Stochastic fiscal policy and the Swedish business cycle

JOURNALOF

Monetary ELSEVIER

Journal of Monetary Economics 38 (1996) 245-268

ECONOMICS

Stochastic fiscal policy and the Swedish business cycle Gunnar Jonsson a, Paul Klein *'b a International Monetary Fund, Washington, DC 20431, USA b Institute for International Economic Studies, Stockholm University, 106 91 Stockholm, Sweden

(Received April 1995; final version received August 1996)

Abstract In this paper we show that fluctuations in distortive taxes can account for some of the key features of the Swedish post-war business cycle. The empirical fit of a simple stochastic growth model is significantly improved when it is amended to include imperfectly predictable fluctuations in payroll taxes, consumption taxes, and government consumption. Indeed, using the simulated method of moments, SMM, we find that, for large sets of conventional moments, models with stochastic fiscal policy cannot be statistically rejected, whereas a model without it is always rejected. Key words: Business cycles; Stochastic growth model; Fiscal policy; Simulated method of moments J E L classification: C32; E32; E62

1. Introduction Is fiscal policy important for the business cycle? In this paper, we claim that it is. Indeed, we argue that fluctuations in distortive taxes and government spending can account for some of the key features of the Swedish post-war business cycle. Since the pioneering articles by Christiano and Eichenbaum (1992), Braun (1994), and McGrattan (1994), it is well known that stochastic fiscal policy * Corresponding author. We thank Torsten Persson, Paul Sfderlind, Anders Warne, an anonymous referee, and the editor for helpful comments. We have also benefited from seminars at the Stockholm School of Economics, lIES, and the 1995 EEA congress in Prague. Financial support from the Jan Wallander and Tom Hedelius Foundation is gratefully acknowledged. The views expressed herein are those of the authors and not necessarily those of the IMF. 0304-3932/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 3 9 3 2 ( 9 6 ) 0 1 2 8 4 - 6

246

G. Jonsson, P. Klein/Journal o f Monetary Economics 38 (1996) 245~68

improves the performance of an equilibrium business cycle model when fitted to U.S. data. In this paper, we consider whether these findings carry over to the case of Sweden. There are strong reasons to think that they might, since Swedish taxes and government spending have changed considerably over time. Between 1947 and 1990, the central government income tax schedule was changed 22 times. And when the schedule was nominally unchanged, inflation and real growth caused an automatic steepening of the effective tax schedule. 1 During the same period, local government taxes, consumption taxes, and payroll taxes have also fluctuated; in particular, they have risen. Also, the Swedish post-war business cycle exhibits a number of empirical regularities or 'stylized facts' which a simple equilibrium business cycle model is unable to reproduce,2 and which can potentially be explained and understood by considering the effects of stochastic fiscal policy. In particular, we stress the following three facts about the Swedish business cycle. First, as in many other European countries but in stark contrast to the United States, private consumption is about as volatile as output, indeed slightly more so. We argue that a major reason for this fact is fluctuating taxes on consumption. A temporarily high tax rate on consumption provides an incentive to postpone consumption to a later date, when the tax rate is likely to be lower. Hence, as the consumption tax fluctuates, so does consumption. Consequently, the inclusion of a fluctuating tax on consumption in a model should increase the predicted volatility of consumption, and thus bring the implications of theory closer to the facts of reality. 3 Second, as in most industrialized countries, labor input is nearly as volatile as output. Our explanation of this is the volatility of taxes on labor. Intuitively, if income and payroll taxes fluctuate over time, it is optimal to work hard when taxes are relatively low and to take time off when they are relatively high. Finally, in Sweden, as in nearly all countries, the correlation (at businesscycle frequencies) between labor input and the real wage is close to zero. 4 This is in sharp contrast to the predictions of the early equilibrium business cycle models which had only technology shocks. Following McGrattan (1994) and Braun (1994), we interpret shocks to income and payroll taxes as shocks to labor

l When measuring the parameters that characterize the empirical income tax schedule, we adjust for inflation and real growth so as to take this 'bracket creep' phenomenon into account. 2 Danthine and Donaldson (1993) provide a thorough survey and discussion of these issues. See also Backus and Kehoe (1992) for some additional evidence. 3 The reason why the high relative volatility of consumption in many countries has so far received relatively little attention is presumably that it is not a feature of the U.S. data. In the spirit of our approach, it is not hard to guess why U.S. consumption is so smooth. Unlike Swedish consumption taxes, U.S. sales taxes are both low and stable over time. For time series of consumption taxes in various countries, see Mendoza, Razin, and Tesar (1994). 4 For a thorough treatment of this issue, see Hassler, Lundvik, Persson, and S6derlind (1992).

G Jonsson, P. Klein~Journal of Monetary Economics 38 (1996) 245 268

247

supply, as opposed to the technology shocks which may be (loosely) interpreted as shocks to labor demand. For obvious reasons, shocks to labor supply tends to bring down the correlation between labor input and the real wage, and bring the model's prediction more in line with the empirical evidence. The approach in this paper can be summarized as follows. In order to study whether stochastic fiscal policy can help explain the character of the Swedish business cycle in a manner consistent with standard economic theory, we build three stochastic general equilibrium models; two with fiscal policy, and, as a benchmark, one without fiscal policy. We then check whether and to what extent the fiscal policy versions perform better than their benchmark rival. This question is addressed in a precise way: By using the simulated method of moments (SMM), we are able to perform statistical tests of our models. The results favor the view that fluctuations in distortive taxes and government consumption are important for the Swedish business cycle. A model which includes payroll taxes, consumption taxes, and government consumption (but no income taxes) systematically and strongly outperforms the standard real business cycle model in reproducing the most salient facts of the Swedish business level. Actually, for large sets of conventional moments the model including these instruments of fiscal policy cannot be statistically rejected, whereas the model without taxes is always rejected at the 5% significance level. In particular, this model performs well with respect to the empirical puzzles described above. This means that the near-zero correlation between labor input and the real wage, as well as the rather high variability of both labor input and consumption relative to that of output, are well reproduced by the model with taxes. However, the model does not improve its performance further when augmented to also include a realistic specification of income taxation. The empirical fit is somewhat worse and the parameter estimates are less reasonable. Nevertheless, even this version of the model strongly outperforms the model without fiscal policy. The remainder of the paper is organized as follows. Section 2 describes the general equilibrium model. The dataset is presented in Section 3, and our methods are described and discussed in Section 4. Section 5 presents the results of SMM estimations and tests. Section 6 discusses the relative contribution of each of the shocks to the total variance of GDP around its trend. Conclusions are drawn and discussed in Section 7.

