Understanding the Relative Price Puzzle

Transcription

1 Understanding the Relative Price Puzzle Lin Liu University of Rochester April 213 Abstract This paper examines the impact of unpredictable monetary policy movements in an economy with both durables and nondurables. The empirical evidence shows that with a contractionary monetary policy shock, identified by a structural VAR with short-run recursive assumption, relative price - price of durables relative to price of nondurables, gradually declines for a couple of years and remains significantly below the steady state even after six years. This sluggish and persistent relative price movement is called relative price puzzle. It contradicts with the prediction from a standard New- Keynesian model, and challenges the conventional wisdom - money neutrality in the long run. In this paper, we suggest misidentification of monetary policy shocks is the key for resolving the puzzle. We show that the identified monetary policy shocks, different from the true exogenous ones, are contaminated by the persistent productivity shocks from durable goods sector, which are in fact responsible for the relative price puzzle. JEL Classification: E52; E31; E32; C32. Key Words: Monetary Policy Shocks, Relative Price, Durable Goods, New-Keynesian Model, Structural VAR. 1 Introduction Compared with nondurable goods and services, durables often exhibit much more volatile cyclical fluctuations. In this paper, we attempt to understand how an unexpected change in I thank Yongsung Chang, Bin Chen, Aditya Goenka, Ryan Michaels and seminar participants at University of Rochester for helpful comments. I am especially very grateful to my advisor Mark Bils for his guidance and encouragement. The usual disclaim applies. Department of Economics, Harkness Hall, University of Rochester, Rochester, NY 14627, USA, linliu@rochester.edu 1

2 monetary policy affects durable and nondurable goods sectors differently. In particular, we focus on the differential price effects of the two sectors relative price, defined as price of durables relative to price of nondurables. Empirical evidence shows that after a contractionary monetary policy shock, identified by a structural VAR with short-run recursive assumption, relative price declines gradually for a couple of years, and remains significantly below the steady state even after six years. The initial drop of durable prices relative to nondurables confirms that durables are more interest sensitive than nondurables. However, the sluggish decrease of the relative price contradicts with a standard New-Keynesian model, which predicts an immediate drop of relative price after an unexpected monetary policy change. Moreover, it is puzzling that the persistent gap between the responses of durable and nondurable prices can hardly be accounted for by any monetary economic models, which assume money neutrality in the long run. This finding is robust to various specifications. And it is consistent with empirical evidence provided in other literature (Bils et al. 23, Boivin et al. 29 and etc.), though from different perspectives, all of which describe a persistent relative price movement. Therefore, we call this sluggish and persistent relative price response to a monetary policy shock as relative price puzzle. To the best of our knowledge, no successful attempts have been made to resolve the puzzle. 1 Bringing more propagation mechanisms into the model we may still be able to address the sluggish decline. The persistent movement, however, can hardly be reconciled in any economic model assuming long-run money neutrality. Therefore, this casts doubt on the authenticity of the identified monetary shocks. We think the identified shocks are not truly exogenous, and may be contaminated by other fundamental economic innovations, which are indeed responsible for the sluggish and persistent shift in the relative price. To shed more light on the identified shocks, we further examine the empirical responses of relative output output of durables relative to nondurables, as the relative price changes brought about by the identified shocks should induce individuals to alter their real behavior correspondingly. We find that relative output drops initially, reverts back and is significantly above the steady state after six years. Although it is as puzzling as the responses of relative price, the dynamics of relative output are in line with the individuals optimal consumption allocation decisions, given the responses of prices. It suggests the shocks causing the relative price puzzle should also be the reason for the relative output movement. In particular, the essential fall in relative price and rise in relative quantity suggest a very persistent supply shock is likely to be the candidate for contaminating the identified shocks. 1 In Boivin et al. (29), in order to check the robustness of their results, a long-run restriction is imposed in the estimation that the responses of sectoral prices must be equal to the aggregate price response at the horizon of four years or ten years after the monetary shocks. However, no explanation is provided with regard to what actually generates the persistent gaps among the responses of sectoral prices. 2

