5. The Stochastic Equations of the US Model 
{Site footnote: This chapter has been updated from Chapter 5 in the
1994
The version of the US model discussed here is dated
November 1, 1996. The "Chapter 5 tables" for this version, which this chapter
heavily relies on, are presented at the end of Appendix A of this
document.}

5.1 Introduction 
The stochastic equations of the US model are specified, estimated, and tested
in this chapter. As noted at the beginning of Chapter 3, extra "theorizing" is
involved in going from theory like that in Chapter 2 to empirical specifications. This
chapter thus uses the theory in Chapter 2 plus additional theory in the specification of
the stochastic equations. The stochastic equations are listed in Table A.3 in Appendix A, and the variables are defined in Table A.2. The construction of the variables is discussed in Chapter 3. There are 30 stochastic equations in the US model. The empirical results for the equations are presented in Tables 5.1 through 5.30 in this chapter, one table per equation except for equations 19 and 29. (There are no tables for equations 19 and 29.) Each table gives the left hand side variable, the right hand side variables that were chosen for the "final" specification, and the results of the tests described in Chapter 4. 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 quarters, 5) adding additional variables, and 6) testing for structural stability. Also, the joint significance of the age distribution variables is examined in the household expenditure and money demand equations. Remember that "adding lagged values" means adding lagged values of all the variables in the equation (including the left hand side variable if the lagged dependent variable is not an explanatory variable). As discussed in Section 4.5, this is a test against a quite general dynamic specification. For the autoregressive test, the notation "RHO=4" will be used to denote the fact that a fourth order autoregressive process was used. It will be seen that only a few of the equations pass all the tests. My experience is that it is hard to find macroeconomic equations that do. If an equation does not pass a test, it is not always clear what should be done. If, for example, the hypothesis of structural stability is rejected, one possibility is to divide the sample period into two parts and estimate two separate equations. If this is done, however, the resulting coefficient estimates are not always sensible in terms of what one would expect from theory. Similarly, when the additional lagged values are significant, the equation with the additional lagged values does not always have what one would consider sensible dynamic properties. In other words, when an equation fails a test, the change in the equation that the test results suggest may not produce what seem to be sensible results. In many cases, the best choice seems to be to stay with the original equation even though it failed the test. My feeling (being optimistic) is that much of this difficulty is due to small sample problems, which will lessen over time as sample sizes increase, but this is an important area for future work. Obviously less confidence should be placed on equations that fail a number of the tests than on those that do not. The chi squared value is presented for each test along with its degrees of freedom. Also presented for each test is the probability that the chi squared value would be whatever it is if the null hypothesis that the additional variables do not belong in the equation is true. These probabilities are labeled "pvalue" in the tables. A small pvalue is evidence against the null hypothesis and thus evidence against the specification of the equation. In the following discussion of the results, a pvalue of less than .01 will be taken as a rejection of the null hypothesis and thus as a rejection of the specification of the equation. [Footnote 1: Using a value of .01 instead of, say, .05 gives the benefit of the doubt to the equations, but, as indicated above, the equations need all the help they can get.] It will be seen that lagged dependent variables are used as explanatory variables in many of the equations. They are generally highly significant, even after accounting for any autoregressive properties of the error terms. It is well known that they can be accounting for either partial adjustment effects or expectational effects and that it is difficult to identify the two effects separately. [Footnote 2: See Fair (1984), Section 2.2.2, for a discussion of this.] For the most part no attempt is made in what follows to separate the two effects, although, as discussed in Chapter 4, the tests of the significance of the led values are tests of the rational expectations hypothesis. In testing for the significance of nominal versus real interest rates, some measure of expected future inflation must be used in constructing real interest rate variables. Two measures were used in the following work: the actual rate of inflation in the past four quarters, denoted p_{4}^{e}, and the actual rate of inflation (at an annual rate) in the past eight quarters, denoted p_{8}^{e}. The price deflator used for this purpose is PD, the price deflator for domestic sales, and so p_{4}^{e}=100[(PD/PD_{4})1] and p_{8}^{e}=100[(PD/PD_{8})^{.5}1]. The significance of nominal versus real interest rates was tested as follows. Consider RMA, the after tax mortgage rate, which is used in the model as the long term interest rate facing the household sector. Assume that p_{8}^{e} is an adequate proxy for inflation expectations of the household sector. If real interest rates affect household behavior, then RMAp_{8}^{e} should enter the household expenditure equations, and if nominal interest rates affect household behavior, then RMA alone should enter. The test of real versus nominal rates was first to estimate the equation with RMA included and then to add p_{8}^{e} and reestimate. If real rates instead of nominal rates matter and if p_{8}^{e} is a good proxy for actual inflation expectations, then p_{8}^{e} should be significant and have a coefficient estimate that equals (aside from sampling error) the negative of the coefficient estimate of RMA . The same reasoning holds for p_{4}^{e}. As will be seen, the p_{4}^{e} and p_{8}^{e} variables were never significant when added to the household expenditure equations, whereas the nominal interest rate variables were, and so the data support the use of nominal over real interest rates. It could be, of course, that the inflation expectations variables are not good approximations of actual expectations and that if better expectations variables were used they would be significant. This is an open question and an area for future research. It will also be seen below that the real interest rate does affect nonresidential fixed investment, although the estimated effect is small and may be the result of data mining. The basic estimation period was 1954:11996:3, for a total of 171 observations. For the AP stability tests, T_{1}, the first possible quarter for the break, was taken to be 1970:1 and T_{2}, the last possible quarter for the break, was taken to be 1979:4. The "break" quarter that is presented in the tables for the AP test is the quarter at which the break in the sample period corresponds to the largest chi squared value. Although not shown in the tables, it was generally the case that the chi squared values monotonically rose to the largest value and monotonically fell after that. A ^{*} after the AP value in the tables denotes that the value is significant at the one percent level. In other words, a ^{*} means that the hypothesis of no break is rejected at the one percent level: the equation fails the stability test. Tests of the Leads Three sets of led values were tried per equation. For the first set the values
of the relevant variables led once were added. For the second set the values led one
through four times were added. For the third set the values led one through eight times
were added, with the coefficients for each variable constrained to lie on a second degree
polynomial with an end point constraint of zero. To see what was done for the third set,
assume that one of the variables for which the led values are used is X_{2i}. Then
for the third set the term added is It may be helpful to review the exact procedure that was followed for the leads test. First, the estimation period was taken to be shorter by one, four, or eight observations. (When values led once are added the sample period has to be shorter by one to allow for the led values; when the values led four times are added the sample period has to be shorter by four; and so on.) The equation with the led values added was then estimated using Hansen's GMM estimator under the appropriate assumption about the moving average process of the error term (zero for leads +1, three for leads +4, and seven for leads +8). The M_{i} matrix discussed in Section 4.3 that results from this estimation was then used in the estimation of the equation without the led values by Hansen's method for the same (shorter) estimation period. The chi squared value is then (SS_{i}^{**}  SS_{i}^{*})/T, as discussed at the beginning of Section 4.5. The results of adding the led values to the stochastic equations are used in Chapter 11, Section 11.6, to examine the economic significance of the rational expectations assumption in the US model.[Footnote 3: This work is an updated version of the material in Section 5 in Fair (1993b).] The question asked in Section 11.6 is: How much difference to the properties of the US model does the addition of the led values make? Two versions of the model are examined. The first consists of the "base" equations in Tables 5.15.30, which have no led values in them. This version is called Version 1. The second version consists of the equations with the third set of led values added (i.e., with F_{1t} and F_{2t} added). This version is called Version 2. The particular variables for which led values were used are mentioned in this chapter in the discussion of each equation. For some equations no led values were tried because none seemed appropriate, and so these equations are the same for both Versions 1 and 2. First Stage Regressors The first stage regressors (FSRs) that were used for each equation are listed in Table A.7 in Appendix A. The choice of FSRs for large scale models is discussed in Fair (1984), pp. 215216, and this discussion will not be repeated here. Autoregressive Errors Each equation was first estimated under the assumption of a first order autoregressive error term, and the assumption was retained if the estimate of the autoregressive coefficient was significant. In one case (equation 4) a second order process was used in the final specification, and in one case (equation 11) a third order process was used. In the notation in the tables "RHO1" refers to the first order coefficient, "RHO2" to the second order coefficient, and "RHO3" to the third order coefficient. Previous Version of the US Model The previous version of the US model is presented in Fair (1984), Chapter 4. The present discussion of the model is self contained, and so this previous material does not have to be read. For the most part the current version of the model is quite similar to the previous version. Three of the main changes are 1) the use of disposable income in the household expenditure equations instead of the wage, price, nonlabor income, and labor constraint variables separately, 2) the use of the age distribution variables, and 3) the different treatment of the interest payment variables of the firm and federal government sectors (equations 19 and 29). In addition, a few more coefficient constraints have been imposed in the current version, and different functional forms have been used in a few cases. 
