 All right. Welcome back, everybody. We're glad to have our first paper session of the morning, afternoon, or evening, depending on your time zone. So my name is Ed Notek. I'm with the Federal Reserve Bank of Cleveland, and this is the Empirical Models Paper Session. We're going to have two great papers today. The first one is going to be Inflation at Risk presented by Francesca Luria from the Board of Governors of the Federal Reserve System. Our second presenter will be Paulo Bonamolo from the Netherlands Central Bank, presenting Long Run Philips Curve as a Curve, the Long Run Output Gap Inflation and Monetary Policy. So as we've done before, just as a reminder, or in case you weren't here yesterday, so please put all of your questions into the chat. We will accumulate them, and at the end of our two papers, we'll have some time for Q&A, or authors can potentially answer those questions at their leisure during the session, but we'll try and keep some of those for the end for Q&A to have a lively discussion. So with that, Francesca, the floor is yours. I can see you and I see your slides, so you are all set. Take it away, please. And you can also hear me, right? I can. You're good to go. So you have 20 to 22 minutes. Thank you. So let me start by saying that I'm particularly indebted to both organizing institutions of this conference. The ECB's Price and Cost Division hosted me for a PhD traineeship during my graduate studies, and Federal Reserve Bank of Cleveland invited me for a job market talk back when I still had no gray hair. And I really enjoyed the conference so far and really very much appreciate the opportunity to present my work among so many important contributions to the literature. Finally, thank you all for attending this session. So today I'm going to present some joint work with David Lopez-Salido, which studies inflation at risk and the usual disclaimer applies. Now, the first question that some of you might ask is why we should care at all about risks to the inflation outlook? Well, since the upheavals of the global financial crisis, the emergence of downside inflation risks have increasingly become a source of macroeconomic concern. And as already extensively discussed at this conference, uncertainty surrounding inflation has widened considerably across the globe since the start of the COVID-19 pandemic. And in fact, risks have shifted from persistently on the downside over the last decade or so to decidedly on the upside in the face of resurging demand and supply shortages as economies reopen. Now, in the presence of these tail risks and as President Draghi's quote on this slide reminds us of the conditional inflation mean may not necessarily portray a complete picture of the overall inflation outlook. So the starting point of our paper was precisely the desire to extend our understanding of these risks. After all, most of the analysis studied the muted response of the conditional mean of inflation, leaving fairly unexplored what happens to other parts of the inflation distribution. And with the literature pointing to quivering Phillips curve linkages, our question was whether there are some macroeconomic factors in the Phillips curve umbrella that are still at work perhaps in the tails of the inflation distribution. To be clear, by inflation distribution I mean the predictive distribution of inflation over the next year. So such a distribution can answer questions like what is the probability that inflation will be above or below say 2% over the next year. So to answer these questions in the paper we estimate a quantum Phillips curve, fairly well understood and microphone the time series framework that allows us to study the risk to the entire distribution of the inflation outlook, coming from both conventional inflation determinants as well as financial conditions. And what we find is that the recent muted response of the conditional mean of inflation to economic condition does not necessarily convey another quit representation of inflation dynamics. Indeed, we find ample variability in the tail risk to inflation, even when we focus on the post 2000 period of stable and law mean inflation. In particular, we find that financial conditions carry substantial and persistent downside risk to inflation. And our findings are consistent with evidence from a non linear DSG model survey data inflation options and the regime switching model of inflation. Some of these results I will present today. So we see our paper is ultimately offering a new empirical perspective to existing macro models and to policymakers by highlighting the presence of a significant relationship between credit conditions and risks to the financial, sorry, to the inflation outlook. Okay, our result that there have been sizable downside risk to the inflation outlook in the last 20 years, mainly accounted for by financial titanings is consistent with the idea that due to amplification mechanisms when financial conditions become tighter firms cut prices disproportionately more on average. And these concepts are related to recent research, some of which cited here, which shows that financial conditions matter also for inflation dynamics. Now, while these studies focus on the model outlook, we extend their insights to the study of the inflation tales. In fact, implications for the tales of the inflation distribution, and the role of financial conditions have been fairly unexplored by the literature in general, although there have been some efforts, and in particular the last two studies, some of which featuring ECB authors take our framework as a starting point and confirm our findings that financial conditions are indeed important determinants of inflation risks and especially of downside risks. Okay, so we now move to characterize these inflation risks. We model the conditional quantiles of average future inflation over the next four quarters as relating linearly to inflation determinants xt across quantiles tau. Notice that the determinant xt can exert nonlinear effects on inflation dynamics if it affects differently the median than the tales. Given data for the right hand side variables, inflation quantiles can be constructed for each point in time t, even beyond the estimation sample, as long as data for xt is available. To obtain the inflation densities, we then fit a flexible skewed t distribution by Azzolini and Capitano on the estimated quantiles, very much like in the paper by Adrian Boyacenco and Giannone. One last thing I would like to bring your attention to is the fact that running a regression of this