3 To provide a complete and more convincing argument, we use a multi-sectoral Dynamic New-Keynesian (DNK) model with sticky prices as the data generating process, incorporating five innovations preference shock, labor supply shock, two sectoral specific productivity shocks, and monetary policy shock. In the model, the effects of true monetary shocks are only transitory. Both relative price and relative output drop immediately and soon converge back to the steady state. In contrast, if we apply to the model generated data the identical econometric method used for empirical observations, we are able to replicate the responses of relative price and relative output to the identified shocks, seen in the empirical findings. This suggests that the identified shocks are far from being close to the true exogenous ones. Further examinations show that the identified shocks are mixed with the persistent and positive sectoral specific productivity shocks from durables, which causes the sluggish and persistent movement of relative price. This paper is related to a large number of literature on the monetary SVAR analysis, which is widely used in monetary economics for evaluating the competing models. We add relative price puzzle to the list of the already well-known counter-intuitive results, for example price puzzle. All of those suggest that the VAR might be misspecified. Rudebusch (1998) provides a sharp critique by claiming that the estimated policy reaction function and the estimated structural shocks have little to do with the policy reaction function and the structural shocks perceived by financial market participants. Using a DNK model, Our findings confirm that the identified shocks have little resemblance with the true monetary shocks. Carlstrom et al. (29) provides an analytic explanation for the misidentification. This paper also sheds light on several works about the sticky price DNK model with durable goods (Barsky et al. 27, Erceg et al. 26 and etc.). Our results call for careful interpretation of the empirical evidence derived from SVAR. For example, we show that the drop of relative price after the identified contractionary monetary shocks has no indication on the relative degree of price stickiness of durables and nondurables. The paper is organized as follows: Section 2 presents empirical evidence on relative price puzzle; Section 3 provides an explanation misidentification of monetary policy shocks; Robustness analysis is shown in Section 4; and Section 5 concludes. 2 The Empirical Evidence In this section, we investigate the effects of monetary policy shocks on the economy with durables and nondurables, particularly focusing on the relative price and relative output. 3

4 2.1 Identification and data For aggregate economic variables, SVAR with short-run assumption seems by and large to produce intuitive results and is consistent with the model predictions (Christiano et al. 1999). Thus, it is widely used in monetary economics for identifying monetary policy shocks. First, Let s briefly review the identifying assumption. 2 Throughout the paper, we use federal funds rate (FFR) R t as the monetary policy instrument. There is a k dimensional vector Z t, partitioned into two blocks: R t and k 1 dimensional vector Z 1t [ including mainly ] output, inflation rate, relative price and relative Z 1t ((k 1) 1) output. That is: Z t =. The structural VAR for the variable Z t is given as: R t (1 1) A Z t = A 1 Z t 1 + A 2 Z t A q Z t q + ν t, where q is the number of lagged variables included, and A, A 1,, A q are k k matrix. ν t is the k dimensional fundamental economic innovations with Eν t ν t = Ω, and the kth element is the monetary policy shock. The short-run assumption places the following zero restrictions on the contemporaneous relationship - the coefficient A : A = [ a 11 ((k 1) (k 1)) a 12 (1 (k 1)) a 22 (1 1) ((k 1) 1) The k 1 dimensional vector of zeros reflects the assumption that monetary policy shocks have no contemporaneous effect on the economy. ]. That is, central bank knows about a measure of current economic activities when setting interest rate, while these activities only respond to monetary policy shocks with one period lag. Even though this assumption is not enough to identify all the elements of A, it is sufficient for our purpose. In practice, we assume A is a low-triangular matrix, and estimate the following equation: Z t = B 1 Z t 1 + B 2 Z t B q Z t q + u t, (1) where u t is one-period ahead forecast error, and we have B i = A 1 A i and u t = (A ) 1 ν t. The data we use are from National Income and Product Accounts (NIPA) including detailed accounts about each component of real GDP and its corresponding price deflator, from which we construct time series of output and price for nondurables and durables, respectively. For the nondurables, we define it as the combination of both nondurable goods and services. In 196, nondurable goods expenditure is of 24.7 of GDP and services is of Over the past few decades, we have seen a dramatic rise in services industry. 2 For more details and theoretical proof, see Christiano et al. (1999). 4

5 By 27, personal consumption from services is three times of the one from nondurable goods. Since nondurable goods consumption is the expenditure on tangible commodities with an average life of no more than 3 years and services is the expenditure on commodities that cannot be stored, in our empirical estimation, we combine both together as the time series data for the nondurables. The total consists of a relative constant share of GDP with 54 in 196 and 61 in For the durables, we have two candidates: one is durable goods consumption - tangible commodities that can be stored and have an average life of at least 3 years, and residential investment - consisting of all private residential structures and of residential equipment that is owned by landlords and rented to tenants. Durable goods expenditure is of 8.5 of GDP in 196 and in 27 remains 8.4. Residential investment consists of 5.4 of GDP in 196 and drops slightly to 4 in 27. Since average housing appliances and equipment last more than 15 years with housing structure last even longer - more than 3 years, while the life expectancy of durable goods mostly consisting of motor vehicles can only span 5 to 1 years, we keep them separate. Most empirical estimation in the paper is done with durable consumption goods, and residential investment is used for robustness check. Relative price is defined as price of durables divided by nondurables, and relative quantity is defined as durable output divided by nondurable output. The data are in quarterly frequency. 2.2 Evidence on relative price puzzle The variables included in the estimations are: output, inflation rate, relative price, relative quantity, commodity price, FFR, Non-borrowed Reserve (NBR), Total Reserve (TR) and M1, presented here in the order of the estimation. 4 Note that in addition to the variables we are interested in and mentioned above, we include few other variables. Commodity price is included for solving the price puzzle the counter-intuitive price increase after a monetary tightening. The monetary aggregates NBR, TR and M1, are included to control for the open market operation. A constant and time trend are also included in the estimation. In the benchmark case, the sample period is from 196 to 27, as the FFR reaches zero lower bound since 28. We uses 4 periods or 1 year lag in the estimation. 95 confidence 3 For the method in how to combine quantity and price deflator for nondurable goods and services, please refer to the data appendix. 4 Output, relative price, relative output, commodity price, NBR, TR and M1 are all in log levels. For the concern of stationarity issue and convenience for matching with model generated data later, we use inflation rate rather than price level, because in the model what is determined is the inflation rate, rather than price level. However, using log price level does not change the qualitative results either. 5