5.2 Household Expenditure and Labor Supply Equations 
The two main decision variables of a household in the theoretical model are
consumption and labor supply. The determinants of these variables include the initial
value of wealth and the current and expected future values of the wage rate, the price
level, the interest rate, the tax rate, and the level of transfer payments. The labor
constraint also affects the decisions if it is binding. In the econometric model the expenditures of the household sector are disaggregated into four types: consumption of services (CS), consumption of nondurable goods (CN), consumption of durable goods (CD), and investment in housing (IHH). Four labor supply variables are used: the labor force of men 2554 (L1), the labor force of women 2554 (L2), the labor force of all others 16+ (L3), and the number of people holding more than one job, called "moonlighters" (LM). These eight variables are determined by eight estimated equations. Consider first the four expenditure equations. The household wealth variable in the model is AA, and the lagged value of this variable was tried in each of the equations. The variable is expected to have a positive sign, and if it did not, which occurred in two of the four equations, it was dropped. The household after tax interest rate variables in the model are RSA, a short term rate, and RMA, a long term rate. RSA was used in the CS equation, and RMA was used in the others. The CS and CN equations are in log form per capita, and the interest rates were entered additively in these equations. The means that, say, a one percentage point change in the interest rate has the same percent change over time in each of the two equations. The CD and IHH equations, on the other hand, are in per capita but not log form, and if the interest rates were entered additively in these equations, the effect of, say, a one percentage point change in the interest rate would have a smaller and smaller percent effect over time on per capita durable consumption and on per capita housing investment as both increase in size over time. Since this does not seem sensible, the interest rate in the CD equation was multiplied by CDA, which is a variable constructed from peak to peak interpolations of CD/POP. Similarly, the interest rate in the IHH equation was multiplied by IHHA, which is a variable constructed from peak to peak interpolations of IHH/POP. Both CDA and IHHA are merely scale variables, and they are taken to be exogenous. These interest rate variables are nominal rates. As discussed above, the inflation expectations variables p_{4}^{e} and p_{8}^{e} were added in the testing of the equations to test for real interest rate effects, and the results of these tests are reported below. The age distribution variables were tried in the four expenditure equations, and they were jointly significant at the five percent level in three of the four, the insignificant results occurring in the IHH equation. They were retained in the three equations in which they were significant. The lagged dependent variable and the constant term were included in each of the four expenditure equations. Regarding the wage, price, and income variables, there are at least two basic approaches that can be taken in specifying the expenditure equations. The first is to add the wage, price, nonlabor income, and labor constraint variables separately to the equations. These variables in the model are as follows. The after tax nominal wage rate variable is WA, the price deflator for total household expenditures is PH , the after tax nonlabor income variable is YNL, and the labor constraint variable, discussed in Chapter 3, is Z. The price deflators for the four expenditure categories are PCS, PCN, PCD, and PIH. Consider the CS equation. Under the first approach one might add WA/PH , PCS/PH, YNL/PH, and Z to the equation. The justification for including Z is the following. By construction, Z is zero or nearly zero in tight labor markets (i.e., when JJ is equal to or nearly equal to JJP , where JJ is the actual ratio of worker hours paid for to the total population and JJP is the potential ratio). In this case the labor constraint is not binding and Z has no effect or only a small effect in the equation. This is the "classical" case. As labor markets get looser (i.e., as JJ falls relative to JJP), on the other hand, Z falls and begins to have an effect in the equation. Loose labor markets, where Z is large in absolute value, correspond to the "Keynesian" case. Since Z is highly correlated with hours paid for in loose labor markets, having both WA and Z in the equation is similar to having a labor income variable in the equation in loose labor markets. The second, more traditional, approach is to replace the above four variables with real disposable personal income, YD/PH. This approach in effect assumes that labor markets are always loose and that the responses to changes in labor and nonlabor income are the same. One can test whether the data support YD/PH over the other variables by including all the variables in the equation and examining their significance. The results of doing this in the four expenditure equations generally supported the use of YD/PH over the other variables, and so the equations reported below use YD/PH. This is a change from the version of the model in Fair (1984), where the first approach was used. The dominance of YD/PH does not necessarily mean that the classical case never holds in practice. What it does suggest is that trying to capture the classical case through the use of Z does not work. An interesting question for future work is whether the classical case can be captured in some other way. It will be seen below that the Z variable does work in the labor supply equations, where it is picking up "discouraged worker" effects when labor markets are loose. Some searching was done in arriving at the "final" equations presented below. Explanatory variables lagged once as well as unlagged were generally tried, and variables were dropped from the equation if they had coefficient estimates of the wrong expected sign in both the unlagged and lagged cases. Also, as noted above, each equation was estimated under the assumption of a first order autoregressive error term, and the assumption was retained if the estimate of the autoregressive coefficient was significant. All this searching was done using the 2SLS technique. Equation 1. CS, consumer expenditures: services The results of estimating equation 1 are presented in Table 5.1. The equation is in real, per capita terms and is in log form. The series for CS is quite smooth, and most of the explanatory power in equation 1 comes from the lagged dependent variable. The disposable income variable has a small short run coefficient (.0704) and a long run coefficient of a little over one [1.09 = .0704/(1.9355)].[Footnote 4: Since the equation is in log form, these coefficients are elasticities.] The short term interest rate is significant. The age variables are jointly significant at the one percent level according to the chi squared value. Remember that the coefficient of AG1 is the coefficient for people 2655 minus the coefficient for those 1625. Similarly, the coefficient of AG2 is the coefficient for people 5665 minus the coefficient for those 1625, and the coefficient of AG3 is the coefficient for people 66+ minus the coefficient for those 1625. The age coefficient estimates for the CS equation suggest that, other things being equal, people 2655 spend less than others (the coefficient estimate for AG1 is negative and the other two age coefficient estimates are positive), which is consistent with the life cycle idea that people in their prime working years spend less relative to their incomes than do others. Consider now the test results in Table 5.1. (Remember that an equation will be said to have passed a test if the pvalue is greater than .01.) Equation 1 passes the lags test[Footnote 5: Remember that for the lags test all the variables in the equation lagged once are added to the equation (except for the age variables). This means that for equation 1 three variables are added: log(CS/POP)_{2}, log[YD/(POP^{.}PH)]_{1}, and RSA_{1}.] and the RHO=4 test. These results thus suggest that the dynamic specification of the equation is fairly accurate. On the other hand, the equation dramatically fails the T test: the time trend is highly significant when it is added to equation 1. This suggests that the trending nature of the CS series has not been adequately accounted for in the equation. None of the other specifications that were tried eliminated this problem, and it is an interesting area for future research. Disposable income was the variable for which led values were triedin the form log[YD/(POP^{.}PH)]and the test results show that the leads are highly significant. This is thus evidence in favor of the rational expectations assumption. The largest chi squared value was for 8 leads. This is the equation that is used for Version 2 in Section 11.6 to examine the sensitivity of the model's properties to the use of the led values. The inflation expectations variables are not significant, which is evidence against the use of real versus nominal interest rates. The additional variables ("Other"), which, as discussed above, are the variables that one might use in place of disposable income, are significant. However, although not shown in the table, the coefficient estimates for the variables are all of the wrong expected sign, and so the version of the equation with these variables added is not sensible. There appears to be too much collinearity among these variables to be able to get sensible estimates. For the "spread" test in Table 5.1 and in the other relevant tables that follow, the current and first three lagged values of the spread between the commercial paper rate and the Treasury bill rate were added to the equation. For this test the estimation period began in 1960:2 rather than 1954:1 because data on the spread were only available from 1959:1 on.[Footnote 6: Whenever an estimation period had to be changed for a test, the basic equation was always reestimated for this period in calculating the chi squared value for the test.] As can be seen, the spread values are not close to being significant, with a pvalue of .7326. Finally, the equation fails the stability test. The AP value is 18.58. The value of lambda for this test is 2.53. The one percent critical AP value for 7 coefficients and lambda equal to 2.75 is 9.50 (see Table 4.1 in Chapter 4). This is also roughly the critical value for lambda equal to 2.53, and so the AP value of 18.58 is considerably above the critical value and thus the stability test is not passed. The largest chi squared value occurred for 1971:4, which is near the beginning of the test period of 1970:11979:4. Equation 2. CN, consumer expenditures: nondurables Equation 2 is also in real, per capita, and log terms. The results are presented in Table 5.2. The asset, disposable income, and interest rate variables are significant in this equation, along with the age variables and the lagged dependent variable. Both the level and change of the lagged dependent variable are significant in the equation, and so the dynamic specification is more complicated than that of equation 1. Again, the age coefficients show that people 2655 spend less than others, other things being equal. The equation passes the T test, but it fails the lags and RHO=4 tests. The variable for which led values were tried is again disposable income, and leads +1 and +8 are significant. The inflation expectations variables are not significant. The additional variables, representing the wage, price, nonlabor income, and labor constraint variables are not significant. Likewise, the spread values are not significant. The equation fails the stability test. The AP value is 19.93, which compares to the one percent critical value (for 9 coefficients) of 11.20 in Table 4.1. The maximum chi squared value occurs for 1973:2. Equation 3. CD, consumer expenditures: durables Equation 3 is in real, per capital terms. One of the explanatory variables is
