On the Dynamic Properties of Asymmetric Models of Real GNP

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 489

November 1994

ON THE DYNAMIC PROPERTIES OF ASYMMETRIC MODELS OF REAL GNP

Allan D. Brunner

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion

Papers (other than an acknowledgement that the writer had access to unpublished material) should be cleared with the author or authors.

ABSTRACT

There is now a substantial body of evidence that suggests business cycles are asymmetric. However, the evidence has been accumulated using a wide array of statistical techniques and, consequently, is based on various definitions of asymmetry. This paper examines several parametric models that have been used to study asymmetries in real GNP. Although these models capture

asymmetries in very different ways, their dynamic properties are remarkably similar.

On The Dynamic Properties of Asymmetric Models of GNP Allan D. Brunner!

I. Introduction

Linear time-series models have been used widely and quite successfully by economists for several decades. These models are based on the classical framework set forth by Box and Jenkins (1976), which assumed a linear model and a Gaussian, homogeneous error distribution. These assumptions, however, place strong restrictions on the time-series behavior of economic variables. Most impcrtantly, they imply several types of symmetric behavior. For example, positive and negative shocks of equal magnitude have symmetric effects on the dependent variable using such a model.

Although there is now a substantial body of evidence that suggests that business cycles are not symmetric, that evidence is based on a variety of statistical models and, implicitly, on a variety of definitions for asymmetry. Initially, evidence of asymmetry was based on nonparametric tests. In a seminal article, Neftci (1984) Proposed a nonparametric test for "steepness" in economic time-series. He concluded that contractions are steeper than expansions for postwar unemployment data, and Rothman (1991) confirmed those results. DeLong and Summers (1986), using an alternative test for steepness, found similar results. Sichel (1993) proposed a test for "deepness" and found evidence in unemployment variables that contractions are deeper than expansions.

Mote recently, the evidence of asymmetries has been based on various parametric models. While these models have properties that overlap somewhat, they can be categorized roughly by the

way they relax the classical assumptions. One category has focused on the nonlinear behavior of the

| The author is an Economist at the Board of Governors of the Federal Reserve System. He

would like to thank Michael Boldin, Simon Gilchrist, Gregg Hess, Sang-Won Lee, Philip Rothman, Dan Sichel, Pravin Trivedi, two anonymous referees, and seminar participants at Indiana University, University of Notre Dame, and the St. Louis Federal Reserve Bank for helpful comments on earlier drafts of this paper. He is also grateful to Ron Gallant and George Tauchen for Providing a copy of their SNP code. The views expressed in this paper do not necessarily reflect those held by the Board of Governors or any member of its staff. The author is responsible for any and all errors.

conditional mean. For example, Terasvirta and Anderson (1991), Potter (1991b) and Beauclry and Koop (1993) have used the threshold autoregressive model to study cyclical asymmetry. Their results generally show that contractions are less persistent than expansions.

More recently, economists have turned their attention to the time-varying properties of higher moments and, in particular, to conditional heteroskedasticity. Again, there are a number of such models, including the autoregressive conditional heteroskedasticity (ARCH) model, the generalized ARCH (GARCH) model, and the exponential GARCH (EGARCH) model. Conditional heteroskedasticity has been found in many economic variables, including employment data, GNP, consumption, investment, inventories, durable and nondurable goods, asset prices, and producer and consumer prices. For examples, see Engle (1982), Bollerslev (1986), Nelson (1991), French and Sichel (1993), and Brunner and Hess (1993).

A final category of models relaxes the Gaussian error distribution assumption. The most general is Gallant and Tauchen’s (1990) seminonparametric (SNP) model, which accommociates arbitrary departures from Gaussianity and conditional heterogeneity. Brunner (1992, 1994) and Hussey (1992) have used SNP models to assess the properties of conditional distributions of several economic variables. Each study found strong evidence that the shape and time-varying characteristics of distributions during contractions are quite different from distributions during expansions.

This paper compares several asymmetric models of real GNP growth. Although these models differ dramatically in the way they model asymmetries, the dynamic properties of the models are remarkably similar. In particular, the most prominent feature of each model is conditional heteroskedasticity: The conditional variance of output increases dramatically during contractionary episodes. In addition, there is some evidence of nonlinear behavior in the conditional mean. Overall, this behavior is analogous to the notions of steepness and deepness that is suggested by nonparametric

definitions of asymmetry.

il. Modelling Conditional Asymmetries

[his section briefly outlines a framework for modeling conditional asymmetries, allowing for departures from linearity, Gaussianity, and homogeneity. This framework nests several models that have been used to study asymmetries, including the SETAR model, the SNP model, the ARCH family of models, and the EGARCH model. These models will be used to investigate conditional asymmetries in next section.