6 bands are presented in all the graphs along with the impulse response functions (IRFs). 5 Figure 1 presents age change of the variables after a temporary monetary tightening. The IRFs are normalized so that the initial rise in FFR is 25 basis points. The responses of FFR, output and inflation rate are consistent with the existing literature. That is, monetary shocks have a transitory impact on the real economy lasting for no more than three years, and with monetary policy tightening federal funds rate increases, output contracts and price drops. Nevertheless, anomalies arise when we look at the relative price. With an initial inertial for one period, relative price declines gradually for a couple of years, and remains significantly one age below the steady state after six years. The sluggish decrease of relative price can not be easily reconciled within a standard New-Keynesian model. Under the assumption that durables are generally more price-flexible than nondurables, empirically shown to be relevant (Bils et al. 24), the model with the degree of price stickiness supported by micro evidence, predicts that the relative price should drop immediately and converge back to the steady state within a year. More importantly, the persistent shift of the relative price is particularly troubling for the economists as the significant gap between the responses of durable and nondurable prices in the long-run seems to suggest that monetary shocks have long lasting real effect on the economy. It seriously challenges the very underpinning of new-classical economics - long-run money neutrality. Thus, we call this sluggish and persistent relative price movement as relative price puzzle. [ Figure 1 about here. ] In order to assess the robustness of our benchmark results, we experiment with changes in various specifications. Similarly to the benchmark, for each specification a nine variable SVAR is estimated with short-run restriction, and the results are shown in the first column of Figure 2. 6 Firstly, to see the results are not sensitive to the number of lagged variables used, in the first row 12 period lags are included in the estimation. The results are close to those in the benchmark. Secondly, the persistent movement of relative price is usually thought as an indication that non-stationary time series data is used in the estimation. Thus, instead of using the variables in log levels, in the second row we use first difference of log levels. The results are qualitatively the same as the benchmark, with smaller responses and wider confidence intervals. Thirdly, we analyze the sub-sample stability of the results. We examine the period - from 1983 to 27, usually thought to be a period with relatively stable monetary policy and economic development. The responses shown in the third row 5 The confidence intervals are computed using a bootstrap Monte Carlo procedure following Christiano et al. (1999). 6 For the sake of conciseness, in Figure 2 we omit the IRFs of output, inflation, FFR and etc., which are quite similar to those in the benchmark case anyway. 6

7 are qualitative similar, but quantitatively smaller than those in the full sample period as expected. Lastly, we use residential investment as durables. The fourth row depicts the responses of relative price price deflator for residential investment relative to price of nondurables. The sluggish movement of relative price is similar to the benchmark, and it tends to go back after fours years. But by six years it is still.5 significantly below the steady state. [ Figure 2 about here. ] The sluggish and persistent responses of relative price to a monetary policy shock have been documented in other literature as well, though from different perspectives. Bils et al. (23) exploits the diversities in the degree of price stickiness for 123 categories of goods and services. Similarly to the econometric method we adopt in the paper, they estimate a seven variables SVAR, and use the estimated innovations as measures for monetary policy shocks. Two categories of responses the flexible-price goods and sticky-price goods are considered in the paper. They find that monetary policy shocks have persistent effects on the relative price. Boivin et al. (29) estimates a factor-augmented VAR (FAVAR) using a large set of macroeconomic indicators and disaggregated prices. To identify monetary policy shocks, they assume that FFR may respond to contemporaneous fluctuations in estimated factors, but that none of the latent common components of the economy can respond within current period to unanticipated changes in monetary policy, which is the FAVAR extension of the short-run restriction we use in the paper. They find that following the monetary shocks most of the disaggregated price tend to decline fairly steadily for a couple of years, which results in relatively persistent sectoral inflation movements. Moreover, after four years the price responses do not all converge to the same level, implying important degrees of long-run money nonneutrality. To the best of our knowledge, no successful attempts have been made to resolve the relative price puzzle. For the response of relative price, bringing more propagation mechanisms into the model we may still be able to address the sluggish decline. The persistent effect on the relative price, however, can hardly be reconciled in any monetary model, assuming long-run money neutrality. We further notice that one common measure for monetary policy shocks is shared across all the estimations mentioned above, and this leads us to question the validity of SVAR with short-run assumption and the authenticity of the identified monetary policy shocks. In addition, Bils et al. (23) also points out that one of the explanations could be monetary policy shocks used in the estimation are correlated with idiosyncratic shocks hitting the sectors, though no further explanations provided. Thus, We think the key for resolving the relative price puzzle is that the identified shocks are not truly exogenous, and may be contaminated by other fundamental economic innovations, which are indeed 7