the lagged stock of durable goods, and the justification for including this variable is as
follows. Let KD^{**} denote the stock of durable goods that would be desired if
there were no adjustment costs of any kind. If durable consumption is proportional to the
stock of durables, then the determinants of consumption can be assumed to be the
determinants of KD^{**}: The results of estimating equation 3 are presented in Table 5.3. The coefficient of the lagged dependent variable is .7177, and so g above is .2823. As discussed in Chapter 3, the depreciation rate, DELD, is equal to .049511. Given these two values and given the coefficient of the lagged stock variable in Table 5.3 of .0091, the implied value of q is .082. This implies an adjustment of the durable stock to its desired value of 8.2 percent per quarter. The age variables are jointly highly significant. The age coefficients show people 2655 spending more, other things being equal, than the others. The pattern here is thus different than the pattern for service and nondurable consumption. Regarding the tests, equation 3 passes the lags test, but it fails the RHO=4 and T tests. The variable for which led values were tried is disposable income, and the led values are highly significant. The inflation expectations variables are significant. The other variables, which are the asset, wage, price, nonlabor income, and labor constraint variables, are also significant, as are the spread values. The equation fails the stability test by a wide margin. In short, equation 3 fails all but the lags test, and so it is clearly an equation for which more work is needed. Equation 4. IHH, residential investmenth The same partial adjustment model is used for housing investment than was used above for durable expenditures, which adds both the lagged dependent variable and the lagged stock of housing to the housing investment equation. For example, the coefficient of the lagged housing stock variable, KH_{1} , is g(DELHq), where DELH is the depreciation rate of the housing stock. The equation is estimated under the assumption of a second order autoregressive error term. The asset, income, and interest rate variables are significant in Table 5.4, as are the lagged dependent variable and the lagged housing stock variable. The coefficient of the lagged dependent variable is .5567, and so g is .4433. As discussed in Chapter 3, the depreciation rate for the housing stock, DELH, is .006716. Given these two values and given the coefficient of the lagged housing stock variable of .0807, the implied value of q is .189. The estimated adjustment speed of the housing stock to its desired value is thus greater than the estimated adjustment speed of the durable goods stock. This is not necessarily what one would expect, and it may suggest that the estimated speed for the durable goods stock is too low. The chi squared test for the age variables shows that the age variables are not jointly significant. This is the reason they were not included in the final specification of the equation. Equation 4 passes the lags, RHO=4, and T tests. The variable for which led values were tried is disposable income, and the led values are not significant. The inflation expectations variables are not significant; the "other" variables are not significant; and the spread values are not close to being significant. Equation 4 thus passes all the chi squared tests, the only expenditure equation of the four to do so. It also passes the stability test, again the only expenditure equation to do so. The next four equations of the household sector are the labor supply equations, which will now be discussed. Equation 5. L1, labor forcemen 2554 [Footnote 8: In Section II in Fair and Dominguez (1991) the age distribution data discussed above were used to examine some of Easterlin's (1987) ideas regarding the effects of cohort size on wage rates. This was done in the context of specifying equations for L1 and L2. I now have, however, (for reasons that will not be pursued here) some reservations about the appropriateness of the specifications that were used, and in the present work the age distribution data have not been used in the specification of equations 5 and 6. This is an area of future research. I am indebted to Diane Macunovich for helpful discussions in this area.] One would expect from the theory of household behavior for labor supply to depend, among other things, on the after tax wage rate, the price level, and wealth. In addition, if the labor constraint is at times binding on households, one would expect a labor constraint variable like Z to affect labor supply through the discouraged worker effect. Equation 5 explains the labor force participation rate of men 2554. It is in log form and includes as explanatory variables the real wage (WA/PH), the labor constraint variable (Z), a time trend, and the lagged dependent variable. The coefficient estimate for the real wage is positive, implying that the substitution effect dominates the income effect, although the estimate is not close to being significant. The coefficient estimate of the labor constraint variable is positive, as expected, but it is also not significant. The coefficient estimate for the time trend is negative and significant. There is a slight negative trend in the labor force participation of men 2554 that does not seem to be explained by other variables, and so the time trend was included in the equation. Equation 5 passes the lags test, but fails the RHO=4 test. The variable for which led values were tried is the real wage [log(WA/PH)], and the led values are not significant. Another test reported in Table 5.5 has logPH added as an explanatory variable. This is a test of the use of the real wage in the equation. If logPH is significant, this is a rejection of the hypothesis that the coefficient of logWA is equal to the negative of the coefficient of logPH, which is implied by the use of the real wage. As can be seen, logPH is not significant. The final chi squared test in Table 5.5 has asset and nonlabor income variables added to the equation. These variables are not significant. Equation 5 passes the stability test. Equation 6. L2, labor forcewomen 2554 Equation 6 explains the labor force participation rate of women 2554. It is also in log form and includes as explanatory variables the real wage, the labor constraint variable, and the lagged dependent variable. The coefficient estimate for the real wage is positive and significant. The coefficient estimate for the labor constraint is positive but not significant. The coefficient estimate for the lagged dependent variable is quite high (.9872). Regarding the tests, the equation passes the lags test, the RHO=4 test, and the T test. The variable for which led values were tried is the real wage (log(WA/PH)), and the led values are not significant. The equation thus does well on these tests. When logPH is added to the equation, for the next test, it is not significant, thus supporting the real wage constraint. For a final chi squared test, asset and nonlabor income variables were added to the equation. These variables are significant, but (although not shown in the table) the coefficient estimates were of the wrong expected sign. One expects the level of assets and nonlabor income to have a negative effect on participation, but the coefficient estimates were positive. The equation fails the stability test. The AP value is 11.90, which compares to the critical one percent value for 4 coefficients of 7.00. Equation 7. L3, labor forceall others 16+ Equation 7 explains the labor force participation rate of all others 16+. It is also in log form and includes as explanatory variables the real wage, the labor constraint variable, an asset variable, the time trend, and the lagged dependent variable. The coefficient estimate for the real wage is positive and significant. The asset variable has a negative coefficient estimate and the labor market tightness variable has a positive one, as expected. The coefficient estimate for the time trend is negative and significant, and so, like L1, L3 appears to have a negative trend that is not explained by other variables. Equation 7 passes the lags test and the RHO=4 test. The variable for which led values were tried is the real wage, and the led values are not significant. When logPH is added to the equation, it is insignificant. The "other" variable that is added is the lagged value of nonlabor income, and it is not significant at the one percent level. The equation passes the stability test. Equation 8. LM, number of moonlighters Equation 8 determines the number of moonlighters. It is in log form and includes as explanatory variables the real wage, the labor constraint variable, and the lagged dependent variable. The coefficient estimate for the real wage is positive, suggesting that the substitution effect dominates for moonlighters, although the variable is not significant. The coefficient estimate for the labor constraint variable is positive and significant. The larger is the labor constraint, the fewer are the number of people holding two jobs. Equation 8 does well in the tests. It passes the lags test, the RHO=4 test, and the T test. The variable for which led values were tried is the real wage, and the led values are not significant. When logPH is added to the equation, it is not significant, and so the real wage constraint is supported. The "other" variables that were added are the lagged value of wealth and the lagged value of nonlabor income, and they are significant. The equation thus fails this test. Finally, the equation passes the stability test. This completes the discussion of the household expenditure and labor supply equations. A summary of some of the general results across the equations is in Section 5.10. 
