5. The Stochastic Equations of the US Model 
This chapter discusses the specification, estimation, and testing of
the stochastic equations of the US model.
It is an update of Chapter 5 in the 1994 book.
It relies heavily on the Chapter 5 tables, which are at the end of this
document. If you are reading this online, it may be helpful to print
the Chapter 5 tables for ease of reference.
Three recent papers are relevant for the material in this chapter. See "Testing the NAIRU Model for the United States" for discussion of price and wage equations like equations 10 and 16 in the context of the NAIRU literature. See "Actual Federal Reserve Policy Behavior and Interest Rate Rules" for further discussion of the interest rate reaction function, equation 30. Finally, see "Is There Empirical Support for the 'Modern' View of Macroeconomics" for further tests of nominal versus real interest rates in consumption and investment equations and for further discussion of the CG equation, equation 25.

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. 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. Some of this difficulty may be 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. 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 for the results in the Chapter 5 tables: 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}. The basic estimation period was 1954:12001:1, for a total of 189 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
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. In Chapter 11, Section 11.6, of the 1994 book the results of adding the led values to the stochastic equations were used to examine the economic significance of the rational expectations assumption in the US model.[Footnote 3: This work was an updated version of the material in Section 5 in Fair (1993b).] The question asked in Section 11.6 was: How much difference to the properties of the US model does the addition of the led values make? Two versions of the model were examined. The first consisted of the "base" equations in Tables 5.15.30 in the book, which had no led values in them. This version was called Version 1. The second version consisted of the equations with the third set of led values added (i.e., with F_{1t} and F_{2t} added). This version was called Version 2. First Stage Regressors
Autoregressive Errors

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 it did in all four equations. 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 a labor constraint variable (the unemployment rate, UR) 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
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 four variables are added: log(CS/POP)_{2}, log[YD/(POP^{.}PH)]_{1}, RSA_{1}, and log(AA/POP)^{1}.] and passes the RHO=4 test. The results regarding the dynamic specification of the equation are thus favorable. 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. Neither of the inflation expectations variables is significant, which is evidence against the use of real over 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 .6468. Finally, the equation fails the stability test. The AP value is 11.43. The value of lambda for this test is 2.34. The one percent critical AP value for 8 coefficients and lambda equal to 2.75 is 10.23 (see Table 4.1 in Chapter 4). This is also roughly the critical value for lambda equal to 2.43, and so the AP value of 11.43 is above the critical value and thus the stability test is not passed. The largest chi squared value occurred for 1974:4, which is roughly the middle of the test period of 1970:11979:4. Equation 2. CN, consumer expenditures: nondurables
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, +4, and +8 are all significant. Although one of the two inflation expectations variables is significant (p_{4}^{e}), its coefficient estimate (not shown) is of the wrong expected sign (the estimate is negative). The additional variables, representing the wage, price, nonlabor income, and labor constraint variables are significant, but as with equation 1, the coefficient estimates for the variables are of the wrong expected signs (not shown). The spread values are not significant. The equation fails the stability test. The AP value is 16.65, 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 1979:3. Equation 3. CD, consumer expenditures: durables
The results of estimating equation 3 are presented in Table 5.3. The coefficient of the lagged dependent variable is .7241, and so g above is .2759 (= 1.0  .7421). As discussed in Chapter 3, the depreciation rate, DELD, is equal to .058218. Given these two values and given the coefficient of the lagged stock variable in Table 5.3 of .0090, the implied value of q is .091. This implies an adjustment of the durable stock to its desired value of 9.1 percent per quarter. The age variables are jointly highly significant. The age coefficients show people 5665 spending more, other things being equal, than the others. The pattern here is thus somewhat 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 not significant. The other variables, which are the wealth, wage, price, nonlabor income, and labor constraint variables, are significant, but again the coefficient estimates are of the wrong expected signs (not shown). The spread values are significant. The equation fails the stability test. In short, equation 3 fails all but the lags test and the inflation expectations test, and so it is clearly an equation for which more work is needed. Equation 4. IHH, residential investmenth
The wealth, 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 .5261, and so g is .4739. As discussed in Chapter 3, the depreciation rate for the housing stock, DELH, is .003860. Given these two values and given the coefficient of the lagged housing stock variable of .0519, the implied value of q is .113. The estimated adjustment speed of the housing stock to its desired value is thus slightly larger than the estimated adjustment speed of the durable goods stock (.113 versus .091). 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 and RHO=4 tests, and it fails the T test. The variable for which led values were tried is disposable income, and the led values are not significant. The inflation expectations variables are significant, but (not shown) their coefficient estimates are of the wrong expected sign (they are negative). The "other" variables are highly significant, but again (not shown) their coefficient estimates are of the wrong expected signs. The spread values are not close to being significant. Equation 4 passes the stability test, the only household 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
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 measure of labor market tightness like the unemployment rate (UR) 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), 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 and is essentially zero. 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 wealth, nonlabor income, and the unemployment rate added to the equation. These variables are not significant. Equation 5 passes the stability test. Equation 6. L2, labor forcewomen 2554
Regarding the tests, the equation does not quite pass the lags test, but it passes the the RHO=4 test and the T test. The variable for which led values were tried is the real wage (log(WA/PH)). Leads +1 and Leads +8 are not significant, but Leads +4 is. 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, wealth, nonlabor income, and the unemployment rate were added to the equation. These variables are not significant. Equation 6 passes the stability test. Equation 7. L3, labor forceall others 16+
