1. Introduction

1. Introduction

1.1 Documentation
1.2 Notation for the ROW Model
1.3 Solution Options
1.4 The Complete MC2 Model and How It is Solved
1.5 Solution Errors
1.6 Changing Stochastic Equations
1.7 Creating Base Datasets
1.8 Treatment of the EMU Regime

1.1 Documentation

The complete documentation for the non-US part of the MC2 model, which is called the ROW model, is in Appendix B and the Chapter 6 tables at the end of this workbook and at the end of Macroeconometric Modeling.

The version of the US model that is part of the MC2 model, which is presented in Macroeconometric Modeling, is the same as the version used for the November 3, 1998, forecast (BASE983). The complete documentation for the US model that is part of the MC2 model is contained in Appendix A and the Chapter 5 tables at the end of this workbook and at the end of Macroeconometric Modeling.

The EMU began January 1, 1999, and this date is beyond any of the estimation periods. None of the estimation is thus affected by the EMU. For forecasting purposes, however, the EMU must be taken into account, and the way in which this was done is discussed in Section 1.8 below. Until this section, the following discussion pertains to the model prior to the EMU regime.

1.2 Notation for the ROW Model

The notation for the variables in the ROW model is presented in Tables B.1 and B.2 in Appendix B. Two letters denote the country (CA for Canada, JA for Japan, etc.), and the abbreviations are given in Table B.1. Up to five letters denote the variable (C for consumption, I for investment, etc.), and the names are given in Table B.2 in alphabetical order. The complete name of a variable for a country consists of the country abbreviation plus the variable name, such as CAC for Canadian consumption, JAI for Japanese investment, etc. The two letters EU denote the European countries in the model that are part of the EMU. These are: AU, FR, GE, IT, NE, FI, BE, IR, PO, SP. (Luxembourg, which is also part of the EMU, is not in the model.)

1.3 Solution Options

There are five choices you can make regarding the solution of the MC2 model.

The prediction period, where the default is 1999-2003.
Whether you want the entire MC2 model solved or just the individual country models by themselves. If you choose the latter, none of the variables in one country affect the variables in any other country. Each individual country model stands alone, and all foreign-sector variables in an individual country model are taken to be exogenous. The default is to solve the entire MC2 model.
Whether or not you want the trade share equations used. If you do not want the trade share equations used, the trade shares are taken to be exogenous and equal to the actual values prior to 1998:1 and to the predicted values in the base dataset (MC2BASE) from 1998:1 on. This trade share option is not relevant if you choose to have the individual country models solved by themselves since in this case the output from the trade share calculations does not affect any model. The default is to use the trade share equations.
The number of within country iterations (denoted LIMITA) and the number of across country iterations (denoted LIMITB). The defaults are 2 for LIMITA and 10 for LIMITB. As discussed below, these options are useful for checking if the model has successfully solved.
Whether or not you want to use the historical errors. The default is to set all the error terms equal to zero. If you use the historical errors and make no changes to any of the exogenous variables, add factors, and coefficients, then the solution values of the endogenous variables will be the actual values---a perfect tracking solution---aside from rounding error. This option can be useful for multiplier experiments, as discussed below.

1.4 The Complete MC2 Model and How It is Solved

The size of the MC2 model is as follows. The US model, which is part of the MC2 model, includes 30 stochastic equations plus one more when it is imbedded in the MC2 model. (This additional equation is discussed below.) There are 39 countries in the ROW model and up to 15 stochastic equations per country. If each country had all 15 equations, there would be a total of 585 (39 x 15) stochastic equations in the ROW model. Because of data limitations, however, not all countries have all equations, and there are in fact 332 stochastic equations in the ROW model. Given the 31 stochastic equations in the US model, there are thus 363 stochastic equations in the MC2 model. There are a total of 1650 coefficients in these equations, counting the autoregressive coefficients of the error terms. In addition, as discussed in Section 6.16 in Macroeconometric Modeling, there are 1050 estimated trade share equations. Not counting the trade share coefficient estimates, all the coefficient estimates for the US model are presented in the Chapter 5 tables, and all the coefficient estimates for the ROW model are presented in the Chapter 6 tables.

