NON-LINEARITIES, STATE-DEPENDENT PRICES AND THE TRANSMISSION MECHANISM OF MONETARY POLICY

A sticky price theory of the transmission mechanism of monetary policy shocks based on state-dependent pricing yields two testable implications that do not hold in time-dependent models. First, large monetary policy shocks should yield proportionally larger initial responses of the price level. Second, in a high trend inﬂation regime, the response of the price level to monetary policy shocks should be larger and real effects smaller. Our analysis provides evidence supporting these non-linear effects in the response of the price level in aggregate US data, indicating state-dependent pricing as an important feature of the transmission mechanism of monetary policy.

The New Keynesian (NK) paradigm is one of the main frameworks for the analysis of business cycle fluctuations and the effects of monetary and fiscal policy, both among academic researchers and policy institutions. While there can be alternative mechanisms for the transmission of monetary policy shocks, price stickiness remains the major reason why monetary policy has effects on real variables in virtually all NK models. This pivotal assumption has been substantiated by numerous empirical studies that show that individual prices do indeed change infrequently (e.g., Bils and Klenow, 2004;Klenow and Kryvtsov, 2008;Nakamura and Steinsson, 2008;Nakamura, 2008;Nakamura and Zerom, 2010;Eichenbaum et al., 2011). Moreover, this literature has demonstrated that micro-founded state-dependent models are able to replicate the empirical distribution of price changes. Unlike time-dependent pricing models, such as Rotemberg (1982) and Calvo (1983), micro-founded state-dependent models of nominal rigidities assume that the individual firm can change its price, subject to an adjustment cost. Hence, the frequency of price changes the number of firms that decide to change their prices following a monetary policy shock and the size of these changes are endogenous (the so-called 'selection effect'; see Golosov and Lucas, 2007). In particular, these models predict that individual prices (i) are more flexible in response to large shocks (e.g., Karadi and Reiff, 2019) and (ii) change more frequently when inflation is higher (e.g., Álvarez et al., 2019).
2 the economic journal Some papers find evidence of state-dependent pricing behaviour by investigating the response of prices at the micro-economic level to a particular event. Hobijn et al. (2006) documented a dramatic increase in restaurant prices in the euro area after the introduction of the euro and explained it through the lens of a menu cost model. Karadi and Reiff (2019) documented that micro-level price responses to three large value-added tax (VAT) changes, which occurred in Hungary in 2004 and 2006, were flexible and asymmetrical with respect to positive versus negative tax changes. 1 Bonadio et al. (2019) exploited the Swiss National Bank's decision to discontinue the minimum exchange rate policy of one euro against 1.2 Swiss francs on 15 January 2015, to analyse the pass-through of the Swiss exchange rate shock into import product prices from the euro area. The empirical literature on exchange rate pass-through is very large. It generally concerns measuring the extent of the exchange rate pass-through, and, while somewhat related, it is not concerned explicitly with state-dependent pricing models. Two notable exceptions areÁlvarez et al. (2017) andÁlvarez and Neumeyer (2020) that compare the observed price dynamics at the micro level after large devaluations to the predictions of an adjustment cost model of price setting. Álvarez et al. (2017) used monthly data on consumer price index (CPI) inflation and the nominal exchange rate from a large number of countries in a panel data analysis.Álvarez and Neumeyer (2020) analysed several episodes involving large changes in the nominal price of inputs in Argentina over 2012-2018 by using micro-level price data for the city of Buenos Aires. Álvarez et al. (2017) is the closest paper to ours because they used aggregate data and they explicitly linked the empirical analysis to the prediction of the state-dependent pricing model that the inflation response should depend on the size of the shock.
Yet, despite the large literature on sticky prices, to the best of our knowledge, there is surprisingly little direct evidence that aggregate prices behave as state-dependent models suggest and thus that (i) and (ii) affect the transmission mechanism of monetary policy in aggregate US data. 2 We aim to fill this gap. If staggered prices are of such paramount importance to the transmission mechanism of monetary policy, and if there is a significant fraction of state-dependent prices, then the two very general predictions (i) and (ii) should also emerge in aggregate data. First, large absolute value monetary policy shocks should lead more firms to adjust their prices, and hence yield a proportionally larger response of inflation, whereas the real effects should be subdued. Second, the frequency of price changes should be an increasing function of underlying levels of inflation, that is, prices should be more flexible in a high trend inflation regime than otherwise. Hence, the higher trend inflation, the larger the response of inflation and the smaller should be the real effects of monetary policy shocks.
These theoretical results are quite intuitive and general since they derive from a broad variety of state-dependent sticky price models in the literature. In particular, Álvarez et al. (2017) showed that the impulse response after a monetary shock is size independent in time-dependent models, whilst it is not in state-dependent models. In this sense, our contribution is really a minimal, firstpass test for state-dependent price theories: if state dependency is pivotal, aggregate data should exhibit these two features. We take these two theoretical predictions to US data between 1969 and 2007, applying smooth local projections (Jordà, 2005;Barnichon andBrownlees, 2019) and1Á lvarez et al. (2006) and Gagnon et al. (2012) are earlier papers that find significant increases in the frequency of price changes in the months with VAT tax changes, consistent with state-dependent pricing models.
2 As said, the closest reference isÁlvarez et al. (2017), who however investigated only prediction (ii) above, used a panel of countries and looked at exchange rate pass-through, interpreting exchange rate movements as cost shocks. Veirman (2009)  non-linearities and state-dependent prices 3 the smooth transition function methodology of Granger and Teräsvirta (1993) together with US monetary policy shocks identified with the narrative method of Romer and Romer (2004). The empirical methodology of this paper is most closely related to the work of Tenreyro and Thwaites (2016), focusing on the differential effects of monetary policy in recessions and expansions. As such, our paper also contributes to the literature about the state-dependent effects of monetary shocks.
Our analysis provides new and statistically significant evidence in favour of state-dependent pricing models in aggregate US data. First, large absolute value shocks have disproportionately larger effects on prices on impact, but are less persistent and have weaker real effects, matching the first theoretical prediction. Second, the impulse response functions (IRFs) in the high and low trend inflation regimes are significantly different for prices and inflation and also in line with the second theoretical prediction of higher price flexibility in the high trend inflation regime. However, for the second prediction, we do not find statistically significant evidence of muted real effects. Importantly, the non-linearity of the impulse response functions is not due to a different feedback of monetary policy to inflation in response to small versus large shocks or in periods of high versus low trend inflation. In the Online Appendix we conduct a comprehensive sensitivity analysis to establish the robustness of these results, regarding different empirical specifications, sub-samples, controls and measures of monetary policy surprises.
Our results are the first (to the best of our knowledge) that point towards a significant presence of state-dependent pricing in the US economy from an aggregate perspective. Our macro evidence is therefore a useful complement to existing micro evidence.

