Sign restrictions in Smooth Transition VAR models. February 14, PRELIMINARY, DO NOT CIRCULATE

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1 Sign restrictions in Smooth Transition VAR models Martin Bruns, Michele Piffer, February 14, PRELIMINARY, DO NOT CIRCULATE Abstract We develop a way of identifying Smooth Transition Vector Autoregressive models using sign restrictions. What makes the identification tractable despite the nonlinearities of the model is the use of a Bayesian approach applied directly on the structural form. We then use the model and show that monetary policy shocks are less effective when the economy is experiencing relatively high levels of uncertainty. Since not only output but also prices display a limited response to monetary shocks when uncertainty is high, the results cannot be driven by the higher price flexibility documented by the literature in periods when uncertainty is high. The results indirectly provide support to the existence of additional mechanisms driving the effectiveness of monetary policy in addition to price stickiness. JEL classification: C32, E52. Keywords: Nonlinear models, set identification, monetary policy shocks, price flexibility. We are thankful to Helmut Lütkepohl for helpful comments and suggestions. Freie Universität Berlin, Department of Economics. German Institute for Economic Research (DIW Berlin). Corresponding author m.b.piffer@gmail.com

2 1 Introduction The economic literature uses structural Vector Autoregressive models (structural VARs) to address a wide range of research questions. Standard applications of these models include the analysis of the effects of monetary policy shocks, fiscal shocks, uncertainty shocks, and others (see Ramey, 216 for a survey). Since the crucial step in these models consists of isolating structural shocks, extensive research has been devoted to develop identification strategies that achieve this step. Popular identification approaches are the recursive identification, identification through sign restrictions, long run restrictions, and others (Kilian, 213). In their linear form, VAR models cannot address research questions that involve nonlinearities. For example, linear structural VARs cannot study to what extent the effects of a shock depend on whether uncertainty is high or low at the point in time in which the shock hits the economy, or on whether the economy is in a recessionary or in an expansionary phase. For this reason, the literature has extended the baseline VAR model to a nonlinear framework, developing, among other models, the Smooth Transition Vector Autoregressive (STVAR) model. STVARs have been widely used in applied work since the work by Anderson and Vahid (1998) and Auerbach and Gorodnichenko (212), who generalized to a multivariate case the earlier contribution by Granger and Terasvirta (1993) on univariate models. While linear VAR models are identified in the literature using a rich battery of candidate identification strategies, STVAR models are currently identified using almost exclusively the recursive identification approach. Indeed, the technicalities involved in the estimation of such nonlinear models make it challenging to use identification strategies that differ from the recursive one. However, since the literature on linear models highlights several limitations of the recursive identification, the almost exclusive use of the recursive identification for STVAR models emerges as an important limitation that needs to be addressed. This paper offers one step towards filling this gap by providing a way of identifying 1

3 STVAR models using sign restrictions. Sign restrictions have been extensively used in applied work and offer a flexible identification approach suitable for many research questions (see for instance Uhlig, 25 for an application to monetary policy shocks and Kilian and Murphy, 212 for an application to oil shocks, both in linear VARs). We provide a tractable algorithm to apply sign restrictions to STVAR models. In our paper, what makes it possible to apply sign restrictions within a STVAR model in a tractable way is the use of a Bayesian approach that directly estimates the structural model. The literature has discussed the fact that it is challenging to summarize the set identified with sign restrictions, because all restricted models are equally consistent with the data (Fry and Pagan, 211, Inoue and Kilian, 213). As argued, for example, by Arias et al. (214) and Baumeister and Hamilton (215) in the context of linear VARs, adopting a Bayesian approach bypasses these concerns because the posterior distribution provides a summary of the models, despite the presence of set identification. We show that if the researcher takes a Bayesian approach to sign restrictions, and if the model is estimated directly in its structural form as in Baumeister and Hamilton (215), applying sign restrictions to a STVAR model is methodologically not more involving than applying sign restrictions to a linear VAR. This makes sign restrictions feasible for STVAR models, enriching the set of identification strategies that can be applied to such models. After proposing a way of implementing sign restrictions within a STVAR model, we show an application to monetary policy shocks. Vavra (214) recently argued that higher levels of uncertainty lead firms to adjust prices more frequently and by a higher magnitude. This, in turn, potentially reduces the effectiveness of nominal shocks, including monetary policy shocks. In his model, when uncertainty increases, two effects unfold. The wait-and-see effect predicts that in the presence of price adjustment costs, higher uncertainty leads firms to postpone costly price adjustments, or to reduce them in magnitude. The volatility effect predicts that higher uncertainty leads firms to face larger shocks, and hence to require larger price adjustments. In the theoretical 2