Table B.1 shows that there are in the ROW model 20 variables per country determined by identities (identities I-10 and I-11 are no longer used), 5 variables per country determined when the countries are linked together, and 20 exogenous variables per country. Counting these variables, various transformations of the variables that are needed for the estimation, and the US variables (but not the trade shares), there are about 4500 variables in the MC2 model.

The way in which the US model is imbedded in the MC2 model is explained in Table B.5. The two key variables that are exogenous in the US model but become endogenous in the overall MC2 model are exports, EX, and the price of imports, PIM. EX depends on X90$_US, which is determined in Table B.4. PIM depends on PM_US, which depends on PMP_US, which is also determined in Table B.4.

Feeding into Table B.4 from the US model are PX_US and M90$A_US. PX_US is determined is the same way that PX is determined for the other countries, namely by equation 11 in Table B.3. In the US case logPX_US - logPW$_US is regressed on logGDPD - logPW$_US. The equation is:

logPX_US - logPW$_US = q(logGDPD - logPW$_US)

This equation is estimated under the assumption of a second order autoregressive error for the 1962:1--1997:4 period. The estimate of q is .920 with a t-statistic of 23.50. The estimates (t-statistics) of the two autoregressive coefficients are 1.46 (19.37) and -.46 (-6.18), respectively. The standard error is .0119. Given the predicted value of PX_US from this equation, PEX is determined by the identity listed in Table B.5: PEX=DEL3*PX_US. This identity replaces identity 32 in Table A.3 in the US model.

M90$A_US, which, as just noted, feeds into Table B.4, depends on M_US, which depends on IM. This is shown in Table B.5. IM is determined by equation 27 in the US model. Equation 27 is thus the key equation that determines the U.S. import value that feeds into Table B.4.

The main exogenous variables in the overall MC2 model are the government spending variables (G). In other words, fiscal policy is exogenous. Monetary policy is not exogenous because of the use of the interest rate and exchange rate reaction functions.

The solution of the MC2 model can now be considered. Because some of the countries are annual, the overall MC2 model is solved a year at a time. A solution period must begin in the first quarter of the year. In the following discussion, assume that year 1 is the first year to be solved. The overall MC2 model is solved as follows:

Given values of X90$, PMP, and PW$ for all four quarters of year 1 for each quarterly country and for year 1 for each annual country, all the stochastic equations and identities are solved. For the annual countries "solved" means that the equations are passed through k₁ times for year 1, where k₁ is determined by experimentation (as discussed below). For the quarterly countries "solved" means that quarter 1 of year 1 is passed through k₁ times, then quarter 2 k₁ times, then quarter 3 k₁ times, and then quarter 4 k₁ times. The solution for the quarterly countries for the four quarters of year 1 is a dynamic simulation in the sense that the predicted values of the endogenous variables from previous quarters are used, when relevant, in the solution for the current quarter.
Given from the solution in step 1 values of E, PX, and M90$A for each country, the calculations in Table B.4 can be performed. Since all the calculations in Table B.4 are quarterly, the annual values of E, PX, and M90$A from the annual countries have to be converted to quarterly values first. This is done in the manner discussed at the bottom of Table B.4. The procedure in effect takes the distribution of the annual values into the quarterly values to be exogenous. The second task is to compute PX$ using equation L-1. Given the values of PX$, the third task is to compute the values of a_ij from the trade share equations---see equation 6.13 in Section 6.16 of Macroeconometric Modeling. This solution is also dynamic in the sense that the predicted value of a_ij for the previous quarter feeds into the solution for the current quarter. (Remember that the lagged value of a_ij is an explanatory variable in the trade share equations.) The fourth task is to compute X90$, PMP, and PW$ for each country using equations L-2, L-3, and L-4. Finally, for the annual countries the quarterly values of these three variables are then converted to annual values by summing in the case of X90$ and averaging in the case of PMP and PW$.
Given the new values of X90$, PMP, and PW$ from step 2, repeat step 1 and then step 2. Keep repeating steps 1 and 2 until they have been done k₂ times. At the end of this, declare that the solution for year 1 has been obtained.
Repeat steps 1, 2, and 3 for year 2. If the solution is meant to be dynamic, use the predicted values for year 1 for the annual countries and the predicted values for the four quarters of year 1 for the quarterly countries, when relevant, in the solution for year 2. Continue then to year 3, and so on.