Theory and Testable Implications
Contrary to early standard time-dependent pricing models (e.g., Fischer, 1977;Taylor, 1979;1980;Calvo, 1983), micro-founded state-dependent pricing models feature endogenous price adjustments triggered by changes in the economic environment. The firms that decide to change their prices by paying the adjustment costs after a monetary shock are those further off from their optimal price ('selection effect'). Hence, since these are not random firms and the sizes of the price changes are relatively large, early studies (Caplin and Spulber, 1987;Golosov and Lucas, 2007) find that the aggregate price level can mimic a flexible price environment. Nonetheless, later studies investigate the robustness of this result to various extensions and show that statedependent models yield a large degree of aggregate price stickiness and are important for the transmission mechanism of monetary policy. 3 First testable implication: the impulse response functions of inflation and output to a monetary policy shock should depend on the size of the shock. If there is a non-negligible fraction of state-dependent price setters, the impulse response should be a non-linear function of the size of the shock. The larger the shock, the larger the number of firms that decide to pay the adjustment costs and change their prices immediately, such that the reaction of the aggregate price level at short horizon is increasing in the size of the monetary policy shock (seeÁlvarez and Lippi, 2014). Moreover, the effect on the price level should be less persistent, because the larger the 4 the economic journal shock, the higher the number of firms that adjust immediately and hence the lower the number of firms that eventually would change their price as the shock tapers off. The real effects of monetary policy shocks, instead, are hump shaped with respect to the size of the shock, because of two counteracting effects. First, larger shocks, ceteris paribus, give rise to stronger real effects, just like in a time-dependent model. Second, larger shocks also increase the number of adjusting firms, strengthening the reaction of the aggregate price level and thus reducing the real effects. For a small shock, the first effect prevails, so that both the impact and the cumulative effect on output is increasing in the size of the shock. For large shocks, the opposite occurs. It follows that sufficiently big shocks should have lower real effects than smaller shocks. 4 Second testable implication: the impulse response functions of inflation and output to a monetary policy shock should depend on the average level of inflation. As outlined in Dotsey et al. (1999) or Costain and Nakov (2011), average inflation affects the frequency of price adjustments in state-dependent models, because it erodes a firm's relative price so that firms adjust prices more frequently. Indeed, the empirical analysis inÁlvarez et al. (2019) provides solid evidence of how the frequency of price changes varies with inflation. 5 This evidence implies different impulse responses to a monetary policy shock in high trend inflation regimes compared to low trend inflation regimes. In particular, we should observe a quicker and less persistent reaction of prices in high inflation regimes. Furthermore, Álvarez et al. (2016) showed that in a large class of sticky-price models, the total cumulative output effect of a small unexpected monetary shock is inversely related to the average number of price changes per year. 6 This theoretical prediction provides the second testable implication.

