6. The Stochastic Equations of the ROW Model
{Site footnote: This chapter has been updated from Chapter 6 in the 1994 book. The "Chapter 6 tables," which this chapter heavily relies on, are presented at the end of Appendix B of this document.}
6.1 Introduction
6.2 Equation 1. M: Merchandise Imports
6.3 Equation 2. C: Consumption
6.4 Equation 3. I: Fixed Investment
6.5 Equation 4. Y: Production
6.6 Equation 5. PY: Price Index
6.7 Equation 6. M1: Money
6.8 Equation 7. RS: Short Term Interest Rate
6.9 Equation 8. RB: Long Term Interest Rate
6.10 Equation 9. E: Exchange Rate
6.11 Equation 10. F: Forward Rate
6.12 Equation 11. PX: Export Price Index
6.13 Equation 12. W: Wage Rate
6.14 Equation 13. J: Employment
6.15 Equation 14. L1: Labor Force-Men; Equation 15: L2: Labor Force-Women
6.16 The Trade Share Equations
6.17 Additional Comments
6.1 Introduction
The stochastic equations of the ROW model are specified, estimated, and tested in this chapter. This chapter does for the ROW model what Chapter 5 did for the US model. Stochastic equations are estimated for 32 countries, with up to 15 equations estimated per country. The equations are listed in Table B.3, and they were briefly discussed in Section 3.3.5. The empirical results are presented in the "Chapter 6 Tables" following Appendix B. Table 6a presents the estimates of the "final" specification of each equation, and Table 6b presents the results of the tests.

The 2SLS technique was used for the quarterly countries and for equations 1, 2, and 3 for the annual countries. The OLS technique was used for the other equations for the annual countries. The 2SLS technique had to be used sparingly for the annual countries because of the limited number of observations. The selection criterion for the first stage regressors for each equation was the same as that used for the US model. Briefly, the main predetermined variables in each country's model were chosen to constitute a "basic" set, and other variables were added to this set for each individual equation. As noted in Chapter 5, the choice of first stage regressors for large scale models is discussed in Fair (1984), pp. 215-216.

The estimation periods were chosen based on data availability. With three exceptions, the periods were chosen to use all the available data. The three exceptions are the interest rate, exchange rate, and forward rate equations, where the estimation periods were chosen to begin after the advent of floating exchange rates. The earliest starting quarter (year) for these periods was 1972:2 (1972).

The tests are similar to those done for the US equations. To repeat from Chapter 5, the basic tests are 1) adding lagged values, 2) estimating the equation under the assumption of a fourth order autoregressive process for the error term, 3) adding a time trend, 4) adding values led one or more periods, 5) adding additional variables, and 6) testing for structural stability. For the annual countries the autoregressive process for the error term was taken to be third order rather than fourth order. Because of this, the notation "RHO+" instead of "RHO=4" is used in the tables in this chapter to denote the autoregressive test. The led values were one quarter ahead for the quarterly countries and one year ahead for the annual countries. This means that no moving average process of the error term has to be accounted for since the leads are only one period. The estimation periods used for the leads test were one period shorter than the regular periods because of the need to make room at the end of the sample for the led values.

One of the additional variables added, where appropriate, was the expected rate of inflation. As discussed in Chapter 5, this is a test of the nominal versus real interest rate specification. For the quarterly countries the expected rate of inflation was taken to be the actual rate of inflation during the past four quarters, and for the annual countries it was taken to be the inflation rate (at an annual rate) during the past two years. This measure of the expected rate of inflation will be denoted pe. This variable was only added to the equations in which an interest rate was included as an explanatory variable in the final specification.

Specification

In Section 3.3.5 the equations of the econometric model were matched to the equations of the theoretical model of Section 2.2. This is a guide for the theory behind the model and in particular for the theory behind the linking together of the countries. Also, subject to data limitations, the specification of the ROW equations follows fairly closely the specification of the US equations, and so the theory in Section 2.1 that is behind the specification of the US model is relevant here.

The extra theorizing that is discussed at the beginning of Chapter 5 is also relevant here. For example, the searching procedure was the same as that used for the US equations. Lagged dependent variables were used extensively to try to account for expectational and lagged adjustment effects, and explanatory variables were dropped from the equations if they had coefficient estimates of the wrong expected sign. Both current and one quarter lagged values were generally tried for the price and interest rate variables for the quarterly countries, and the values that gave the best results were used. The equations were initially estimated under the assumption of a first order autoregressive error term, and the autoregressive assumption was retained if the estimate of the autoregressive coefficient was significant.

Data limitations prevented all 15 equations from being estimated for all 32 countries. Also, some equations for some countries were initially estimated and then rejected for giving what seemed to be poor results.

