to stabilize the economy.
A change in money results to an initial increase of the output for the first three quarters, as we can see from the graph (Figure 8), where money is presented by the DM1. After that, it starts declining sharply, leading to a much lower level of output.
On the other hand, when the change in money is presented by the DM2 (Figure 9) level of money, the initial response of the output is to decrease slightly for the first quarter after which it increases for the next quarter and decreases again. The volatility of the output is much higher compared to the previous graph (Figure 8). Interestingly, after the sixth quarter it increases again and by the eighth quarter it is stabilized to a higher level of output. The IRF graphs of DMZM (Figure 10) and DM2M (Figure 11) are not only similar with each other, but also with DM2 (Figure 9). A basic difference is that these changes result to a lower volatility of the output, after the fourth quarter, leading to a steady increase of the level of output. In case of DM2M level of money, the result is steeper.
Similar to DM2’s IRF are also the results for the DM3 (Figure 12) level. The output slightly recedes from the first until the second quarter. From that quarter to the third, it increases.
23
Between the third and the fifth quarter, the output decreases sharply again. There is a small increase from the eighth until the ninth quarter after which it seems to stay stable. Similar to the previous graphs there is some volatility on the reaction of the output after the monetary shock.
Interesting are the results for the DM4 (Figure 13) level of monetary aggregate, where the effect of the shock on money fades away quicker than in previous cases. Initially, there seems to be no effect on the output. Only after the second quarter there is an increase which by the third quarter has turned into a decline on the level of the output. From the fifth quarter the output stays constant. Similar are the results for the DM4M (Figure 14) level. The output presents a small increase for the first quarter and is not stable, as it was previously on the DM4 level. The changes are similar for the next four quarters. At the fifth quarter, it stays constant, in a higher level this time.
Figure 8 -.0100 -.0075 -.0050 -.0025 .0000 .0025 .0050 .0075 .0100 1 2 3 4 5 6 7 8 9 10 Response of Y to DM1 Innovation Figure 9 -.008 -.004 .000 .004 .008 .012 .016 1 2 3 4 5 6 7 8 9 10 Response of Y to DM2 Innovation
24 Figure 10 -.01 .00 .01 .02 .03 .04 1 2 3 4 5 6 7 8 9 10 Response of Y to DMZM Innovation Figure 11 -.01 .00 .01 .02 .03 .04 1 2 3 4 5 6 7 8 9 10 Response of Y to DM2M Innovation Figure 12 -.012 -.008 -.004 .000 .004 .008 .012 .016 1 2 3 4 5 6 7 8 9 10 Response of Y to DM3 Innovation
25 Figure 13 -.006 -.004 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DM4 Innovation Figure 14 -.008 -.004 .000 .004 .008 .012 1 2 3 4 5 6 7 8 9 10 Response of Y to DM4M Innovation
Following the IRF analysis, the results from the Forecast Error Variance Decomposition are presented for the period 1967Q1 until 1980Q1 (Table 4).
For the DM1 level and compared to the results of the previous part it is obvious that the percentage of the variation of the output is higher than it was before (9.14% for the first quarter and 25.95% at tenth quarter). Also compared to the DM2 level of the same time period, DM1 does a better job in explaining the movements of the output throughout the entire forecast horizon.
For the broad levels of Divisia Monetary Aggregates the explanation power over the movements of the output is lower, than in the previous period.