8 responsible for the sluggish and persistent shift in the relative price. 2.3 Further examination on relative output To shed more light on the properties of the identified shocks, we further examine the empirical responses of relative output output of durables relative to nondurables, as the relative price changes brought about by the identified shocks should induce individuals to alter their real behavior correspondingly. We find that in the benchmark case shown in Figure 1, relative output drops, reaching the trough after one year, and then starts converging back to the steady state. Four years after the shock, relative output is significantly above the steady state, and remains so even after six years. The robustness checks with various specifications similar to those for the relative price are provided in the second column of Figure 2. Clearly, along with a sluggish decline of relative price, which is significantly below the steady state in the long-run, relative output drops, converges back around one year after the shocks and is significantly above the steady state in the long-run. Interestingly, Bils et al. (23) also find that monetary policy shocks have persistent effects on the relative quantity, which moves in the direction opposite to the responses of the relative price few years after the shocks. Boivin et al. (29) find that sectors in which prices fall the most tend to be sectors in which quantities fall the least, that is, after few years relative price and relative quantity tend to respond in the opposite direction. Although it is as puzzling as the relative price movement, the dynamics of relative output are in line with the individuals optimal consumption allocation decisions, given the responses of prices. From individual s maximization problem, we know consumption goods are allocated such that marginal rate of substitution between them equals to user cost. In the context with nondurables only, user cost is just the relative price. However, with durables in addition to relative price user cost consists of real rate of durables, which is the sum of nominal interest rate and expected capital loss. To be more specific, we define the change in relative use cost uc t as: ûc t = ˆp t + 1 (1 τ)r + δ + c π D,e [(1 τ) ˆR t ˆπ D,e t ], (2) whereˆdenotes age change, p is relative price, r is real interest rate, π p,e is expected inflation rate of relative price and δ is depreciation rate. Making use of this formula and the estimated empirical IRFs, in Figure 3 we present the response of user cost. 7 We use the realized expected capital gain or loss as the substitute for the expected ones. So the fact that relative price declines sluggishly in the first couple of 7 Taking into account tax and maintenance cost in deriving the user cost does not change the qualitative results in the Figure. 8

9 years and remains flat afterwards suggests a huge expected capital loss for durables initially, which vanishes after few years. The responses of user cost consist of two parts: one is the first item in equation (2) the responses of relative price, and the other is the second item in equation (2) the sum of interest rate rise and expected capital loss. As the result of interest rate rise and expected capital loss, use cost increases in the first few years. Then the drop in relative price dominates the responses of interest rate and expected inflation rate, which causes user cost to go down in the long-run. The movement of user cost explains both the decrease in relative output in the short-run and increase in the long-run, suggesting the responses of relative output is consistent with those of relative price. Therefore, we think the reason behind relative price puzzle should also be the cause for relative output puzzle. In the short-run, as the result of rise in both interest rate and expected capital loss of durables, user cost increases which cause relative output to drop, while in the long run, user cost is dominated by the persistent decline of relative price, which lead the relative output to increase. The short-run dynamics may suggest the identified shocks do consist of an adverse demand shock, likely a contractionary monetary shock. In particular, the essential fall in relative price and rise in relative quantity suggest a very persistent supply shock is likely to be the candidate for contaminating the identified monetary policy shocks. [ Figure 3 about here. ] 3 An Explanation Misidentification of Monetary Policy Shocks To provide a complete and more convincing argument, in this section we use a multi-sectoral DNK with sticky prices as our laboratory or data generating process. In the model, the effects of monetary policy shocks are transitory relative price drops immediately and converges back to the steady state within a year. But if we apply SVAR with short-run identifying assumption to the model generated data, exactly identical to the way we treat the empirical observations, we are able to replicate the sluggish and persistent response of relative price. And the identified shocks are found to be contaminated by persistent sector specific productivity shocks from durable goods sector, which are in fact accountable for the persistent relative price puzzle. Few terminologies are clarified here. The empirical IRFs are those derived by applying SVAR with short-run assumption to the empirical observations, seen in the previous section. The theoretical IRFs are those derived directly inside the model irrespective of the specifications of other shocks, and most often seen in the literature, for example Christiano et al. 9