One difference between the US and ROW models to be aware of is that the asset variable A for each country in the ROW model measures only the net asset position of the country vis-a-vis the rest of the world; it does not include the domestic wealth of the country. Also, the asset variable has been divided by PY.YS before it was entered as an explanatory variable in the equations. (PY is the GDP index and YS is potential GDP.) This was done even for equations that were otherwise in log form. As discussed in Section 3.3.3, the asset variable is off by a constant amount, and so taking logs of the variable is not appropriate. Entering the variable in ratio form in the equations allows the error to be approximately[Footnote 1: If the level error, say A-, is in A and not in A/(PY.YS), then including the latter variable in the equation means that it is not A- but A-/(PY.YS) that is part of the equation, and A-/(PY.YS) is not constant. This is what is meant by the error being only approximately absorbed in the estimate of the constant term.] absorbed in the estimate of the constant term. This procedure is, of course, crude, but at least it somewhat responds to the problem caused by the level error in A.

Because much of the specification of the ROW equations is close to that of the US equations, the specification discussion in this chapter is brief. Only the differences are emphasized, and the reader is referred to Chapter 5 for more detail regarding the basic specifications.

The Tables

The construction of Tables 6a and 6b is as follows. All the coefficient estimates in an equation are presented in Table 6a, along with the estimated standard error (SE) of the equation and the Durbin-Watson (DW) statistic. The sample period is also presented for each equation. The test results are presented in Table 6b. To save space, only the p-values are presented for each test except the stability test. As in Chapter 5, an equation will be said to pass a test if the p-value is greater than .01. For the stability test the AP value is presented along with the degrees of freedom and the value of lambda. Some of the values of lambda for the annual countries are 1.0, which means that only one possible break point was specified. This was done because of the short sample periods. The AP value has a * in front of it if it is significant at the one percent level, which means that the equation fails the stability test.

There are obviously a lot of estimates and test results in this chapter, and it is not feasible to discuss each estimate and test result in detail. The following discussion tries to give a general idea of the results, but the reader is left to pour over the tables in more detail if desired.

Previous Version of the ROW Model

The previous version of the ROW model is presented in Fair (1984), Chapter 4. Again, as with the US model, the present discussion of the model is self contained, and so this previous material does not have to be read. More changes have been made to the ROW model since 1984 than have been made to the US model. Some of the main changes are the following. First, the number of countries (not counting the United States) for which structural equations are estimated is now 32 rather than 42, and the trade share matrix is now 45 x 45 rather than 65 x 65. The model was cut in size to lessen problems caused by poor data. Second, OECD data were used whenever possible rather than IFS data. The OECD has better NIPA and labor data than is available from the IFS data. Third, annual data were used for countries in which only annual NIPA data existed. In the previous version, quarterly data were constructed for all the countries by interpolating the annual data. Fourth, wage, employment, and labor force equations were added to the model (equations 12-15). Fifth, estimates of the capital stock of each country were made, and the capital stock variable was used in the investment equation. Finally, as for the US model, a few more coefficient constraints were imposed.

The basic structure of the ROW model has, however, remained the same between the previous version and the current version, and some of the discussion in the following sections is similar to the discussion of the previous version in Sections 4.2.5 and 4.2.6 in Fair (1984).

6.2 Equation 1. M: Merchandise Imports
Equation 1 explains the real per capita merchandise imports of the country. The explanatory variables include price of domestic goods relative to the price of imports, the short term or long term interest rate, per capita income, the lagged value of real per capita assets, and the lagged dependent variable. The variables are in logs except for the interest rates and the asset variable. Equation 1 is similar to equation 27 in the US model. The three main differences between the equations are 1) the U.S. asset variable was not significant in equation 27 and so was dropped from the equation, 2) the import variable includes all imports in equation 27 but only merchandise imports in equation 1, and 3) the income variable is disposable personal income in equation 27 and total GDP in equation 1.

The results in Tables 6a and 6b for equation 1 show that reasonable import equations seem capable of being estimated for most countries. The coefficient estimate for income is of the expected sign for all countries, and many of the estimates of the coefficients of the relative price variable and the interest rate variables are significant. 9 of the 31 equations fail the lags test (at the one percent level), 5 fail the RHO+ test, and 6 fail the T test. 15 of 30 fail the stability test. The led value of the income variable was used for the leads test, and only 2 of 31 were significant at the one percent level. The expected inflation variable is relevant for 10 countries, [Footnote 2: Remember that the expected inflation variable is relevant if an interest rate appears as an explanatory variable in the equation.] and it is only significant for 1. [Footnote x: Multicollinearity problems prevented this test from being computed for FI.] For the countries in which the relative price variable was used, the log of the domestic price level was added to test the relative price constraint. The constraint was rejected (i.e., logPY was significant) in only 3 of the 19 cases.