4 contribution by Vavra (214), the latter effect dominates, potentially lowering the effectiveness of monetary policy shocks. Having developed an approach to apply sign restrictions to the STVAR model, we study how monetary shocks, which are frequently identified in the literature using sign restrictions, propagate to the economy depending on the level of uncertainty in the economy. We estimate a small-scale model and report evidence that monetary shocks have a smaller effect on output when uncertainty is in the top quartiles rather than in the lower quartiles, confirming the prediction by Vavra (214) on output. However, at the same time we document that also prices respond less to a monetary shock that hits the economy when uncertainty is high. This partly calls into question the mechanism discussed above through price flexibility, suggesting the presence of an additional general equilibrium mechanism that makes the monetary shock less effective on output despite the lower response of prices. The empirical results call into question a series of recent papers that build on the intuition by Vavra (214), including for example Bachmann et al. (213) and Baley and Blanco (215). With regard to VAR applications, Aastveit et al. (213) and Pellegrino (214) use an interacted VAR model to study how monetary shocks interact with uncertainty. Compared to their contribution, the STVAR model used in our paper allows the study of nonlinearities also on the impact effect of the shocks, and not only on the autoregressive component of the model. The result that prices respond to monetary shocks by less when uncertainty is high is consistent with Aastveit et al. (213) and Pellegrino (214), further suggesting the need for future research to clarify the interaction between uncertainty, monetary policy and price flexibility. Our paper also relates to Tenreyro and Thwaites (216) and Weise (1999), who study the interaction between monetary policy and the different phases of the business cycle. The paper is organized as follows. Section 2 develops the estimation of the model in a general framework, referring to the appendix for the detailed derivations of the posterior. Section 3 shows the application to monetary policy shocks. Section 4 3

5 concludes. 2 The model This section discusses STVAR models in a general framework. We start by outlining the model. We then discuss the prior distribution and the posterior derivation. 2.1 The structural form We write the STVAR model in structural form as y t = g(z t 1 )Π 1 x t 1 + (1 g(z t 1 ))Π 2 x t 1 + B t ɛ t, B t = g(z t 1 )B 1 + (1 g(z t 1 ))B 2, (1a) (1b) g(z t 1 ) = The k 1 vector y t ɛ t N(, I), (1c) 1, γ >. (1d) 1 + e γ(z t 1 c) contains the variables of the model, the (kp + 1) 1 vector x t 1 = (y t 1,..., y t p, 1) contains the p lags of the variables of the model and the regressor for the constant terms, and g(z t 1 ) indicates a scalar transition function explained below. The coefficients for the autoregressive components of the models (as well as for the constant terms) are contained in the k (kp + 1) matrices Π 1 and Π 2. The k k matrix B t from equation (1b) contains the impulse vectors that map structural shocks into reduced form innovations at time t. The k structural shocks, contained in ɛ t, are assumed normally distributed as in equation (1c). We adopt the normalization V (ɛ t ) = I k, in accordance to which the structural shocks have a time invariant variance-covariance matrix, which is set to the identity matrix. 1 1 The invariance of V (ɛ t ) over time is without loss of generality and is not intended as a restriction on the shocks driving the data. This can be seen by rewriting the data generating process as u t = B t ɛ t with V ( ɛ t ) = D t, D t time varying and diagonal. Model (??) is then obtained through the normalization u t = B t ɛ t = B t Mt 1 M t ɛ t = B t ɛ t, with M t = Dt.5, B t = B t Mt 1 and ɛ t = M t ɛ t. By 4

6 The evolution of the parameters of the model is ruled by the logistic transition function g(z t 1 ) in equation (1d). g(z t 1 ), which takes values in the interval [, 1], drives the model as a time varying linear convex combination of models (2a) and (2b), with y t = Π 1 x t 1 + B 1 ɛ t ɛ t N(, I), (2a) y t = Π 2 x t 1 + B 2 ɛ t ɛ t N(, I). (2b) Model (1) moves between equations (2a) and (2b) depending on the evolution of the transition variable z t 1. In the analysis in this paper, z t 1 is a function of {y t 1, y t 2,..., y t m } and is hence endogenous to the system through the last m observations of y t, excluding the contemporaneous observations. Given γ >, g(z t 1 ) is a decreasing function in z t 1. The slope parameter γ and the location parameter c control for the shape of the transitions and the midpoint of the transition, respectively. 2 In the limit case of z t 1 and z t 1, the system converges to the linear models (2a) to model (2b). The restriction that the transition variables is the same for equation (1a) and (1b) can be abandoned, as in Teräsvirta et al. (214) and in Bolboaca and Fischer (215). We specify the smooth transition directly in structural form rather than in the reduced form by applying the smooth transition directly on the matrix B t as in equation (1b), rather than using Ω t = g(z t 1 )Ω 1 + (1 g(z t 1 ))Ω 2 with Ω t = B t B t, as in Auerbach and Gorodnichenko (212). Specifying the evolution directly on B t is more construction, V (ɛ t ) = I. Impulse responses to a one standard deviation shock to variable j are then obtained through the impulse vector B t,j dt,j B t,j 1, with B t,j and B t,j the j column of B t and B t, respectively, and d t,j the j, j element of D t. Accordingly, we can assess the effect of one standard deviation shocks without restricting the standard deviation to be time invariant. Whether a change over time in the response to a one standard deviation shock is driven by a different impact effect, a different size of the shock given or a combination of the two remains unidentified. 2 Higher values of γ imply more abrupt transitions from one regime to the other, given a variation in z t 1. Higher values of c imply that the first regime receives more and the second regime receives less weight. In the limit case of γ, the model converges to the Threshold VAR with c representing the threshold beyond which an abrupt regime switch occurs. 5