Empirical Methodology
In this section we describe the data and the empirical methodology used to estimate smooth impulse responses and conduct inference. We test the two predictions by analysing the presence of non-linearities in the impulse response functions to a monetary policy shock for large and small shocks, and during high and low trend inflation. Our empirical methodology follows a growing body of literature employing local projections (Jordà, 2005) to account for the response to non-linear terms and state dependency in empirical impulse responses (e.g., Auerbach and Gorodnichenko, 2012a,b;Caggiano et al., 2014;Furceri et al., 2016;Tenreyro and Thwaites, 2016;Ramey and Zubairy, 2018).

Data
The monthly sample for our response variables runs from 1969m1 up to 2007m12, excluding the most recent financial crisis, where monetary policy has been very different and the zero-lower bound on nominal interest rates has been binding. We analyse three response variables: output, 4Á lvarez and Lippi (2014) showed that the monetary shock that maximises the cumulated effect on output (i.e., the area under the impulse response function) is about one-half of the SD of price changes.  showed that a similar effect occurs in a model with state-dependent prices and wages. 5 Figures 5 and 6 therein show that the frequency of price changes do not react much for levels of annual inflation up to 5%, then it starts accelerating and finally it increases linearly for values of annual inflation above 14% with an elasticity of about two-thirds. This is in line with Sheshinski and Weiss (1977)'s adjustment cost model with no idiosyncratic shocks. 6 The same holds in a model in which nominal rigidities in both wages and prices are state dependent (see Figure 11 of , and in a model that allows for temporary price changes, because firms can set a price plan, rather than a fixed price as in the standard adjustment cost model (see Figure 7 ofÁlvarez and Lippi, 2020). C 2021 Royal Economic Society.
Downloaded from https://academic.oup.com/ej/advance-article/doi/10.1093/ej/ueab049/6302374 by guest on 28 August 2021 non-linearities and state-dependent prices 5 inflation and the nominal interest rate. The series for output is the industrial production index, for inflation, we use personal consumption expenditure (PCE) inflation and, for the nominal interest rate, we use the effective federal funds rate, all from the Federal Reserve Bank of St. Louis Database. 7 The main shock variable used in this analysis is based on the narrative analysis of Romer and Romer (2004). They identified monetary policy surprises by using a narrative approach to infer the intended federal funds rate at every Federal Open Market Committee meeting from 1969 onwards. By regressing changes of this intended rate on Greenbook forecasts they derived a measure of monetary policy surprises that is arguably exogenous to the Fed's information set about the future state of the economy. We utilise this methodology and the extended shock sample until December 2003 by Tenreyro and Thwaites (2016).

Smooth Local Projections
We estimate local projection coefficients using the recently developed methodology by Barnichon and Brownlees (2019) to improve accuracy and inference over the standard least squares approach. 9 With this technique, the impulse response coefficients, i.e., all shock-dependent coefficients, are modelled as linear combinations of B-spline basis functions. One can then estimate the coefficients of these linear combinations using generalised ridge estimation, with a penalty parameter that selects the degree of shrinkage. When the shrinkage parameter is close to zero, the estimation yields the standard least squares estimates. Conversely, if the parameter is high, the impulse response is converging to a smooth limit polynomial distributed lag model. We follow Barnichon and Brownlees (2019) and select the shrinkage parameter using k-fold cross validation (Racine, 1997).
We also follow Barnichon and Brownlees (2019) on conducting inference. In particular, in order to take into account potential autocorrelation and heteroskedasticity, we estimate the variance of the coefficients using a modified Newey and West (1987) estimator, corrected for the penalty parameter. We use the resulting variance matrix to construct confidence intervals and t-statistics in the ordinary way. We provide further details on this and the estimation technique as a whole in the Online Appendix. 6 the economic journal

Results
In this section we present the main results regarding the two testable implications above.