6.3 Equation 2: C: Consumption
Equation 2 explains real per capita consumption. The explanatory variables include the short term or long term interest rate, per capita income, the lagged value of real per capita assets, and the lagged dependent variable. The variables are in logs except for the interest rates and the asset variable. Equation 2 is similar to the consumption equations in the US model. The two main differences are 1) there is only one category of consumption in the ROW model compared to three in the US model and 2) the income variable is total GDP instead of disposable personal income.

The results in Tables 6a and 6b for equation 2 are of similar quality as the results for equation 1. The interest rate and asset variables appear in many of the equations in Table 6.2a, and so interest rate and wealth effects on consumption have been picked up as well as the usual income effect.

Most of the tests in Table 6.2b for equation 2 are passed. 12 equations fail the lags test, [Footnote 3: Multicollinearity problems prevented the lags test from being computed for AU.] 6 fail the RHO+ test, and 14 fail the stability test. The led value of the income variable was used for the leads test, and it is significant in only 4 cases. The expected inflation variable is relevant for 17 countries, and it is only significant for 1.

6.4 Equation 3: I: Fixed Investment
Equation 3 explains real fixed investment. It includes as explanatory variables the lagged value of investment, the current value of output, and the short term or long term interest rate. The variables are in logs except for the interest rates. Equation 3 differs from the investment equation 12 for the US. In the initial work a capital stock series was constructed for each country, and equation 5.26 in Chapter 5 was estimated for each country. However, this did not produce sensible results for most countries. Typically, the coefficient estimate of the current change in output term seemed much too large. Part of the problem was probably poor estimates of each country's capital stock. The capital stock series were thus not used, and the simpler equation just mentioned was estimated for each country.

The results for equation 3 in Table 6a show that most of the estimated income coefficients are significant and that many of the estimated interest rate coefficients are. The test results in Table 6b are mixed. The lags test is not passed in 14 of the 24 cases, and the RHO+ test is not passed in 14 of the cases. The dynamic properties are thus not well captured in a number of the cases. The T test is not passed in 8 of the 24 cases, a slightly better showing. The led value of output was used for the leads test, and in only 2 cases was the led value significant. Equation 3 fails the stability test in 11 cases. In only 1 of the 12 relevant cases is the price expectations variable significant.

6.5 Equation 4: Y: Production
Equation 4 explains the level of production. It is the same as equation 11 for the US model, which is equation 5.22 in Chapter 5. It includes as explanatory variables the lagged level of production, the current level of sales, and the lagged stock of inventories.

The value of q presented in Table 6a for equation 4 is one minus the coefficient estimate of lagged production. Also presented in the table are the implied values of a and b. The parameters q, a, and b are presented in equations 5.19-5.21. a and q are adjustment parameters. For the quarterly countries q ranges from .517 to .884 and a ranges from .035 to .399. For the annual countries q ranges from .492 to 1.000 and a ranges from .027 to .596. For the United States q was .692 and a was .454.

Equation 4 does well in the tests in Table 6b except for the stability test. 2 of the 19 equations fail the lags test, 1 fails the RHO+ test, 1 fail the T test, and 3 fail the leads test. The led value of sales was used for the leads test. The equation fails the stability test in 9 of the cases.

As was the case for equation 11 in the US model, the coefficient estimates of equation 4 are consistent with the view that firms smooth production relative to sales, and so these results add support to the production smoothing hypothesis.

6.6 Equation 5: PY: Price Index
Equation 5 explains the GDP price index. It is the same as equation 10 for the US model except for the use of different demand pressure variables and the addition of a time trend. It includes as explanatory variables the lagged price level, the price of imports, the wage rate, a demand pressure variable, and a time trend.

Up to 6 demand pressure variables were tried per country in an attempt to estimate the nonlinearity between the level of the unemployment rate or output to the price level that seems likely to exist at low levels of the unemployment rate or high levels of output. Two functional forms were tried for the unemployment rate (when data on the unemployment rate existed for a country). In addition, two other activity variables, both measures of the output gap, were tried in place of the unemployment rate. Two functional forms were also tried for each gap variable. Let ut denote the unemployment rate, and let ut' = ut-umin, where umin is the minimum value of the unemployment rate in the sample period (t=1, ..., T). The first form tried was linear, namely Dt = ut'. The other was Dt = 1/(ut'+ .02). For the second form Dt is infinity when ut' equals -.02, and so this form says that as the unemployment rate approaches 2.0 percentage points below the smallest value it reached in the sample period, the price level approaches infinity. The smaller is a, the more nonlinearity there is near the smallest value of the unemployment rate reached in the sample period. For the first output-gap variable, the potential output series, YS, was used. Define the gap, denoted GAPt, as (YSt - Yt)/YSt, where Yt is the actual level of output, and let GAPt' = GAPt - GAPmin, where GAPmin is the minimum value of GAPt in the sample period. For this variable the first form was linear, and the other was Dt = 1/(GAPt' + .02). For the second output-gap variable, a potential output series was constructed by regressing, over the sample period, logYt on a constant and t. The gap GAPt is then defined to be logYt^ - logYt^, where logYt^ is the predicted value from the regression. The rest of the treatment is the same as for the first output-gap variable. Two functional forms for the unemployment rate and two each for the output-gap variables yields 6 different variables to try. In addition, each variable was tried both unlagged and lagged once separately, giving 12 different variables. This searching was done under the assumption of a first order autoregressive error term, and the autoregressive coefficient was estimated along with the other coefficients. In addition, three other variables were added to the equation during the searching: the price level lagged twice, the wage rate lagged once, and the price of imports lagged once. The demand pressure variable chosen for the "final" equation was the one with the coefficient estimate of the expected sign and the highest t-statistic. No variable was chosen if the coefficient estimates of all the demand pressure variables were of the wrong sign.