10 25. The simulated IRFs are those derived by applying to the model generated data the identical econometric method used for the empirical IRFs, known as Sims-Cogley-Nason approach (Kehoe, 26). It requires full specifications of all shock processes in the model. 3.1 The model The DNK model is used here because it is the simplest and most commonly used model in which monetary policy shocks do affect the economy. The model setup is very close to Barsky et al. (27), Bils et al. (212) and etc. In addition to monetary policy shocks, we incorporate four other structural innovations in the model. Here we briefly sketch the model and more details are contained in the appendix. There are two sectors: nondurable goods sector and durable goods sector, which are exactly the same in the firms setup. Within each sector, there are many final goods firms and intermediate goods firms. Final goods firms are perfect competitive with intermediate goods as input, while intermediate goods firms are monopolistic competitive, each of which produces a differentiable intermediate good. Capital and labor are used in producing intermediate goods, and the technology is subject to a productivity shock ξt i (i = C for nondurables or i = D for durables). Both factors are mobile across all the firms and sectors. For simplicity, we assume capital is in fixed supply. Due to price stickiness, as in Calvo (1983), with probability θ i price can not be reoptimized. There are a continuum of households. Each household derives utility from consuming both nondurable C t and durable goods D t, and disutility from working. Households supply labor N t with wage setting following Erceg et al. (2), which is directly analogous to Calvo pricing. That is, with probability θ W, wage remains unchanged. We assume utility function is separable in both consumption and labor supply, and there is full insurance. So despite of difference in individual choices of wage and labor supply, each household has the same consumption level and we retain the representative household setup. The following parametric utility functions are used: u(c t, D t ) = [ ( ρ 1 σ ρ ψc σ 1 v(n t ) = exp(ξt N )φ η η+1 η + 1 N η t + (1 ψ) exp(ξ P t )D t, ρ 1 ρ t ) ρ ] σ 1 σ ρ 1 where ξ P t is the preference shock and ξ N t is the labor supply shock, similar to the wage markup shock in Smets and Wouters (27). In addition, durable goods depreciate at the rate δ. For the durable investment I t, households incur a adjustment cost 1 It Φ( 2 D t 1 δ) 2 D t 1. Central bank adopts the Taylor rule and adjusts nominal interest rate R t responding to 1

11 the deviations of inflation rate π t and total output Y t from the targeted levels: log(r t / R) = b R log(r t 1 / R) + (1 b R )(b π log(π t / π) + b Y log(y t /Ȳ )) + ξr t, where denotes the steady state value, b R is the interest rate inertial coefficient b π and b Y are the responding coefficients for inflation rate and output, respectively. ξt R is the monetary policy shock. All shocks are orthogonal to each other and follow AR(1) processes given as: ξ i t = ρ i ξ i t 1 + ɛ i t (i = C, D, P, N, R), (3) where the innovations ɛ i t are i.i.d and follow normal distribution with zero mean and standard deviation σ i. In equilibrium, both goods market and labor market clear. We solve the model by first detrending all the nominal variables and then linearising around the steady state. To see the results do not depend on some peculiar parameter values, we follow the standard literature. The simulation is done for a quarterly frequency. Discount rate β is chosen so the annual interest rate is 3.5%. We simplify the utility function by setting ρ = 1 and σ = 1, and choose ψ so that in the steady state non-durable goods production consists of 75% of GDP. We set frisch labor supply elasticity η to be one, and φ is chosen to target the labor supply N = 1 3. Depreciation rate δ for durable goods is chosen so the annual depreciation rate is 5%, and the adjustment cost parameter Φ =.455/δ following Bils et al. (212). We normalize the total capital supply to be one, and the capital share is.35. Elasticity of substitution for intermediate goods or labor supply are set so the markup is 1%. For the monetary policy rule, following the convention, the interest rate inertial coefficient b R equals to.95, and the responding coefficients for inflation rate and output are 1.5 and.5, respectively. For the degree of price stickiness, we choose θ C =.5 and θ D =.25, that is, prices of durables are more flexible than those of nondurables. For the wage stickiness we set θ W =.8. The auto-correlation of monetary policy shocks is zero and standard deviation is.12%. These are enough for us to derive the theoretical IRFs of monetary policy shocks, which are independent of other shock processes. Figure 4 depicts the theoretical IRFs of monetary policy shocks. After a temporal contractionary monetary policy shock, interest rate increases by.1 and gradually declines over time. As a result, output and inflation drop initially by.65 and.2, respectively. It takes inflation rate half a year to converge back to the steady state, while output takes around four years due to the assumption of both wage stickiness and adjustment cost. These are also the reasons why it takes a bit longer for relative output to converge back to the steady state than relative price. Because prices of durables are assumed to be more flexible than those of nondurables, with an adverse demand shock durable goods price drop more 11

12 than nondurables, causing relative price to decline. One year after the shock, most firms have already adjusted their prices and relative price converges back to the steady state. For the output, sticky wage and adjustment cost are incorporated in the model so that durable output drops more than nondurables, causing relative output to decrease immediately. Then it converges back gradually. Comparing the theoretical IRFs with the empirical ones, they give the correct predictions about the signs of all the responses. Interest rate goes up, quantities decline and so do prices. Moreover durables are more interest sensitive than nondurables. However, if we look closely to the dynamics of relative price, we notice in the model the effect of monetary shocks on relative price is extremely short lived, lasting for less than one year, which is by no means close to the dynamics seen in the empirical IRFs. This is particularly challenging for New- Keynesian model, and all other monetary models assuming long-run money neutrality in general. Therefore, we think one of possible explanations is that the empirical IRFs are not the responses of the true exogenous monetary shocks, but contaminated ones. [ Figure 4 about here. ] 3.2 Misidentification of monetary policy shocks What causes the monetary policy shocks to be misidentified? Which fundamental innovations are in fact responsible for the relative price puzzle? Can a SVAR with short-run assumption truly recover the effects of a monetary shock in a DNK model? To better answer these questions, we first provide the VAR representation of the DNK model. The state space form of the equilibrium is given as: S t = F 1 S t 1 + F 2 ɛ t Z t = F 3 S t 1 + F 4 ɛ t. S t is an n dimensional vector of possibly unobserved state variables. Z t is a k dimensional vector of variables observed by the economists. We assume that output, inflation rate, relative price, relative output and interest rate are observed, and k = 5. ɛ t is a k dimensional vector of fundamental economic innovations with ɛ t = [ɛ C t ɛ D t ɛ P t ɛ N t ɛ R t ], defined in equation (3). Note we intentionally choose the number of innovations to be equal to the number of observables, because if there are more shocks than observables we can never possibly recover the true monetary shocks. The matrix F i (i = 1, 2, 3 and 4) are dictated by the model specifications and parameter values. As shown in Fernandez-Villaverde et al. (27), since 12