7 convenient for the purpose of our paper because it allows using prior beliefs directly on B a and B b, which are the parameters that reflect the identifying restrictions. Our specification should be viewed as an approximation of the representation by Auerbach and Gorodnichenko (212), since the smooth transition on B t does not imply a smooth transition of Ω t. Model (1) can be rewritten in compact form as y = Zδ + u, (3) where δ = (δ 1, δ 1) = (vec(π 1 ), vec(π 2 ) ) equals the vectorization of the autoregressive parameters of the model. The likelihood function of the model is then rewritten as p(y δ, B, γ, c) = (2π) T n 2 Ω(B, γ, c) 1 2 e 1 2 (y Zδ) Ω(B,γ,c) 1 (y Zδ), (4) where B = [B 1, B 2 ] is a k 2k matrix including the impulse vectors B 1 and B 2. The derivation of equations (3) and (4) is available in Section 5.1 of the appendix. 2.2 Prior and posterior distributions The model is composed of three types of parameters. δ captures the autoregressive parameters of the model, B captures the impulse vectors of the model mapping structural shocks into reduced form shocks, and γ, c govern the transition function. The prior distribution is specified as p(δ, B, γ, c) = p(δ) p(b) p(γ, c). (5) The assumption that δ, B and γ, c are independent is taken for notational simplicity and can be relaxed. The normality assumed for the structural shocks suggests the normality for p(δ), as standard in Bayesian linear VARs. For this reason, we build the analysis on a joint 6

8 normal prior distribution for δ, using δ N(µ δ, V δ ). (6) The hyperparameters µ δ and V δ can be chosen, for example, to replicate the Minnesota prior. On the contrary, the analysis is more general with regard to p(b) and p(γ, c). Flexibility on p(b) is particularly important, because it is through p(b) that identifying restrictions are imposed. p(b) can include a combination of zero and sign restrictions, in accordance to the researcher s prior beliefs on the structure behind the data. The joint posterior distribution p(δ, B, γ, c Y ) can be explored more conveniently after decomposing it into p(δ, B, γ, c Y ) = p(b, γ, c Y ) p(δ B, γ, c, Y ). (7) As shown in detail in subsection 5.2 of the appendix, it holds that p(b, γ, c Y ) p(b) p(γ, c) Ṽδ 1 2 Ω(B, γ, c) 1 2 e 1 2 {y Ω(B,γ,c) 1 y µ Ṽ 1 δ µ} (8) δ B, γ, c, Y N( µ δ, Ṽδ), (9) with Ṽ δ = [V 1 δ + Z Ω(B, γ, c) 1 Z] 1 (1) µ δ = Ṽδ [V 1 δ µ δ + Z Ω(B, γ, c) 1 y]. (11) Hence, p(δ, B, γ, c Y ) can be explored using a posterior simulator for p(b, γ, c Y ) and drawing δ B, γ, c, Y from a normal distribution. In the application discussed in the rest of the paper we study the response to a monetary shock using nonlinear impulse responses (Koop et al., 1996). Having sampled 7

9 the posterior distribution for the parameters of the model, constructing nonlinear impulse responses does not pose additional challenges. We discuss the concept of nonlinear impulse responses in subsection 5.3 of the appendix. 3 An application to monetary policy shocks The model developed in Section 2 is particularly suitable to study monetary policy shocks. In fact, sign restrictions were initially discussed in the literature with regard to monetary policy shocks. We apply the model from Section 2 to study the theoretical prediction from Vavra (214). According to Vavra (214), higher levels of uncertainty lead firms to adjust prices more frequently and by more. This, in turn, increases the aggregate price flexibility in the economy, leading to a reduced effectiveness of monetary policy, as predicted for example by the New Keynesian model (Bachmann et al., 213). In this section we document that the data provide support for this prediction only in part. 3.1 Data For the baseline specification of the model we use four variables: the log difference of industrial production, the log difference of the consumer price index, the VXO and the effective federal funds rate. We choose the first three variables to improve the comparability of the analysis with a large battery of existing small-scale monetary VAR models. In the baseline specification, uncertainty is measured with the VXO. We use monthly data for the period 1987M1 through 29M12. As transition variable z t 1 we use a standardized moving average of the endogenous variable VXO. In the baseline specification, the moving average at time t is computed using the 3 months until t, i.e. z t 1 = (V XO t 1 + V XO t 2 + V XO t 3 )/3. We use two lags for the estimation of the model, setting p = 2 in equation (1a). 8