Once the demand pressure variable was chosen, three further specification decisions were made based on the estimates using the chosen demand pressure variable. The first decision is whether the current wage rate or the lagged wage rate should be included in the final specification, the second is whether the current import price or the lagged import price should be included, and the third is whether the autoregressive assumption about the error term should be retained. For each of the first two decisions the variable with the higher t-statistic was chosen provided its coefficient estimate was of the expected sign, and for the third decision the autoregressive assumption was retained if the autoregressive coefficient estimate was significant at the 5 percent level. If when tried separately both the current wage rate and the lagged wage rate had coefficient estimates of the wrong sign, neither was used, and similarly for the current import price and the lagged import price.

Once the final specification of equation 5 was chosen for each country, various tests were performed on it.

The results in Table 6a for equation 5 show that the price of imports is significant in most of the cases. Import prices thus appear to have important effects on domestic prices for most countries.

A demand pressure variable (denoted DP in Table 6a) appears in 18 of the 27 equations. When the functional form of the demand pressure variable is linear, the coefficient estimate is expected to be negative, and otherwise it is expected to be positive. Although not directly shown in the table, the linear form was chosen for CA, JA, FR, GE, IT, ST, NO, SW, CO, and TH. For the measure of demand pressure, the unemployment rate was used for FR, FI, DE, and NO, the first gap variable was used for AU, ST, AS, SW, GR, SP, CO, and TH, and the second gap variable was used for the rest (CA, JA, GE, IT, KO, and BE). The variable was lagged once (as opposed to unlagged) for CA, FR, GE, ST, AS, and KO.

Equation 5 does well in the tests in Table 6b except for the stability test. The first lags test is passed in 19 of 26 cases, as is the second lags test. [Footnote x: Multicollinearity problems prevented the lags tests from being computed for AU.] Only 5 equations fail the RHO+ test. The led value of the wage rate was used for the leads test. The wage rate appears in 18 equations, and of these 18 equations, only 4 fail the leads test. On the other hand, 16 of 26 equations fail the stability test.

6.7 Equation 6: M1: Money
[Footnote 6: Money demand equations are estimated in Fair (1987) for 27 countries, and the results in this section are essentially an update of these earlier results.]

Equation 6 explains the per capita demand for money. It is the same as equation 9 for the US model. The same nominal versus real adjustment specifications were tested here as were tested for US equation 9 (and for the US equations 17 and 26). Equation 6 includes as explanatory variables one of the two lagged money variables, depending on which adjustment specification won, the short term interest rate, and income.

The estimates of equation 6 in Table 6a show that the nominal adjustment specification won in 15 of the 20 cases, and so this hypothesis continues its winning ways. Table 6b shows that equation 6 does well in the tests. Only 2 of the 20 equations fails the lags test, 5 fail the RHO+ test, 7 fail the T test, and 6 fail the stability test. The nominal versus real (NvsR) test results in the table simply show that adding the lagged money variable that was not chosen for the final specification does not produce a significant increase in explanatory power. [Footnote x: Multicollinearity problems prevented the N vs R test from being performed for IT, VE, and ID.]

6.8 Equation 7: RS: Short Term Interest Rate
Equation 7 explains the short term (three month) interest rate. It is interpreted as the interest rate reaction function of each country's monetary authority, and it is similar to equation 30 in the US model. The explanatory variables that were tried (as possibly influencing the monetary authority's interest rate decision) are 1) the rate of inflation, 2) two demand pressure variables, ZZ and JJS, 3) the first two lagged values of the asset variable for the quarterly countries and the current and one year lagged value of the asset variable for the annual countries, 4) the German short term interest rate (for the European countries only), and 5) the U.S. short term interest rate. The change in the asset variable is highly correlated with the balance of payments on current account, and so putting in the two asset variables is similar to putting in the balance of payments. The U.S. interest rate was included on the view that some monetary authorities' decisions may be influenced by the Fed's decisions. Similarly, the German interest rate was included in the (non German) European equations on the view that the (non German) European monetary authorities' decisions may be influenced by the decisions of the German central bank. The two asset variables were included on the view that monetary authorities may be influenced in their policy by the status of their balance of payments.