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" variables that are added are the wealth and nonlabor income, and these two variables are not significant. Equation 7 passes the stability test. The overall test results for equation 7 are thus quite strong. Equation 8. LM, number of moonlighters
Equation 8 passes the lags test and the RHO=4 test. It fails 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 significant, which is evidence against the real wage constraint. (The real wage constraint was, however, still retained.) The "other" variables that were added are wealth and nonlabor income, and they are significant. The equation thus fails this test. Equation 8 also fails 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. 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 17 countries for which the demand for money equation (equation 6) is estimated, the nominal adjustment hypothesis dominates in 12.] Equation 9. MH, demand deposits and currencyh
The income variable and the time trend are significant in equation 9, but the interest rate is not. The coefficient estimate of the time trend is negative. Other things being equal, there is a negative trend in MH that does not seem capable of being explained with standard economic variables. 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 not jointly significant at the one percent level. Although not shown, the coefficient estimates of the age variables 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. [Footnote 11: In Fair and Dominguez (1991), Table 3, the sign pattern of the age distribution variables was as expected, although 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 test, but it fails the RHO=4 and stability tests. For the stability test the largest chi squared value occurred for 1979:2. In general, the results for equation 9 are not very good, and more work is needed. Fortunately, the equation does not play a large role in the model. Equation 17. MF, demand deposits and currencyf
The transactions variable is significant in the equation, but the interest rate variable is not. The test results show that the lagged dependent variablethat pertains to the real adjustment specification [log(MF/PF)_{1}] is insignificant. The equation passes the lags test and the T test. It fails the RHO=4 test and the stability test. As with equation 9, the overall results for equation 17 are not very good, but the equation does not play a large role in the model. Equation 26. CUR, currency held outside banks
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 not significant at the one percent level, which supports the nominal adjustment specification. The equation passes the lags, RHO=4, T, and stability tests. 
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
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 price levels of other firms. The time trend in equation 10 is meant to pick up any trend effects on the price level not captured by the other variables. Adding the time trend to an equation like 10 is similar to adding the constant term to an equation specified in terms of changes rather than levels. The time trend will also pick up any trend mistakes made in constructing LAM. If, for example, LAM_{t} = LAM^{a}_{t} + a_{1}t, where LAM^{a}_{t} is the correct variable to subtract from the wage rate variable to adjust for potential productivity, then the time trend will absorb this error. Turning to Table 5.10, all the explanatory variables in equation 10 are significant and all the tests are passed except the stability test. The variable for which led values were tried is the wage rate variable, and the led values are not significant. The last two chi squared tests in Table 5.10 have output gap variables added. When each of these variables is added, it is not significant and (not shown) the unemployment rate retains its significance. The unemployment rate thus dominates the output gap variables. Overall, the results for equation 10, which is a key equation in the model, are good. For further discussion of this equation and a comparison of it to a NAIRU equation, see Fair (1998). Equation 11. Y, productionf
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 and Leads +8 are not significant. [Footnote 13: Collinearity problems prevented Leads +4 from being calculated for equation 11.] This is evidence against the hypothesis that firms have rational expectations regarding future values of sales. The spread values are not significant. Equation 11 passes 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. This work has in turn been challenged in Fair (1989) as being based on faulty data. The results in this paper, which use data in physical units, suggest 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. KK, stock of capitalf
The estimated equation for KK is based on the following two
equations:
Combining equations (5.24) and (5.25) yields:
The results of estimating equation (5.26), which is equation 12 in the model, are in Table 5.12. The coefficient estimate of logKK_{1} implies that q is .0111, and the coefficient estimate of logKK_{2} implies that q is .0212. Although not shown, these two estimates of q are significantly different from each other, and so the constraint in equation (5.26) is not supported. The adjustment process is probably more complicated than that specified in equation (5.25). The current output variable is significant in Table 5.12, although the lagged output variables are not. The interest rate coefficient is negative, as expected, but it only has a tstatistic of 1.38. I have found it very difficult over the years to obtain significant cost of capital effects in investment equations, and the current results are no exception. Equation 12 passes all the tests, including the stability test. The variable used for the led values was output, and it is interesting to see that the future output values 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. The overall test results for equation 12 are thus quite good. In earlier specifications of equation 12 the dependent variable was investment (IKF) rather than the capital stock, and one of the explanatory variables was the amount of excess capitial on hand. In the theoretical model the amount of excess capital on hand affects the current investment decision. These earlier empirical specifications did not hold up well over time, which in part may be due to the difficulty of getting good estimates of excess capital. As just discussed, the current specification of equation 12 seems fairly good. It is especially encouraging that it passes the stability test. Equation 13. JF, number of jobsf
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 .0533 (= .1005 + .0472), which means that 5.33 percent of the amount of excess labor on hand is eliminated each quarter. Equation 13 fails 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 led values are not significant. This is again evidence against the hypothesis that firms have rational expectations with respect to future values of output. Equation 14. HF, average number of hours paid per
jobf
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 further evidence against the hypothesis that firms have rational expectations with respect to future values of output. Equation 15. HO, average number of overtime hours paid per
jobf
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
A constraint was 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. All four coefficient estimates are significant. The chi squared test results show that the real wage restriction discussed above is not rejected by the data. The equation passes the lags test, but fails the RHO=4 test. The final chi squared test in the table has the unemployment rate 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. Equation 16 passes the stability test. 