5.3 Money Demand Equations 
[Footnote 9: The material in this section on the test of real versus nominal
adjustment is taken from Fair (1987). The tests of the age distribution variables in
equation 9 is taken from Fair and Dominguez (1991).] In the theoretical model a household's demand for money depends on the level of transactions, the interest rate, and the household's wage rate. High wage rate households spend less time taking care of money holdings than do low wage rate households and thus on average hold more money. With aggregate data it is not possible to estimate this wage rate effect on the demand for money, and in the empirical work the demand for money has simply been taken to be a function of the interest rate and a transactions variable. However, the age distribution variables have been added to the household money demand equation, and, as discussed below, this may pick up a wage rate effect. The model contains three demand for money equations: one for the household sector, one for the firm sector, and a demand for currency equation. Before presenting these equations it will be useful to discuss how the dynamics were handled. The key question about the dynamics is whether the adjustment of actual to desired values is in nominal or real terms. Let M_{t}^{*}/P_{t} denote the desired level of real
money balances, let y_{t} denote a measure of real transactions, and let r_{t}
denote a short term interest rate. Assume that the equation determining desired money
balances is in log form and write If the adjustment of actual to desired money holdings is in real terms, the
adjustment equation is A test of the two hypotheses is simply to put both lagged money variables in the equation and see which one dominates. If the real adjustment specification is correct, log(M_{t1}/P_{t1}) should be significant and log(M_{t1}/P_{t}) should not, and vice versa if the nominal adjustment specification is correct. This test may, of course, be inconclusive in that both terms may be significant or insignificant, but I have found that this is rarely the case. This test was performed on the three demand for money equations, and in each case the nominal adjustment specification won. The nominal adjustment specification has thus been used in the model.[Footnote 10: The nominal adjustment hypothesis is also supported in Fair (1987), where demand for money equations were estimated for 27 countries. Three equations were estimated for the United States (versions of equations 9, 17, and 26) and one for each of the other 26 countries. Of the 29 estimated equations, the nominal adjustment dominated for 25, the real adjustment dominated for 3, and there was 1 tie. The nominal adjustment hypothesis is also supported in Chapter 6. Of the 19 countries for which the demand for money equation (equation 6) is estimated, the nominal adjustment hypothesis dominates in 13.] Equation 9. MH, demand deposits and currencyh Equation 9 is the demand for money equation of the household sector. It is in per capita terms and is in log form. Disposable income is used as the transactions variable, and the after tax three month Treasury bill rate is used as the interest rate. The equation is estimated under the assumption of a first order autoregressive error term. The short run income elasticity of the demand for money in Table 5.9 is .0504, and the long run elasticity is .369 = .0504/(1.0.8633). The interest rate variable is significant. The age distribution variables were added to the equation to see if possible differences in the demand for money by age could be picked up. The Chi Square (AGE) value in Table 5.9 shows that the age distribution variables are jointly significant. The coefficient estimates, however, were large and not necessarily sensible, implying, for example, that people 5665 hold much more money, other things being equal, than anyone else. The age variables were thus not included in the final specification. The coefficients for the age variables show that people 2655 hold more money, other things being equal, than do others, which is as expected if people 2655 have on average higher wage rates than the others. The age variables are not, however, jointly significant at the five percent level, and so not much confidence should be placed on this result. [Footnote 11: In Fair and Dominguez (1991), Table 3, the sign pattern of the age distribution variables was as expected, but the variables were not jointly significant.] The test results show that the lagged dependent variable that pertains to the real adjustment specificationlog[MH/(POP^{.}PH)]_{1}is insignificant. As discussed above, this supports the nominal adjustment hypothesis. Equation 9 passes the lags and T tests, but it fails the RHO=4 and stability tests. For the stability test the largest chi squared value occurred for 1979:2. Equation 17. MF, demand deposits and currencyf Equation 17 is the demand for money equation of the firm sector. The results for this equation are presented in Table 5.17. The equation is in log form. The transactions variable is the level of nonfarm firm sales, XFA, and the interest rate variable is the after tax three month Treasury bill rate. The tax rates used in this equation are the corporate tax rates, D2G and D2S, not the personal tax rates used for RSA in equation 9. The income and interest rate variables in the equation are of the right signs, but are not significant. Again, the test results show that the lagged dependent variable that pertains to the real adjustment specification [log(MF/PF)_{1}] is insignificant. The equation passes the lags test, the T test, and the stability test. It fails the RHO=4 test. Equation 26. CUR, currency held outside banks Equation 26 is the demand for currency equation. It is in per capita terms and is in log form. The transactions variable that is used is the level of nonfarm firm sales. The interest rate variable used is RSA, and the equation is estimated under the assumption of a first order autoregressive error term. The results are presented in Table 5.26. All the variables in the equation are significant. The test results show that the lagged dependent variable that pertains to the real adjustment specificationlog[CUR/(POP^{.}PF)]_{1}is significant at the one percent level. Although not shown in the table, the lagged dependent variable that pertains to the nominal adjustment specification is also significant, and so the test is inconclusive. The equation passes the lags and T tests, but it fails the RHO=4 test. 