The DW statistic is not included in Table 6a for equation 7 because of space constraints. The results for equation 7 show that the inflation rate is included in 9 of the 24 cases, a demand pressure variable in 10 cases, the asset variables in 11 cases, the German rate in 7 cases, and the U.S. rate in 9 cases. There is thus evidence that monetary authorities are influenced by inflation, demand pressure, and the balance of payments. The signs of the coefficient estimates of the asset variables (negative for the first and positive for the second) suggest that an increase (decrease) in the balance of payments has a negative (positive) effect on the interest rate target of the monetary authority.

Equation 7 does very well in the tests in Table 6b. Only 1 of the 24 equations fail the lags test, only 2 fail the RHO+ test, only 2 fail the T test, and only 6 fail the stability test. This is quite a strong showing.

6.9 Equation 8: RB: Long Term Interest Rate
Equation 8 explains the long term interest rate. It is the same as equations 23 and 24 in the US model. For the quarterly countries the explanatory variables include the lagged dependent variable and the current and two lagged short rates. For the annual countries the explanatory variables include the lagged dependent variable and the current and one lagged short rates. The same restriction was imposed on equation 8 as was imposed on equations 23 and 24, namely that the coefficients on the short rate sum to one in the long run.

The test results in Table 6b for equation 8 show that the restriction that the coefficients sum to one in the long run is supported in 15 of the 18 cases. The equation does very well in the other tests. Only 3 equations fail the lags test, only 2 fail the RHO+ test, only 3 fail the T test, and only 5 fail the stability test. The led value of the short term interest rate was used for the leads test, and it is not significant at the one percent level in any of the 18 cases. As noted in Chapter 5, my experience with term structure equations like equation 8 is that they are quite stable and reliable, which the results in Table 6b support.

6.10 Equation 9 E: Exchange Rate
Equation 9 explains the country's exchange rate: E for the non European countries plus Germany and H for the non German European countries. E is a country's exchange rate is relative to the U.S. dollar, and H is a country's exchange rate relative to the German mark. An increase in E is a depreciation of the country's currency relative to the dollar, and an increase in H is a depreciation of the country's currency relative to the mark. The theory behind the specification of equation 9 is discussed in Chapter 2. See in particular the discussion of the experiments in Section 2.2.6 and the discussion of reaction functions in Section 2.2.7. Equation 9 is interpreted as an exchange rate reaction function.

The equations for E and H are the same, where U.S. variables are the base for the E equations and the German variables are the base for the H equations. The following discussion will focus on E. The equation for E is base on the following two equations.

(6.6) E* = ea0[(1 + RS/100)/(1 + USRS/100)]).25a1(PY/USPY)

(6.7) E/E-1} = (E*/E-1)q

E is the exchange rate, PY is the country's domestic price index, USPY is the U.S. domestic price index (denoted GDPD in the US model), RS is the country's short term interest rate, and USRS is the U.S. short term interest rate (denoted simply RS in the US model). [Footnote 8: RS and USRS are divided by 100 because they are in percentage points rather than percents. Also, the interest rates are at annual rates, and so a1 is multiplied by .25 to put the rates at quarterly rates. For the annual countries, the .25 is not used.] Equation 6.6 states that the long run exchange rate, E*, depends on the relative price level, PY/USPY, and the relative interest rate, (1 + RS/100)/(1 + USRS/100). The coefficient on the relative price level is constrained to be one, which means that in the long run the real exchange rate is assumed merely to fluctuate as the relative interest rate fluctuates. Equation 6.7 is a partial adjustment equation, which says that the actual exchange rate adjusts q percent of the way to the long run exchange rate each period.

The use of the relative price level in equation 6.6 is consistent with the theoretical model in Chapter 2. In this model a positive price shock led to a depreciation of the exchange rate. (See experiments 3 and 4 in Section 2.2.6.) In other words, there are forces in the theoretical model that put downward pressure on a country's currency when there is a relative increase in the country's price level. Because equation 6.6 is interpreted as an exchange rate reaction function, the use of the relative price level in it is in effect based on the assumption that the monetary authority goes along with the forces on the exchange rate and allows it to change in the long run as the relative price level changes.

Similarly, the use of the relative interest rate in equation 6.6 is consistent with the theoretical model, where a fall in the relative interest rate led to a depreciation. (See experiments 1 and 2 in Section 2.2.6.) Again, the assumption in equation 6.6 is that the monetary authority goes along with the forces on the exchange rate from the relative interest rate change.

