The Asymmetric Effects of Monetary Policy: A Nonlinear Vector Autoregression Approach

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This paper investigates whether monetary policy has "asymmetric" effects on output and prices using a nonlinear vector autoregression approach. The model that is estimated is consistent with a wide variety of structural models, in particular, models incorporating asymmetric nominal rigidity. Impulse response functions generated by this model are used to shed light on three questions that have been the focus of a great deal of research in recent years: do positive and negative monetary shocks have asymmetric effects? do the effects of monetary shocks vary over the business cycle? do the effects of monetary shocks vary disproportionately with the size of the shock? Methodologically, the paper extends the literature on nonlinearities in univariate time series to the multiple equation case.(1)

A number of economic theories imply that monetary policy may have asymmetric effects. Morgan (1993) argues that asymmetry is a feature of many widely accepted economic models, including the standard Keynesian model with "Keynesian" and "Classical" regions of the aggregate supply curve, the liquidity trap theory, credit constraint models, and menu cost models. Kandil (1995) points to asymmetric wage indexation and price adjustment as possible causes of asymmetric effects of monetary policy. Asymmetric "real" rigidities such as irreversibility of investment (see, for example, Abel and Eberly 1994) may also be a source of asymmetry.

The evidence for asymmetry in the effects of monetary shocks is mixed. Cover's (1992) finding that positive and negative monetary shocks have asymmetric effects was supported by DeLong and Summers (1988), Morgan (1993), Kandil (1995), Karras (1996), and Thoma (1994). On the other hand, Ravn and Sola (1996) find that positive and negative monetary shocks have symmetric effects once the regime shift in monetary policy in 1979 is controlled for. Garcia and Schaller (1995) find that monetary shocks have stronger effects during recessions than booms, although Evans (1986) finds no evidence for this type of asymmetry. Thoma (1994) finds that negative monetary shocks have stronger effects on output during high-growth periods than low-growth periods, while the effects of positive monetary shocks do not vary over the business cycle. Ravn and Sola (1996) find evidence for asymmetry in the effects of large versus small monetary shocks.

This paper uses a more general approach to modeling asymmetry than that used in the papers cited above. Cover (1992), Kandil (1995), DeLong and Summers (1988), Morgan (1993), and Thoma (1994) estimate reduced-form output equations that can be interpreted as threshold autoregressions where the switching variable is the money supply or the unanticipated component of the money supply. Many other choices of switching variable can be defended on the basis of economic theory, however. This paper follows Beaudry and Koop (1993) and Thoma (1994) in using the economy's position in the business cycle (here proxied by the growth rate of real output) as a switching variable. The change in the inflation rate is also considered. In addition, other studies allow the coefficient on the monetary variable to change in response to realizations of the switching variable, but constrain coefficients on other variables to be constant. This paper constructs a simple aggregate demand--aggregate supply model in structural form and shows that in general such a model will have a reduced form in which all coefficients vary across realizations of the switching variable. The approach used here allows for this more general pattern of switching. Finally, the model estimated in this study allows for a smooth transition between regimes, whereas previous studies assume discrete regime shifts.

The paper makes three contributions to the literature on the asymmetric effects of monetary policy. First, it demonstrates that structural models incorporating asymmetry may suggest different choices of switching variables than those used in previous studies, and require more coefficients to be subject to switching in the reduced form. Second, the paper shows how asymmetries may be modeled in a very general way using a logistic smooth transition vector autoregression (LSTVAR) model. This is a multiple equation extension of the logistic smooth transition autoregression model described in Terasvirta and Anderson (1992). Statistical tests reject linearity in favor of the LSTVAR specification. Third, the paper finds that while monetary shocks have dramatically different effects depending on the state of the economy, there is no evidence to support the hypothesis that positive and negative monetary shocks have different effects. There is some evidence that large and small shocks have different effects, and that positive and negative monetary shocks have different effects when the shocks are large.

Section 1 presents a model that introduces asymmetry through asymmetric price rigidity and describes how its reduced form may be represented by an LSTVAR model. Section 2 reports tests for linearity in a standard three-variable vector autoregression, where the alternative hypothesis is an LSTVAR model like that constructed in section 1. Section 3 describes the estimation strategy and reports on further tests of the validity of the LSTVAR model. Section 4 presents impulse response functions from the LSTVAR model. Section 5 concludes.

1. MODELING ASYMMETRY IN A VECTOR AUTOREGRESSION

This section develops a simple structural model incorporating asymmetric nominal rigidities. The model has a reduced-form (nonlinear) VAR representation that encompasses the reduced-form models estimated by researchers investigating the asymmetric effects of monetary shocks. However, the model suggests alternatives to the choice of the money supply as the switching variable. This structural model motivates the empirical work in the sections that follow.