5.4 The Main Firm Sector Equations 
In the maximization problem of a firm in the theoretical model there are five
main decision variables: the firm's price, production, investment, demand for employment,
and wage rate. These five decision variables are determined jointly in that they are the
result of solving one maximization problem. The variables that affect this solution
include 1) the initial stocks of excess capital, excess labor, and inventories, 2) the
current and expected future values of the interest rate, 3) the current and expected
future demand schedules for the firm's output, 4) the current and expected future supply
schedules of labor facing the firm, and 5) the firm's expectations of other firms' future
price and wage decisions. In the econometric model seven variables were chosen to represent the five decisions: 1) the price level of the firm sector, PF, 2) production, Y, 3) investment in nonresidential plant and equipment, IKF, 4) the number of jobs in the firm sector, JF, 5) the average number of hours paid per job, HF, 6) the average number of overtime hours paid per job, HO, and 7) the wage rate of the firm sector, WF. Each of these variables is determined by a stochastic equation, and these are the main stochastic equations of the firm sector. Moving from the theoretical model of firm behavior to the econometric specifications is not straightforward, and a number of approximations have to be made. One of the key approximations is that the econometric specifications in effect assume that the five decisions of a firm are made sequentially rather than jointly. The sequence is from the price decision, to the production decision, to the investment and employment decisions, and to the wage rate decision. In this way of looking at the problem, the firm first chooses its optimal price path. This path implies a certain expected sales path, from which the optimal production path is chosen. Given the optimal production path, the optimal paths of investment and employment are chosen. Finally, given the optimal employment path, the optimal wage path is chosen. Equation 10. PF, price deflator for XFA Footnote 12: The material on the level versus change specification in this section is taken from Fair (1993a).] Equation 10 is the key price equation in the model, and the results for this equation are in Table 5.10. The equation is in log form. The price level is a function of the lagged price level, the wage rate inclusive of the employer social security tax rate, the price of imports, and a demand pressure variable. The equation is estimated under the assumption of a first order autoregressive error term. The lagged price level is meant to pick up expectational effects, and the wage rate and import price variables are meant to pick up cost effects. The demand pressure variable, log[(YS_{1}Y_{1})/YS_{1}+.04] , is the log of the percentage gap between potential and actual output plus .04. (Remember that YS is the potential value of Y.) This functional form implies that as actual output approaches four percent more than potential, the demand pressure variable approaches minus infinity, which implies that the price level approaches plus infinity. This functional form effectively prevents actual output from ever exceeding potential output by more than four percent. The demand pressure variable is lagged one quarter in equation 10 because this gave slightly better results than did the use of the variable unlagged. An important feature of the price equation is that the price level is explained by the equation, not the price change. This treatment is contrary to the standard Phillipscurve treatment, where the price (or wage) change is explained by the equation. Given the theory outlined in Chapter 2, the natural decision variables of a firm would seem to be the levels of prices and wages. For example, the market share equations in the theoretical model have a firm's market share as a function of the ratio of the firm's price to the average price of other firms. These are price levels, and the objective of the firm is to choose the price level path (along with the paths of the other decision variables) that maximizes the multiperiod objective function. A firm decides what its price level should be relative to the pricelevels of other firms. Fortunately, it is possible to test whether the price level or price change
specification is better. Let p denote the log of the price level, let w denote the log of
the wage rate, and let D denote the level of some demand pressure variable. The price
equation in level form is It is not possible to nest 5.15 into 5.16 or vice versa, but they can each be
nested in a more general equation. This equation is Equation 10 in the model is like equation 5.15 above except that the price of imports is also an explanatory variable. This is a variable like w in that it is assumed to pick up cost effects. The results in Table 5.10 show that all the explanatory variables are significant. The test results are as follows. First, to test the level specification, the lagged values of the wage rate and price of importslog[WF(1+D5G)]_{1} and logPIM_{1}are added. As can be seen, these two variables are not significant, and so the level specification is supported over the more general specification. The change specification, on the other hand, is not supported, as can be seen in the last chi squared test in Table 5.10. When the lagged values of the price level, the wage rate, and the price of imports are added to the change form of the price equation, they are highly significant, with a chi squared value of 63.32. The change form is thus strongly rejected.[Footnote 13: The level versus change specification was also tested in Fair (1993a) for 40 disaggregate price equations. The results were somewhat mixed, but overall slightly favored the level specification. The disaggregate results thus provide some support for the current aggregate estimates. Also, as will be seen, the results in Chapter 6 of estimating price equations for different countries strongly support the level specification over the change specification. As discussed later in this chapter, the result that the level specification is supported over the change specification has important implications for the long run properties of the economy.] Equation 10 passes the lags test, the RHO=4 test, and the T test. The variable for which led values were tried is the wage rate, and the led values are not significant. The test results next show that the unemployment rate lagged once is significant when added to the equation. Although not shown in the table, the addition of the unemployment rate makes the demand pressure variable insignificant. The next test shows that the simple percentage gap variable lagged once, [(YSY)/YS]_{1}, is significant. Although not shown in the table, the addition of this variable also makes the demand pressure variable insignificant. The functional form chosen for the demand pressure variable is thus not supported by the data. The best results (e.g., best fit) are obtained when no nonlinearity is introduced. This situation is unsatisfactory from a theoretical perspective in that one expects that there is some degree of demand pressure beyond which prices rise faster for a further increase in demand pressure than they did before this degree was reached. The problem is that the U.S. economy has not experienced enough periods in which demand pressure was very high to allow one to estimate adequately how prices behave in these extreme periods. The large price increases in the 1970s were primarily cost driven, and so they are of no help for this problem. The way that this problem has been handled in the model is simply to use the nonlinear functional form described above even though simpler forms do somewhat better. This problem is, of course, an important area for future work, and it suggests that any policy experiments with the model that push the economy to very high levels of activity should be interpreted with considerable caution. The behavior of prices at very high activity levels has probably not been very accurately estimated. Finally, the price equation fails the stability test. The largest chi squared value occurs for 1972:2, near the beginning of the first oil price shock. Equation 11. Y, productionf The specification of the production equation is where the assumption that a firm's decisions are made sequentially begins to be used. The equation is based on the assumption that the firm sector first sets it price, then knows what its sales for the current period will be, and from this latter information decides on what its production for the current period will be. In the theoretical model production is smoothed relative to sales. The reason
for this is various costs of adjustment, which include costs of changing employment, costs
of changing the capital stock, and costs of having the stock of inventories deviate from
some proportion of sales. If a firm were only interested in minimizing inventory costs, it
would produce according to the following equation (assuming that sales for the current
period are known): The estimated production equation is based on the following three assumptions: Equation 11 passes the lags, RHO=4, and T tests. The variable for which led values were used is the level of sales, X, and Leads +1, but not Leads +8 is significant. [Footnote 14: Collinearity problems prevented Leads +4 from being calculated for equation 11.] There is thus some support for the hypothesis that firms have rational expectations regarding future values of sales. The spread values are not significant at the five percent level. Finally, the equation fails the stability test. The estimates of equation 11 are consistent with the view that firms smooth production relative to sales. The view that production is smoothed relative to sales has been challenged by Blinder (1981) and others, and this work has in turn been challenged in Fair (1989) as being based on faulty data. The results in Fair (1989) using physical units data for specific industries suggests that production is smoothed relative to sales. The results using the physical units data thus provide some support for the current aggregate estimates. Equation 12. IKF, nonresidential fixed investmentf Equation 12 explains nonresidential fixed investment of the firm sector. It is based on the assumption that the production decision has already been made. In the theoretical model, because of costs of changing the capital stock, it may sometimes be optimal for a firm to hold excess capital. If there were no such costs, investment each period would merely be the amount needed to have enough capital to produce the output of the period. In the theoretical model there was no need to postulate explicitly how investment deviates from this amount, but for the empirical work this must be done. The estimated investment equation is based on the following three equations: Both of these variables are multiplied by IKFA, which is a variable constructed by peak to peak interpolations of IKF. Since IKF has a trend and TXCR and RB' do not, one would expect a given change in TXCR or RB' to have an effect on IKF that increases over time, and multiplying both variables by IKFA is a way of accounting for this. IKFA is exogenous; it is merely a scale variable. How can the use of the cost of capital variables be justified? In the theoretical model the cost of capital affects investment by affecting the kinds of machines that are purchased. If the cost of capital falls, more capital intensive machines are purchased and investment expenditures increase. For the empirical work, data are not available by types of machines, and approximations have to be made. The key approximation is the postulation of the production function 3.1 in Chapter 3. This production function is one of fixed proportions in the short run. Technical change and changes in the cost of capital relative to the cost of labor affect over time the q and u coefficients in the equation, and these are accounted for through the peak to peak interpolations discussed in Chapter 3. KKMIN in equation 93 in the model is determined using MUH, the peak to peak interpolation of u^{.