2. The general equilibrium model Our model economy consists of infinitely many identical households, each of which faces a tax system whose structure closely resembles that of post-war Sweden. A representative household chooses leisure and consumption in each

248

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996)245-268

period so as to maximize oo ~t

E0 E

(2.1)

fl U(ct, h i ) ,

t=0

where /~ is the subjective discount factor. The period utility function, U, is assumed to be o f the conventional form

U(ct, ht)

=

lim [c~(1 ~-~

-ht)l-~] 1-~1 - ~

1

'

(2.2)

where ct is consumption o f the single good and ht is the share o f available time spent in paid employment. The parameter ct is the weight attached to consumption relative to leisure, whereas a is the coefficient o f relative risk aversion as well as the reciprocal of the intertemporal elasticity o f substitution. The flow budget constraint facing the household is (1 + r t ) c t + it = f;t - atfibt ' + ~kt + trt ,

(2.3)

where z~ is the consumption tax (the distinction between a sales tax and a value added tax being irrelevant here), it is (private) investment, -Vt is the taxable income o f the household, and 6 is the depreciation rate of capital kt. The term 6kt is not included in pre-tax income )3t since depreciation is tax-deductible, s The variables at and bt describe the income tax schedule. In particular, bt may be interpreted as the degree of progressivity o f the income tax schedule; thus bt = 0 represents a poll tax, bt = 1 a proportional tax, and bt > 1 a progressive tax. In the version of the model without income taxes, a t = 0 for all t. Finally, trt is a lump-sum transfer from the government. Apart from consumption taxes, an innovation of our paper is that we capture the progressivity o f income taxation by using an isoelastic specification of the tax schedule rather than a linear one. The reason for choosing an iso-elastic specification o f the income tax schedule rather than a linear one is (apart from enhanced realism) the following. A linear tax schedule in the model would be characterized by a single variable: its slope. This slope's empirical counterpart would, presumably, be the average marginal tax rate paid by households. But the issue that we are mainly concerned with is whether households respond in various ways to exogenous changes in fiscal policy variables. Thus, it would be problematic if some o f these variables could be controlled to any extent by a

5 It should be noted that the model contains a deterministic labor-augmenting rate of technical change 7, and that the model has been stationarized (growth-adjusted) to take care of this. This implies that the effective subjective discount factor is /~ = 13(1 + 7)~(1-'7) where fl has the usual interpretation as the weight the household attaches to next period's utility. Thus, convergence of the maximand requires that /~ < 1, but not that fl < 1. It also means that output, Yt, (and similarly for the other nonstationary variables) should be interpreted as output per capita at time t divided by (1 + y)t. This method of stationarizing a growth model is described in detail in Hansen and Prescott (1995).

G. Jonsson, P. Klein~Journal o f Monetary Eeonomics 38 (1996)245-268

249

single household. But in a progressive tax system, the marginal tax rate depends on the quantity of labor supplied by the individual household. In contrast, for our Swedish data, the marginal tax rate divided by the average tax rate (the elasticity of the tax schedule) is nearly constant along the tax schedule. At least this holds for income levels which are neither very low nor very high. Thus, the variables at and bt are exogenous in the desired sense. The representative household's taxable income is given by 1 Yt -- 1 + z~ wtht + (rt -- 6 ) k t ,

(2.4)

wt = A z t (1 - O ) K ° t H t - °

(2.5)

r t = AztOK°t-lHt

(2.6)

where

and 1-0

.

Eqs. (2.5) and (2.6) are profit maximization conditions of a representative finn in a perfectly competitive industry which produces the economy's single good using a Cobb-Douglas production function, i.e., according to yt = A K Ot H tl o , where y t is output, z t is the exogenously given level of technology, and 0 is the capital share of income. In Eq. (2.4) we encounter the payroll tax z~v. Given the presence of this tax, it is important to keep in mind that wt is the unit labor cost of an employer. Meanwhile, rt is the pre-tax return on capital, kt. A is a constant designed to bring the average level of growth-adjusted pre-tax household income in line with the data. The reason why the value of this scale parameter matters is, of course, that the tax system is progressive: the higher the level of income, the higher is the marginal tax rate. Note that the wage as well as the retum on capital is exogenously given to the household, and that the return on period t capital is received in period t. Capital letters denote population averages. The law of motion for the capital stock is given by (1 + y)k,+~ = (1 - 6)kt + i,.

(2.7)

Note that each unit of growth-adjusted period t capital corresponds to (1 + ;,) 1 units of growth-adjusted period t + 1 capital. (y is a deterministic labor-augmenting rate of technical change, see Footnote 5.) 6 The government's budget constraint is "/7W

tr, + gtY, = "~;C t

~-

~ w t H

, ~- atYbt ' ,

(2.8)

t

6 Notice that the model economy we consider is closed. Since there is just one good, there is rio international trade, and international borrowing is not allowed.

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996)245~68

250

where gt is the ratio of (useless) government consumption to output. Since technology exhibits constant returns to scale, we have yt = wtht ÷ rtkt, where yt is output. Note that due to the existence of a payroll tax, the income of the household, -fit, is not equal to output, Yr. Since there is no government debt, any surplus or deficit is immediately transferred lump-sum to the households. We assume that the lump-sum transfer, trt, is determined residually. In this economy, then, the household's problem is to solve max Eo (h,,k,+l)~o

fl U(ct, ht) ,

(2.9)

t=0

subject to (2.3)-(2.8), and where the household also takes into consideration the stochastic process governing the evolution of the fiscal policy variables, as well as the aggregate decision rules Ht = H(bt, z~, z~, Yt, zt, g t ) and Kt+l = K~(bt, z~, z~, gt, zt, Kt). These matter to the household, because they determine the stochastic price processes. The functions H and K r are given to and known by each household. Corresponding to the aggregate decision rules we have the individual household's decision rules ht-~ h(bt, z~, z~, gt, zt, Kt, kt) and kt+l = U(bt, z~, z~, gt, zt, Kt, kt). We impose, of course, as market equilibrium conditions, that the aggregate decision rules are compatible with the household's decision rules. 7'8 Thus, finding the equilibrium is essentially a matter of solving a fixed point problem: each household's strategy (decision rule) must be optimal given the (average) strategy of all the other households.