This completes the discussion of the solution of the model. In the above notation LIMITA is k₁ and LIMITB is k₂. I have found that going beyond k₁=2 and k₂=10 leads to very little change in the final solution values, and so these are the default values.

1.5 Solution Errors

If you ask to solve the MC2 model and the model does not solve, you will get an error message to that effect. When this happens your dataset will not have been changed. You will need to make less extreme changes and try again. The model has not been solved when there is an abnormal abort, which usually means that the program has tried to take the log of a negative number.

Because the MC2 model (unlike the US model) is not iterated until convergence (because k₁ and k₂ are fixed, it may be the case that the program finishes normally (no abnormal abort) but that the model did not really solve. If you are concerned about this, there is one check that you can perform, which is to increase LIMITA and LIMITB. If the model has correctly solved, it should be the case the increasing LIMITA and LIMITB has a very small effect on the solution values. You can thus increase LIMITA and LIMITB and see if the output values change much. If they do not, then you can have considerable confidence that the model has been solved correctly. The maximum values of LIMITA and LIMITB that you are allowed are 12 and 21, respectively.

1.6 Changing Stochastic Equations

There are four changes you can make to any of the 363 stochastic equations:

Drop (or add back in) an equation. When an equation is dropped, the variable determined by the equation is taken to be exogenous, and it can be changed if desired. The default values for the variable are the historical values when they exist and forecast values from the base dataset otherwise.
Take an equation to begin after the beginning of the basic prediction period. When an equation begins later than the basic prediction period, the variable determined by the equation is taken to be exogenous for the earlier period, and it can be changed if desired. The default values for the variable are the historical values when they exist and forecast values from the base dataset otherwise. For quarterly countries the period that you want the equation to begin is a quarter, not a year. You can, for example, have an equation begin in 1999:3 when the basic prediction period is 1999-2003.
Add factor an equation, where the add factors can differ for different periods. For quarterly countries the add factors are for individual quarters, not years.
Change any of the 1650 coefficients in the equations. Unlike for the US model alone, however, you cannot add variables to the equations.

1.7 Creating Base Datasets

If you ask the program to solve the MC2 model for any period beginning 1999 or later and you make no changes to the coefficients and exogenous variables, the solution values for the endogenous variables will simply be the values that are already in MC2BASE. If, on the other hand, you ask the program to solve the model for a period beginning earlier than 1999, where at least some actual data exist, the solution values will not be the same as the values in MC2BASE because the model does not predict perfectly (the solution values of the endogenous variables are not in general equal to the actual values). It is thus very important to realize that the only time the solution values will be the same as the values in MC2BASE when you make no changes to the exogenous variables and coefficients is when you are solving beginning 1999 or later.

If you want to work with the MC2 model for a period for which actual data exist, you should read Section 2.6 of The US Model Workbook carefully. The same considerations apply to the MC2 model as apply to the US model. In other words, Section 2.6 is as relevant for the MC2 model as it is for the US model. This discussion will not be repeated here. The only new thing to note is that whenever you are making comparisons, you must treat the trade share equations the same for all runs---either use the trade share equations for all the relelvant solutions or take the trade shares to be exogenous for all of them.

An alternative to doing what Section 2.6 discusses should be done when working with a period for which actual data exist is to use the historical errors. When using the historical errors, the base prediction path is just the path of the actual values, and so any changes made are off of the actual path.

1.8 Treatment of the EMU Regime

As noted above, there are 10 countries in the model that are part of the EMU: AU, FR, GE, IT, NE, FI, BE, IR, PO, and SP. These 10 countries together will be denoted EU. Each of these countries has an estimated interest rate reaction function (equation 7), and each country except FI and SP has an estimated long term interest rate equation (equation 8). In addition, GE has an estimated exchange rate equation where the exchange rate explained is the DM/$ rate, and each of the other countries has an estimated exchange rate equation where the exchange rate explained is the local currency/DM rate (equation 9).

For the EMU regime, which begins in 1999:1, the interest rate reaction functions of the 10 countries except GE were dropped from the model. For GE the coefficient estimates of the interest rate reaction function were retained but the right hand side GE variables were replaced with the respective EU variables. This equation is interpreted as the interest rate reaction function of the European Central Bank (ECB). What this means is that the ECB is assumed to have the same policy rule as the Bundesbank did except that the rule reacts to the overall EU variables and not just to the GE variables.