10 3.2 Prior Distributions The prior distribution for δ is set following the Minnesota prior, δ N(µ δ, V δ ). (12) We follow Bańbura et al. (21) for the selection of the hyperparameters µ δ, centering the normal prior around a white noise process for the variables entering in log differences, and around the random walk without drift for the other variables. As for V δ, we apply Bayesian shrinkage, setting the hyperparameters of the variances as in Canova (27). Regarding the parameters γ, c in the transition function, we follow Gefang and Strachan (21) and set the prior for the location parameter c as c U(c L, c H ), (13) with c L and c H equal to the 1th and 9th percentile of the transition variable z t 1, respectively. We then depart from Gefang and Strachan (21) and use γ U(γ L, γ H ), (14) rather than a γ distribution. We set γ L =.5 and γ H = 1 to ensure that γ is sufficiently far away from zero, a value at which neither Π 1 and Π 2 nor B 1 and B 2 are separately identified, and sufficiently high to approximate also very fast transitions in the spirit of threshold VAR models. Before discussing the prior p(b) it is convenient to rewrite B 1 and B 2 as B 1 = [b 1, B 1], (15) B 2 = [b 2, B 2]. (16) 9

11 The vectors b 1 and b 2 indicate the impulse vector to a monetary shock in the extreme case 1 and 2, respectively. The vectors in B1 and B2 gather the impulse vectors to shocks that are not identified in the model. We aim to identify the monetary shock using sign restrictions, which we impose using the prior distribution p(b). Following, Faust (1998) and Benati (28), we impose the restrictions that a monetary shock that decreases the federal funds rate increases industrial production and inflation. On the contrary, we leave the response of the VXO restricted. Regarding the remaining impulse vectors B1 and B2, we require that no vector in B1 and B2 satisfies the sign restrictions associated with the monetary shock. To make these prior restrictions on B operative, we use the following univariate independent normal priors β e,ij N(µ e,ij, V e,ij ), (17) where β e,ij is the element in row i and column j of the matrix B r related to the extreme case r, with r = 1, 2. For each e, i, j we calibrate µ e,ij and V e,ij as follows. We depart from a linear VAR model with the same variables and number of lags as in the STVAR model, and estimate the variance-covariance matrix Σ of the reduced form shocks. Given Σ = BB, it can be shown that Σ constraints the elements of B according to the equation b ij [ σ ii, + σ ii ] (see Piffer, 216). Hence, we take the set ( β i L, β i H ) = ( 1 σ ii, 1 σ ii ) as a realistic set where to allocate most of the mass of the prior for β e,ij. Given this we set µ e,i1 and V e,ij as follows: if β e,i1 is not sign restricted, µ e,i1 = and V e,i1 = 5 σ ii. This implies approximately 95% of the probability mass in the set ( β i L, β i H ); if β e,i1 is restricted to be positive (or negative), µ e,i1 = 2.5 σ ii (or µ e,i1 = 2.5 σ ii ) and calibrate V e,i1 to ensure that 95% of the prior probability mass is in the space (, β i H ) (or ( β i L, )). if β e,ij for j 2, draw from a normal distribution identical to the unrestricted β e,i1 discussed above and then discard the draw if any column vector of B1 or 1

12 Fed funds rate VXO CPI Ind. production B 2 satisfies the sign restrictions imposed on the monetary shock; Figure 1: Prior distribution for B Note: Prior distribution for B 1 and for B 2. The first column reflects the sign restrictions imposed to identify the monetary policy shock. The remaining columns show draws from mean-zero normal distributions, after discarding draws in order to ensure tat the sign pattern in the first column is not repeated. The above prior distribution on B, which is shown in Figure 4, is novel to the literature. It has the advantage of providing a flexible approach to the prior of impulse vectors, while allowing for the imposition of sign restrictions in a straightforward manner. It uses the linear model as an approximation of where in the parameter space the impulse vectors might be, and then ensures that the sign restrictions are imposed in a sufficiently agnostic way to allow the data to update the prior distribution for B 1 and B 2 potentially in different ways. 11