13 the model satisfies the regularity conditions, 8 it admits the following VAR representation: Z t = H 1 Z t 1 + H 2 Z t H j Z t j + + F 4 ɛ t, (4) where H j = F 3 [F 1 F 2 F 1 4 F 3 ] j 1 F 2 F 1 4. That is, the current value Z t is the sum of all the past values and the current period shocks. In practice, however, the reduced form VAR equation (1) is estimated. That is, only finite number of lagged variables are included and in order to identify monetary policy shocks, the matrix F 4 is assumed to be low-triangular. All these could lead to misidentification. To examine whether the misidentified monetary shocks can be one of the reasons for generating relative price puzzle, we run a five variable VAR on the model generated data including output, inflation rate, relative price, relative output and interest rate. To give the DNK model best chance to match the data, the parameter values for the shock processes, other than monetary policy shock, are chosen by minimizing the distance between the simulated IRFs and empirical IRFs for relative price, relative output and interest rate. 9 estimated parameters are presented in Table 1, and the results are shown in the first row of Figure 5. For the moments to be matched, the simulated IRFs pretty much lie within the confidence bands of the empirical counterparts. We deliberately choose not to match with the responses of output and inflation rate. The reason is that, for inflation rate, the empirical responses are dampened by incorporating commodity prices in the estimation, which we do not think is the real cause for the price puzzle. We think the inflation rate does rise more dramatically and significantly as what is shown in the Figure 5, and the reason behind it is also the misidentification of the monetary shocks. The responses of output are used as the check for the performance of the estimation. Surprisingly, the simulated output responses are completely in line with the empirical ones. This boosts our confidence on the estimated parameters. To have a clear comparison, we present the simulated IRFs along with the theoretical IRFs in the second row of Figure 5. The The dynamics of interest rate are similar to each other rise immediately, decline over time and vanish by three years. Nevertheless, for other variables the responses are quite different from each other. For the theoretical ones, all the variables jump immediately, while for the simulated ones, all the variables remain fixed in the initial period due to the identifying assumption that monetary policy shock 8 The regularity conditions are that F 4 is nonsingular and the eigenvalues of F 1 F 2 F 1 4 F 3 are strictly less than one in modulus. They are satisfied with the parameter values used in the paper. 9 Given parameter values, we generate 5 time series data sets with the equal length as the empirical observations. The first 2 periods are discarded. For each dataset, we apply SVAR with short-run assumption and generate IRFs. The simulated IRFs is the average of all IRFs from these datasets. Please refer to Christiano et al. (26) for more details. For the GMM estimation using the IRFs as the moments, please refer to Christiano et al. (25). 13

14 Table 1. The Estimated Parameters for the Shock Processes Shocks Auto-Correlation (ρ i ) Standard Deviation (σ i ) Preference shock ( ɛ P t ).18.5 labor supply shock ( ɛ N t ) Productivity shock in nondurables (ɛ C t ) Productivity shock in durables ( ɛ D t ) has no contemporaneous effect on the economy. Moreover, in the theoretical case, relative price converges back to the steady state within one year. In contrast, we see a sluggish and persistent shift of relative price in simulated case. Similarly, the simulated relative output responses remain significantly deviated from the steady state while the theoretical responses vanish by four years. In addition, compared with a decrease of inflation rate in theoretical case, the simulated inflation rate rises dramatically and declines over time. The theoretical IRFs describe responses of variables to the pure exogenous monetary policy shocks, and the simulated IRFs depict dynamics of variables to the identified monetary policy shocks. The sharp contrast between the two seems to suggest the identified shocks are far from being close to the true exogenous ones, and this misidentification is the reason for the relative price puzzle. Comparing the theoretical VAR representation of the DNK model equation(4) with the practically estimated reduced-form VAR equation(1), we see there are several factors causing misidentification incorrect identifying assumption and truncation bias. The identified monetary shocks are approximately a linear function of all fundamental innovations and all the past lagged variables that are not included in the estimation. To further investigate which fundamental innovations are indeed responsible for the relative price puzzle, we regress the identified shocks, denoted by ɛ R t, on all the exogenous innovations: ɛ R t = β + β 1 ɛ C t + β 2 ɛ D t + β 3 ɛ P t + β 4 ɛ N t + β 5 ɛ R t + γ 1 Z t γ 8 Z t 12 + ϖ t. (5) ( 19) (195) (25.8) (11.6) (12.2) We also control for the lagged variables from the previous two years to previous three years (Z t 5 Z t 12 ), since we have already taken into account one year lag in the SVAR estimation. β i is a scalar and γ i is a k-dimensional row vector. 1 We present the estimators for β, along with the t-statistics. The R 2 of the estimation is.99, and without controlling for the lagged variables, the R 2 is.88, indicating the fundamental innovations alone account for a 1 We run the estimation for each model generated data set, and the statistics reported in the paper are the average of the statistics from all the data sets. 14