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 (5.38) DF/DF_{1}=(DF^{*}/DF_{1})^{q} then the equation to be estimated is (5.39) log(DF/DF_{1}) = qlog(pie/DF_{1}) Equation 18 in Table 5.18 is the estimated version of equation 5.39. The level of after tax profits in the notation of the model is PIEFTFGTFS. The estimate of q is .0274, which implies a fairly slow adjustment of actual to desired dividends. 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 passed. The equation passes the lags test, the RHO=4 test, and the T test. 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. Equation 18 fails the stability test. Equation 20. IVA, inventory valuation adjustment
Equation 21. CCF, capital consumptionf
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 .0687 in Table 5.21 says that capital consumption allowances are taken on 6.87 percent of new investment in the current quarter, then 6.40 percent [.0687(1.0687)] of this investment in the next quarter, then 5.96 percent [.0687(1.0687)^{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. Although the restriction is rejected, it is still retained in the equation. The equation passes the lags and RHO=4 tests, and it fails the T test. The results of the last chi squared test in the table show that the constant term is significant at the one percent level in the equation. Again, although the restriction of no constant term is rejected, the constant term is not included in the equation. 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 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. Overall, the results for equation 22 are not very good, but fortunately the equation plays a small role in the model. Equation 23. RB, bond rate; Equation 24. RM, mortgage rate
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, RHO=4, T, and stability tests. 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. 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 fit of equation 25 is not very good, and the profit variable is not quite significant. The change in the bond rate is significant, however, which provides some link from interest rates to stock prices in the model. The equation passes 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 profits. 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 expenditures on consumption and investment, 2) the private, nonfarm price deflator (a price deflator for domestically produced goods) relative to the import price deflator, 3) a time trend, 4) the lagged dependent variable, and 5) four dummy variables to account for two dock strikes. The results are in Table 5.27. All the coefficient estimates are significant and of the expected signs. The equation fails the lags and RHO=4 tests. The variable for which led values were tried is the per capita expenditure variables, and the led values are not significant. The last chi square test in Table 5.27 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 .1589 and the coefficient for logPF is .2717. 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. 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. The equation is clearly one where further 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
The results in Table 5.28 show that all the coefficient estimates are significant. The equation passes the RHO=4 and T tests, and it fails the lags and stability tests. Equation 30. RS, three month Treasury bill rate
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 unemployment rate, 3) the change in the unemployment rate, 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 variables for which led values were tried are inflation and the unemployment rate. The led values are not significant. The AP stability test could not be run because of the use of the dummy variable, but an alternative stability test was performed. The hypothesis that the coefficients are the same in the 1954:11979:3 period as in the 1982:42001:1 period was tested using a Wald test. The results in Table 5.30 show that the hypothesis is not rejected. The Wald statistic is 10.81, which has a pvalue of .2127. 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, the unemployment rate, the change in the unemployment rate, 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 procedure used here is a follows. Let RQ denote a weighted average of the current value of the
short term interest rate (RS)
and current and past values of the long term rate (RB):
The estimates of equation 19 are presented in Table 5.19. The estimate of the interest rate variable is significant. The first chi squared test is of the hypothesis that the two coefficients sum to one. The hypothesis is not rejected. The equation passes the RHO=4 test, but it fails the lags, T, and stability tests. Equation (5.41) was also estimated for the federal government, where INTG replaces INTF and AG replaces AF. This is equation 29 in the model. In this case the equation was not estimated under the assumption of an autoregressive error term, although the restriction that the two coefficients sum to one was retained. The estimates of equation 29 are presented in Table 5.29. Again, the interest rate variable is significant. The first chi squared test shows that the coefficient restriction is rejected. The equation also fails the lags and RHO=4 tests. It passes the T test and the stability test. Equations 19 and 29 are important in the model because when interest rates change, interest payments change, which changes household income. As discussed above, it is difficult to model this link. Although the overall results for equations 19 and 29 are not strong, as rough approximations they may not be too bad. 
5.10 Additional Comments 
The following is a discussion of some of the results that pertain to
sets of
equations.