1.1 A Structural Model Incorporating Asymmetry

The model grafts asymmetry onto a simple aggregate supply-aggregate demand framework similar in spirit to the models in Cover (1988a, b). No attempt is made to model the sources of asymmetry from microeconomic foundations. Consider first a simple flexible price, neoclassical model. Potential output growth is given by

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the growth rate of potential output in period t, [y.sub.0] is a constant, and [[Theta].sub.t] is interpreted as a technology shock with E([[Theta].sub.t]) = 0. Aggregate demand is given by a quantity theory equation augmented with a general lag structure:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [m.sub.t] is growth in money supply, [p.sub.t] is the rate of inflation, [y.sub.t] is equilibrium output growth, [X.sub.t] = ([y.sub.t], [p.sub.t], [m.sub.t])' is the vector of endogenous variables, and [[Eta].sub.t] is a nonmonetary aggregate demand or price shock. The constant term, [y.sub.0], is set to the same value as the constant in equation (1), implying that aggregate demand and potential output grow at the same rate. The model is completed with the specification of a money supply rule:

(3) [m.sub.t] = [m.sub.0] - [Phi][y.sub.t] - [Pi][p.sub.t] + B(L)[X.sub.t-1] + [[Mu].sub.t],

where [[Mu].sub.t] is a monetary shock with E([[Mu].sub.t]) = 0.

In a flexible price world, [p.sub.t] adjusts so that output demanded is equal to potential output. Therefore in full employment equilibrium

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When nominal rigidities are present, prices may deviate temporarily from the level given in equation (2):

(5) [p.sub.t] = [Alpha]([z.sub.t])[p.sub.t-1] + (1 - [Alpha]([z.sub.t]))[p*.sub.t] = (1 - [Alpha]([z.sub.t]))[[m.sub.t] + 1/[Delta] A(L)[X.sub.t-1] + 1/[Delta] ([[Eta].sub.t] - [[Theta].sub.t])] / (1 - [Alpha]([z.sub.t])L),

where [z.sub.t] is a "switching variable" that represents the state of the economy and [Alpha]([Z.sub.t]) is a price-stickiness parameter that may vary according to [z.sub.t]. Asymmetric effects of monetary shocks may arise in this model through the state-dependent price-stickiness parameter.

The structural model represented by equations (2), (3), and (5) can be expressed in matrix form as

(6) [X.sub.t] = [X.sub.0] + [C.sub.0] [X.sub.t] + C(L)[X.sub.t-1] + D(L)[[Epsilon].sub.t],

where [X.sub.0] = ([y.sub.0], 0, [m.sub.0])', [[Epsilon].sub.t] = ([[Theta].sub.t], [[Eta].sub.t], [[Mu].sub.t])', [C.sub.0] = [0 -[Delta] [Delta] 0 0 1 - [Alpha]([Z.sub.t]) -[Phi] -[Pi] 0]

and C(L) and D(L) are complicated polynomials in the lag operator. The reduced form of this model is

(7) [X.sub.t] = [(I - [C.sub.0]).sup.-1] [X.sub.0] + [(I - [C.sub.0]).sup.-1] C(L)[X.sub.t-1] + [(I - [C.sub.0]).sup.-1] D(L)[[Epsilon].sub.t]

where [(I - [C.sub.0]).sup.-1] is given by

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1.2 Economic Interpretation of the Structural Model

Equation (7) looks like a standard vector autoregression, except that each of the estimated reduced-form coefficients is allowed to vary according to the state of the economy as represented by the switching variable [z.sub.t] The model does not derive the source of asymmetry from first principles, but it is worthwhile thinking about what the sources might be and how they would be represented in this model. The theoretical arguments that are commonly cited as motivation for asymmetry (for example, by Morgan 1993 and Kandil 1995) suggest a number of candidates for the switching variable. For example, if workers and firms resist downward movements in nominal variables, then an appropriate choice of switching variable is the rate of price or wage inflation. This is also the appropriate switching variable in the context of the model in Ball, Mankiw, and Romer (1988), in which high rates of inflation reduce nominal rigidity. Ball and Mankiw (1994) and Tsiddon (1993) present models in which symmetric menu costs in the presence of trend inflation produce downward price rigidity. The switching variable suggested by these models is unanticipated inflation. Fuhrer and Moore (1995) show that the U.S. economy is more aptly characterized by a model incorporating rigid inflation as opposed to a rigid price level. The analog to the downward price level rigidity argument in an environment of sticky inflation is to assume that workers and firms resist changes in the rate of growth of nominal variables, for example, in the case of nominal wage indexation. In this case, the change in the rate of inflation …