15 large part of variations in the identified shocks. Moreover, we find that the identified shocks are mostly correlated with labor supply shock and the correlation is.9. The correlations with productivity shocks from nondurable and durable goods sector are.12 and.5, respectively. The identified shocks are only negatively correlated with the preference shocks and the correlation is.9. Surprisingly, the identified shocks bear little resemblance to the true exogenous monetary innovations with correlation as low as.5. The identified monetary shocks are contaminated by all exogenous shocks, but which one is responsible for the persistent relative price movement? Making use of the estimated parameters, in the first row of Figure 6 we provide the contribution of each fundamental innovation to the impulse responses of the identified shocks, along with the summation of all of them, that is the contribution from all exogenous shocks. If the identified shocks only consist of the exogenous shocks, the IRFs from all exogenous shocks (the solid line) should start from the origin, except for interest rate, due to the identifying assumption. Here, because the lagged variables affect the identified shocks, and hence the estimated IRFs, they do not start exactly from the origin, but very close to the simulated IRFs in Figure 5. This is another piece of evidence suggesting that the contamination by the exogenous shocks is the reason for the misidentification, rather than the lagged variables caused by truncation bias. All the IRFs are normalized so that the initial rise of interest rate caused by all exogenous shocks (the solid line) is 25 basis points. In the first row of Figure 6, for all variables the responses caused by preference shocks is too small to be discerned in the graphs. For relative price, the IRFs caused by labor supply shocks, nondurable productivity shocks and monetary policy shocks all vanish within three years. The persistent shift of relative price is entirely caused by productivity shock from durable goods sector. The sluggish movement of relative price is because of the fact that the responses caused by durable productivity shocks converge to the steady state much slowly, compared with the responses caused by other innovations. And all these are due to the fact that durable productivity shocks are highly persistent. Similarly, for the relative output, during the first three years the negative responses from all other innovations outweigh the positive response from the durable productivity shocks, which cause it to decline. Over time, when the negative responses converge back to the steady state, the positive responses resulting from the persistent durable productivity shocks dominate and relative output overshoots the steady state. Interestingly, they do not have large impact on the aggregate variables output and inflation. For the inflation rate, it is clear that labor supply shocks are solely responsible for the price puzzle. So we agree that price puzzle arises due to the misidentification of monetary policy shocks. But what causes the misidentification? Sims (1992) suggests missing variables in a VAR model is responsible for the misidentification, and it is argued that commodity 15

16 price resolves the price puzzle because it contains information that helps the Fed forecast inflation. However, as shown by Hanson (24) there is little correlation between an ability to forecast inflation and an ability to resolve the price puzzle. In contrast, here we show that the incorrect identifying strategy causes the misidentification. In addition, this finding is also consistent with the one in Smets and Wouters (27) that inflation development are mostly driven by wage mark-up shocks, which correspond to labor supply shocks here. 11 Therefore, we show that the identified contractionary monetary policy shocks are contaminated by the positive and persistent productivity shocks from durable goods sector, which are responsible for the relative price puzzle. We also find that labor supply shocks are the reason behind the price puzzle. The misidentification arises due to the incorrect identifying strategy. The short-run recursive assumption used in a SVAR assumes that the matrix F 4 in equation (4) is low-triangular, while in the DNK model it is dictated by model specifications and parameter values. From the theoretical IRFs in Figure 4 that all economic variables adjust immediately to the monetary shocks indicates that the matrix F 4 is certainly not low-triangular. That is, the short-run identifying assumption assumes monetary policy shocks have no contemporaneous effect on the economy, while in the model no single shock has this property. Thus, with the incorrect identifying assumption, SVAR picks up the mix of all fundamental economic innovations, the weighted combination of which satisfies the short-run assumption. Hence, the estimated IRFs of the identify shocks are the weighted combination of the impulse responses from all these shocks. 12 So far, we use the DNK model as the data generating process and with various realized exogenous shocks, we have different data sets with equal length to the empirical observations. Then we look at the average of impulse responses from all these data sets the simulated IRFs. What about the confidence intervals of the simulated IRFs? As we know from Chari et al. (24), due to small sample bias the confidence bands of the simulated IRFs are too wide to provide useful inference. However, with much larger simulated data sets, it reduces the bias and provides a much clearer inference, shown in the second row of Figure The confidence bands reflect the uncertainty across the draws of the exogenous shocks. To avoid singularity problem in the estimation, a small measurement error is added in the simulated data as well. Judging from the responses of output, inflation and interest rate, the monetary effects are transitory and last for three years. But the significant responses of relative price and relative output suggest monetary nonneutrality in the long-run. All these are very much in line with the empirical evidence we provide in section 2, and we know by now it is the 11 In fact, it is stated in Smets and Wouters (27) that inflation developments are mostly driven by the price mark-up shocks in the short-run and the wage mark-up shocks in the long-run. However, we do not have price mark-up shocks in the paper. 12 See Carlstrom et al. (29) for an analytic explanation of misidentification of monetary shocks in a textbook NK model. 13 We simulate 5 data sets with length of 25 periods each. 16