Equations 6.6 and 6.7 imply that

(6.8) log(E/E-1) = qa0 + qa1(.25)log[(1 + RS/100)/(1 + USRS/100)] + q[log(PY/USPY) - logE-1]

The restriction that the coefficient of the relative price term is one can be tested by adding logE-1 to equation 6.8. If the coefficient is other than one, this variable should have a nonzero coefficient. This is one of the tests performed in Table 6b.

The equations for the European countries (except Germany) are the same as above with H replacing E, GERS replacing USRS, and GEPY replacing USPY, where GERS is the German short term interest rate and GEPY is the German domestic price index.

Exchange rate equations were estimated for 25 countries. For a number of countries the estimate of the coefficient of the relative interest rate variable was of the wrong expected sign, and in these cases the relative interest rate variable was dropped from the equation. Also, for five countries---JA, Au, IT, NE, and UK---the estimate of q in equation 6.8 was very small ("very small" defined to be less than .025), and for these five countries the equation was reestimated with q constrained to be .025.

The estimates of equation 9 in Table 6a do not provide much support for the use of the relative interest rate variable in the equation. The variable is included in only 7 of the equations, and it is never statistically significant. The variable had the wrong sign (and was almost always insignificant) for the other countries. Two of the countries for which the variable is included are Japan and Germany, which are important countries in the model, and so in this sense the relative interest rate variable is important. It will be seen in Chapter 12 that some of the properties of the model are sensitive to the inclusion of the relative interest rate in the exchange rate equations. Given that the relative interest rate is not significant in either the Japanese or German equation, the properties that are sensitive to the inclusion must be interpreted with considerable caution. This is discussed more in Chapter 12.

The unconstrained estimates of q in the equation vary from .020 to .301 for the quarterly countries and from .062 to .552 for the annual countries. A small value for q means that it takes considerable time for the exchange rate to adjust to a relative price level change.

Equation 9 does well in the tests in Table 6b. The restriction discussed above that is tested by adding logE-1 to the equation is only rejected in 3 of the 25 cases. Only 4 of the 25 equations fail the lags test, 10 fail the RHO+ test, 3 fail the T test, and 4 fail the stability test. It is encouraging that so few equations fail the stability test. The key German exchange rate equation passes all the tests, as does the Japanese equation.

Since equation 9 is in log form, the standard errors are roughly in percentage terms. The standard errors for a number of the European countries are quite low, but remember that these are standard errors for H, not E. The variance of H is much smaller than the variance of E for the European countries.

Exchange rate equations are notoriously hard to estimate, and given this, the results in Tables 6a and 6b do not seem too bad. The test results suggest that most of the dynamics have been captured and that the equations are fairly stable. However, many of the key coefficient estimates have t-statistics that are less than two in absolute value.

6.11 Equation 10 F: Forward Rate
Equation 10 explains the country's forward exchange rate, F. This equation is the estimated arbitrage condition, and although it plays no role in the model, it is of interest to see how closely the quarterly data on EE, F, RS, and USRS match the arbitrage condition. The arbitrage condition in this notation is

F/EE = [(1 + RS/100)/(1 + USRS/100)].25

In equation 10, logF is regressed on logEE and .25log(1 + RS/100)/(1 + USRS/100). If the arbitrage condition were met exactly, the coefficient estimates for both explanatory variables would be one and the fit would be perfect.

The results in Table 6a for equation 10 show that the data are generally consistent with the arbitrage condition, especially considering that some of the interest rate data are not exactly the right data to use. Note the t-statistic for Germany of 6998.01!

6.12 Equation 11 PX: Export Price Index
Equation 11 explains the export price index, PX. It provides a link from the GDP index, PY, to the export price index. Export prices are needed when the countries are linked together (see Table B.4 in Appendix B). If a country produced only one good, then the export price would be the domestic price and only one price equation would be needed. In practice, of course, a country produces many goods, only some of which are exported. If a country is a price taker with respect to its exports, then its export prices would just be the world prices of the export goods. To try to capture the in between case where a country has some effect on its export prices but not complete control over every price, the following equation is postulated:

(6.11) PX = PYq(PW$.E)1-q

PW$ is the world price index in dollars, and so PW$.E is the world price index in local currency. Equation 6.11 thus takes PX to be a weighted average of PY and the world price index in local currency, where the weights sum to one. Equation 11 was not estimated for any of the major oil exporting countries, and so PW$ was constructed to be net of oil prices. (See equations L-4 in Table B.4.)