}H^{}. If, as seems quite likely, the effects of cost of capital changes on firms' decisions are not completely captured through the peak to peak interpolations, then adding cost of capital variables to equation 5.23 (and thus equation 5.26) may be warranted. For example, when the cost of capital falls, KKMIN may underestimate the desired amount of capital, and at least part of this error may be picked up by adding cost of capital variables to the equation. The estimate of q in equation 12 is .0175, which says that gross investment adjusts 1.75 percent to its desired value each quarter. The implied value of a_{0} is .3429 = .0060/.0175, which says that 34.29 percent of the amount of excess capital on hand is desired to be eliminated each quarter. The change in output terms have coefficient patterns that are sensible, and the first two terms are significant. The tax credit variable has a tstatistic of 1.01, and the real bond rate lagged three times has a tstatistic of 1.32. The tax credit variable is thus not significant, although its coefficient estimate is of the expected sign, and the same is true for the bond rate variable. I have found it very difficult over the years to obtain significant cost of capital effects in equation 12, and the current results are the best that I have managed to do. Even here the lag of three quarters for the bond rate seems a little long, but shorter lags gave poorer results. The results may thus be spurious and merely the result of data mining, but they are retained because it is embarrassing not to have cost of capital effects in the investment equation. Equation 12 fails the lags test, but it passes all the others, including the stability test. The variable used for the led values was the change in output, and it is interesting to see that the future output changes are not significant. This is evidence against the hypothesis that firms have rational expectations with respect to future values of output. Note also in Table 5.12 that the constant term is not significant. According to equation 5.26 there should be no constant term in the equation, and the results bear this out. The chi squared test for the addition of the constant term is not significant. The spread values are also not significant. Equation 13. JF, number of jobsf The employment equation 13 and the hours equation 14 are similar in spirit to the investment equation 12. They are also based on the assumption that the production decision is made first. Because of adjustment costs, it is sometimes optimal in the theoretical model for firms to hold excess labor. Were it not for the costs of changing employment, the optimal level of employment would merely be the amount needed to produce the output of the period. In the theoretical model there was no need to postulate explicitly how employment deviates from this amount, but this must be done for the empirical work. The estimated employment equation is based on the following three equations: The second addition is accounting for what seemed to be a structural break in the mid 1970s. When testing the equation for structural stability, there was evidence of a structural break in the middle of the sample period, with the largest chi squared value occurring in 1977:2. Contrary to the case for most equations that fail the stability test, the results for equation 13 suggested that the break could be modeled fairly simply. In particular, the coefficient of the change in output did not appear to change, but the others did. This was modeled by creating a dummy variable, DD772, that is one from 1977:2 on and zero otherwise and adding to the equation all the explanatory variables in the equation (except the change in output) multiplied by DD772 as additional explanatory variables. The results in Table 5.13 show that the main difference before and after 1977:2 is in the estimate of the time trend coefficient. The other variables with DD772 in them are not significant. All the other varibles in the equation are significant. The coefficient of the excess labor variable for the period after 1977:2 is .0947 (= .0821  .0126), which means that 9.47 percent of the amount of excess labor on hand is eliminated each quarter. Equation 13 does not pass the lags test, where the chi squared value is quite large. Experimenting with various specifications of this equation reveals that it is very fragile with respect to adding lagged values in that adding these values changes the values of the other coefficient estimates substantially and in ways that do not seem sensible. The equation also fails the RHO=4 test. The variable for which led values were tried is the change in output variable, and the values led four quarters (but not one and eight) are significant at the one percent level. Equation 14. HF, average number of hours paid per jobf The hours equation is based on equations 5.28 and 5.29 and the following
equation: Equation 14 passes the lags and RHO=4 tests. The variable for which led values were tried is the change in output variable, and the led values are not significant. This is contrary to the case for equation 13, where the values led four quarters are significant. Equation 15. HO, average number of overtime hours paid per jobf Equation 15 explains overtime hours, HO. One would expect HO to be close to zero
for low values of total hours, HF, and to increase roughly one for one for high values of
HF. An approximation to this relationship is The coefficient estimates are significant in equation 15. The equation passes the lags, RHO=4, and T tests. It also passes the stability test. The equation thus seems to be a reasonable approximation to the way that HO is determined, although the estimate of the autoregressive coefficient of the error term is quite high. Equation 16. WF, average hourly earnings excluding overtimef Equation 16 is the wage rate equation. It is in log form. In the final specification, WF was simply taken to be a function of a constant, time, the current value of the price level, and the first four lagged values of the price level and the wage rate. Labor market tightness variables like the unemployment rate were not significant in the equation. The time trend is added to account for trend changes in the wage rate relative to the price level. Its inclusion is important, since it along with some of the lags identifies the price equation, equation 10. Equation 16 is estimated under the assumption of a first order autoregressive error. Constraints were imposed on the coefficients in the wage equation to ensure that
the determination of the real wage implied by equations 10 and 16 is sensible. Let p=logPF
and w=logWF. The relevant parts of the price and wage equations regarding the constraints
are The results for equation 16 (using 2SLS) are presented in Table 5.16. The wage rate lagged four times is nearly significant, and this is the reason for the use of four lags even though lags 2 and 3 are not significant. The time trend is highly significant, which is picking up a trend in the real wage. The chi squared test results show that the real wage restrictions discussed above are rejected by the data at the one percent level. In spite of this, the real wage restriction has been retained. The equation passes the lags test, but fails the RHO=4 test. The final chi squared test in the table has the unemployment rate lagged once added as an explanatory variable, and it is not significant. As noted above, no demand pressure variables were found to be significant in the wage equation. Finally, the equation fails the stability test, with the largest chi squared value occurring in 1972:3. 
5.5 Other Firm Sector Equations 
Equation 18. DF, dividends paidf Let pie
denote after tax profits. If in the long run firms desire to pay out all of their after
tax profits in dividends, then one can write DF^{*} = pie, where DF^{*} is
the long run desired value of dividends for profit level pie. If it is assumed that actual
dividends are partially adjusted to desired dividends each period as Because of the assumption that DF^{*} = pie, the coefficient of log(PIEFTFGTFS) is restricted to be the negative of the coefficient of logDF_{1} in equation 18. If instead DF^{*} = pie^{g}, where g is not equal to one, then the restriction does not hold. The first test in Table 5.18 is a test of the restriction (i.e., a test that g= 1), and the test is failed. The equation passes the lags test, but it fails the RHO=4 and T tests. The next test in the table shows that the constant term is not significant. The above specification does not call for a constant term, and this is supported by the data. Finally, the equation fails the stability test. Equation 20. IVA, inventory valuation adjustment In theory IVA = (PP_{1})V_{1}, where P is the price of the good and V is the stock of inventories of the good. Equation 20 in Table 5.20 is meant to approximate this. IVA is regressed on a constant and (PXPX_{1})V_{1}, where PX is the price deflator for the sales of the firm sector. The equation is estimated under the assumption of a first order autoregressive error term. As an approximation, the equation seems fairly good. It passes all but RHO=4 test, including the stability test. Equation 21. CCF, capital consumptionf In practice capital consumption allowances of a firm depend on tax laws and on current and past values of its investment. Equation 21 in Table 5.21 is an attempt to approximate this for the firm sector. PIK^{.}IKF is the current value of investment. The use of the lagged dependent variable in the equation is meant to approximate the dependence of capital consumption allowances on past values of investment. This specification implies that the lag structure is geometrically declining. The restriction is also imposed that the sum of the lag coefficients is one, which means that capital consumption allowances are assumed to be taken on all investment in the long run. There are two periods, 1981:11982:4 and 1983:11983:4, in which CCF is noticeably higher than would be predicted by the equation with only log[(PIK^{.}IKF)/CCF_{1}] in it, and two dummy variables, D811824 and D831834, have been added to the equation to account for this. This is, of course, a crude procedure, but the equation itself is only a rough approximation to the way that capital consumption allowances are actually determined each period. Tax law changes have effects on CCF that are not captured in the equation. Regarding the use of the two dummy variables, if CCF is larger than usual in the two subperiods, which the coefficient estimates for the two dummy variables suggest, then one might expect CCF to be lower at some later point (since capital consumption allowances can be taken on only 100 percent of investment in the long run). No attempt, however, was made to try to account for this in equation 21. The coefficient estimate of .0587 in Table 5.21 says that capital consumption allowances are taken on 5.87 percent of new investment in the current quarter, then 5.52 percent [.0587(1.0587)] of this investment in the next quarter, then 5.20 percent [.0587(1.0587)^{2}] in the next quarter, and so on. The first chi squared test in Table 5.21 is a test of the restriction that the sum of the lag coefficients is one. This is done by adding logCCF_{1} to the equation. The results show that the restriction is rejected at the one percent level. The equation fails the lags and T tests, and it passes the RHO=4 test. The results of the last chi squared test in the table show that the constant term is not significant at the one percent level in the equation. This is as expected since the above specification does not call for a constant term. The stability test was not performed because of the use of the dummy variables. 