17 result of misidentification of monetary policy shocks. 4 Robustness analysis In this section, we provide several robustness analysis. For the sake of conciseness, only the dynamics of relative price and relative output are reported. 1. Estimation with various specifications. Several factors cause the true exogenous monetary policy shocks to be misidentified. One of these is truncation bias, emphasized in Chari et al. (28). In the previous section, we argue that it is the incorrect identifying strategy rather than the truncation bias, from the large R 2 of the estimation in equation (5) and the derived IRFs in Figure 6. To further strength the argument, the robustness analysis is provided to show the relative price puzzle remains even when more lagged variables are included in the estimation. So we estimate a SVAR with 12 periods lags, and the results are shown in Figure 7 along with the baseline. The dynamics are qualitatively similar with a little dampened responses. But, relative price and relative output are still.3 and.6 age deviated from the steady state six years after the shocks. The empirical observations contain more or less some degree of measurement errors. To check the results are robust to these, for each simulated data set, we add a measurement error for each variable, except interest rate which is unlikely to be measured with error. We assume the measurement error follows a normal distribution with mean zero and standard error equal to 5% of standard deviation of the corresponding simulated data. The correlation of the time series data with and without measurement error is.99. The results are shown in Figure 7, and they are very close to the baseline case. In particular, relative price drops immediately after the shock. However, if we run the same estimation of equation (5) with the simulated data containing measurement errors, the R 2 reduces from.99 to.56, suggesting the identified shocks now consist a large part of measure errors. [ Figure 7 about here. ] 2. Comovement of durables and nondurables. The paper focus on the responses of relative output. What about the dynamics of the output for durables and nondurables separately? We know that in the model, when the economy is hit by a positive productivity shock from durable goods sector, nondurable output decreases and durable output increases. This may potentially compromise the results in the paper. The reason is that the identified shocks are contaminated by these productivity shocks, which suggest we should observe that durables and nondurables also move in the opposite direction in the data. However, the empirical evidence clearly shows after a monetary 17

18 shock there is a comovement between durables and nondurables. The empirical IRFs are shown in the first row of Figure Both durable and nondurable drop after the identified monetary shocks, and revert back to the steady state at around one and half years. To see our results are in line with the empirical evidence, we derive the impulse responses of durables and nondurables by using the estimators from equation (5) and the theoretical IRFs. The second row of Figure 8 depicts the contribution of each fundamental innovation to the responses of durables and nondurables. The solid line clearly mimics the comovement in the empirical findings, with durables responding more than nondurables. In particular, similar to the empirical evidence in the first row, durables overshoot the steady state after four years as the result of the persistent productivity shocks, while at the same time nondurables converge back to the steady state. This confirms that it is the positive productivity shocks from the durables, which cause the persistent shift in the relative price and relative output, rather than possibly some negative productivity shocks from the nondurables. Because we could generate the same relative movement through a very persistent but negative productivity shock from nondurable goods sector. However, this would imply that nondurable output stays significantly below the steady state in the long run, while durable output converges back to the steady state, which is not true. Thus, the further examinations about durable and nondurable output confirm the results in the paper. [ Figure 8 about here. ] 3. Relative degree of price stickiness for durables and nondurables. In the previous section, following the literature, we assume prices of durables are more flexible than those of nondurables, which is not without controversy. Here we show how our results change as we vary the relative degree of price stickiness. The standard DNK model predicts a close link between the relative degree of price stickiness and the dynamics of relative price after a monetary policy shock (Barsky et al., 27). In the first row of Figure 9, we present the theoretical IRFs from one unit increase of a monetary policy shock under various assumptions of relative degree of price stickiness. We can see that if durable goods are more price-flexible than nondurables, relative price goes down after a monetary tightening; if durables are more price sticky than nondurables, relative price goes up; and if durables are as price sticky (or flexible) as nondurables, it remains unchanged. For all these scenarios, relative output always drops. For the comparison, in the second row of Figure 9, we also present the theoretical IRFs from one unit increase of a productivity shock from durable sector. Regardless of relative degree of price stickiness, relative price decreases and relative output increases. 14 We estimate a 11 variable VAR with outputs and prices for durables and nondurables, instead of relative price and relative output. 18