13 3.3 Results To study how the effects of a monetary policy shock depend on the level of uncertainty, we first divide the sample period into four subperiods, depending on the value of the transition variable. In particular, having constructed the transition variable on the VXO, we compute the four quartiles of the transition function and classify each period in the sample according to whether it refers to a period of low uncertainty (first quartile), relatively low uncertainty (second quartile), relatively high uncertainty (third quartile) and high uncertainty (fourth quartile). We then compute nonlinear impulse responses using the approach by Koop et al. (1996), separating among the four different quartiles as the period in which the shock hits the economy. The classification of the sample into quartiles of uncertainty is displayed in Figure 2. The solid line shows the transition variable z t 1, while the shaded areas indicate quartiles of z t 1. Higher levels of uncertainty are indicated with darker shadings. The transition variable captures the rise in uncertainty following some well-known events: the Black Monday in 1987, the First Gulf War in 199, the Asian crisis in 1997, the Russian default in 1998, the 9/11 attacks in 21, the second Gulf War in 23 and the recent financial crisis starting in 28. Figure 3 shows the non-linear impulse response functions to a monetary policy shock. The four different columns represent the quartile of the transition function indicating the period in which the shock hits the economy. The four different rows of the figure indicate the variables of the model. As an illustration, the impulse responses on the first column of the figure report the pointwise median and 68 % confidence band across shocks that hit the economy at randomly selected starting points, as long as they were periods in which the transition variables fell within the first quartile. The shock which is scaled such that the federal funds rate decreases by 1 basis points. A thorough discussion of the construction of the nonlinear impulse responses is available in Section 5.1 of the appendix. Figure 3 provides partial support for the theoretical findings by Vavra (214). In 12

14 Figure 2: Transition variable, moving average on the VXO 4 3 1st quartile 2nd quartile 3rd quartile 4th quartile z t Note: The sample is 1987M1-29M12. The transition variable, z t, is computed as the standardized 3 month moving average of the VXO. Darker shadings indicate a time period corresponding to a higher quartile of the transition variable. particular, a monetary shock that hits the economy in a period in which uncertainty is low or relatively low has significantly stronger effects on economic activity than a shock that hits the economy in a period in which uncertainty is relatively high or high. The order of magnitude of the difference between the response from the first to the fourth quartile is around 1% points, equivalent to around 7% of the maximum increase in industrial production in the period in which monetary shocks have the highest effect. However, the results show that prices respond more to a shock that hits when uncertainty is high rather than low. Put differently, it cannot be the higher flexibility in prices to explain the lower effectiveness of a monetary shock when uncertainty is high, because the response of log of CPI is lower when uncertainty is high, not higher. A similar result is also found in Aastveit et al. (213) and Pellegrino (214). 13

15 Figure 3: Impulse response to an expansionary monetary shock Note: Non-linear IRFs to a monetary policy shock, calibrated to decrease the federal funds rate by 1 basis points. The columns represent the quartiles of the transition variable for the period in which the shock is simulated to hit the economy. Solid think red line: pointwise posterior median. Shaded region: 68% confidence bands. Thin blue lines: 68% bands from a linear model using the same prior distribution as the STVAR model. Figure 4 compares the nonlinear impulse responses documented in Figure?? with the responses estimated using a linear VAR. For comparability, the linear VAR is 14

16 Figure 4: Comparison to the linear model Note: Non-linear IRFs to a monetary policy shock, calibrated to decrease the federal funds rate by 1 basis points. The columns represent the quartiles of the transition variable for the period in which the shock is simulated to hit the economy. Solid think red line: pointwise posterior median. Shaded region: 68% confidence bands. Thin blue lines: 68% bands from a linear model using the same prior distribution as the STVAR model. estimated using Bayesian techniques, after employing as prior beliefs the prior distributions used for the extreme cases 1 and 2 in the nonlinear model, and using the 15

17 same number of lags. Figure 4 shows evidence supporting the presence of nonlinearities in the response to a monetary shock. While the impulse response in the linear model tends to be inside the confidence bands from the nonlinear models, there are significant differences between the linear and the nonlinear impulse responses. We notice also that the response to the federal funds rate differ considerably depending on whether a linear or a nonlinear model is used, with the federal funds rate displaying an endogenous sudden contractionary monetary policy within 3 months after the shock. 16

18 4 Conclusions While VAR models are identified using a wide range of identifications strategies, Smooth Transition VAR models are typically identified using the recursive identification approach. This paper contributes to the literate in two ways. First, we develop a tractable way of identifying Smooth Transition VAR models using sign restrictions. Second, we apply the model to study if the effects of monetary policy shocks depend on the level of uncertainty that prevails when the shock hits the economy. The Bayesian approach followed in this paper makes the implementation of sign restrictions tractable despite the presence of nonlinearities, offering a valid alternative to the recursive identification. We find that higher levels of uncertainty reduce the effectiveness of monetary policy shocks. In particular, we find that a monetary shock that hits the economy when uncertainty is in the top quartiles affects industrial production significantly more than when a comparable shock hits the economy when uncertainty is in the bottom quartiles. This suggests that monetary policy looses its grip on the real economy at times in which monetary policy might be needed the most, namely when the current state of the economy is associated with levels of uncertainty that already exert detrimental effects on the economy. The results support the findings by Weise (1999), Lo and Piger (25) and Tenreyro and Thwaites (216), who study how the effects of monetary policy shocks interact with the business cycle. While the results of our paper provide evidence favouring the theoretical prediction of the model by Vavra (214), it suggests that the mechanism discussed in his work might be overturn by other mechanisms, which are yet to be uncovered. In particular, Vavra (214) argues higher uncertainty leads firms to adjust prices more frequently and by more. This, in turn, increases price flexibilities, which reduces the effectiveness of nominal shocks, in light to the theoretical predictions of the New Keynesian paradigm. This mechanism has received considerable attention in the literature, for example in the works by Bachmann and Sims (212) and Baley and Blanco (215). We find that 17