3. Data

The data sample consists of 40 annual observations from 1951 to 1990. 9 Fig. 1 plots the per capita values (in natural logs) of GDP, private consumption, private investment, worked hours, and the hourly product wage, all in real terms. GDP and consumption are calculated at factor cost, i.e., excluding indirect taxes and subsidies. The series over consumption includes durables purchases. This partly accounts for the rather high volatility of consumption as we define it. The main reason for not excluding durables purchases is that these goods, of course, are subject to the consumption tax too. And we want to study whether fluctuations in the consumption tax (including various excise fees) are sufficient to explain

7More precisely, we require that H(bt, r~', z~, yt, zt, Kt)----h(bt, z~, "r~, gt, zt, Kt, Kt) and K'(bt, z~, ~, 9t, zt, Kt)--k'(bt, "r~, z~, 9t, zt, K,, K,). 8 The household also faces a no-Ponzi-game condition which means that the limit of the net present discounted value o f assets must, with probability 1, be nonnegative where the discount rate equals the after-tax interest rate. 9 For an account of our data sources, see Appendix A.

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996)245--268 GDP

251

Consumption

4.5

3.8 r

4.8 3,7

4,4

3.8

42

8.5

4.0

3.4 3.3

3.8

3.8

3.8

3.1

3.~'fiO 1954 1955 1962 1965 1970 1874 1978 1965 1986 1990 8"P950 1854 1956 1962 1966 1970 1974 1978 1982 1966 1890 Private [nveetment

Worked Hours

3.6

6,86

3,3

6,82 6,78

3,0

6,74 6,70

~.4 2.1

6.66 6.82

Y

1.~ ~o

1954 1956 1988 1966 1970 1974 1978 1988 1986 1990

6'5P950 1084 1958 1082 1968 1070 1974 1978 1982 1956 1990

Hourly Wage 4.8

4.4

4.0

3,5

3,2

f

2'[~60 18'54 19'58' 19'62 19'66 18'70' 19'74' 19'7e' 19'82' 19~56 1990

Fig. 1. Natural logs of per capita variables.

the relative volatility of consumption. Moreover, any classification of goods into durables and nondurables is bound to be controversial. It should also be mentioned that inventory investment is not included in the empirical data series on investmentJ ° This is because inventories do not feature in the model, so there is no hope of reproducing its time series properties. The fiscal policy variables are plotted in Fig. 2. A summary of the method of estimating the degree of progressivity of the income tax schedule is provided

10This implies that the accounting identity y = c + i + g does not hold in the empirical series. But since Sweden is an open economy, it would not hold in any case.

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996)245~68

252

Progre~sivity of Income Tax

Consumption Tax

2.0

0.8

1.8 0.6 1.6

/ 0.4

1,4 O,~

1.2

1.t~'g"0"

1956

1950

1965

1970

1976

1980

1985

1990

1955

Payroll Tax

1960

1965

1970

1976

1950

1955

1990

1955

1990

Government Consumption/Gl)P 0.6

0.8

0.6 0.4

0.4 ~...-..._,f

f

0.2

0.~ r

f J 0.~ 50

1955

1960

I965

1970

1975

1989

1985

1990

0'~950

1965

1960

1956

1970

1975

1990

Fig. 2. Fiscal policy variables.

in Appendix A. The approach basically involves fitting a curve with constant elasticity to a function that relates income earned to income tax paid, using a least squares criterion. From the figure one can notice that the tax system has been progressive throughout the period (bt > 1). To measure the consumption tax, we simply divide indirect tax revenues (minus subsidies) by private consumption expenditures (net of taxes and subsidies)J 1 Payroll taxes, which consist mainly of compulsory social insurance contributions, are measured in a similar way. To some extent, social insurance contributions may generate claims on future transfer payments in the form of pensions etc. from the government in proportion to the contributions paid. These contributions would, then, not distort behavior in the way that an ordinary tax does. Ideally, one would like to separate, for each year, the insurance premium/savings part of payroll taxes from the distortive tax part. But since this would raise difficult conceptual issues as well as involve an extremely cumbersome data collection

11 The consumption tax includes both VAT (moms) as well as various other indirect taxes and excise fees, such as taxes on alcohol, motor vehicles, horse race betting, etc. This explains why the consumption tax has been, for example, above 30% since the early eighties, although the rate of VAT has been lower than that during the same period.

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996)245-268

253

exercise, we avoid such a separation and treat the entire payroll tax as a tax in the ordinary distortive sense. ~2

4. Method

4.1. Solution algorithm and filterin 9 In this paper we have chosen the conventional method of approximating the original problem around its steady state by a linear-quadratic problem. This approximation of the problem can then be solved rather swiftly. We should perhaps stress that the linear-quadratic approximation is around the natural logs of all the variables except for the tax variables bt, ~ , and ~ ' . For data to be amenable to estimation, it must somehow be rendered stationary. We follow the convention in the literature and use the HP-filter, introduced into economics by Hodrick and Prescott (1980). In order to capture what we usually mean by the business cycle (i.e., fluctuations with a period of 3-8 years), we set the smoothness parameter, 2, to 100. In our dataset this implies that real GDP exhibits about 8 cycles during the studied 40-years interval.J3 After this subsection, all nonstationary variables as well as hours worked per head should be interpreted as having been submitted to HP-filtration. Note that all the fiscal policy series are treated as stationary. Consequently they are not filtered. 14

4.2. The V A R model In the general equilibrium model fiscal policy is treated as an exogenous vector-valued stochastic process. This process is modelled as a V A R ( p ) where the parameters are estimated by least squares. Using the parameter estimates, we then estimate the remaining parameters using SMM. The choice of order p was settled using a likelihood ratio test. The hypothesis p = 1 could not be rejected against the alternative hypothesis p = 2 at the 5% significance level.

12In fact, we think that this is a reasonable approximationof the facts. Especially since the 1950s when a new pension plan was introduced,the Swedish social insurancesystem has been characterized by a rather complex and tenuous relation between social insurance contributionsand receipts. 13We have also performed the simulations with 2 equal to 20 and 200, respectively. These values imply that the business cycle (for real GDP) is about 4 years and 8 years, respectively, ibr Swedish post-war data. The results from these simulationsare similar to those reported in Section 5 (where 2 is set to 100). In particular, the model that includes fiscal variablesalways outperforms the benchmark model. 14The results are qualitativelysimilar if the fiscal policy series are filtered. The main effect of filtering is that the absolutemagnitudesof the correlationswhich includefiscal policy variablesbecome greater.

254

(7. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996) 245-268

Since we have no data on the capital stock, the (stationarized) Solow residual, zt, cannot be measured. Hence, p and a~ given in lnzt+l = plnzt + et+l,

/3t+ 1 ~

N(0, o'~) ,

(4.1)

are estimated by SMM along with the preference and technology parameters of the general equilibrium model. The following VAR(1 ) model for the fiscal variables government consumption, degree of income tax progressivity, consumption tax, and payroll tax is used:IS I ln gt+l In at+l

.[ _cbt+l

= [o ¢p]

Tt+ 1

"1 In gt btlnat

+

~t+l ,

~ t + l "~ N ( 0 , 2;) .