Non-linear local projections
In order to test the size dependence of impulse responses, we consider the non-linear local projection which is estimated for h = 0, 1, . . . , H . We set H = 48, which corresponds to an impulse response horizon of four years. By y t+h we denote the variable of interest, in our case either the industrial production index, PCE inflation or the federal funds rate. The e t are the narrative Romer and Romer (2004) shocks. By w t,k we denote the kth control variable and by v t+h the estimation error, possibly heteroskedastic and serially correlated. The set of control variables includes up to two months of lags of industrial production, PCE inflation and the federal funds rate. Moreover, we follow Ramey (2016) and include contemporaneous values of the industrial production index and PCE inflation. This is equivalent to assuming recursiveness between the three different variables of interest since inflation and industrial production can contemporaneously affect the federal funds rate but not vice versa. 10 While the coefficient β h captures the linear component, the coefficient ζ h on the absolute value interaction term accounts for non-linearities in the impulse response function due to the size of the shock. The interaction term, (e t · |e t |), magnifies the size of the shock, but it keeps the same sign of the shock. Hence, contrary to a simple quadratic term, it isolates the pure effect of a change in the size of the shock. If ζ h has the same sign of β h then the non-linear interaction term amplifies the linear effects of the impulse response. On the contrary, if ζ h has the opposite sign to β h , it counteracts the linear impulse response, possibly even tilting the overall effect from one sign to another for large enough shocks. Whenever ζ h = 0, the impulse response function is linear with respect to the shock size.
The main coefficient of interest to test our first theoretical prediction is therefore ζ h . Let y t signify prices, and assume a large monetary contraction, i.e., a positive value of e t . Large monetary policy shocks should induce a more price-flexible impulse response function of the price level and inflation. If firms exhibit state-dependent pricing, we should see a negative ζ h at small horizons as we expect β h to be close to zero or negative. This would mean that firms decrease prices quicker and so prices decline disproportionately at small horizons. Furthermore, we would then expect to see a positive ζ h at larger horizons, weakening the price response, as more firms have already changed prices earlier. Consequently a combination of a negative ζ h at small horizons, as more firms change prices right away, and a positive ζ h at larger horizons, as persistence is lower due to earlier price changes, would speak in favour of state-dependent pricing as a valid aggregate propagation mechanism of monetary policy shocks.