19 the data suggests a different response of aggregate prices. We find that prices increase by more following a monetary expansion that hits the economy when uncertainty is relatively low rather than relatively high, partly contradicting the mechanism by Vavra (214). The effects documented on the response of prices are partly consistent with the empirical work by Aastveit et al. (213) and by Pellegrino (214). Understanding the interaction between uncertainty, price flexibility and the effectiveness of monetary policy is an important requirement for the understanding of how monetary policy is transmitted to the economy. More research is required on this field to uncover how, and to what extent, price dynamics contribute to reduce the effectiveness of monetary policy in periods of high uncertainty. 18

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21 Canova, F. (27). Methods for applied macroeconomic research, Volume 13. Princeton University Press. Faust, J. (1998). The robustness of identified VAR conclusions about money. In Carnegie-Rochester Conference Series on Public Policy, Volume 49, pp Elsevier. Fry, R. and A. Pagan (211). Sign restrictions in structural vector autoregressions: A critical review. Journal of Economic Literature 49 (4), Gefang, D. and R. Strachan (21). Nonlinear impacts of international business cycles on the uk a bayesian smooth transition var approach. Studies in Nonlinear Dynamics and Econometrics 14 (1), 2. Granger, C. W. and T. Terasvirta (1993). Modelling non-linear economic relationships. OUP Catalogue. Inoue, A. and L. Kilian (213). Inference on impulse response functions in structural VAR models. Journal of Econometrics 177 (1), Kilian, L. (213). Structural vector autoregressions. In N. Hashimzade,, and M. A. Thornton (Eds.), Handbook of Research Methods and Applications in Empirical Macroeconomics. Cheltenham. Kilian, L. and D. P. Murphy (212). Why agnostic sign restrictions are not enough: understanding the dynamics of oil market VAR models. Journal of the European Economic Association 1 (5), Koop, G., M. H. Pesaran, and S. M. Potter (1996). Impulse response analysis in nonlinear multivariate models. Journal of econometrics 74 (1), Lo, M. C. and J. M. Piger (25). Is the response of output to monetary policy asymmetric? evidence from a regime-switching coefficients model. Journal of Money, Credit, and Banking 37 (5),

22 Lütkepohl, H. (25). New introduction to multiple time series analysis. Springer Science & Business Media. Pellegrino, G. (214). Uncertainty and monetary policy in the us: A journey into non-linear territory. University of Verona, mimeo. Pesaran, H. H. and Y. Shin (1998). Generalized impulse response analysis in linear multivariate models. Economics letters 58 (1), Piffer, M. (216). Bayesian model comparison for sign restrictions in SVAR models. Ramey, V. A. (216). Macroeconomic shocks and their propagation. Technical report, National Bureau of Economic Research. Tenreyro, S. and G. Thwaites (216). Pushing on a string: US monetary policy is less powerful in recessions. American Economic Journal: Macroeconomics. Teräsvirta, T., Y. Yang, et al. (214). Specification, estimation and evaluation of vector smooth transition autoregressive models with applications. Research Paper 8. Uhlig, H. (25). What are the effects of monetary policy on output? results from an agnostic identification procedure. Journal of Monetary Economics 52 (2), Vavra, J. (214). Inflation dynamics and time-varying volatility: New evidence and an ss interpretation. The Quarterly Journal of Economics 129 (1), Weise, C. L. (1999). The asymmetric effects of monetary policy: A nonlinear vector autoregression approach. Journal of Money, Credit and Banking,

23 5 Appendix 5.1 Derivation of the likelihood function The model from equation (1) in section 2 can be rewritten as y t = Πw t + u t, (18) with Π = [Π 1, Π 2 ] of dimensions k 2m with m = kp + 1, k the number of variables in the model and p the number of lags in the model, w t = [g(z t 1, γ, c)(y t 1,..., y t p, 1), (1 g(z t 1, γ, c))(y t 1,..., y t p, 1)] of dimension 2m 1 and u t = [g(z t 1, γ, c)b 1 + (1 g(z t 1, γ, c))b 2 ]ɛ t of dimension k 1. Starting from equation (18), rewrite the model in compact form as Y = ΠW + U, (19) with Y = [y 1,..., y t,..., y T ] of dimensions k T, W = [w 1,..., w t,..., w T ] of dimensions 2m T and U = [u 1,..., u t,..., u T ] of dimensions k T. The compact form stacks all T observations of each of the k variables next to each other. It facilitates deriving the vectorized version. We make use of the formula vec(abc) = (C A) vec(b) (see Lütkepohl, 25, mathematical appendix) and obtain y = Zδ + u, (2) with y = vec(y ) of dimension T k 1, Z = (W I k ) of dimension T k 2mk, u = vec(u) of dimension T n 1 and δ = vec(π) of dimension 2mk 1. Define Σ(B, g(z t 1, γ, c)) = B t B t, (21) given B t = [g(z t 1, γ, c)b 1 + (1 g(z t 1, γ, c))b 2 ]. (22) 22