Let Ay, denote the growth rate of real GNP, and let X,, denote a vector containing the history

of Ay,. Consider the following framework for modelling y,:

Ay, = f(x.) * 9,72,

o, = AC x,,) (1)

Zz, ~ BC %,, ) In equation (1), the conditional mean, the conditional variance, and the error distribution are statedependent functions of X,.;- In this framework, f(-) allows for possible nonlinearities in the conditional mean, h(-) permits conditional heteroskdasticity, and g(-) permits more general forms of non- Gaussianity and conditional heterogeneity. By contrast, if f(-) is a linear function of X,.y> h() is timeinvariant, ancl g(-) is Gaussian and time-invariant, the model in equation (1) is a standard autoregressive time-series model. This framework nests several models that permit departures from linearity, homogeneity, and

Gaussianity. The remainder of this section outlines a few of these models that will be used in the next

section to study the asymmetric properties of real GNP.

SETAR models. Although many nonlinear models have been developed, the threshold

autoregressive model has been used quite successfully to study business cycle asy mmetries.” Potter (1991b), for example, has introduced the self-exciting threshold autoregressive (SETAR) model. The SETAR model relaxes both the linearity and homoskedasticity assumptions by allowing the parameters of the autoregressive model to switch between various states. The switches are driven by the value of the current state vector (x,_,). The SETAR(k,d,p) model has the following general form: P Ay, = > 0; ;°4 Yj + 9° if Ay, , € A; (=1,..4) 2 (2) z, ~ N(O,1) where p denotes the number of autoregressive lags; 6; is a constant; k denotes the number of possible states; d denotes the specific lag, Ay,.4, that drives the regime shifts; and A; denotes the range of values for Ay,4 that are associated with regime i, i=1],..k. ARCH-type Models. In a seminal article, Engle (1982) introduced the ARCH model, which models the conditional variance as a function of lagged, squared forecast errors. Bolerslev (1986) extended the ARCH model -- called the generalized ARCH (GARCH) model -- to permit the effects of an increase in conditional variance to decay slowly over time. One drawback to the ARCH and GARCH models is that both positive and negative forecast errors lead to an increase in conditional variance. Brunner and Hess (1993), using state-dependent models of conditional variance (SDM-V), relaxed this symmetry condition and allowed the conditional variance to be an asymmetric function of either lagged levels of the dependent variable or lagged forecast errors.2 The AR(p) model with

SDM-V(k,I,m) errors can be written as follows:

2 Fora complete treatment of nonlinear models, see Priestley (1988, 1989). Hamilton’s (1989) switching regime model has also been used extensively to study business cycle asymmetries. That model does not fit into the framework in this paper, however, since the model depends on an unobservable, exogenous variable, rather than the history of the dependent variable.

3 Nelson’s (1991) EGARCH model allows for asymmetric effects of forecast errors on the conditional variance but does not nest ARCH and GARCH models.

4

Pp Ay, ~ » O,°Ay,; + 6, °%, j=l

k I m (3) 2 2 2 2.2 5, = G + Bi C As Ny)” + 2, Baj "(25 Yj)” + 2, B31 JF JF Jz

z, ~ N(O,1)

t

SNP Models. Gallant and Tauchen’s (1990) seminonparametric (SNP) models are able to accommodate arbitrary departures from both Gaussianity and homogeneity. The SNPRX(p,K,,K,)

model can be written as follows:

p Ay, = » 8,-Ay,; + 9, °%, iP

; p (4) 0, = [0 + > B; "A y,] jal

% ~ 80%.)

Gallant andl Tauchen model g(-) as a Hermite polynomial expansion of a Gaussian density, which can approximate any general departures from normality. The degree of the polynomial is K7. In addition,

the parameters of the polynomial are allowed to be K,-degree polynomials in x,_, in order to capture

more general forms of heterogeneity. See Gallant and Tauchen (1990) for more technical details.*

III. Asymmetric Models of Real GNP Th's section of the paper describes the results of estimating three asymmetric models of real GNP -- the SETAR model, the SNPRX model, the SDM-V model. The data are 175 quarterly

observatiors of U.S. real GNP growth in $1982 from 1947 to 1990. An "optimal" model in each

4 Gallant and Tauchen (1990) used the absolute value of AY,.j in their specification of h(-), which imposes symmetry with respect to Ay,;. Brunner (1992) found the asymmetric relationship in equation (4) to be important in stabilizing the conditional heterogeneity found in real GNP.

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category was chosen on the basis of several model selection criteria.>

The optimal specifications are the SETAR(2,2,2) model, the SNPRX(2,2,2) model, and an AR(2) model with SDM-V(0,1,!) errors. A simple AR(2) model will serve as the benchmark for the analysis.

Table | presents the results of several diagnostic tests that were performed on the residuals from each selected model. These tests are designed to detect model misspecification related to linearity, Gaussianity, and homogeneity. The first two tests are standard tests for detecting evidence of non-Gaussianity -- see Greene (1990). The remaining tests are Lagrange multiplier (LM) tests of the type suggested by Engle (1982), Breusch and Pagan (1978), White (1982), Newey (1985) and Tauchen (1985). The LM tests can be divided into misspecification tests for the conditional mean and the conditional variance. For mean tests, the residuals from each model were regressed on lagged values of the residuals (serial correlation), and on lagged values and lagged squared-values of real GNP growth rates (level effects). Likewise, for variance tests, squared residuals were regressed on lagged values of squared residuals (ARCH effects), and on lagged values and lagged squared-values of real GNP growth rates (level effects).