Coefficient estimates
The resulting coefficients from estimating local projection (1) for PCE inflation, industrial production and the federal funds rate are reported in the panels of Figure 1. The solid black line plots the coefficients of the linear term,β h , and the dash-dot green line plots those of the non-linear absolute value interaction term,ζ h , together with their 90% confidence interval bands.
The linear terms in the projection deliver a familiar picture in the top panel. After a positive (contractionary) monetary policy shock, the linear coefficients yield an initially muted response of inflation followed by significant decline thereafter. There is an initial positive response that is however not significant on impact and marginally so for just a few months in the initial year. Hence, the response displays a slight price puzzle, if we were to consider only the linear effect.
The non-linear effects implied by the absolute value interaction coefficients are supporting the theoretical predictions. Initially, theζ h coefficients are negative, counteracting the price puzzle as the shock size increases and indicating a quicker negative response of inflation on impact. Moreover, after about two years, the coefficients on the non-linear term turn positive (and increase), meaning that inflation responds less to a large shock at long horizons. Hence, the evidence suggests that inflation has a stronger reaction at short horizons and a weaker one at long horizons. This is consistent with theoretical models of state-dependent pricing as, for large shocks, more firms adjust immediately and hence less adjust later on. Most importantly, the confidence interval shows that the coefficients are statistically significant for most of the horizons.
The other two panels report the results for the linear coefficient in the industrial production and federal funds rate local projection. These results are in line with the theoretical implications from the literature. The linear output coefficient starts to fall (after a small, positive, but not significant, initial response), reaching its trough two and a half years after the shock, and then recovers. The linear federal funds rate coefficient exhibits a hump-shaped response and remains positive for more than two years after the shock before turning negative.
Furthermore, the green dash-dot line in the middle panel shows that the output response supports the predictions of state-dependent pricing models. The coefficients on the non-linear term in (1) are negative at the beginning, counteracting the small positive output response, and then turn positive at longer horizons. As such, theζ h estimate counteracts the linear response for both short and long horizons, flattening the overall response of output. While the top panel shows that large monetary policy shocks predict a higher degree of price flexibility, the middle panel shows that larger monetary policy shocks have weaker real effects. Again, the confidence band indicates that the non-linear interaction term is statistically significant.
It is crucial to note that these results are not due to a stronger response of monetary policy, because the coefficients of the federal funds rate on the non-linear term in the bottom panel are negative for most of the short to medium horizons, suggesting a proportionally weaker response of monetary policy to a larger shock.
In sum, prices exhibit a non-linear, size-dependent impulse response function, reacting strongly at short and weakly at long horizons for large shocks. Coherently, the output response seems to be smaller, and monetary policy feedback does not seem to drive the above results. Hence, we interpret these findings as evidence in favour of state-dependent prices as an important propagation mechanism of monetary policy, as larger shocks induce more firms to change prices early, thus reducing the real effects of a monetary shock. non-linearities and state-dependent prices 9 3.1.3. Impulse response functions In order to further assess and clarify the importance of the non-linear effect, Figure 2 compares IRFs to different shock sizes of all three headline variables over a four year horizon. It depicts the impulse responses for a 25 (dashed blue line), a 100 (solid black line) and a 200 (dashdot green line) basis point (bp) shock, where each impulse is standardised by dividing it by the respective shock size. The standardised IRFs clearly visualise that inflation responds more strongly to a larger shock at short horizons, but then the reaction is less persistent, so that the response is weaker at long horizons, as theory would predict. First, the larger the shock, the quicker inflation decreases. Second, for a large enough shock, the initial price puzzle on impact tends to disappear. 11 The response of output to small and large shocks also supports the theoretical prediction. The standardised IRFs show that the trough in output is smaller relative to the size of the shock, consistent with the behaviour of the responses of inflation. Recall that in state-dependent models there are two opposite effects on output. First, the larger the shock, ceteris paribus, the larger the response of output. This standard effect is the only one also present in time-dependent pricing models. Second, the larger the shock, the greater the number of firms that adjust the price; hence, the larger the response of inflation and the smaller the response of output. This second effect is absent in time-dependent models. Therefore, the output response to a large shock is proportionally flatter in state-dependent models, because of this second effect that counteracts the first one. By showing the response relative to the size of the shock, the standardised IRFs isolate the second effect, thus revealing whether there is a significant effect coming from state-dependent pricing. Finally, the standardised IRFs also highlight the effect of state-dependent pricing both on the scale, i.e., decreasing real impact as the shock gets larger, and on the timing, i.e., arriving sooner with larger shocks, of the response of output and inflation to the policy shocks.
To strengthen our point, Table 1 shows the cumulative effect of a monetary policy shock on inflation (i.e., the PCE deflator) and on output for small and large shocks, standardised by the size of the shock, and their significance levels, for different horizons: 1, 12, 24, 36 and 48 months. 12 Coherently with the theoretical predictions, prices move significantly downwards on impact for large shocks, while they do not for small shocks. Because of this lagged and inertial behaviour in the case of small shocks, the cumulative response of the price level after four years is almost doubled, relative to the size of the shock, compared with the response to a large shock. The cumulative response of output reflects that of the price level. The initial response of output is sharper for large shocks, but the cumulative drop in output after four years is about 80% proportionally larger for small shocks.
Finally, the last row in Figure 2 again illustrates that the empirical results are not due to the different behaviour of monetary policy after a large shock. The standardised IRFs of the Fed funds rate is milder for larger shocks, relative to shock size, and hence, if anything, it would play against our results. 13 11 Figure H5, discussed in Online Appendix C.3, depicts the unscaled (i.e., non-standardised) impulse responses for a 25 basis point shock and a 200 basis point shock and their 90% confidence intervals calculated with the delta method, respectively (see the Online Appendix for details). The response of inflation to a 200 basis point shock is firstly not significantly different from zero and then significantly negative, while it is positive for some months for a 25 basis point shock (even if only marginally significant). Consequently, a sufficiently large shock counteracts the small linear coefficient and switches the sign of the overall impulse response of inflation, removing a potential price puzzle. 12 We simply cumulate the point estimates from the standard least squares estimation and statistics using the delta method. 13 Moreover, the response of the federal funds rate is statistically different (in the first year) between the 25 versus 200 basis points shock (see Figure H5 in the Online Appendix).  Notes: Newey-West SEs in parentheses. * * * , * * , * indicate significance at the 1%, 5% and 10% levels, respectively. Figure 2 and Table 1 reinforce our previous empirical results on the significant size-dependent effects of monetary policy shocks. Large shocks induce firms to change prices early on and thus reduce the real effects of such a monetary shock, in accordance with our first theoretical prediction.