26
Table 4 1967q1 - 1980q1
Forecast Error Variance Decomposition of Y Forecast Horizon Y P O M R DM1 2 73,67 9,04 7,21 9,14 0,95 6 37,68 10,30 13,59 16,21 22,22 10 37,46 14,04 11,55 11,00 25,95 DM2 2 84,87 0,35 13,51 1,27 0,00 6 49,89 3,29 19,66 7,14 20,02 10 72,43 4,53 7,92 9,08 6,04 DM2M 2 80,86 1,02 15,58 2,53 0,02 6 51,33 4,83 15,76 11,83 16,26 10 59,43 0,99 6,63 30,98 1,98 DMZM 2 80,74 1,02 15,86 2,35 0,03 6 51,37 4,90 15,31 12,41 16,00 10 59,35 0,96 6,80 30,98 1,91 DM3 2 90,58 0,630 7,15 1,62 0,02 6 47,67 11,91 12,91 9,77 17,74 10 75,07 8,11 4,24 6,61 5,96 DM4 2 93,12 0,25 6,02 0,02 0,59 6 56,72 7,67 9,72 5,85 20,04 10 77,00 7,78 3,63 2,50 9,09 DM4M 2 88,78 1,45 7,83 1,91 0,03 6 49,08 10,67 13,04 9,94 17,27 10 76,96 7,08 3,97 6,48 5,51
27
Among the broad levels the one that stands above the other, is the DM3 level of aggregation on the tenth quarter which is equal to 6.6%. This is slightly higher than the DM4M which is almost 6.5% and the DM4 (2.5%). Contrary to that, when the sixth forecast horizon is considered the percentages are better, implying that on early quarters they perform better.
iii. Impulse Response Function and Forecast Error Variance Decomposition from 1980Q1 to 2013Q2
When interpreting the results for this period, it must not be overlooked that in the meantime the recent economic crisis burst. This created many anomalies and difficulties for the monetary policy authorities to stabilize the economy and to make a solid plan that could reverse the twist in the economy.
The response of the output to a onetime shock in money, denoted by the narrow level of Division Monetary Aggregate DM1 is presented on Figure 15. It is obvious from the IRF that the shock causes no reaction in the output, at least for the first two quarters. After that, there is a positive reaction in the output where it slightly increases until the middle of the fifth quarter. Until the sixth quarter there is a very small fall. Finally, from the sixth quarter and up to the end of the forecast horizon, there is a steady increase in the output.
When the DM2 (Figure 16) is used in the estimation of the model the response of output is instantly positive followed by a quick increase for the next three quarters. After the initial reaction it seems to be more or less stable with almost insignificant changes.
In this specific time period, the previous IRF graph (Figure 16) is similar to the IRFs when the DM2M (Figure 17) and DMZM (Figure 18) levels are included.
Positive is the initial response of the output when the DM3 (Figure 19) is included, at least for the first three quarters where it elevates quickly. However, after the initial increase the output gradually decreases. At the tenth quarter the level of the output is still higher than the initial.
The initial increase of the output after the shock on money, denoted by the DM4 (Figure 20) level, lasts for almost three quarters. This is followed by a constant decrease in the output’s level till the sixth quarter. After the sixth quarter, the output is lower than it was before the
28
shock. This states that the initial positive effect of money shock in output results to a negative effect leading to a worst point than initially.
Noteworthy are the results when the broad level of money aggregate DM4M (Figure 21) is used. The response of the output is very different from the DM4. As expected, innovation on money, for the first three quarters, increases the level of the output. After that the effect is turning to be negative and it starts to shrink. By the end of the tenth quarter the output becomes almost equal to the initial level.
Figure 15 -.004 -.002 .000 .002 .004 .006 .008 .010 1 2 3 4 5 6 7 8 9 10 Response of Y to DM1 Innovation Figure 16 -.004 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DM2 Innovation
29 Figure 17 -.004 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DM2M Innovation Figure 18 Figure 19 -.004 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DM3 Innovation -.004 -.002 .000 .002 .004 .006 .008 1 2 3 4 5 6 7 8 9 10 Response of Y to DMZM Innovation
30 Figure 20 -.008 -.006 -.004 -.002 .000 .002 .004 .006 1 2 3 4 5 6 7 8 9 10 Response of Y to DM4 Innovation Figure 21 -.006 -.004 -.002 .000 .002 .004 .006 1 2 3 4 5 6 7 8 9 10 Response of Y to DM4M Innovation
The Forecast Error Variance Decomposition of the output is presented on Table 5.
The results suggest that among the narrow levels of the monetary aggregates, DMZM level is the one that surpasses the other levels. DMZM explains 8.56% of the movement of the output followed by the DM2M level, with 8.39%. The percentages of the variation of the output explained from the DM1 and DM2 are equal and near to 7%.
On the broad levels the explanation of output variations are smaller. The DM3 level does the best in explaining the changes of output, equal to 6.6%. The DM4M is the next best, while DM4 explains only the 1.56% of the movement.