The results of the diagnostic tests for the simple AR(2) model suggest that real GNP growth is not well captured by a linear model with Gaussian, homogeneous errors. There is statistically significant evidence of kurtosis, as well as marginally-significant evidence of nonlinearities in the conditional mean and of time-varying conditional variance. The SETAR model, which moclels heteroskedasticity and nonlinearities in the conditional mean, provides somewhat mixed results. Although the model removes any evidence of nonlinearity in the conditional mean, there is still significant evidence of non-Gaussianity and marginal evidence of heterogeneity.

The SNPRX model, which models both non-Gaussianity and heterogeneity, appears to pass all

> The model selection process is described in a supplemental appendix available from the author

upon request. An EGARCH model was also estimated and provided nearly identical to those for the SDM-V model.

of the diagnostic tests. There is a very marginal amount of ARCH left in the residuals, however, as evidenced by the p-value at the fourth lag. That could be the result of the way the SNP framework models the time-varying conditional heteroskedasticity, as a function of lagged output rather than lagged forecast errors. The AR(2) model with SDM-V errors also appears to perform well, even though it reiaxes only the heteroskedasticity assumption.

Table 2 presents other criteria for evaluating the asymmetric models of real GNP growth. The number of estimated parameters and the value of the negative log-likelihood function are listed in the first and second rows of the table, respectively. Since each model nests the AR(2) model, likelihood ratio tests for this restriction are reported in the third row of the table. Conditional symmetry can be rejected at tie 1% significance level in all cases. The final rows of the table present two standard criteria for comparing non-nested models. Minimizing the Akaike information criteria (AIC) results in selecting the SNPRX model. The Schwarz criterion, which is known to be more conservative than the AIC in small samples, selects the SETAR model (the smallest of the asymmetric models).

The results presented so far are somewhat inconclusive. While there is no doubt that real GNP has asymmetric properties, it is unclear what the optimal specification for asymmetry should be. The SNPRX. model is the only model that is both suggested by a model selection criterion (the AIC) and passes tie battery of diagnostic tests. The SDM-V model passes the diagnostic tests, but finishes second-to-last using either model selection criteria. Although the SETAR model is chosen by the Schwarz criteria, it does not pass all of the diagnostic tests. Rather than choose an optimal model at this stage, the next section takes a closer look at the asymmetric properties of these models using

analytical tools discussed in Potter (1991a), Brunner (1992), and Gallant, Rossi, and Tauchen (1993).

IV. Asymmetric Properties of Real GNP Nonlinearities. The SETAR(2,2,2) specification entails an AR(2) model, with the parameters of the mode! switching between two sets of values based on whether output growth two periods in the

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past was negative or positive. The parameter estimates (and their standard errors) are:

Ay, = 1.57 + .31 AYy,4 + .20 AY,» + 3.50 z if Ay,» >0O (2.8) (©.9) (1.6) (19.7) (5) Ay, = -161+ 44 Ay,,- 79 Ay,, + 4822, ifAy,. <0 (-1.3) (2.0) (-1.9) 7 (10.8)

and indicate both a nonlinear conditional mean and conditional heteroskedasticity.

In order to examine the nonlinear properties that are implied by the SETAR model, Figures la through le plot responses of both y,,; and 6,,; to impulses of various magnitudes -- z, = +2. +1, -1, and -2.© The impulse response functions in Figure 1a have been conditioned on output growth having been at its unconditional mean for the two preceding quarters (Ay,.; = Ay, = 3.2 percent). Although the responses of the level of GNP to each shock are fairly similar, there is some evidence cf nonlinearities: Note that the responses to positive shocks are fairly gradual, while the effects of a negative shock accumulate somewhat more quickly. By contrast, the responses of the conditional standard deviation are radically different. Negative shocks lead to strong increases in uncer-ainty about future values of output, since these shocks put GNP growth near zero. Positive shocks have little impact on the conditional variance, however, because the model has been conditioned on output growth being greater than zero.

Figure 1b shows the responses when output growth has been about | standard deviation below its unconditional mean (-1.1 percent) in two preceding periods. In this case, there appears to be more evidence of nonlinearities. With respect to the level of GNP, even bad news leads quickly <o expansionary growth after a couple of quarters, as a result of the negative second autoregressive

coefficient in equation (5). In addition, similar to the results by Beaudry and Koop (1993), positive

© The impulse response functions were simulated in RATS 4.0, using 10,000 replications. The simulations are based only on estimated model parameters and ignore any additional structure implied by the diagnostic tests presented in Table 1.

shocks arz much more persistent than negative shocks. Indeed, the cumulative effect of a negative shock is not much greater than the initial impulse.