Smooth transition local projections
We use smooth transition local projections to test whether the impulse responses after a monetary policy shock are different in high and low inflation regimes. Auerbach and Gorodnichenko (2012b) and Tenreyro and Thwaites (2016) popularised this method, and we follow their approach to a large extent. The impulse response of the variable of interest y t at horizon h in state s = HI, LO 14 to a unitary structural shock e t is the estimated coefficient β s h in for h = 0, 1, . . . , H . Again, our set of controls {w t,k } K k=1 includes the contemporaneous values of industrial production and PCE inflation and up to two months of lags of industrial production, PCE inflation and the effective federal funds rate. Here F(z t ) is a smooth transition function that indicates the state of the economy (Granger and Teräsvirta, 1993). We use a logistic function with the form If state-dependent prices are an important aggregate propagation mechanism, we would expect β HI h to be statistically significantly more negative than β LO h for the response of the price level and inflation, especially at short horizons. Prices should be more flexible and so react both more quickly and strongly to monetary policy shocks in a high inflation regime.
In the main specification of our smooth transition function local projection, i.e., (2), the state variable z t represents smoothed PCE inflation, so we take a twenty-four month centred moving average (MA) to capture trend inflation. 15 We set γ = 5 as this gives an intermediate degree of regime switching intensity. This is relatively standard in the literature and also fits our inflation data well. Finally, c corresponds to the 75th percentile of the historical trend inflation distribution. This is equivalent to assuming that about 70% of the time trend inflation is classified as negligible (i.e., F(z t ) ∈ [0, 0.1]) and 30% of the time there is some trend inflation (i.e., F(z t ) ∈ (0.1, 1]). Figure 3 displays the resulting smooth transition function, F(z t ). The solid black line, measured on the left vertical axis, shows PCE inflation on an annual basis. The green dashed line, measured on the right vertical axis, depicts the smooth transition function based on our MA-filtered measure of PCE inflation. The period of the Great Inflation from around 1974 to 1983 is characterised 15 The moving average is our benchmark smoothing procedure. However, the results from the local projections are very similar with a HP-filter (λ = 14,400) smoothing procedure; see the Online Appendix. by two pronounced spikes of inflation of up to 11%. The smooth transition function reaches 1 around these two peaks and stays above 0.4 for the entire period of the Great Inflation, classifying the latter period mostly as a high inflation regime. We take this to be a reasonable approximation for periods of high and low trend inflation in the United States. 16

Coefficient estimates
The panels of Figure 4 display the coefficient from estimating (2) for each response variable (by row). The first column shows the point estimates of theβ i h coefficients for the linear (solid black line), the high inflation (dash-dot green line) and low inflation (dashed blue line) regimes. Columns two and three depict the impulse responses conditional on the high inflation and low inflation regimes, respectively, with their 90% confidence intervals. The last column displays the t-statistic that tests the null of equality of the high and low inflation regime coefficients, i.e., β HI h =β LO h , where the grey area represents the 90% z values. A positive value means that the 16 This calibration is also in accordance withÁlvarez et al. (2019), who showed that the frequency of price changes starts increasing significantly from annual inflation rates of 5%. Our smooth transition function indicates a value of approximately 0.5 with such an annualised inflation rate. 14 the economic journal Notes: The last rows report the t-statistic with the null hypothesis of the two coefficients being equal. Newey-West SEs are in parentheses. * * * , * * , * indicate significance at the 1%, 5% and 10% levels, respectively.
high inflation response is larger, whereas a negative value of the t-statistic indicates the opposite. First, the linear terms for PCE inflation in the top-left panel show the familiar picture in the literature. PCE Inflation declines eventually, after an initial positive response, which however is not statistically significant. Second, inflation in a high inflation regime declines right away on impact after a contractionary monetary policy shock. On the contrary, the impulse response function in a low inflation regime exhibits a price puzzle for about one year, which is marginally statistically significant only for a few months. This suggests that in this regime firms are not willing to change price as frequently, so the price level stays persistently around zero for a longer period. Moreover, at long horizons the response is smaller in a high trend inflation regime. Again, this is consistent with the idea that the effect is less persistent in a high inflation regime, because more firms adjust on impact after the shock. Third, the last column shows that the responses of inflation in a high and low inflation regime are statistically significantly different both at short and at long horizons. We interpret these results as evidence in favour of state-dependent price models as key propagation mechanisms of monetary policy shocks, because they predict a faster, and less persistent, reaction to a monetary disturbance in a high trend inflation regime. This is exactly what the impulse response functions show. The first panel in the second row shows that the IRFs for output exhibit the usual hump-shaped dynamics. Output reacts with a larger delay in a low inflation regime compared to a high inflation regime, but this reaction is stronger and reaches a trough after two years that is roughly twice as deep as that in a high inflation regime. The difference in the IRFs between the two regimes however is not statistically significant, so that we do not find evidence for our second theoretical implication regarding output. Table 2 displays similar results for the cumulated IRFs of inflation and output at different horizons. Prices drop from the outset in a high inflation regime, while the reaction is sluggish in a low inflation regime. However, the latter is more persistent, such that it eventually catches up and overtakes the cumulative drop in a high inflation regime, so that the cumulative drop after four years is 50% higher. The difference in the initial response, up to two years, is statistically significant, providing supporting evidence for state-dependent pricing. This is not the case for the cumulative response of output, which is neither significant for low inflation regimes nor statistically different between low and high inflation regimes. Again, the point estimates are in line with the theoretical prediction, but the SEs get so large (especially for the low inflation regime, as also evident form Figure 4) that these differences are not significant.
Finally, the panels in the third row show that the interest rate increases after a monetary policy shock and stays positive for about two years, in the linear case and low inflation regime, while for only one year in the high inflation regime. Thus, monetary policy initially reacts differently to a shock in the two regimes, and these differences are statistically significant for the first two years. In a high inflation regime, the nominal interest rate initially reacts more, but then it decreases much faster than in a low inflation regime. While one might argue that this pattern may explain the initially quicker reaction of the price level in the high inflation regime, prices in the low regime react considerably more sluggishly even though the interest rate is positive for a longer period of time. Indeed, the last column shows that, for most of the IRFs, the coefficients in the low inflation regime are larger than those in a high inflation regime, perhaps signalling a stronger endogenous feedback of monetary policy in response to the shock (as also evident from the IRFs in column one of Figure 4). On the one hand, it would be hard to argue that the different monetary policy behaviour is driving the different price responses between the high and low inflation regimes. On the other hand, the fact that the path of the federal funds rate after the initial monetary contraction is different across the two regimes blurs the comparison, possibly explaining why the evidence for the differences in output responses is not statistically significant.
To conclude, we find evidence in favour of state-dependent prices regarding our second testable implication with respect to the behaviour of inflation. Our results show that in a high trend inflation regime, inflation declines right away after a policy shock, and there is no price puzzle, as theory predicts. Moreover, inflation is more persistent in a low inflation regime, despite the interest rate staying positive for a longer amount of time. Regarding the response of output, however, the point estimates are coherent with the theoretical prediction, but the differences between the high and low inflation regimes are not statistically significant.