5.6 Financial Sector Equations 
The stochastic equations for the financial sector consist of an equation
explaining member bank borrowing from the Federal Reserve, two term structure equations,
and an equation explaining the change in stock prices. Equation 22. BO, bank borrowing from the Fed The variable BO/BR is the ratio of borrowed reserves to total reserves. This ratio is assumed to be a positive function of the three month Treasury bill rate, RS, and a negative function of the discount rate, RD. The estimated equation also includes a constant term and the lagged dependent variable. The coefficient estimates of RS and RD in Table 5.22 are positive and negative, respectively, as expected. The equation passes the lags and T tests, and it fails the RHO=4 test. It also fails the stability test. Equation 23. RB, bond rate; Equation 24. RM, mortgage rate The expectations theory of the term structure of interest rates states that long term rates are a function of the current and expected future short term rates. The two long term interest rates in the model are the bond rate, RB , and the mortgage rate, RM. These rates are assumed to be determined according to the expectations theory, where the current and past values of the short term interest rate (the three month bill rate, RS) are used as proxies for expected future values. Equations 23 and 24 are the two estimated equations. The lagged dependent variable is used in each of these equations, which implies a fairly complicated lag structure relating each long term rate to the past values of the short term rate. In addition, a constraint has been imposed on the coefficient estimates. The sum of the coefficients of the current and lagged values of the short term rate has been constrained to be equal to one minus the coefficient of the lagged long term rate. This means that, for example, a sustained one percentage point increase in the short term rate eventually results in a one percentage point increase in the long term rate. (This restriction is imposed by subtracting RS_{2} from each of the other interest rates in the equations.) Equation 23 (but not 24) is estimated under the assumption of a first order autoregressive error term. The results for equations 23 and 24 are presented in Tables 5.23 and 5.24, respectively. The short rates are significant except for RS_{1} in equation 24. The test results show that the coefficient restriction is not rejected for either equation. Both equations pass the lags, T, and stability tests, and equation 24 passes the RHO=4 test. The results for both term structure equations are thus strong. My experience with these equations over the years is that they are quite stable and reliable. During most periods they provide a very accurate link from short rates to long rates. The variable for which led values were tried was the short term interest rate (RS), and the chi squared tests show that the led values are not significant at the one percent level except for Leads +4 for equation 24. [Footnote 14: Collinearity problems prevented Leads +8 from being calculated for equation 23.] This is thus at least slight evidence against the bond market having rational expectations with respect to the short term interest rate. The test results also show that the inflation expectations variables, p_{4}^{e} and p_{8}^{e}, are not significant in the equations. Equation 25. CG, capital gains or losses on corporate stocks held by h The variable CG is the change in the market value of stocks held by the household sector. In the theoretical model the aggregate value of stocks is determined as the present discounted value of expected future after tax cash flow, the discount rates being the current and expected future short term interest rates. The theoretical model thus implies that CG should be a function of changes in expected future after tax cash flow and of changes in the current and expected future interest rates. In the empirical work the change in the bond rate, <>RB, was used as a proxy for changes in expected future interest rates, and the change in after tax cash flow, <>(CFTFGTFS), was used as a proxy for changes in expected future after tax cash flow. Equation 25 in Table 5.25 is the estimated equation. The fit of equation 25 is not very good, and the cash flow variable is not significant and has a coefficient estimate of the wrong expected sign. The change in the bond rate is significant, however, which provides some link from interest rates to stock prices in the model. The equation fails the lags, RHO=4, T, and stability tests. The variables for which led values were tried are the change in the bond rate and the change in after tax cash flow. The led values are not significant. For the final chi squared test <> RS, the change in the short term rate, was added under the view that it might also be a proxy for expected future interest rate changes, and it is not significant. 
5.7 The Import Equation 
Equation 27. IM, Imports The import equation is in per capita terms and is in log form. The explanatory variables are 1) per capita real disposable income, 2) the private, nonfarm price deflator (a price deflator for domestically produced goods) relative to the import price deflator, 3) the long term after tax interest rate lagged one quarter, 4) the lagged dependent variable, and 5) four dummy variables to account for two dock strikes. The results are in Table 5.27. The short run income elasticity of imports is .3791, and the long run elasticity is 2.73 [.3971/(1.8545)], both fairly high. The coefficient estimate for the relative price term is positive as expected. The coefficient estimate for the long term interest rate is negative as expected. Many chi squared tests were performed for the import equation. It passes the lags test, but fails the RHO=4 and T tests. The variable for which led values were tried is disposable income, and the led values are not significant except for Leads +4. The inflationary expectations variables, p_{4}^{e} and p_{8}^{e}, are highly significant, but (although not shown in the table) their coefficient estimates are of the wrong expected sign. The next test in the table adds logPF to the equation, which is a test of the restriction that the coefficient of logPF is equal to the negative of the coefficient of logPIM. The logPF variable is highly significant, and so the restriction is rejected. Although not shown in the table, when logPF is added to the equation, the coefficient for logPIM is .1082 and the coefficient for logPF is .1972. The results thus suggest that the level of imports responds more to the domestic price deflator than to the import price deflator. This is contrary to what one expects from theory. It does not seem sensible that in the long run the level of imports rises simply from an overall rise in prices. Therefore, the relative price constraint was imposed on the equation, even though it is strongly rejected by the data. The next test adds the level of per capita nonfarm firm saleslog[(XFA)/POP]to the equation to see if it better explains imports than does disposable income. The chi squared values is not significant, and so on this score the sales variable does not have independent explanatory power. On the other hand (not shown in the table), the tstatistic on the sales variable was higher (1.48) than the tstatistic on the disposable income variable (0.57). The sales variable thus dominates the disposable income variable in this sense. The consequences of using the sales variable in place of the disposable income variable are examined in Section 11.3.4. The "other" variables that were added for the next test, which are asset, real wage, labor constraint, and nonlabor income variables, are highly significant, but (not shown in the table) they all have coefficient estimates of the wrong expected sign. Finally, the spread values are highly significant. The stability test was not performed for the import equation because of the use of the dummy variables. Experimenting with the import equation reveals that it does much better in the tests if the relative price restriction is not imposed. In other words, when logPF is added, the equation does much better. So in summary, the import equation passes the lags and leads tests, but it fails the others. It is clearly an equation in which future research is needed. 
5.8 Government Sector Equations 
There is one stochastic equation for the state and local government sector,
explaining unemployment insurance benefits, UB. There are two stochastic equations for the
federal government sector, one explaining interest payments, INTG, and one explaining the
three month Treasury bill rate, RS . The equation explaining RS is interpreted as an
interest rate reaction function of the Federal Reserve. The equations for UB and RS are
discussed in this section, and the equation for INTG is discussed in the next section. Equation 28. UB, unemployment insurance benefits Equation 28 is in log form and contains as explanatory variables the level of unemployment, the nominal wage rate, and the lagged dependent variable. The inclusion of the nominal wage rate is designed to pick up the effects of increases in wages and prices on legislated benefits per unemployed worker. The equation is estimated under the assumption of a first order autoregressive error term. The results in Table 5.28 show that the coefficient estimates are significant except for the estimate of the constant term. The equation passes the lags, RHO=4, and T tests, and it fails the stability test. Equation 30. RS, three month Treasury bill rate A key question in any macro model is what one assumes about monetary policy. In the theoretical model monetary policy is determined by an interest rate reaction function, and in the empirical work an equation like this is estimated. This equation is interpreted as an equation explaining the behavior of the Federal Reserve (Fed). In one respect, trying to explain Fed behavior is more difficult than, say, trying to explain the behavior of the household or firm sectors. Since the Fed is run by a relatively small number of people, there can be fairly abrupt changes in behavior if the people with influence change their minds or are replaced by others with different views. Abrupt changes are less likely to happen for the household and firm sectors because of the large number of decision makers in each sector. Having said this, however, only one abrupt change in behavior appeared evident in the data, which was between 1979:4 and 1982:3, and, as will be seen, even this change appears capable of being modeled. Equation 30 is the estimated interest rate reaction function It has on the left hand side RS. This treatment is based on the assumption that the Fed has a target bill rate each quarter and achieves this target through manipulation of its policy instruments. The right hand side variables in this equation are variables that seem likely to affect the target rate. The variables that were chosen are 1) the rate of inflation, 2) the degree of labor market tightness, 3) the percentage change in real GDP, and 4) the percentage change in the money supply lagged one quarter. What seemed to happen between 1979:4 and 1982:3 was that the size of the coefficient of the lagged money supply growth increased substantially. This was modeled by adding the variable D794823^{.}PCM1_{1} to the equation, where D794823 is a dummy variable that is 1 between 1979:4 and 1982:3 and 0 otherwise. The estimated equation also includes the lagged dependent variable and two lagged bill rate changes to pick up the dynamics. The signs of the coefficient estimates in Table 5.30 are as expected, and the equation passes all of the tests! The results thus seem good for this equation. The stability test was not run because of the use of the dummy variable. The variables for which led values were tried are the inflation variable, the labor market tightness variable, and the percentage change in real GDP, and the led values are not significant. Equation 30 is a "leaning against the wind" equation in the sense that the Fed is predicted to allow the bill rate to rise in response to increases in inflation, labor market tightness, real growth, and money supply growth. As just noted, the results show that the weight given to money supply growth in the setting of the bill rate target was much greater in the 1979:41982:3 period than either before or after. 