Figure 1c shows the responses when the SETAR model has been conditioned on a very good state-of-the-world -- GNP growth has been one standard deviation above its unconditional mean (about 7.4 percent growth). The responses of GNP are nearly identical to those shown in Figure la. In addition, since output growth is almost always positive after any shock in this state, uncertainty about future valies of output growth is not affected by new information.

Non-Gaussianity. The SNPRX(2,2,2) model was chosen as the optimal SNP specification, which indicates statistically significant departures from both Gaussianity and heterogeneity. Figure 2 shows twc possible densities from the SNPRX model. The density to the right is conditioned on 7.4 percent growth in two previous quarters, the density to the left is conditioned on minus 1.1 percent in the two previous quarters. While the density on the right -- corresponding to a contraction in GNP -has a larger conditional variance, both densities appear to be fairly Gaussian. Table 3 presents a more comprehensive analysis of possible departures from Gaussianity. Each row corresponds to a different set of conditioning information (previous values), ranging from 13.9 percent growth to minus 7.5 percent in two previous quarters. The values in columns 2 through 9 are cumulative probabilities found in the tails of the densities relative to various critical values. For reference, the last number in each column (at the bottom of the table) corresponds to the probability value associated with a standardized Gaussian distribution.’ These results are consistent with the features of the SNPRX model discussed earlier (Figure 2): After correcting for heteroskedasticity, the SNPRX model is fairly Gaussian when growth has been close to its unconditional mean. Departures from Gaussianity appear

only when growth has been very strong or very weak.

’ The numbers in the table are analogous to the cumulative values used in Kolmogrov-Smirnov tests; see Bradley (1968, pg. 296).

Heterogeneity. The SNPRX(2,2,2) specification also indicate a departure from homogeneity. The results presented in Table 3, however, suggest that the model is primarily capturing conditional heteroskedasticity and not higher-order forms of heterogeneity. Further evidence is presented in Figure 3, which plots the model’s impulse response functions using the same conditions as in Figure 1. As with the SETAR model, there is some evidence of nonlinearity in the conditional mean and a sharp increase in the conditional standard deviation during contractionary episodes.

Since heteroskedasticity appears to be the most important feature of the SETAR and SNPRX

models, it could be more efficient to use a model with this distinct feature. Recall that an AR(2) model with SDM-V(0,1,1) errors was chosen as the optimal specification within the SDM-V class of

models. The parameter estimates for this model are:

Ay, = 158 + .28 Ay, + 7 Ay,, + 0,2, (3.5) (3.0) (1.9) (6) o? = 1.237 + 317(z.,- 95 )? + 83 of, (1.4) (2.4) (-1.7) (10.8)

Note that with this specification of the conditional variance, uncertainty decreases for some positive shocks and increases sharply for all negative shocks. As before, uncertainty is fairly autocortelated, and the effects of "news" take several periods to die out. |

Figure 4 shows the impulse response functions based on an AR(2) model with SDM-V(C,1,1) errors. Since Ay, is a linear function of x,_, and 6, is not a function of x,., in this model, the impulse response functions are impervious to all sets of conditioning information. Moreover, responses to the level are symmetric with respect to the magnitude of the impulse; that is, a +2 shock has the exact opposite response as a -2 shock. Finally, as suggested by the previous results, the impulse response functions for the conditional variance are remarkably similar to impulse responses using the SETAR

and the SNPRX models. Still, although negative shocks increase uncertainty about future values of

10

output growth, positive shocks have little impact on the conditional variance. The small impact of positive "news" is similar to the behavior in Figures la and 3a, however, and presumably reflects the

fact that the SDM-V model averages across all values of Xp

V. Conclusion

This paper compares several asymmetric models of real GNP growth. While it is not clear which model is best for capturing asymmetries in real GNP, each asymmetric model studied in this paper exhibits the same dominant feature: During contractionary phases of the business cycle, the conditional variance of forecasts increases dramatically. In addition, there is some evidence of

nonlinearities in the conditional mean, as evidenced by the SETAR model.

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REFERENCES

Anderson, H.M., and T. Terasvirta (1991), " Modeling Nonlinearities in Business Cycles Using Smooth Transition Autoregressive Models,: unpublished manuscript.

Beaudry, P., and G. Koop (1993), "Do Recessions Permanently Change Output," Journal of Monetary Economics, 31:149-163.

Bollerslev, T. (1986), "Generalized Autoregressive Conditional Heteroskedasticity," Journal of Econometrics, 31:307-327.

Box, G.E.P., and G.M. Jenkins (1976), Time Series Analysis, Forecasting and Control, Holden-Day, San Francisco.

Bradley, J.V. (1968), Distribution-Free Statistical Tests, Prentice-Hall, Inc., Englewood, New Jersey.

Breusch, T. and A. Pagan (1978), "A Simple Test for Heteroskedasticity and Random Coefficient Variation," Econometrica, 46:1287-1294.