Robustness
In this section we report the results of two particularly important robustness exercises regarding our local projection estimates. We conduct comprehensive robustness checks on many other dimensions, which we confine to the Online Appendix and we briefly summarise below.

Stability
It is crucial for our analysis to be relatively confident that the non-linear and state-dependent dynamic behaviour of the impulse response functions of output and inflation is not driven by a feedback effect of monetary policy. 17 To show that monetary policy feedback plays a limited role with respect to large and small shocks, we check if inflation coefficients with respect to 17 As discussed in Subsection 3.1, several of our results seem to contradict this possibility. For example, monetary policy seems to react weakly after a large shock, whereas prices react by more at the beginning of the horizon. The same applies to a large extent to the test of the second implication in Subsection 3.2 as the funds rate stays positive for longer in the regime where prices seem to be more sticky. These patterns seem to indicate that the dynamics of inflation and output are not defined by the reaction of monetary authorities only. 16 the economic journal Table 3. Estimated Hansen (1992)  We apply the Hansen (1992) stability test to the coefficients in the local projection (1) for PCE inflation. This test has locally optimal power and needs no a priori assumption concerning the breakpoint. Furthermore, this test is robust to heteroskedasticity, a potential concern in this analysis. Table 3 shows the results for the two coefficients of interest and the joint test for parameter stability at 1, 3, 6, 12, 24 and 36-month horizons for our PCE inflation local projection. 18 We cannot reject the null of individual parameter constancy for neither the linear nor the absolute value interaction coefficients at these horizons (with the exception of the linear coefficient at the 36-month horizon). However, the joint test statistic does indicate a rejection of the null that all parameters in the local projection are constant. This suggests that, even though the dynamic feedback of monetary policy via lagged control values may have changed throughout time, the shape and non-linearity of the impulse response after a monetary policy shock stay relatively constant throughout the sample. The same results hold for the linear and interaction coefficients when the dependent variable is either the industrial production or the Fed funds rate (see the Online Appendix). 19