(4.2)

c

• z~

Rather than estimate the constant, v, we equated the unconditional mean /~ = ( I - ~ p ) - l o of the process to the sample mean of the empirical series and estimated the system without a constant. The estimation results show that all fiscal policy variables are quite highly autocorrelated, and that there is a relatively high negative covariance between In at and bt. For the existence of a steady state (which we must have in order to use the solution algorithm we have chosen), we require that the exogenous state vector is stationary. For this it suffices that all the eigenvalues of ~p lie strictly inside the unit circle in the complex plane. Fortunately, this is the case for our estimates of ¢p. 4.3. S M M

Heuristically, simulated method of moments estimation consists in choosing those parameter values that produce the best match between empirical and simulated moments. 16 To test the model, we use the fact that sample moments are functions of time averages, and hence asymptotically normally distributed. Thus, multiplying the vector of differences between empirical and simulated moments around the inverse of a consistent estimate of its asymptotic variance matrix yields a test statistic which is asymptotically distributed as Z2 with degrees of freedom equal to the number of moments minus the number of parameters being estimated. When interpreting the test statistics (and the associated p-values) it is worth keeping in mind what it is that is being tested. The null hypothesis is not, as in our reading of Christiano and Eichenbaum (1992) and Braun (1994), that a

15 For the model without income taxes, we estimate a system that does not include In at a n d J6 F o r details, see A p p e n d i x B.

bt.

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996) 245-268

255

certain subset of the model's moments are equal to the corresponding empirical sample moments (the latter regarded as fixed numbers). Rather, the hypothesis is that a certain subset of the model's (true, not simulated) moments are equal to the (true, not sample) moments of the stochastic process of which our observed macroeconomic time series are but a single realization. In other words, our population is not the actual history, but, as is the convention in statistics, the set of possible histories. Thus, sampling uncertainty affects not only our parameter estimates, but also our sample standard deviations and correlations which should be seen as estimates of their population counterparts. ~7

4.4. Choice of moments In all estimations, we use only standard deviations, relative standard deviations, contemporaneous correlations, and first-order autocorrelations. The main reason for this restriction is that the set of moments would otherwise be too large, and that the moments we do use are the easiest to interpret. Also, the relative variability of the real wage is never considered in estimation, simply because our model strongly exaggerates the volatility of the real wage. In this dimension, the model clearly fails. Gomme and Greenwood (1992) deal rather successfully with this problem by assuming that wages are set so as to provide risk-averse employees optimal insurance. Our model, however, does not contain this feature and therefore fails to account for the rather low volatility of the real wage that we observe in the data. We estimate the models with six different sets of moments. The first set (set 1) is chosen so as to highlight the contrast between the models with fiscal policy and the model without fiscal policy. In particular, it contains the three moments we discussed in the introduction: the relative variability of hours worked and of consumption to that of output, and the correlation between hours worked and the real wage. The following nine moments were included in set 1:

corr(lnht, lnht_l), corr(ln yt,ln yt_l), corr(lnct,lnct_l), corr(lnht, ln yt), corr(lnht, lnct), corr(lnht, lnwt)

.

(4.3)

To see whether the model with fiscal policy could outperform its rival on a larger set of moments, we included in set 2 all (relative) standard deviations

17 A rather subtle problem arises in the estimation of the model with progressive income taxes. The problem is that the value of the scale parameter A that delivers the empirically correct average level of pre-tax household income depends on how the other parameters are chosen. Hence it typically happens that when one minimizes with respect to ct, r, etc., the average pre-tax household income becomes too low or too high, so that A must be adjusted. But once A has been adjusted, the values of ct, fl may no longer be optimal. The natural solution to this problem is, of course, an iterative procedure.

256

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996)245-268

except that of the real wage, all contemporaneous correlations, and all first-order autocorrelations of the endogenous variables. 18 Third, the models with fiscal policy are estimated with sets of moments including the exogenous variables. The reason for doing this is to see whether the mechanism we use to explain fluctuations of and comovements between hours, output, consumption, investment, and real wages is the right one, i.e., whether private behavior reacts to fiscal shocks in the way that our model predicts. Thus, in addition to the moments included in set 1 and 2, sets 3 to 6 include a wide range of contemporaneous correlations between endogenous and exogenous variables.

5. Results of the S M M estimations

In this section we report the parameter estimates and the goodness-of-fit measures from the SMM procedure. As already mentioned, the first and second sets of moments include no fiscal policy variables. This makes the comparison of the models with and without taxes straightforward. In Section 5.3 we report the results where the sets of moments are extended to also include correlations between endogenous variables and exogenous fiscal policy variables. We call the model without fiscal policy model I, the model with government consumption, payroll taxes, and consumption taxes model II, whereas the model with all five fiscal policy instruments (gt, t~, t~V,at, bt) is called model III. 5.1. S e t 1

Focusing on nine of the most common moments the results are as follows. The parameter estimates for model I are (d, /~, 0, d, 3, fi, 62) = (0.055, 0.685, 0.59, 0.30, 0.15, 0.905, 0.00031),

(5.1) whereas for model II the estimates are (& /~, 0, d, 6, t~, 62) = (0.330, 0.985, 0.32, 3.00, 0.20, 0.990, 0.00026).

(5.2) Finally, for model III, the parameter estimates are (a, /~, O, 6, 3, fi, 6~) = (0.700, 0.995, 0.09, 1.90, 0.26, 0.795, 0.00022).

(5.3) 18We also excluded moments including the capital stock, since we do not have data on the Swedish capital stock.

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996) 245~68

257

It can immediately be noticed that the parameter estimates for the model without fiscal policy, model I, are unreasonable, indicating that this model is misspecified. In particular, the estimated reciprocal of the intertemporal elasticity of substitution, ~, is implausibly low, implying a very high willingness to substitute leisure and consumption over time. Also, the estimated discount factor, /~, is too low. In contrast, all the estimated parameters for model II are very close to those that would have been obtained via a conventional calibration procedure. 19 The estimates for model III are also more reasonable than for the model without fiscal policy (model I) with respect to/~ and & However, under this specification, $ is quite high, implying an average proportion of time spent in paid employment of 58%. It can also be noted that the estimate of p, the persistence of technology, is unusually low. The results from the simulations of the moments and the tests for the goodness of fit are reported in Table 1. We immediately see that the models with fiscal policy variables strongly outperform the model without taxes. For model II, all nine moments are closer to their empirical values in the model which includes distortive taxes, whereas for model III, eight out of nine moments are closer to their empirical counterparts. In particular, allowing for stochastic fiscal policy decreases the model's predicted correlation between hours and real wages in a way that makes the prediction quite close to the empirical correlation. Moreover, the estimations of the variability of both worked hours and consumption (relative to that of output) are much closer to their empirical values when fiscal policy variables are included in the model. Indeed, the models with fiscal policy cannot be rejected at standard significance levels, whereas the model without it is rejected at the 5% significance level. 5.2. Set 2