Brunner, A.D. (1992), "Conditional Asymmetries in Real GNP: A Seminonparametric Approach," Journal of Business and Economic Statistics, 10:65-72.

Brunner, A.D. (1993), "Testing for Conditional Asymmetries in the Goods Market Using Sem: nonparametric Models," unpublished manuscript.

Brunner, A.D. and G.D. Hess (1993), "Are Higher Levels of Inflation Less Predictable? A State Dependent Conditional Heteroskedasticity Approach," Journal of Business and Economic Statistics.

DeLong, B.J. and L. Summers (1986), "Are Business Cycles Symmetrical?" American Business Cycles: Continuity and Change, edited by Robert Gordon, NBER and University of Chicago

Press.

Engle, R. (1982), "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation, Econometrica, 50:987-1008.

French, M. and D. Sichel (1993). "Cyclical Patterns in the Variance of Economic Activity," Journal of Business and Economic Statistics.

Gallant, A.R. and G. Tauchen (1990), "Seminonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications," Econometrica, 57:1091-1120.

Gallant, A.R., P.E. Rossi, and G. Tauchen (1993), "Nonlinear Dynamic Structures," Econometrica, 61:871-907.

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Greene. W. H. (1990), Econometric Analysis, New York: McMillan Publishing Company.

Hamilton, J.D. (1989), "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, 57:357-384.

Hussey, R. (1992), "Nonparametric Evidence on Asymmetry in Business Cycles Using Aggregate Employment Time Series," Journal of Econometrics.

Neftci, S. (1984), "Are Economic Time Series Asymmetric over the Business Cycle?" Journal of Political Economy.

Nelson, D. (1991). "Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica, 59:347-370.

Newey, W. K. (1985), "Maximum Likelihood Specification Testing and Conditional Moment Tests," Econometrica, 53:1047-1071.

Potter, S.M. (1991a), "Nonlinear Impulse Response Functions," unpublished manuscript. Potter, S.M. (1991b), "A Nonlinear Approach to U.S. GNP," unpublished manuscript.

Priestley, M.B. (1988), Non-linear and Non-stationary Time Series Analysis, Academic Press, San Diego.

Priestley, M.B. (1989), "State-Dependent Models: A General Approach to Non-linear Time Series Analysis," Journal of Time Series Analysis, 1:47-71.

Rothman, P. (1991). "Further Evidence on the Asymmetric Behavior of Unemployment Rates Over the Business Cycle," Journal of Macroeconomics, 13:291-298.

Sichel, D.E. (1993), "Business Cycle Asymmetry: A Deeper Look," Economic Inquiry.

Tauchen, G. (1985), "Diagnostic Testing and Evaluation of Maximum Likelihood Models," Journal of Econometrics, 415-443.

White, H. (1982). "Maximum Likelihood Estimation of Misspecified Models," Econometrica, 50:1-25.

Table 1. Diagnostic Tests on Standardized Residuals From Various Models of Real GNP Growth (all values are significance levels)

Model Specification

Diagnostic Lag Length AR(2) SETAR(2,2,2) SNPRX(2,2,2) SDM-V(0,1,1) Skewness - .78 .07 30 33 Kurtosis - .02 <.01 52 47 Serial Corr. 1 .78 .84 61 .70 in Mean 2 .68 88 87 83 3 .40 .65 .63 .63 4 71 97 81 .90 Level Effects 1 42 81 49 54 in Mean 2 18 94 61 4] 3 11 35 .40 28 4 33 59 .74 .64 ARCH Errors 1 30 .63 21 48 2 25 .89 45 .78 3 31 45 14 50 4 .63 95 .70 92 Level Effects 1 16 12 .83 .96 in Variance 2 .09 .26 83 .67 3 16 18 88 .90 4 mol 85 71 49

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Table 2. Summary Information for Various Models of Real GNP Growth

Mode! Specification

AR(2) SETAR(2,2,2) SNPRX(2,2,2) SDM-V(0,1,1) Number of 4 6 23 7 Para‘neters -L.L.F. Value 478.2 469.3 446.1 472.2 Symmetry Test - <.001 <.001 01 AIC 482.2 475.3 469.1 479.2 Schwarz 488.5 484.7 505.2 490.2

ea

Note: The test for symmetry is a likelihood ratio test, comparing each asymmetric model to the simple AR(2) model.

Table 3. Cumulative Probability Under Standardized SNPRX(2,2,2) Densities

Area Relative to Various Critical Values

Previous Values < -2.33 < -1.65 < -1.28 < -0.84 > 0.84 > 1.28 > 1.65 > 2.33

13.9 .00 .00 .02 11 15 11 .09 .07 11.8 .00 .02 .06 15 10 .04 .03 .03 9.6 01 .04 .09 18 17 .07 .03 01 7.4 01 .05 .10 20 20 .09 .04 01 5.3 .O1 .06 10 .20 21 10 .04 .00 3.2 01 05 10 .20 .20 .10 .04 01 1.0 01 05 10 20 20 09 04 01 -L.1. 01 — O05 - . 10 19 19 .09 .04 01 -3.2 01 .05 10 19 19 .08 .03 01 -5.4 01 ~ 05 .09 19 18 .08 .03 01 -7.5 01 .04 .09 19 17 .07 .03 01

Gaussian

Density 01 .05 10 .20 .20 10 .05 01

Note: Each row denotes a density that is conditioned on two lags of GNP growth with values shown in column one. Critical values are standard deviations relative to the conditional mean, based on the conditional mean and variance of the corresponding density.