31 Table 5 1980q1-2013q4
Forecast Error Variance Decompostion of Y Forecast Horizon Y P O M R DM1 2 98.54 0.89 0.01 0.01 0.54 6 71.44 12.82 0.89 1.02 13.83 10 54.45 17.24 1.19 7.04 20.09 DM2 2 96.71 1.69 0.00 1.59 0.003 6 67.87 20.52 0.42 6.31 4.88 10 57.26 27.65 0.58 7.04 7.47 DMZM 2 97.13 1.00 0.01 1.70 0.16 6 73.76 15.25 0.07 7.38 3.55 10 59.82 23.90 1.45 8.39 6.44 DM2M 2 97.06 1.21 0.00 1.73 0.00 6 69.37 18.92 0.15 7.34 4.23 10 55.22 30.09 0.60 8.56 5.53 DM3 2 96.96 1.245 0.02 1.74 0.03 6 71.05 13.46 2.45 7.47 5.56 10 58.88 17.93 4.75 6.62 11.82 DM4 2 97.81 1.50 0.04 0.64 0.01 6 71.81 18.11 2.89 1.82 5.38 10 57.80 24.40 5.54 1.56 10.69 DM4M 2 97.14 1.21 0.08 1.65 0.00 6 71.08 14.42 3.34 5.83 5.33 10 57.52 20.30 6.39 4.38 11.42
32
Summing up the previous analysis for the period 1980Q1 until 2013Q2, it is obvious that in this case the narrow levels of Division Monetary Aggregate are superior and more informative about the movements of the output relative to the broader levels. From the narrow levels DMZM is better and from the broad the DM3.
8. Conclusion
We can now conclude that there is a basic difference in the performance of Divisia Monetary Aggregates for different time periods. A basic difference that really outstand from the results is that when all the data are included in the model, as it happens in the first case with quarterly data expand from 1967Q1 until 2013Q2, the broad levels of Divisia Monetary Aggregates outperform, in general, the narrow levels. Especially, the DM4M level is better while the variations of DM1 explain the least of the output variations. The results in our case are opposite to the results that are presented on Gogas et al. (2013) and Schunk (2011), that in both cases the narrow levels of Divisia Monetary Aggregates outperform the broad levels. On the other hand, as it is presented on tables 4 and 5, when the time period is separated in two parts, the narrow Divisia Monetary Aggregates outperform the broad levels. The first (1967Q1-1980Q1) stands for the period when monetary aggregates were used by monetary policy authorities in the construction of the policy. The second (1980Q1-2013Q2) stands for the period after the switch of the monetary policy to the control of the short-term interest rate as a tool for the construction of monetary policy. Moreover, in the case of 1967Q1 to 1980Q1 the DM2M and DMZM, as it has already been mentioned, are equally good in explaining the changes of output. From 1980Q1 until 2013Q2 the DM2M is the best, followed by DMZM and in both cases the DM4 level performs worse than the other aggregates.
It is possible, according to the results, that changes on the regime of monetary policy to have no great effect on which level of aggregation has more informative power with respect to the output. It is interesting though, that while in the two separate time periods the narrow levels dominate the broad levels, when all the data are included in the SVAR the broad levels exceed the narrow levels. This can be triggered by the fact that since 1967 there have been major changes in the economy. Moreover, the rapid evolution and innovation in the financial sector along with the recent global economic crisis can explain the break of the linkage between money and macroeconomic variables. We must also keep in mind that for the
33
estimation of the SVAR for 1967Q1 until 1980Q1 the length of the lag is equal to six, while in the others is equal to ten, which can be a possible explanation for the different results. The results could be different if in place of the Oil-price index another price index was used. They can also be affected by a change on the econometric model or even if the time periods where even more. For example by not taking into account data from the recent global financial crisis could have a significant effect on the results of the last period. In other words it could possibly lead to more straightforward results about which level of the Divisia Monetary Aggregates has more explanation power with respect to the output.