Equation 6.11 was estimated in the following form:

(6.12) logPX - log(PW$.E) = q[logPY - log(PW$.E)]

The restriction that the weights sum to one and that PW$ and E have the same coefficient (i.e, that their product enters the equation) can be tested by adding logPY and logE to equation 6.12. If this restriction is not met, these variables should be significant. This is one of the tests performed in Table 6b.

Equation 11 was estimated for 28 countries. For 7 of the countries the estimate of q was greater than 1, and for these cases the equation was reestimated with q constrained to be 1. When q is 1, there is a one to one link between PY and PX. Equation 11 was estimated under the assumption of a second order autoregressive error term. The results in for equation 11 in Table 6a show that the estimates of the autoregressive parameters are generally large.

Equation 11 does reasonably well in the tests in Table 6b. The restriction discussed above is rejected in 10 of the 28 cases. The equation fails the RHO+ test in only 2 cases. Multicollinearity problems prevented the stability test from being performed for 5 countries (GE, NE, FI, DE, and PO). Of the 23 remaining cases, the equation fails the stability test in 3 of them.

It should be kept in mind that equation 11 is meant only as a rough approximation. If more disaggregated data were available, one would want to estimate separate price equations for each good, where some goods' prices would be strongly influenced by world prices and some would not. This type of disaggregation is beyond the scope of the present work.

6.13 Equation 12: W: Wage Rate
Equation 12 explains the wage rate. It is similar to equation 16 for the US model. It includes as explanatory variables the lagged wage rate, the current price level, the lagged price level, a demand pressure variable, and a time trend. Equation 16 of the US model included three further lags of the wage rate and price level, which equation 12 does not. Also, equation 16 of the US model does not include any demand pressure variables because none were significant. The same restriction imposed on the price and wage equations in the US model is also imposed here. Given the coefficient estimates of equation 5, the restriction is imposed on the coefficients in equation 12 so that the implied real wage equation does not have the real wage depend on either the nominal wage rate or the price level separately. (See the discussion of equations 5.35, 5.36, and 5.37 in Section 5.4.)

The same searching for the best demand pressure variable was done for the wage equation as was done for the price equation. This searching was done without imposing the coefficient restriction in (11) and under the assumption of a first order autoregressive error term.

The estimates of equation 12 in Table 6a show some support for the demand pressure variables having an effect on the wage rate. A demand pressure variable (denoted DW) appears in 11 of the 15 equations. Again, when the functional form of the demand pressure variable is linear, the coefficient estimate is expected to be negative, and otherwise it is expected to be positive. Although not directly shown in the table, the linear form was chosen for AU, GE, IT, FI, DE, GR, and SP. For the measure of demand pressure, the unemployment rate was used for GE, IT, FI, and DE, and the second gap variable was used for the rest (AU, NE, UK, KO, SW, GR, and SP). (The first gap variable was not chosen for any of the countries.) The variable was lagged once (as opposed to unlagged) for AU, GE, IT, NE, and FI.

The test results in Table 6b show that the real wage restriction is rejected in only 2 of the 15 cases. 3 equations fail the lags test, 5 fail the RHO+ test, and 9 fail the stability test. The test results are thus good except for the stability results.

6.14 Equation 13: J: Employment
Equation 13 explains the change in employment. It is in log form, and it is similar to equation 13 for the US model. It includes as explanatory variables the amount of excess labor on hand, the change in output, the lagged change in output, and a time trend. Equation 13 for the US model does not include the lagged change in output because it was not significant. On the other hand, US equation 13 includes terms designed to pick up a break in the sample period, which equation 13 does not, and it includes the lagged change in employment, which equation 13 does not.

Most of the coefficient estimates for the excess labor variable are significant in Table 6a for equation 13, which is at least indirect support for the theory that firms at times hold excess labor and that the amount of excess labor on hand affects current employment decisions. Most of the change in output terms are also significant.

Equation 13 fails the lags test in 4 of the 14 cases, and it fails the RHO+ test in 6 cases. The led value of the change in output was used for the leads tests, and it was only significant in 1 case. The equation fails the stability test in 6 cases.

6.15 Equation 14: L1: Labor Force-Men; Equation 15: L2: Labor Force-Women
Equations 14 and 15 explain the labor force participation rates of men and women, respectively. They are in log form and are similar to equations 5, 6, and 7 in the US model. The explanatory variables include the real wage, the labor constraint variable, a time trend, and the lagged dependent variable.

The labor constraint variable is significant in most cases for equations 14 and 15 in Table 6a, which provides support for the discouraged worker effect. There is only modest support for the real wage. It appears in only 2 of the 6 cases for equation 14 and in 3 of the 4 cases for equation 15. When the real wage appeared in the equation, the log of the price level was added to the equation for one of the tests to test the real wage restriction. Table 6b shows that the log of the price level was not significant in any of the cases.