Figure 1. Impulse Response Functions Using the SETAR Model

(a) y(t-1) = y(t-2) = 3.2 percent

2 1 > @ Oo ao} oO -1 -2 0 4 8 12 16 0 4 8 12 16 Number of Quarters Ahead Number of Quarters Ahead

(b) y(t-1) = y(t-2) = -1.1 percent

0 4... 8 12 16 0 4 8 12 16 Number of Quarters Ahead Number of Quarters Ahead

(c) y(t-1) = y(t-2) = 7.4 percent

2 +2 S.d. srceeeeres 4-1 dd, 1 -1s.d.

-2 s.d.

Std. Dev. ro)

0 4 8 12 16 0 4 8 12 16 Number of Quarters Ahead Number of Quarters Ahead

l6a

Probability

0.15

0.10

0.05

0.00

Figure 2. Conditional Densities Using the SNPRX Model

t-1) = -1.1 percent

—- y(t = y(t-1) = 7.4 percent —— yi=y

-10 0 10 Real GNP Growth

16b

20

Figure 3. Impulse Response Functions Using the SNPRX Model

(a) y(t-1) = y(t-2) = 3.2 percent

0 4 8 12 16 0 4 8 12 16 Number of Quarters Ahead Number of Quarters Ahead

(b) y(t-1) = y(t-2) = -1.1 percent

a

6) 4 8 12 16 (0) 4 8 12 16 Number of Quarters Ahead Number of Quarters Ahead

(c) y(t-1) = y(t-2) = 7.4 percent

—

pocccnecorerenwcecoconccoscccccercccccccccccconccccces. woos . o oo

0 4 8 12 16 0 4 8 12 16 Number of Quarters Ahead Number of Quarters Ahead

l6c

eooore™

Figure 4. Impulse Response Functions Using the SDM-V Model

ecccccccscncconencscocosaccoseccncsococosoorosooooscs eoos:

4 8 12 16 ) 4 8 12 Number of Quarters Ahead Number of Quarters Ahead

16d

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Supplemental Appendix

This appendix describes how specifications of the SNPRX, the SDM, and the EGARCH models were obtained for real GNP growth rates. The SETAR specification used in the paper is nearly identical to the specification discussed in Potter (1991b). Tables Al-A3 present detailed exploraticns of the likelihood surface for each of the models. To illustrate the selection process, consider the results shown in Figure Al. Each row of the table corresponds to a different SNPRX(p,K,,K,) model. The first three columns of the table describe characteristics of the model. The total number of estimated parameters for the model is listed in column four. The fifth column contains tne value of the average objective function, evaluated at the optimum.

The p-values contained in columns 6 through 8 correspond to the p-values of a chi-square statistic for a test that compares the model in that row to its SNP successor for that column. For example, the p successor of an SNPRX(1,2,1) model is an SNPRX(2,2,1) model. The chi-square statistic for comparing the two specifications is (2)-(171):(1.29797 - 1.22769) = 24.04 with (14 - 9) = 5 degrees of freedom. Since the corresponding p-value is less than 0.01, the SNPRX(1,2,1) model is easily rejected in favor of the SNPRX(2,2,1) model. Likewise, the p-value for comparing an SNPRX(2,2,1) to an SNPRX(2,2,2) model -- the Ky successor -- is 0.01, which provides considerable evidence towards rejecting the SNPRX(2,2,1) in favor of an SNP(2,2,2).

Based on p-values, the results in Table Al can be summarized as follows. First, the appropriate lag length is 2. Second, there is some evidence of departures from Gaussianity (K7>0). When p=2, a quadratic polynomial is preferred to no polynomial at the 1% level, but cannot be rejected ir favor of a cubic polynomial. Third, there is also strong evidence of conditional heterogeneity (Ky:>0) -- at the 1% level, the SNPRX(2,2,0) model is easily rejected in favor of an

SNPRX(2,2,1) model, and an SNPRX(2,2,1) models is rejected in favor of an SNPRX(2,2,2) model.