34
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Appendix - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to Y - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to P - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to O - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to DM1 - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to R - .01 .00 .01 .02 2 4 6 8 1 0 Response of P to Y - .01 .00 .01 .02 2 4 6 8 1 0 Response of P to P - .01 .00 .01 .02 2 4 6 8 1 0 Response of P to O - .01 .00 .01 .02 2 4 6 8 1 0 Response of P to DM1 - .01 .00 .01 .02 2 4 6 8 1 0 Response of P to R - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to Y - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to P - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to O - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to DM1 - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to R - .02 - .01 .00 .01 .02 .03 2 4 6 8 1 0 Response of DM1 to Y - .02 - .01 .00 .01 .02 .03 2 4 6 8 1 0 Response of DM1 to P - .02 - .01 .00 .01 .02 .03 2 4 6 8 1 0 Response of DM1 to O - .02 - .01 .00 .01 .02 .03 2 4 6 8 1 0 Response of DM1 to DM1 - .02 - .01 .00 .01 .02 .03 2 4 6 8 1 0 Response of DM1 to R - 1 .0 - 0 .5 0 .0 0 .5 1 .0 1 .5 2 4 6 8 1 0 Response of R to Y - 1 .0 - 0 .5 0 .0 0 .5 1 .0 1 .5 2 4 6 8 1 0 Response of R to P - 1 .0 - 0 .5 0 .0 0 .5 1 .0 1 .5 2 4 6 8 1 0 Response of R to O - 1 .0 - 0 .5 0 .0 0 .5 1 .0 1 .5 2 4 6 8 1 0 Response of R to DM1 - 1 .0 - 0 .5 0 .0 0 .5 1 .0 1 .5 2 4 6 8 1 0 Response of R to R
37 ` - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to Y - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to P - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to O - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to DM2 - .01 2 - .00 8 - .00 4 .00 0 .00 4 .00 8 .01 2 2 4 6 8 1 0 Response of Y to R - .01 0 - .00 5 .00 0 .00 5 .01 0 .01 5 2 4 6 8 1 0 Response of P to Y - .01 0 - .00 5 .00 0 .00 5 .01 0 .01 5 2 4 6 8 1 0 Response of P to P - .01 0 - .00 5 .00 0 .00 5 .01 0 .01 5 2 4 6 8 1 0 Response of P to O - .01 0 - .00 5 .00 0 .00 5 .01 0 .01 5 2 4 6 8 1 0 Response of P to DM2 - .01 0 - .00 5 .00 0 .00 5 .01 0 .01 5 2 4 6 8 1 0 Response of P to R - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to Y - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to P - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to O - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to DM2 - .10 - .05 .00 .05 .10 .15 2 4 6 8 1 0 Response of O to R - .02 - .01 .00 .01 .02 2 4 6 8 1 0 Response of DM2 to Y - .02 - .01 .00 .01 .02 2 4 6 8 1 0 Response of DM2 to P - .02 - .01 .00 .01 .02 2 4 6 8 1 0 Response of DM2 to O - .02 - .01 .00 .01 .02 2 4 6 8 1 0 Response of DM2 to DM2 - .02 - .01 .00 .01 .02 2 4 6 8 1 0 Response of DM2 to R - 0 .5 0 .0 0 .5 1 .0 2 4 6 8 1 0 Response of R to Y - 0 .5 0 .0 0 .5 1 .0 2 4 6 8 1 0 Response of R to P - 0 .5 0 .0 0 .5 1 .0 2 4 6 8 1 0 Response of R to O - 0 .5 0 .0 0 .5 1 .0 2 4 6 8 1 0 Response of R to DM2 - 0 .5 0 .0 0 .5 1 .0 2 4 6 8 1 0 Response of R to R
38 - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to Y - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to P - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to O - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to DM2M - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to R - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to Y - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to P - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to O - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to DM2M - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to R - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to Y - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to P - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to O - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to DM2M - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to R - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DM2M to Y - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DM2M to P - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DM2M to O - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DM2M to DM2M - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DM2M to R - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to Y - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to P - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to O - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to DM2M - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to R
39 - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to Y - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to P - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to O - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to DMZM - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to R - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to Y - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to P - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to O - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to DMZM - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to R - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to Y - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to P - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to O - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to DMZM - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to R - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DMZM to Y - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DMZM to P - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DMZM to O - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DMZM to DMZM - .0 4 - .0 2 .0 0 .0 2 .0 4 2 4 6 8 10 Response of DMZM to R - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to Y - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to P - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to O - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to DMZM - 1.0 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to R