24 Since V (u) = Ω(B, γ, c) with Ω(B, γ, c) = diag(σ(b, g(z, γ, c)),..., Σ(B, g(z t 1, γ, c)),.., Σ(B, g(z T 1, γ, c))), (23) it holds that u N(, Ω(B, γ, c)) and hence p(u) = (2π) T n 2 Ω(B, γ, c) 1 2 e 1 2 u Ω(B,γ,c) 1u. (24) Constructing the likelihood function from this distribution gives p(y δ, B, γ, c) = (2π) T n 2 Ω(B, γ, c) 1 2 e 1 2 (y Zδ) Ω(B,γ,c) 1 (y Zδ), (25) which coincides with equation (4) from subsection Derivation of the posterior distribution Consider the prior distributions p(b, γ, c) and δ N(µ δ, V δ ). distribution equals The joint posterior p(y δ, B, γ, c) p(δ, B, γ, c Y ) =p(δ)p(b, γ, c) p(y ) =(2π) n 2 with p(y ) defined as p(y ) = γ joint posterior distribution as Vδ 1 2 e 1 2 (δ µ δ ) V 1 δ (δ µ δ)... p(b, γ, c) p(y ) 1... (26) (2π) T n 2 Ω(B, γ, c) 1 2 e 1 2 (y Zδ) Ω(B,γ,c) 1 (y Zδ), c δ B p(y δ, B)dcdγdBdδ. We aim to rewrite the p(δ, B, γ, c Y ) = p(δ B, γ, c, Y ) p(b, γ, c Y ), (27) 23

25 and to exploit analytical results for p(δ B, γ, c, Y ). To do so, rewrite first the joint posterior distribution as p(δ, B, γ, c Y ) = k e 1 2 [ (δ µ δ ) V 1 δ (δ µ δ )+(y Zδ) Ω(B,γ,c) 1 (y Zδ) ], (28) with k a term that is constant in δ. Factorize the relevant terms in the exponent of the above expression as (δ µ δ ) V 1 δ (δ µ δ ) + (y Zδ) Ω(B) 1 (y Zδ) = (29) (δ µ δ ) Ṽ 1 δ (δ µ δ ) + y Ω(B) 1 y µ δṽ 1 δ µ δ + µ δ V 1 δ µ δ, (3) with Ṽ δ = [V 1 δ + Z Ω(B, γ, c) 1 Z] 1, (31) µ δ = Ṽδ [V 1 δ µ δ + Z Ω(B, γ, c) 1 y]. (32) Hence, the joint posterior distribution can be equivalently expressed as p(δ, B Y ) =(2π) n 2 Ṽ δ 1 2 e 1 2 (δ µ 1 δ ) Ṽδ (δ µ δ)... p(b, γ, c) Ṽδ 1 2 Ω(B, γ, c) 1 2 e 1 2 {y Ω(B,γ,c) 1 y µ Ṽ 1 δ µ}... (33) V δ 1 2 p(y ) 1 (2π) T n 2 e µ δ V 1 δ µ δ. It follows that δ B, γ, c, Y N( µ δ, Ṽ 1 δ ), (34) while the kernel of p(b, γ, c Y ) satisfies 24

26 log(p(b, γ, c Y )) log(p(b, γ, c)) log( Ṽδ ) 1 2 log( Ω(B, γ, c) ) 1 2 {y Ω(B, γ, c) 1 y µ Ṽ 1 δ µ}. (35) Equation (35) can be used to sample from p(b, γ, c Y ) using a Metropolis-Hastings algorithm, as for example in Baumeister and Hamilton (215). 5.3 Computing impulse responses Impulse responses are used in the literature to study the evolution of the variables of a model in response to a shock that hits the model. While impulse responses can be viewed as the outcome of different types of conceptual experiments, in the present paper we view impulse responses as a tool addressing the following thought experiment: Given some initial point in time, how will the evolution of the variables of the model differ depending on whether, in addition to some other shocks, the economy is also hit by an additional structural shock of interest? We now develop this thought experiment and outline how it is affected by the joint combinations of nonlinearities and set identification. In a stationary linear SVAR model of the type y t = Πx t + Bɛ t, (36) with x t = (1, y t 1,..., y t p ) and with known parameter values, the impulse responses φ i,h of y t at horizons h =,..., H to a shock of size δ to variable i can be computed following six steps: 1. select a starting point x, which in the linear model consists of the unitary vector for the constant terms and the values for p vectors of the lagged dependent 25