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Based on p-values, the optimal model is an SNPRX(2,2,2) model.®

The final two columns of Table Al present two alternative methods of model selecttou which place more weight on selecting a parsimonious model than does selecting according to p-values. The first criterion is the Akaike information criterion (AIC). The AIC adds a "penalty" of pg/N to the objective function. Minimizing the AIC also results in selecting the SNPRX(2,2,2) specification. The second criterion is the Schwarz criterion which adds a penalty of pg:-log(N)/2N to the objective function. The Schwarz criterion, which is known to be more conservative than the AIC in small samples, selects the SNPRX(2,2,0) model. Since the SNPRX(2,2,0) and SNPRX(2,2,1) models did not pass the diagnostic tests described in the text and the SNPRX(2,2,2) did, the optimal specification for real GNP appears to be the SNPRX(2,2,2) model.

The SDM-V and the EGARCH models, shown in Tables A2 and A3, were chosen in a similar fashion. As before, the general rule was to select the most parsimonious model that was suggested by

a model selection criterion and that passed the diagnostic tests.

8 Although larger models could have been fit. it did not seem prudent to do so as the SNPRX(2,2,2) model already involves a saturation ratio of about 7 observations per parameter

18

Table Al. SNPRX Models of Real GNP, 1947:II - 1990:IV (171 effective observations)

00

p-values Model Selection

P K, Ky Po Sx (8) P K, Ky AIC Schwarz ——_—— 4x War

] 0 0 4 1.34634 .02 1] 1.36973 1.40648 1 2 0 6 1.33353 <.01 01 .01 1.36862 1.42374 1 2 1 9 1.29797 <.01 .03 1.35060 1.43328 ] 2 2 12 1.27122 <.01 1.34140 1.45163 1 3 0 7 1.31556 <.01 1.35650 1.42080 2 0 0 6 1.32105 ll <.01 1.35614 1.41126 2 2 0 8 1.27891 .67 85 01 1.32569 1.39918 2 2 1 14 1.22769 .10 01 1.30956 1.43817 2 2 2 23 1.15670 05 1.29120 1.50248 2 3 0 9 1.27881 .67 1.33144 1.41412 3 0 0 8 1.30818 .01 1.35496 1.42845 3 2 0 10 1.27657 85 .O1 1.33505 1.42691 3 2 l 19 1.20089 01 1.31200 1.48654 3 2 2 37 1.09593 1.31230 1.65219 3 3 0 11 1.27647 1.34080 1.44185

SSS

Ncte: P indicates number of autoregressive lags in the conditional mean and the conditional variance. Kz>0 indicates departure from Gaussianity and Kx>0 indicates departure from homogeneity. Pg indicates the number of estimated parameters. The log-likelihood function values -- Sx(8) -- are not directly comparable to the values for the SDM or EGARCH models, since the objective functions are slightly different.

Table A2. SDM-V Models of Real GNP, 1947:II - 1990:1V (171 effective observations)

DO evaues Model Ssleciion

P K LM Po s\(8) ~~ P- K~—OMaL SMSC Schware eee ET

] 0 0 0 3 2.80571 .08 1.00 .40 2.82325 2.85081

] 0 I 0 5 2.80042 07 1.00 <01 <0O1 2.82966 2.87559

1 0 2 0 7 2.76349 .08 .20 23 2.80443 2.86873

1 1 0 0 5 2.80571 .08 40 2.83495 2.88088

] 1 1 0 7 2.80042 .08 <.01 2.81136 2.90566

] 1 2 0 9 2.75395 10 2.80658 2.88926

1 0 1 ] 6 2.77189 05 12 2.80698 2.86210

] 0 ] 1 8 2.75933 .10 2.80611 2.87960

2 0 0 0 4 2.79671 05 1.00 .39 2.82010 2.85685

2 0 ] 0 6 2.79114 .03 1.00 <01 <01 2.82623 2.88134

2 0 2 0 8 2.75434 02 94 3] 2.80112 2.87461

2 1 0 0 6 2.79671 .O1 39 2.83180 2.88692

2 ] 0 8 2.79114 .03 <.01 2.83792 2.91141

2 ] 2 0 10 2.75395 .02 2.81282 2.90468

2 0 I 1 7 2.76112 07 12 2.80206 2.86636

2 0 2 1 9 2.75137 03 2.80400 2.88668

3 0 0 0 5 2.78588 28 25 2.81512 2.86105

3 0 ] 0 7 2.77787 1.00 <01 <0O1 2.81881 2.88311

3 0 2 0 9 2.73744 1.00 1.00 2.79007 2.87275

3 1 0 0 7 2.77835 92 2.81929 2.88359

3 l l 0 9 2.77787 <.01 2.83050 2.91318

3 ] 2 0 1] 2.73744 2.80177 2.90282

3 0 ] ] 8 2.75175 09 2.79853 2.87203

3 0 2 1 10 2.73744 2.79592 2.88778

SSS

Note: P indicates the number of autoregressive parameters in the conditional mean. K and L denote the number of lags of output growth and residuals, respectively, in the conditional variance. M

indicates whether the conditional variance was allowed to be autore of estimated parameters. The log-likelihood function values -- Sy(8) the values for the SNPRX model, since the objective functions are sli

20

gressive. pg indicates the number -- are not directly comparable to ghtly different.