Let GERSOLD be the interest rate determined by equation 7 for GE prior to the EMU regime, and let GERSEMU be the interest rate determined by the ECB equation just mentioned. In the coding of the model the German interest rate (GERS) is taken to be (1 - EMU)*GERSOLD + EMU*GERSEMU, where EMU is a dummy variable that takes on a value of 1 from 1999.1 on and 0 otherwise. This treatment allows the estimated equations for both GERSOLD and GERSEMU to be included in the coding of the model.

Let ITRSOLD be the (short term) i nterest rate determined by equation 7 for IT prior to the EMU regime, and let ITRSEMU be the short term IT rate in the EMU regime. ITRSEMU is taken to be GERSEMU + (ITRSOLD_1998:4 - GERSOLD_1998:4). In other words, the Italian short term interest rate in the EMU regime is taken to be the rate determined by the ECB equation (which is GERSEMU) plus the difference in 1998:4 between the Italian rate and the German rate. The same procedure was followed for the other 8 countries. The differences used are: 0.00 for AU, 0.00 for FR, 1.49 for IT, -0.18 for NE, 0.23 for FI, 0.08 for BE, 3.37 for IR, 1.93 for PO, and 2.11 for SP. In future revisions of the model these differences will probably disappear, but for present purposes they were retained. The coding of the model is the same for these countries as it is for Germany. For example, the short term IT interest rate (ITRS) is taken to be (1 - EMU)*ITRSOLD + EMU*ITRSEMU.

A similar procedure was followed for the long term interest rate as for the short term rate. For the EMU regime, the long term interest rate equations (equation 8) of the 8 countries except GE that have such equations were dropped from the model. The equation for GE was retained as is.

Let ITRBOLD be the interest rate determined by equation 8 for IT prior to the EMU regime, and let ITRBEMU be the long term IT rate in the EMU regime. ITRBEMU is taken to be GERB + (ITRBOLD_1998:4 - GERB_1998:4). In other words, the Italian long term interest rate in the EMU regime is taken to be the rate determined by the German equation (which is GERB) plus the difference in 1998:4 between the Italian rate and the German rate. The same procedure was followed for the other 6 countries. The differences used are: 0.33 for AU, 0.30 for FR, 0.87 for IT, 0.84 for NE, 1.30 for BE, 2.22 for IR, and 1.21 for PO. Again, in future revisions of the model these differences will probably disappear, but for present purposes they were retained. The coding of the model is the same for the long term rate as it is for the short term rate. For example, the long term IT interest rate (ITRB) is taken to be (1 - EMU)*ITRBOLD + EMU*ITRBEMU.

Finally, a similar procedure was followed for the exchange rate as for the short term interest rate. For the EMU regime, the exchange rate equations for the 10 countries except GE were dropped from the model. For GE the coefficient estimates of the exchange rate equation were retained but the right hand side GE variables were replaced with the respective EU variables. This equation is interpreted as the exchange rate equation for the euro. What this means is that the euro is assumed to be determined in the same way that the DM was determined except that the explanatory variables are overall EU variables and not just GE variables.

Let GEEOLD be the exchange rate determined by equation 9 for GE prior to the EMU regime, and let GEEEMU be the interest rate determined by the euro equation just mentioned. In the coding of the model the German exchange rate (GEE) is taken to be (1 - EMU)*GEEOLD + EMU*GEEEMU.

Let ITHOLD be the exchange rate determined by equation 9 for IT prior to the EMU regime, and let ITHEMU be the IT exchange rate in the EMU regime. ITHEMU is taken to be ITHOLD_1998:4. In other words, the Italian exchange rate (relative to the DM) in the EMU regime is taken to be the rate that existed in 1998:4. The same procedure was followed for the other 8 countries. The 1998:4 rates used are: 7.03 for AU, 3.351 for FR, 985.5 for IT, 1.127 for NE, 3040.9 for FI, 20.63 for BE, 380.4 for IR, 101.1 for PO, and 84.43 for SP. The coding of the model is the same for the exchange rate as it is for the interest rates. For example, the IT exchange rate (ITH) is taken to be (1 - EMU)*ITHOLD + EMU*ITHEMU.