27 variables; 2. draw a vector of random shocks at each point h and summarize these pseudo structural shocks in {ɛ O h }H h= ; 3. compute pseudo shocks {ɛ I h }H h= that augment {ɛo h }H h= with a shock of magnitude δ to equation i by setting ɛ I h = ɛo h + e iδ, h =, and ɛ I h = ɛo h, h >, given e i the k 1 selection vector of zeros with value 1 in row i; 4. compute pseudo reduced form shocks {u O h }H h= and {ui h }H h= as ui h = BɛI h and u O h = BɛO h, respectively; 5. generate pseudo data {y O h }H h= and {yi h }H h= using x as the starting point and recursively feeding into the model the pseudo shocks {u O h }H h= and {ui h }H h=, respectively; 6. compute impulse responses as φ h = yh I yo h, h =,..., H, and collect them into the k H + 1 matrix Φ i, or more formally Φ i ( x, {ɛ O h }, δ). As already discussed in the literature (Lütkepohl, 25, section 2.3), in a linear model it holds that Φ i ( x, {ɛ O h }, δ) = Φ iδ. Put it differently, impulse responses do not depend on the past (the initial point x), nor on the present and the future (the pseudo shocks {ɛ O h }H h= ), and they are a scalar multiple of δ. Hence, in a linear SVAR, the effects of a shock do not depend on when the shock hits the economy, nor on which other shocks hit the economy in addition to the shock of interest. Thanks to these properties, impulse responses can be computed with the convenient close form solution provided by the moving average representation of the model. The independence of the impulse responses from the past, present and future, as well as their proportionality to the shock of interest, do not generally hold in a nonlinear setting. For example, in the STVAR model with an endogenous transition variable discussed in this paper and widely used in the literature, the size of the shock given, the starting point and the pseudo shocks affect the endogenous evolution of the system, 26

28 and, through g(z t 1 ), affect how past values map into future values. This means that in a nonlinear model the effect of a shock is not necessarily proportional to the shock given, and it depends, in general, on the point in time in which the shock is given, and on which other shocks hit the economy within the horizon considered. Impulse responses are still computed as outlined above, adjusting for the specific nonlinear model used. The heterogeneity in the responses according to initial points and present and future shocks is then typically summarized using the non-linear impulse response functions proposed by Koop et al. (1996), which average numerically across initial points and present and future shocks. 3 In this paper, the use of a Bayesian approach makes it particularly tractable to compute nonlinear impulse responses. In particular, define t a selection of period of interest, for example periods if a particularly acute recessions, periods in which the transition function g(z t 1 ) was above or below a threshold value, or even a single period of interest. We address the question how will the evolution of the variables of the model differ depending on whether, in addition to some other shocks, the economy is also hit in period τ t by an additional structural shock i of size ɛ?. To do so we follow the following steps: 1. select a random period τ t from a uniform distribution and compute the corresponding starting point x and the corresponding value of the transition variable z; 2. draw pseudo shocks {ɛ O h }H h= and compute ɛi h = ɛo h + e i ɛ, h =, and ɛ I h = ɛo h, h >, given e i the k 1 selection vector of zeros with value 1 in row i; 3. select a random extraction δ i, B i from the posterior distribution p(δ, B Y ); 3 We do not use the term Generalized Impulse Response Function (GIRF) because there is some disagreement in the literature about the dimension along which the generalization is being carried out. While some researchers use this term to express that this concept can be applied to linear as well as non-linear models, others, e.g. Pesaran and Shin (1998), argue that GIRFs are more general than traditional impulse response functions even in the linear case. We will use the term nonlinear impulse responses in order to avoid confusion. 27

29 4. given B i, z and hence B h= from equation (1b), compute the pseudo reduced form shocks u O h= = B h= ɛ O h= and ui h= = B h= ɛ I h= ; 5. generate pseudo data y O h= and yi h= using x as the starting point and feeding into the model the pseudo reduced form shocks u O,H h= and ui h=, respectively; 6. update z h in light of the generated pseudo data yh= O and yi h=, compute B h=1 and the pseudo reduced form shocks u O h=1 = B h=1ɛ O h=1 and ui h=1 = B h=1ɛ I h=1, and generate pseudo data yh=1 O and yi h=1 ; 7. repeat the last step until the pseudo data are generated for all horizons H; 8. compute impulse responses as φ h = yh I yo h, h =,..., H; 9. repeat the entire process drawing a new initial period, a new set of pseudo structural shocks and a new posterior extraction for the parameters. 28