The above conclusions are unchanged when we study how the models perform with respect to 10 additional moments. The results from the estimation of the 19 moments and the tests for the goodness of fit are reported in Table 2. Again we note that the models with taxes (models II and III) strongly outperform the model without taxes (model I). The additional moments are also in general closer to their empirical values in the tax model. Actually, the moment predictions of the models with taxes are closer to their empirical values for 15 (model II) and 14 (model III) moments, respectively, of the 19 moments. Together this again implies that models II and III cannot be rejected at the 5% significance level, 2° whereas the model without taxes is rejected at any reasonable significance level.

t9 The exception is 3, which is higher than what is usually assumed. However, if 3 is fixed at 0.10, the model can still not be rejected. 2oHowever, model II1 can be rejected at the 10% significance level.

258

G. Jonsson, P. Klein~Journal o f Monetary Economics 38 (1996) 245-268

Table 1 Estimation of moments and test for goodness of fit, set 1: 9 moments No-tax model

Tax models

Moment

Model I

Model II

°'In yt aln ht/aJn y~ °'In ct/O'ln Yt

0.022 0.36 0.79

0.018 0.57 1.12

corr(ln corr(ln corr(ln corr(ln corr(ln corr(ln

0.40 0.63 0.77 0.70 0.43 0.44

0.53 0.53 0.52 0.58 0.30 0.01

0.44 0.46 0.50 0.57 0.51 -0.09

X2(2)

8.02

0.56

2.00

p-value

0.02

0.76

0.37

ht, In h t - 1) yt, In Y t - 1) ct, In c t - 1) ht, In yt ) ht, In ct ) ht, In wt )

Model III 0.017 0.65 1.13

Empirical 0.016 0.64 1.07 0.56 0.45 0.48 0.52 0.26 0.00

Table 2 Estimation of moments and test for goodness of fit, set 2 : 1 9 moments

Moment

No-tax model

Tax models

Model I

Model II

Model III

0.016 0.82 1.03 2.43

0.012 0.87 1.37 2.74

°'In yt °'In h,/trln Yt °'In ct/aln yt Oln i,/O'lny,

0.020 0.33 0.74 2.92

corr(ln ht, In h t - 1) corr(ln yt, In Y t - 1) corr(ln ct, In c t - 1) con'(In it, In it_ 1) corr(ln wt, In w t - j ) corr(ln ht, In yt ) corr(ln ht, In ct) corr(ln ht, In it) corr(ln ht, In wt ) corr(ln yt, In ct) corr(ln yt, In it) corr(ln yt, In wt) corr(ln ct, In it ) corr(ln ct, In wt ) corr(ln it, In wt )

0.36 0.54 0.69 0.38 0.67 0.79 0.56 0.98 0.60 0.95 0.90 0.97 0.72 1.00 0.76

0.53 0.53 0.51 0.43 0.48 0.79 0.57 0.56 -0.05 0.75 0.71 0.58 0.74 0.46 0.40

0.47 0.54 0.50 0.47 0.49 0.76 0.66 0.38 -0.15 0.68 0.64 0.51 0.77 0.16 0.48

100.01

16.89

20.87

0.00

0.15

0.05

;(2(12) p-value

Empirical 0.016 0.64 1.07 2.40 0.56 0.45 0.48 0.49 0.65 0.52 0.26 0.47 0.00 0.54 0.63 0.52 0.33 0.44 0.21

G. Jonsson, P. Klein~Journal o f Monetary Economics 38 ( 1 9 9 6 ) 2 4 5 ~ 6 8

259

Table 3 Estimation of moments and tests for goodness of fit, sets 3 to 6: 13, 27, 21, and 31 moments Moment

Model II

Model II

Model III

Model lII

Empirical

crtnv, aLnh,/gin y.~ °-Incl/°'In y, O'ln it/gin yt

0.018 0.58 1.09

0.018 0.78 0.95 2.31

0.016 0.61 1.22

0.017 0.84 1.14 1.93

0.016 0.64 1.07 2.40

corr(ln ht, In h t_ 1) corr(ln yt, In Y t - 1 ) corr(ln ct, In ct-l ) corr(ln ht, In yt ) corr(ln ht, In cl ) corr(ln ht, In wt ) corr(In it, In it_ 1) corr(In wt, In w t - I ) corr(In ht, In it ) corr(ln yt, In ct ) corr(ln Yt, In it ) corr(ln yt, In wt ) corr(ln ct, In it ) corr(ln ct, In wt ) corr(ln it, In wt )

0.46 0.52 0.56 0.64 0.40 0.08

corr(ln ht, In ,qt ) corr(ln ht, z~) corr(ln ht, zt') corr(ln ht, bt ) corr(ln yt, In 9t ) corr(ln Yt, r~") corr(ln yt, r~') corr(ln yt, bt ) corr(ln ct, In 9t) corr(ln ct, r~t") corr(ln ct, r~') corr(ln ct, bt ) corr(ln it, In yt ) corr(ln il, z~ ) eorr(ln it, z~') corr(ln it, bt)

O.17

0.54 0.52 0.51 0.77 0.54 -0.01 0.44 0.48 0.61 0.75 0.77 0.63 0.70 0.51 0.46

0.45 0.51 0.51 0.48 0.40 -0.14

0.46 0.51 0.51 0.70 0.58 -0.19 0.45 0.47 0.51 0.69 0.88 0.57 0.72 0.27 0.63

-0.12 -0.38

0.11 -0.28 0.04

0.11 -0.28 -0.05

-0.09 -0.24

-0.05 -0.17 -0.02

-0.07 -0.22 -0.03

-0.29 - 0.22

-0.20 0.19 -0.11

-0.21 -0.24 -0.09

0.03 -0.32

0.03 -0.22 0.17

0.02 -0.24 -0.20

0.04

-0.38

-0.27

z2(6,20,14 or 24)

6.06

28.3l

11.37

53.43

p-value

0.42

0.10

0.66

0.00

5.3.