5.9 Interest Payments Equations 
Equation 19. INTF, interest paymentsf; Equation 29. INTG, interest
paymentsg INTF is the level of net interest payments of the firm sector, and INTG is the same for the federal government. Data on both of these variables are NIPA data. AF is the level of net financial assets of the firm sector, and AG is the same for the federal government. Data on both of these variables are FFA data. AF and AG are negative because the firm sector and the federal government are net debtors, and they consist of both short term and long term securities. The current level of interest payments depends on the amount of existing securities issued at each date in the past and on the relevant interest rate prevailing at each date. The link from AF to INTF (and from AG to INTG) is thus complicated. It depends on past issues and the interest rates paid on these issues. A number of approximations have to be made in trying to model this link, and the following is a discussion of the procedure used here. Consider the federal government variables first. The difference AGAG_{1}
is the net change in the value of securities of the federal government between the end of
the previous quarter and the end of the current quarter. The value of new securities
issued by the federal government during the current quarter is this difference plus
the value of old securities that came due during the current quarter. It is first assumed
that the government issues two kinds of securities, a "short term" security,
where the short term is defined to be one quarter, and a "long term" security,
which is taken to be of length k quarters, where k is to be estimated. It is next assumed
that q percent of the net change in the value of securities in a quarter (i.e., of
AGAG_{1}) consist of long term issues, with the rest consisting of short
term issues. In addition, it is assumed that the long term securities that expire during
the quarter are replaced with new long term securities. Let BG denote the value of long
term securities issued during the current quarter by the federal government. Then the
above assumptions imply that : It is next assumed that the government pays an interest rate RS on its short
term securities, where RS is the three month Treasury bill rate, and an interest rate RB 
c on its long term securities, where RB is the AAA bond rate and c is a constant parameter
to be estimated. c is subtracted from RB because the government generally pays less than
the AAA bond rate on its bonds. Given these assumptions, the interest payments of the
federal government are:[Footnote 17: In the notation in this equation BG_{0} is
the same as BG. Similarly, for the firm sector BF_{0} is the same as BF.] Using equations 5.40 and 5.41, the aim of the estimation work is to find values of k, q, c that lead to a good fit, i.e., that lead to the predicted values of INTG from equation 5.41 being close to the actual values. This work takes as given the actual values of RS, RB, and AG. The estimation period was 1952:11996:3, which is the period for which data on AG exist. The estimation procedure was as follows. First, given a value for k and a value for q, the value of BG for 1952:1 was taken to be equal to (1/k)qAG_{1952:1}. In addition, the k1 values of BG before 1952:1 were taken to be equal to this value. Given these values, values of BG for 1952:2 through 1996:3 can be generated using equation 5.40 and the given values of k and q. Second, if values of RB for the computations in equation 5.41 were needed before 1947:1, which is the first quarter for which data on RB exist, they were taken to be equal to the 1947:1 value. (Values before 1947:1 are needed if k is greater than 20 quarters.) Third, given the values of k and q and the above computations and given a value for c, equation 5.41 can be used to obtain predicted values of INTG for the 1952:11996:3 period, from which a root mean squared error (RMSE) can be computed. The entire procedure can then be repeated for a different set of values of k, q, and c, and another RMSE computed. A program was written to search over different sets of values of k, q, and c and print out the RMSE for each set. The set that was chosen for the model is k = 18, q = .70, and c = .3. Equation 5.40 above is equation 56 in the model, and equation 5.41 is equation 29. These two equations are presented in Table A.3 in Appendix A using the values of k, q, and c chosen. A similar procedure was followed for the interest payments of the firm sector, INTF, with two differences. First, c was taken to be zero, which means that firms are assumed to pay interest rate RB on their long term securities. Second, between 1981:3 and 1991:2 INTF grew faster than seemed consistent with the values of the interest rates and AF. No values of k and q could be found that gave sensible fits for this period. To account for this unexplained growth, a dummy variable, TI, was constructed that was 0 through 1981:2, 1 in 1981:3, 2 in 1981:4, ... , 40 in 1991:2, and 40 after 1991:2. The term g^{.}TI was then added to the equivalent of equation 5.41 for the firm sector, where g is a coefficient to be estimated. The searching procedure for the firm sector thus consisted in searching over values of k, q, and g. The set of values that was chosen for the model is k = 34, q = .50, and g = .24. Equation 5.40 above for the firm sector is equation 55 in the model, and equation 5.41 for the firm sector is equation 19. These two equations are also presented in Table A.3 using the values of k, q, and g chosen. Although the above specification is obviously only a rough approximation of the links from interest rates, AG, and AF to interest payments, it does tie changes in these variables to changes in interest payments in a way that is not likely to deviate substantially in the long run from the true relationship. In other words, interest payments change as interest rates change and as AG and AF change in a way that seems unlikely to drift too far from the truth.[Footnote 18: In previous specifications of equations 19 and 29 the interest payments variables were regressed on interest rates and the value of securities. The equations were usually in log form and usually included the lagged dependent variable. For example, one version of equation 29 had logINTG regressed on a constant, logINTG_{1}, log(AG), logRS, and logRB. These types of equations provide a slightly better fit than the procedure discussed above, but they have poor dynamic properties. With no restrictions imposed, the predicted interest payments from the equations tend to drift away from sensible values, sensible in the sense of being consistent with the predicted values of interest rates and security issues.] It will be seen in Section 11.7 that equation 29 has an effect on the effectiveness of monetary policy in the model. As the size of the federal government debt (AG) increases, the change in interest payments of the federal government for a given change in interest rates increases in absolute value. Since households hold much of the debt, the change in interest revenue of the household sector for a given change in interest rates is getting larger in absolute value as the size of the debt increases. This income effect on households is thus increasing over time and, as will be seen, is now offsetting more of the substitution effect of a change in interest rates than it did earlier. Finally, although equations 19 and 29 have not been estimated in a usual way, they are still stochastic equations in the sense that the predicted values from the equations do not in general equal the actual values. (Equations 55 and 56 are, however, identities because BF and BG have simply been constructed using the equations.) With respect to the notation for the model in equation 4.1 in Chapter 4, both equations 19 and 29 have in general nonzero values of u_{it}. For 3SLS and FIML estimation and the stochastic simulation work below, where an estimate of the covariance matrix of all the errors in the model is needed, the error terms for equations 19 and 29 have been used after adjusting for heteroskedasticity. The variance of the error term in equation 19 was assumed to be proportional to (AF+10)^{2}, and the variance of the error term in equation 29 was assumed to be proportional to AG^{2}. This means that equation 19 is divided through by AF+10 and that equation 29 is divided through by AG before computing the error terms to be used in estimating the covariance matrix.[Footnote 19: AF is close to zero for the first few quarters of the estimation period, and this is the reason for adding 10 to it.] This means that uncertainty from equations 19 and 29 is taken into account in 3SLS and FIML estimation and in the stochastic simulation work even though they are not estimated in a traditional way. 
5.10 Additional Comments 
The following is a discussion of some of the results that pertain to sets of
equations.