40 - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to Y - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to P - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to O - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to DM3 - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to R - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to Y - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to P - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to O - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to DM3 - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to R - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to Y - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to P - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to O - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to DM3 - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to R - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM3 to Y - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM3 to P - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM3 to O - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM3 to DM3 - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM3 to R - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to Y - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to P - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to O - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to DM3 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to R
41 - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to Y - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to P - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to O - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to DM4 - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to R - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to Y - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to P - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to O - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to DM4 - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to R - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to Y - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to P - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to O - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to DM4 - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to R - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of DM4 to Y - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of DM4 to P - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of DM4 to O - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of DM4 to DM4 - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of DM4 to R - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to Y - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to P - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to O - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to DM4 - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to R
42 - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to Y - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to P - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to O - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to DM4M - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 .0 12 2 4 6 8 10 Response of Y to R - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to Y - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to P - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to O - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to DM4M - .0 10 - .0 05 .0 00 .0 05 .0 10 .0 15 2 4 6 8 10 Response of P to R - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to Y - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to P - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to O - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to DM4M - .1 0 - .0 5 .0 0 .0 5 .1 0 .1 5 2 4 6 8 10 Response of O to R - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM4M to Y - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM4M to P - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM4M to O - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM4M to DM4M - .0 2 - .0 1 .0 0 .0 1 .0 2 .0 3 2 4 6 8 10 Response of DM4M to R - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to Y - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to P - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to O - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to DM4M - 0.5 0.0 0.5 1.0 1.5 2 4 6 8 10 Response of R to R
43 - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to Y - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to P - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to O - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to DM1 - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to R - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to Y - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to P - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to O - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to DM1 - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to R - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to Y - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to P - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to O - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to DM1 - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to R - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 2 4 6 8 10 Response of DM1 to Y - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 2 4 6 8 10 Response of DM1 to P - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 2 4 6 8 10 Response of DM1 to O - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 2 4 6 8 10 Response of DM1 to DM1 - .0 12 - .0 08 - .0 04 .0 00 .0 04 .0 08 2 4 6 8 10 Response of DM1 to R - 2 - 1 0 1 2 2 4 6 8 10 Response of R to Y - 2 - 1 0 1 2 2 4 6 8 10 Response of R to P - 2 - 1 0 1 2 2 4 6 8 10 Response of R to O - 2 - 1 0 1 2 2 4 6 8 10 Response of R to DM1 - 2 - 1 0 1 2 2 4 6 8 10 Response of R to R
44 - .0 3 - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to Y - .0 3 - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to P - .0 3 - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to O - .0 3 - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to DM2 - .0 3 - .0 2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of Y to R - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to Y - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to P - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to O - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to DM2 - .0 1 .0 0 .0 1 .0 2 2 4 6 8 10 Response of P to R - .3 - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to Y - .3 - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to P - .3 - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to O - .3 - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to DM2 - .3 - .2 - .1 .0 .1 .2 2 4 6 8 10 Response of O to R - .0 8 - .0 4 .0 0 .0 4 2 4 6 8 10 Response of DM2 to Y - .0 8 - .0 4 .0 0 .0 4 2 4 6 8 10 Response of DM2 to P - .0 8 - .0 4 .0 0 .0 4 2 4 6 8 10 Response of DM2 to O - .0 8 - .0 4 .0 0 .0 4 2 4 6 8 10 Response of DM2 to DM2 - .0 8 - .0 4 .0 0 .0 4 2 4 6 8 10 Response of DM2 to R - 2 - 1 0 1 2 3 2 4 6 8 10 Response of R to Y - 2 - 1 0 1 2 3 2 4 6 8 10 Response of R to P - 2 - 1 0 1 2 3 2 4 6 8 10 Response of R to O - 2 - 1 0 1 2 3 2 4 6 8 10 Response of R to DM2 - 2 - 1 0 1 2 3 2 4 6 8 10 Response of R to R