Sets3

0.56 0.45 0.48 0.52 0.26 0.00 0.49 0.65 0.47 0.54 0.63 0.52 0.33 0.44 0.21 -0.09 0.12 0.01 0.13 0.06 0.14 0.02 -0.14 -0.01 0.08 0.04 0.08 -0.07 0.09 -0.10 -0.04

to6

I n o r d e r to s t u d y i f t h e e c o n o m y

r e a c t s to fiscal s h o c k s in t h e w a y p r e d i c t e d

b y t h e m o d e l , w e a l s o r e p o r t t h e r e s u l t s f r o m e s t i m a t i o n s o f m o d e l s II a n d III w h e r e s e v e r a l a d d i t i o n a l m o m e n t s , r e l a t e d to fiscal p o l i c y v a r i a b l e s , are i n c l u d e d .

260

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996) 245-268

The estimated moments and the tests for goodness o f fit are reported in Table 3. While the fit is far from perfect, the predicted correlations between the endogenous variables and the tax variables are reasonably close to their empirical counterparts. In particular one can notice that model II is able to predict the small positive correlation between output and government consumption. Moreover, the correlations between the consumption tax and consumption on the one hand, and investment on the other, have the expected signs and are fairly well predicted by both models II and III. This pattern is consistent with the argument that the household postpones consumption (and saves more) when the consmnption tax is temporarily high. For model III, it can also be noted that the negative correlation between the degree o f income tax progressivity, bt, and hours as well as the negative correlation between b t and consumption are fairly well predicted by the model. Finally, one should note that although the model quantitatively exaggerates the negative correlation between some fiscal variables and the endogenous variables, the models can still avoid rejection for large sets o f moments. Only when all 31 moments are included can model III be statistically rejected. In results not reported in the tables, we find that model II is also rejected when subjected to such a large set o f moments. Hence, there is a clear upper limit to the number o f moments our models are able to handle simultaneously, but, by the standards o f the real business cycle literature, this upper limit is quite high.

6. Impulse responses and variance decompositions What fraction o f fluctuations o f real GDP can be accounted for by shocks to productivity? This is, o f course, one o f the most frequently asked questions in the general equilibrium business cycle literature. Although it is not the main objective o f this paper to answer this question, it may be o f some interest to see what answers our models give to it. Our approach to this issue is to exploit the variance decomposition method introduced by Sims (1972). This approach requires us to orthogonalize the disturbance vector via a Cholesky decomposition, where the order o f the variables in general matters for the results. 21 The practical significance o f the ordering is that a shock to a variable is allowed to have contemporaneous effects only on the variable itself and on those variables which succeed it in the ordering. Perhaps arbitrarily, we chose to order the exogenous variables in the following way: ln zt, ln gt, lnat, bt, z~, "c~v.22 el Partly for this reason, lngram et al. (1994) are sceptical towards this kind of approach, and indeed raise doubts as to whether there is any satisfactory way of answering the question: 'What fraction of GDP fluctuations is explained by productivity shocks?' 22 Since we have in any case ruled out any correlation between the technology shock and the tax variables (whether contemporaneous or at any lead or lag) the position of the technology shock should not matter for the assessment of the contribution of technology shocks.

G. Jonsson, P. Klein~Journal

o/" Monetary Economics 38 (1996) 245-268

261

Using this ordering, we carried out impulse-response analysis for models II and III. We analyzed the effects over time of a one-standard-deviation shock at t = 1 to each of the exogenous processes on the natural log of output's deviation from its deterministic trend. The results from the impulse-response analysis for model II are found in Fig. 3 and for model III in Fig. 4. The parameters used in these simulations are taken from the estimation of moment set 1. For model 11, the impulse responses come close to what one might expect. For example, consider the effects of a one-period shock to the consumption tax. Because such a tax lowers the purchasing power of an hour's worth of gainful employment, hours worked go down in the first period, bringing output down below trend. The second effect of the consumption tax, however, is to encourage saving. This leads to an accumulation of capital above trend, bringing output above trend as well, after which the economy slowly goes back to the steady state. The dynamic responses of model III are less intuitive, perhaps partly because contemporaneous correlations blur the distinctions between the shocks. From the impulse-response analysis, a decomposition of the variance follows in a standard way. We report the results from models II and II1, using the parameter values from moment sets 1 and 2, in Table 4. The main result is that the fraction of the variance of real GDP around its trend attributed to productivity shocks is, perhaps predictably, much greater for model II than for model III. Indeed, in model II the productivity shocks account for more than 85% of the total variance Technology Shock

Government Consumption Shock

0.025

0.01o

0.015

0.005 0.005

o.ooo -0.000

-0.005

-0.015

-0,025

lo

2o

3o

40

6o

8o

7o

80

oo

1oo

-O.OLO

1o

zo

Consumption Tax Shock 0.010

O.OLO

0005

o.oo5

0,000

o.ooo

-0.005

-0.005

-0.010

lO

20

30

40

50

60

70

00

30

40

eo

60

ro

80

90

~oo

80

90

100

Payroll Tax Shock

90

100

-0,010

/ 10

20

00

Fig. 3. I m p u l s e r e s p o n s e s for m o d e l II.

40

60

60

70

G. Jonsson, P. Klein~Journal o f Monetary Economics 38 (1996)245-268

262

Government Consumption Shock

Technology Shock 0.025

0.02

[

0.015

0,005

0.01

\

0.00 -0.005 -O.0l

-0.015

-0,026

~o

2o

30

4o

60

80

~o

eo

eo

too

-0.02

0.010

0.001

0.006

O.OOi

0.000

-0.001

-0.005

lo

20 30 40 50 60

70

eo

so

ioo

-O.OIO,

0.010

0.005

0.005

0,000

0.000

-0.006

-0.006

20

30

40

50

60

70

60

70

80

90 100

80

?0

80

90 100

60

90 100

Payroll Tax Shock

0.010

10

50

, , , , , , 1 0 2 0 3 0 4 0 6 0 6 0

Consumption Tax Shock

-0.010

40

Progressivity Shock

Shock to 'lna' 0.010

--O.O|l

00

90

tO0

-0.OLO

tO 20

30

40 50

60

?0

Fig. 4. Impulse responses for model III.

in output, w h e r e a s the c o r r e s p o n d i n g figure for m o d e l III is b e l o w 25%. 23 This suggests that our answers to the question p o s e d in the b e g i n n i n g o f this section is highly model-specific, and should therefore be treated with caution. H o w e v e r , one robust result is that a m o n g the fiscal policy variables, it is fluctuations in the

23 One important reason for this difference is that the persistence of technology, given by the estimate of the p parameter, is much higher for model II than for model III. This means, of course, that the unconditional variance of technology is much higher for the former than for the latter.