Excluding the NBR Targeting Period
Coibion (2012) and others have suggested that the exclusion of the nonborrowed reserves (NBR) targeting period between October 1979 and September 1982 can account for a difference in results between the Romer and Romer (2004) and VAR approaches. Critically, the largest absolute monetary shock values lie in this period and may thus play a significant role for the conclusion 18 For a test on an individual coefficient, we can reject the null of parameter constancy at the 5% significance level, if the relevant test statistic is larger than the asymptotic critical value of 0.47. The null hypothesis is that each coefficient in (1) is constant and the respective distribution is non-standard and depends on the number of parameters tested for stability. The intuition is that, under the null hypothesis, the cumulative sums of the first-order conditions from the estimation will have mean zero and wander around zero. However, under the alternative hypothesis of parameter instability, these first-order conditions will not be mean zero for parts of the sample and so the test statistic will be large, leading us to reject the null hypothesis (see the Online Appendix). 19 These results substantiate the visual impression from the plots of the recursive estimation of the local projection coefficients displayed in the Online Appendix. on our first implication. In order to account for this suggestion, we exclude this part of the shock sample, modifying the non-linear local projection equation (1) by interacting the linear and non-linear coefficients with a time dummy that takes a value of 1 for the sample between October 1979 and September 1982 and 0 otherwise. As in Figure 2, the three rows in Figure 5 display the IRFs for this specification for PCE inflation, industrial production and the Fed funds rate, respectively, for both the coefficients of the linear term,β h (solid green line), and those on the non-linear absolute value interaction term,ζ h (dashed blue line). The impulse response coefficients are qualitatively similar to the benchmark results. The results, however, change somewhat both in terms of magnitude and, especially, in terms of significance, particularly so for the interaction coefficients. This is hardly surprising. The reduced sample does not include the large shocks of the NBR targeting period, and it features low sample variation with values mostly below 1. Hence, the non-linear effects of large shocks are more difficult to identify, inducing wide confidence intervals.

Other Robustness Checks
In the Online Appendix we present comprehensive robustness checks. We provide plots of the recursive estimates of the local projection coefficients in (1), to further visually inspect their stability. Besides, results are robust to a different specification of the non-linear terms in (1) that includes a squared and a cubed term for the shock value, rather than the absolute value interaction term.
Results for the smooth transition local projections (2) are robust to: (1) changes in γ and c; (2) using Hodrick-Prescott-filtered inflation for z; (3) using a model-based trend inflation measure from Ireland (2007).
Finally, the estimates of the coefficients in both (1) and (2) are also robust to: (1) using the CPI instead of the PCE; (2) using quarterly data with GDP as the output measure; (3) including as controls (i) the commodity price index, (ii) the corporate bond credit spread by Gilchrist and Zakrajsek (2012) to control for financial frictions, (iii) proxies for fiscal policy using either measure of excess returns on stocks of military contractors from Fisher and Peters (2010) or the exogenous tax changes from Romer and Romer (2010); (4) different measures of shocks obtained from non-linear models from (i) the Romer and Romer (2004) regression using the smooth transition function; (ii) a smooth transition VAR; (5) including leads and lags of the shocks to control for potential autocorrelation and misspecification as suggested by Alloza et al. (2019).

Conclusion
The assumption of sticky prices lies at the very centre of the current workhorse model for the analysis of business cycle fluctuations and, particularly, monetary policy effects. The literature features two types of sticky price models: time-dependent and state-dependent prices. A sticky price theory of the transmission mechanism of monetary policy shock based on state-dependent pricing yields two testable implications that do not hold in time-dependent models; the impulse response function of the aggregate price level and inflation should be more flexible both after a large shock and during high trend inflation regimes. Employing the methodology of local projections, we tested these predictions on aggregate US data. We found some evidence in favour of state-dependent models of price stickiness rather than time-dependent ones. With regards to the response to large shocks, the coefficient of the absolute value interaction shock projections matched our theoretical prior, both in terms of output and inflation. When the NBR targeting period of US monetary policy, between October 1979 and September 1982, is taken out of the sample, our results lose statistical significance, because too little variation is present to identify the non-linear effect and the significance bands become very wide. The empirical investigation during large trend inflation regimes also showed that inflation reacts significantly more quickly to a monetary policy shock in times of high trend inflation. In this case, the evidence for output is not significant; however, the point estimates behave according to the theoretical predictions. Furthermore, the Online Appendix shows that the results are robust to a very large variety of robustness tests.
Our results are the first (to the best of our knowledge) that point towards a significant presence of state-dependent pricing in the US economy using aggregate data. Our macro evidence is a useful complement to the large empirical literature on individual firm price micro data. Hence, 'prices are sticky after all'-just as recent literature has shown (Kehoe and Midrigan, 2015)-but less so when shocks are large or inflation is high. These results are in line with what would be expected if state-dependent pricing played a significant role in the US economy. This supports the theoretical implication that the frequency of changing prices is, at least to some extent, endogenous to the economic environment, as inÁlvarez et al. (2017). So, although the Calvo (1983) model may work quite well for 'normal' times, when considering situations where high trend inflation is present or large shocks are likely, a state-dependent sticky price framework that accounts for these phenomena seems more appropriate (as in, e.g., Costain and Nakov, 2011;Alvarez and Lippi, 2014;Álvarez et al., 2016;.