Table A3. EGARCH Models of Real GNP, 1947:II - 1990:IV (171 effective observations)

p-values Model Selection P L M Po Sy(8) P L M AIC Schwarz 1 0 0 3 2.8057] .08 35 2.82325 2.85081 1 ] 0 5 2.79964 .09 .03 <.01 . 2.82888 2.87481 1 2 0 7 2.77818 .04 01 2.81912 2.88342 1 1 1 6 2.77125 .07 12 2.80634 2.86146 1 2 l 8 2.75861 .05 2.80539 2.87888 2 0 0 4 2.79671 .05 39 2.82010 2.85685 2 | 0 6 2.79120 .03 01 <.01 2.82629 2.88141 2 2 0 8 2.76563 02 .03 2.81241 2.88590 2 1 l 7 2.76164 .08 .09 2.80258 2.86688 2 2 1 9 2.74778 .09 2.80041 2.88309 3 0 0 5 2.78588 34 2.82146 2.86739 3 | 0 7 2.78586 01 <.01 2.82680 2.89110 3 2 0 9 2.75577 .03 2.80840 2.89108 3 1 1 8 2.74988 .25 2.79666 2.87015 3 2 1 10 2.74187 2.80035 2.89221

Note: P indicates the number of autoregressive parameters in the conditional mean. L denotes the number of lags of residuals in the conditional variance. M indicates whether the conditional variance was allowed to be autoregressive. pg indicates the number of estimated parameters. The loglikelihood function values -- s,(8) -- are not directly comparable to the values for the SNPRX model, since the objective functions are slightly different.

IFDP Number

489

488

487

486

485

484

483

482

481

480

479

478

477

International Finance Discussion Papers

Titles

1994 On The Dynamic Properties of Asymmetric Models of GNP

A distributed block approach to solving near-block-diagonal systems with an application to a large macroeconometric model

Conditional and Structural Error Correction Models

Bank Positions and Forecasts of Exchange Rate Movements

Technological Progress and Endogenous Capital Depreciation: Evidence from the U.S. and Japan

Are Banks Market Timers or Market Makers? Explaining Foreign Exchange Trading Profits

Constant Returns and Small Markups in U.S. Manufacturing

The Real Exchange Rate and Fiscal Policy During the Gold Standard Period: Evidence from the United States and Great Britain

The Debt Crisis: Lessons of the 1980s for the 1990s

Who Will Join EMU? Impact of the Maastricht Convergence Criteria on Economic Policy Choice and Performance

Determinants of the 1991-93 Japanese Recession: Evidence from a Structural Model of the Japanese Economy

On Risk, Rational Expectations, and Efficient Asset Markets

Finance and Growth: A Synthesis and Interpretation of the Evidence

Autho-(s)

Allan D. Brunner

Jon Faust Ralph Tryon Neil R. Ericsson

Michael P. Leahy Robert Dekle John Ammer

Allan D. Brunner

Susanto Basu John G. Fernald

Graciela L. Kaminsky Michael Klein Graciela L. Kaminsky

Alfredo Pereira

R. Sean Craig

Allan D. Erunner Steven B. Kamin

Guy V.G. Stevens Dara Akbarian

Alexander Galetovic

Please address requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the Federal Reserve System, Washington, D.C. 20551.

22

IFDP Number

476

475

474

473

472 47]

470

469

468

467

466

465

464 463

International Finance Discussion Papers

Trade Barriers and Trade Flows Across Countries and Industries

The Constancy of Illusions or the Illusion of Constancies: Income and Price Elasticities for U.S. Imports, 1890-1992

The Dollar as an Official Reserve Currency under EMU

Inflation Targeting in the 1990s: The Experiences of New Zealand, Canada, and the United Kingdom

International Capital Mobility in the 1990s

The Effect of Changes in Reserve Requirements on Investment and GNP

International Economic Implications of the End of the Soviet Union

International Dimension of European Monetary Union:

Implications For The Dollar

European Monetary Arrangements: Implications for the Dollar, Exchange Rate Variability and Credibility

Fiscal Policy Coordination and Flexibility Under European Monetary Union: Implications for Macroeconomic Stabilization

The Federal Funds Rate and the Implementation of Monetary Policy: Estimating the Federal Reserve’s Reaction Function

Understanding the Empirical Literature on Purchasing Power Parity: The Post-Bretton Woods Era

Inflation, Inflation Risk, and Stock Returns

Are Apparent Productive Spillovers a Figment of Specification Error?

23

Author(s)

Jong-Wha Lee Phillip Swagel

Jaime Marquez

Michael P. Leahy John Ammer Richard T. Freeman Maurice Obstfeld

Prakash Loungani Mark Rush

William L. Helkie David H. Howard Jaime Marquez

Karen H. Johnson

Hali J. Edison Linda S. Kole

Jay H. Bryson

Allan D. Brunner

Hali J. Edison Joseph E. Gagnon William R. Melick

John Ammer

Susanto Basu John S. Fernald