G. Jonsson, P. Klein~Journal o f Monetary Economics 38 (1996) 245--268

263

Table 4 Variance decomposition, fraction of the variance of output attributed to various exogenous shocks

Inzt In gt In at

Model I1 Parameters from estimation of

Model Ill Parameters from estimation of

Moment set 1

Moment set 1

Moment set 2

97% 2%

88% 8%

1% 3%

23% 52% 4% 13% 6% 2%

l 1% 64% 5% I 1% 7% 2%

0% 1% 100%

100%

100%

100%

bt

r~ r~" Total

Moment set 2

ratio of government consumption to output that contributes most to the variance of output. 24

7. Conclusions and discussion The main conclusion in this paper is that fiscal shocks do indeed matter for the Swedish business cycle. The introduction of stochastic fiscal policy significantly improves the empirical fit of a simple stochastic growth model. In the course of making this point, we show that the fluctuations in distortive taxes provide a plausible explanation for several empirical puzzles. In contrast to a simple stochastic growth model, the model that takes fiscal policy shocks into account predicts a close-to-zero correlation between the product real wage and labor input, as well as a high variability in both worked hours and consumption (relative to that of output). These three predictions are all consistent with Swedish post-war data. It is perhaps worth stressing that the results in this paper are substantive, and not purely mechanical. The extension of a general equilibrium model does not automatically improve its empirical performance in the way that, for example, adding regressors to a regression model automatically increases R 2. Indeed, an extended model may easily perform worse than a simple one, as the example of model lII illustrates. Nevertheless, the problem of modelling income taxes has not been resolved in a fully satisfactory way. When the model is augmented to include a realistic specification of the income tax schedule, the volatility of labor input becomes

24 Indeed, this result seems to be robust to reorderings of the variables. In particular, if In at and In g~ switch places, In gt remains by far the greatest contributor to the variance of output, in spite of the fact that the unconditionalvariance of In at is much higher than that of In gt-

264

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996) 245-268

severely exaggerated unless the share of time spent in paid employment (a dimension in which the model is not formally evaluated) is set counterfactually high. One way of interpreting this result is that marginal income taxes are hard to measure. Since we ignore the possibility of measurement errors, these errors are treated as true fluctuations. The model then exaggerates the volatility of tax rates, and this has to be compensated for via implausible parameter values. However, when we experiment with lower values of the volatility of the variables that characterize the income tax schedule, the results do not improve much at all. Significant improvements are achieved only when average income taxes are lowered. Nor is our decision to model the progressivity of the tax schedule the root of the problem. The results from simulations of a model with linear income taxes (not reported) are qualitatively similar and quantitatively worse. The question may arise as to whether the tests for accepting or rejecting the model have any power, given the small number of observations (40) and the possible instability over time of some sample correlations. One answer is that the tests are evidently powerful enough to reject the model without fiscal policy at any reasonable significance level. Another answer is that in other studies these tests (or similar ones) usually reject the models; see, e.g., Lundvik (1992), Christiano and Eichenbaum (1992), and Braun (1994). In particular, Braun, who has less data points than we do, can avoid rejection only by including labor indivisibility as well as fiscal policy, and by confining his attention to no more than eight moments at a time. Nevertheless, a problem is that we do not estimate our parameters simultaneously. The OLS parameter estimates from the VAR system are treated as the true parameter values in the estimation of the preference and technology parameters by SMM. Ideally, we should have included the variance and autocovariance of the fiscal variables as moments and estimated ~0 and 2; as part of the SMM procedure. Unfortunately, that would involve so many moments and parameters that SMM estimation is no longer practically feasible. 25 A source of shocks that we have not considered in our models is movements in international variables. A natural extension would be to consider an open economy model with fiscal policy. However, both Lundvik (1992) and Hassler (1993) show that fluctuations in international variables seem to have only a small impact on fluctuations in Swedish macroeconomic variables. Another, more interesting, extension of our work would be to combine stochastic fiscal policy with home production in the manner of McGrattan, Rogerson, and Wright (1993). Such a combination would be particularly useful given our focus on a fluctuating consumption tax levied on consumer durables purchases, but not on investment in market capital. Nevertheless, despite our model's shortcomings, it is striking how much we are able to explain by amending a basic model to include fiscal policy variables.

25This problem is discussed further in Yaron (1996).

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996) 245~68

265

Appendix A: Data A. 1. I n c o m e t a x e s

See Klein (1995) for details on the collection of data on the income tax schedule. For the purposes of interpreting the results in this paper the following should suffice. When reconstructing the income tax schedule, we use statistics on how much income tax Ti was paid on average by those who earned an income in an interval [Yi' Yi+l] "26"27 Approximating the income earned by each individual or household in each interval by )~i = O3i + )3i+1)/2, we get a finite number o f pairs O3i, ~), each associated with the number o f individuals or households (assessments) in interval i. We then fit a curve to these observations. In particular, we run a weighted least square regression of In Ti on ln)3i, where the weights are given by the number o f assessments in each band. We then measure bt as the estimated slope from this regression. Meanwhile, in order to adjust for growth, In at is given by lnat = constantt + t(slopet - 1)ln(1 + 7). The consumption tax is estimated for each year by relating the amount of total tax revenues from consumption to the amount o f total private consumption, whereas the payroll tax for each year is found by relating the total compensation of employees to the total amount of employers contributions to social security, private pensions, etc. The data is, of course, available upon request. A.2. Sources

For legislation pertaining to income taxation, we consulted 'Svensk Frrfattningssamling', various years and issues. For local income tax rates, we used 'A.rsbok f r r Sveriges kommuner' (Central Bureau of Statistics, Stockholm, various years). For data on the distribution of income for the years 1952 71, we used 'Skattetaxeringarna samt frrdelningen av inkomst och frrmrgenhet' (Central Bureau o f Statistics, Stockholm, various years). The source o f all figures for 1972-91 is 'Riksrevisionsverkets taxeringsstatistiska undersrkning' (Central Bureau of Statistics, Stockholm, various years). For consumption tax revenues we used 'Statens Finanser' and 'Utfallet av statens finanser' (Riksrevisionsverket, Stockholm, various years). All other variables are taken from the annual Swedish national accounts (Central Bureau of Statistics, Stockholm, various years).

26 Slightly abusing the notation, )3 is here defined as in Eq. (2.3), except that it is not growth-adjusted. 27 Strictly speaking, there exist no statistics relating income earned to income tax paid for the period 1951-1971. For those years we instead consult the relevant legislation in order to determine what the relation should have been between income and taxes.