The EU variables that are used in the model from 1999:1 on are:

EUY=GEY+AUY/(AUE90/GEE90)+FRY/(FRE90/GEE90)+ITY/(ITE90/GEE90) +NEY/(NEE90/GEE90)+FIY/(FIE90/GEE90) +BEYQ/(BEE90Q/GEE90)+IRYQ/(IRE90Q/GEE90)+POYQ/(POE90Q/GEE90) +SPYQ/(SPE90Q/GEE90)
EUYS=GEYS+AUYS/(AUE90/GEE90)+FRYS/(FRE90/GEE90)+ITYS/(ITE90/GEE90) +NEYS/(NEE90/GEE90)+FIYS/(FIE90/GEE90) +BEYSQ/(BEE90Q/GEE90)+IRYSQ/(IRE90Q/GEE90)+POYSQ/(POE90Q/GEE90) +SPYSQ/(SPE90Q/GEE90)
EUYY=GEPY*GEY+(AUPY*AUY)/AUH+(FRPY*FRY)/FRH+(ITPY*ITY)/ITH +(NEPY*NEY)/NEH+(FIPY*FIY)/FIH +(BEPYQ*BEYQ)/(BEEQ/GEE)+(IRPYQ*IRYQ)/(IREQ/GEE) +(POPYQ*POYQ)/(POEQ/GEE)+(SPPYQ*SPYQ)/(SPEQ/GEE)
EUS=GES+AUS/AUH+FRS/FRH+ITS/ITH+NES/NEH+FIS/FIH +BESQ/(BEEQ/GEE)+IRSQ/(IREQ/GEE)+POSQ/(POEQ/GEE)+SPSQ/(SPEQ/GEE)
EUPY=EUYY/EUY
EUZZ=(EUYS-EUY)/EUYS
EULPYZ=LOG(EUPY/USPY)
EULEA=EULPYZ-GELE(-1)
EUPCPY=((EUPY/EUPY(-1))**4-1.)
EUSZ=EUS/(EUPY*EUYS)
EUSZZ=LOG((1+EUSZ)/(1+((USUSS/1000)/(USPXX*USYS))))

A Q at the end of a variable for the annual countries (BE, IR, PO, and SP) means that the variable is quarterly. For these four countries, quarterly data on Y, YS, and S were obtained by dividing the relevant annual observation by 4. Quarterly data on PY, E, and E90 were obtained by taking each quarterly observation to be the relevant annual observation.

EUY is real output of EU, EUYS is potential real output, EUYY is nominal output, EUS is the current account, EUPY is the output price deflator, and EUZZ is the output gap. The remaining variables are variables used in the EU interest rate and exchange rate equations, which are disucssed next. Note that EUY and EUYS are in units of 1990 DM and that EUYY is in units of current DM. The variable FRE90/GEE90 is the Franc/DM rate in 1990 and the variable FRH is the Franc/DM nominal rate (and similarly for the other countries).

The EU interest rate reaction function is

GERSEMU = -0.1056638602 + 0.7571426373*GERS(-1) -22.0977014888*EUZZ + 0.2123142018*USRS + 0.1807386342*EUPCPY

This is the same equation as the GE interest rate reaction function except that EUZZ replaces GEZZ and EUPCPY replaces GEPCPY. See the discussion of the GE equation 7 in Chapter 6 in Macroeconometric Modeling.

The EU exchange rate equation is

GELE1EMU = -0.4023993473 + 0.0636035421*EULEA -2.4298460083*GELRSZ -0.4889293528*EUSZZ(-1), RHO1=0.2451951156

where GELE1EMU is LOG(GEEEMU/GEE(-1)) and GELRSZ is .25*LOG((1+GERSEMU/100.)/(1+USRS/100.)). The other two variables in the equation are defined above. RHO1 means that the equation is estimated under the assumption of first order serial correlation of the error term. This is the same equation as the GE exchange rate equation except that EULEA and EUSZZ replace the corresponding GE variables. See the discussion of the GE equation 9 in Chapter 6 in Macroeconometric Modeling.

The software allows you to change the EU interest rate and exchange rate equations. The "country" that you will click is EU. Remember that these equations are only relevant from 1999:1 on. Also remember that the equations that have been dropped for the 9 countries from 1999:1 on are not part of the model from 1999:1 on. They only matter prior to this period.