Table 6b also shows that equation 14 fails the lags test in 1 of the 6 cases and the RHO+ test in 5 cases. Equation 15 fails no lags tests out of 4 and 2 RHO+ tests. Both equations do poorly in the stability test. Equation 14 fails all but 1 of the cases, and equation 15 fails them all.

6.16 The Trade Share Equations
As discussed in Chapter 3, aij is the fraction of country i's exports imported by j, where i runs from 1 to 44 and j runs from 1 to 45. The data on aij are quarterly, with observations for most ij pairs beginning in 1960:1.

One would expect aij to depend on country i's export price relative to an index of export prices of all the other countries. The empirical work consisted of trying to estimate the effects of relative prices on aij. A separate equation was estimated for each ij pair. The equation is the following:

(6.13) aijt = bij1 + bij2aijt-1 + bij3(PX$it/SUM44k=1akjtPX$kt) + uijt (t = 1, ..., T)

PX$it is the price index of country i's exports, and SUM44k=1akjtPX$kt) is an index of all countries' export prices, where the weight for a given country k is the share of k's exports to j in the total imports of i. (In this summation k=i is skipped.)

With i running from 1 to 44, j running from 1 to 45, and not counting i=j, there are 1936, (= 44 x 44) ij pairs. There are thus 1936 potential trade share equations to estimate. In fact, only 1280 trade share equations were estimated. Data did not exist for all pairs and all quarters, and if fewer than 26 observations were available for a given pair, the equation was not estimated for that pair. A few other pairs were excluded because at least some of the observations seemed extreme and likely suffering from measurement error. Almost all of these cases were for the smaller countries.

Each of the 1280 equations was estimated by ordinary least squares. The main coefficient of interest is bij3, the coefficient of the relative price variable. Of the 1280 estimates of this coefficient, 81.3 percent (1041) were of the expected negative sign. 40.7 percent had the correct sign and a t-statistic greater than two in absolute value, and 63.8 percent had the correct sign and a t-statistic greater than one in absolute value. 4.4 percent had the wrong sign and a t-statistic greater than two, and 9.8 percent had the wrong sign and a t-statistic greater than one. The overall results are thus quite supportive of the view that relative prices affect trade shares.

The average of the 1041 estimates of bij3 that were of the right sign is -.0137. bij3 measures the short run effect of a relative price change on the trade share. The long run effect is bij3/(1-bij2), and the average of the 1041 values of this is -.0768.

The trade share equations with the wrong sign for bij3 were not used in the solution of the model. The trade shares for these ij pairs were taken to be exogenous.

It should be noted regarding the solution of the model that the predicted values of aijt, say, ^aijt, do not obey the property that SUM44i=1^aijt = 1. Unless this property is obeyed, the sum of total world exports will not equal the sum of total world imports. For solution purposes each ^aijt was divided by SUM44i=1^aijt = 1, and this adjusted figure was used as the predicted trade share. In other words, the values predicted by the equations in 6.13 were adjusted to satisfy the requirement that the trade shares sum to one.

6.17 Additional Comments
The following are a few general remarks about the results in this chapter.
  1. The strong rejection of the change form of the price equation in Table 6b is an important result. As discussed in point 11 in Section 5.10, this has important implications for the long run properties of the model. The significance of the import price index in the price equations is also important. This shows how price levels in different countries affect each other.
  2. The results of estimating the demand for money equation in Table 6a provide further support for the nominal adjustment hypothesis over the real adjustment hypothesis. See also point 5 in Section 5.10.
  3. The U.S. interest rate is significant or nearly significant in 12 of the interest rate reaction functions in Table 6a. This is evidence that the Fed influences the economies of other countries by influencing other countries' interest rates. It will be seen in Chapter 12 that this is an important link. Similarly, the German interest rate is significant or nearly significant in 6 of the interest rate reaction functions.
  4. A key question for the exchange rate equation in Table 6a is whether one can trust the inclusion of the relative interest rate variable in the equations. The verdict is not yet in on this question.
  5. The excess labor variable is significant in most of the estimates of equation 13 in Table 6a, which adds further support to the theory that firms at times hoard labor.
  6. As was the case for the US model, the results support the use of nominal interest rates over real interest rates. In very few cases is the inflation expectations variable significant.
  7. There is little support for the use of the led values and thus little support for the rational expectations hypothesis. The led values are significant at the one percent level in only 16 of the 155 cases in which they were tried.
  8. The equations in general do well for the lags, T, and RHO+ tests. For the lags test there are 62 failures out of 256 cases (24 percent); for the T test there are 39 failures out of 192 cases (20 percent); and for the RHO+ test there are 72 failures out of 286 cases (25 percent). These results suggest that the dynamic specification of the equations is reasonably good. The results are not as good for the stability test, where there are 113 failures out of 272 cases (42 percent). More observations are probably needed before much can be done about this problem.