266

G. Jonsson, P. Klein~Journal o f Monetary Economics 38 (1996)245-268

Appendix B: SMM The following brief summary is based on, but slightly extends, the work of Lee and Ingram (1991). The idea behind SMM is very similar in spirit to that behind GMM. Suppose we have a model which is fully specified except for an unknown parameter vector fl E Nk, and which can be used to simulate K independent /-dimensional sequences (Yt,i(fl))t=l, ~ N i = 1,2 . . . . . K, and have an observed /-dimensional sequence (xt)f=l. Suppose n = NK/T > 1. 28 Without loss of generality, let the mean of these series be 0. We now want to (i) estimate the model's parameters, and (ii) test the model. We begin by specifying a 'raw' moment function v : Nt ~ ~1(l+1)/2 defined by

vech(xtx~).

v(xt) =

(B. 1 )

Thus, v (xt) is just the vector of all possible products between the elements of arranged in suitable order. Define

xt

T VT~'~ 1 Z v(x,) ' t=l

(B.2)

N =

1 Z v(yt,i(fl))

(B.3)

t=l

If we wanted only variances and covariances as moments, the above would sufrice. But since it is easier to interpret standard deviations, relative standard deviations and correlation coefficients, we go one step further and define a function m : Nl(t+l)/2 ___, Nj. Since sample standard deviations and correlation coefficients are fimctions of the raw moments in vr and VN,i(fl), this is all we need. Finally, define mr = m(vr),

(B.4)

mN, i ( f l ) = m(VN, i(/~)) ,

(B.5)

mg(fl) = -~1 ~-'~mN,i(fl)

(B.6)

i=l

If the model is true, then, for some fl E R k, we should presumably have plim[mr] = T--~oc

plim [ m N ( f l ) ] = m ,

(B.7)

N, K---*oo

28 The reason for simulating K independent sequences of length N rather than a single sequence of length NK is the gain in efficiency. This gain is large because of the high degree of persistence in our simulated series.

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996) 245 268

267

where m E EJ is a nonrandom vector of population moments. This motivates the following estimator of fl (the SMM estimator): /3 = argmin[mr - m N ( ~ ) ] t ~ ' [ m T -- mN(/~)] ,

(B.8)

where W is some, possibly random, positive definite matrix such that plimW = W, where W is nonrandom. Then, under various technical assumptions, 29 /~ is consistent and asymptotically normal. The optimal choice of W is the inverse of the asymptotic variance matrix of (mr - m N ( [ } ) ) , i.e., [~m(vr) ( ! ) l+

s

~ ] - '

,

(B.9)

where S is the asymptotic variance matrix of yr. A suitable estimator of S is given by Newey and West (1987) as S = Ro ~- Z

1

i=t

(Ri-~Ri)

p~-l

(B. IO)

'

where p is some suitable integer-valued strictly increasing function of T and 1

Ri = ~

T Z

(V(Xt) -- VT)(V(Xt--i)--- UT) t.

(n. l l)

t-i+l

To test the model, we use the fact that if the weighting matrix is chosen optimally, then, as T ~ oc (keeping n fixed), we have

T[mT- - mN(fl)]'lTV[mT --

mN(]~)]

L x2(j -- k ) ,

(B.12)

where j is the number of moments and k is the number of estimated parameters. 3° In each estimation, the K random sequences (et,i)tUl and (¢t,i)tUt are fixed. As a reasonable compromise between speed and efficiency, we set N = 300 and K = 10. In practice, minimization of (B.8) is done by a grid search where each parameter takes on three different values. In order to determine where to search for the minimizing parameters, i.e., in order to find the centre and the steps of the grid, the effects of large variations for each parameter at a time was considered before the final grid search was performed.

29 In particular, our series must be stationary and ergodic. 3o We are aware of the possibility that the small sample distribution of our test statistic may deviate significantly from its asymptotic distribution. Ideally, this issue should be settled by performing a Monte Carlo study. However, since a single estimation of our model takes several hours, it would be extremely time-consuming to perform such a study.

268

G. Jonsson, P. Klein~Journal of Monetary Economics 38 (1996)245-268

References Backus, D. and P. Kehoe, 1992, Historical properties of business cycles, American Economic Review 82, 865-888. Braun, R., 1994, Tax disturbances and real economic activity in the postwar United States, Journal of Monetary Economics 33, 441-462. Christiano, L. and M. Eichenbaum, 1992, Current real-business-cycle theories and aggregate labormarket fluctuations, American Economic Review 82, 430 450. Danthine, J.P. and J.B. Donaldson, 1993, Methodological and empirical issues in real business cycle theory, European Economic Review 37, 1-35. Gomme, P. and J. Greenwood, 1992, On the cyclical allocation of risk, Research report 9205 (Department of Economics, University of Western Ontario, London). Hansen, G., and E. Prescott, 1995, Recursive methods for computing equilibria of business cycle models, in: T. Cooley, ed., Frontiers of business cycle research (Princeton University Press, Princeton, NJ). Hassler, J., 1993, Are they swinging together? A measure of linear comovements with an application to Swedish and foreign business cycles, Seminar paper no. 531 (Institute for International Economic Studies, Stockholm University, Stockholm). Hassler, J., P. Lundvik, T. Persson, and P. S6derlind, 1992, The Swedish business cycle: Stylized facts over 130 years, Monograph series no. 21 (Institute for International Economic Studies, Stockholm University, Stockholm). Hodrick, R. and E. Prescott, 1980, Post-war U.S. business cycles: An empirical investigation, Working paper no. 451 (Carnegie Mellon University, Pittsburgh, PA). Ingram, B.F., N.R. Kocherlakota, and N.E. Savin, 1994, Explaining business cycles: A multiple-shock approach, Journal of Monetary Economics 34, 415-428. Klein, P., 1995, Swedish income taxes 1951-1990, Mimeo. (Stockholms University, Stockholm). Lee, B. and B. Ingram, 1991, Simulation estimation of time-series models, Journal of Econometrics 47, 197-205. Lundvik, P., 1992, Business cycles and growth, Monograph series no. 22 (Institute for International Economic Studies, Stockholm University, Stockholm). Mendoza, E.G., A. Razin, and L.L. Tesar, 1994, Effective tax rates in macroeconomics: Cross-country estimates of tax rates on factor incomes and consumption, Journal of Monetary Economics 34, 297-323. McGrattan, E.R., 1994, The macroeconomic effects of distortionary taxation, Journal of Monetary Economics 33, 573-601. McGrattan, E.R., R. Rogerson, and R. Wright, 1993, Household production and taxation in the stochastic growth model, Staff report 166 (Federal Reserve Bank of Minneapolis, MN). Newey, W.K. and K.D. West, 1987, A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica 55, 703-708. Sims, C., 1972, Money, income and causality, American Economic Review 62, 540-552. Yaron, A., 1996, Liquidity shocks and international asset pricing, Mimeo. (Carnegie Mellon University, Pittsburgh, PA).