type is akin to a direct forecast as opposed to an iterated forecast where the dependent variable would be pi t plus one, and where one would iterate the one step ahead prediction to obtain multi horizon forecasts. Now, as discussed in my paper with Dario Caldara, Dario Cascaldi Garcia and Pablo Cuba Barda on Markov switching models and growth at risk, in simulation direct and iterated forecast delivered the same results when they share the same VAR data generating process. In empirical data, both the direct and iterated model perform well in terms of predictive scores, and most importantly perhaps in the context of density forecasts in terms of coverage and correct calibration of the predictive density. All right, so to give you a glimpse into our augmented quantum Phillips curve model, I list here the variables conditional on which we are estimating the quantiles of average four quarter ahead for CPI inflation in the United States. The first set of conditioning variables directly relates to inflation. Okay, so these are average inflation over the previous four quarters and long term inflation expectations from consensus economics. To preserve the notion that inflation persistently deviates from inflation expectations we impose the homogeneity constraint and prices by constraining the two coefficients on these inflation determinants to some of two one. We also condition on measures of labor market slack through the unemployment gap of relative prices through import or all prices and no financial conditions through the Gilchrist-Zakrashek credit spread. So in the paper we look at how the importance of risk factors changed across time. And indeed, the two distinct subsamples emerge when characterizing the determinants of the inflation distribution in the United States. The first period running from 1973 to 1999 notably covers the OPEC shocks, the subsequent folk or disinflation in the early stages of the Great Moderation, whereas the second sub-sample running roughly from 2000 onwards is characterized by large movements and credit spreads progressively well anchored inflation expectations but subdued inflation pressures. Now in the figure presented here you can see the estimated quantile slopes for each inflation determinant for our two sub-sample. So each box stands for an inflation determinant and the blue, red and yellow bars are respectively for the 10th, 50th and 19th quantile regression coefficient. Now as you can appreciate in the middle right box over time, long run inflation expectations, which we here treat as exogenous, became the decisive inflation determinant. And this is consistent with the analysis, for instance, in Blanchard, Summers and Chirruti of a time varying parameter version of a Phillips curve model. Now I want to pause here for a second to recognize the fact that to an applied econometrician like me, it did not appear all that surprising that in the last sub-sample when regressing a relatively constant variable that of average future inflation on another relatively constant variable that of inflation expectations, one would find a strong relationship between the two. Well it turns out that if you decompose inflation into a cycle and the trend component, for instance say using the Elmar Martens 2016 restart model, inflation expectations became more important over time in terms of forecasting precisely for their predictive content of the more slow moving trend component of inflation. Now one important point that we also make in the paper is that it would however be misleading to dismiss the roles, the role of other factors. In fact credit conditions, the box at the bottom and to a lesser extent labor market outcomes, the upper left box are key drivers of the asymmetry in the inflation distribution in this more modern quantum Phillips curve for the last sub-sample. While inflation expectations only have a symmetric effect. Now focusing on the credit spread, you can see that there is a substantial sub-sample instability in its link to the inflation outlook. The first sub-period is characterized by relatively small variations in credit spreads in a period of high and volatile inflation, reason why the actual contribution of credit spreads to the inflation outlook in that period is smaller. And from 2000 onward, low and stable inflation has coexisted with substantial variation in credit spreads, especially as we know during the global financial crisis. Now focusing on this most recent sub-sample, a novel result is that the link between the inflation outlook and financial conditions is asymmetric. In that an increase in credit spreads is associated with a larger increase of downside risk than in a reduction of upside risks. And this is reminiscent of the results for the GDP growth outlook in Adrian Boyarchenko and Giannone and confirmed by other papers that they previously cited. Notice that all coefficients are statistically significant and that ANOVA tests on the equality of the slope coefficients across quantas reject the null hypothesis of equality between the slopes on the 10th and the 50th quanta. I now want to provide some theoretical grounding for our results. So in general, our findings can be rationalized by non-linear models featuring amplification mechanisms such that the relationship between credit spreads and inflation is more pronounced during bad times. Our findings can, for instance, be replicated by applying our quanta regression framework to simulated data generated from the Gertler, Kiyotaki and Priscipino model. Their model, just as a reminder, is a fully micro-founded non-linear DSGE model which features the possibility of a severe financial crisis through a bank run. And there are two equilibria in their model, one with and one without a financial panic. So when shocks are small, the economy fluctuates around the standard equilibrium. In contrast, a big negative shock pushes the economy into a bank run equilibrium. Now, combined with a sunspot shock, it triggers a financial panic and bank net worth collapses. Banks are forced to sell assets, which ultimately disrupts firms borrowing. Consequently, economic activity drops substantially more than in the equilibrium without a bank run. So we simulate the model using the original calibration of the D parameters and of the capital quality shock process. And in order to generate a rare financial crisis, we calibrate the process for the sunspot shock such that the bank equilibrium arises after a big negative shocks of above two standard deviations. We simulate this model a thousand times and store the inflation rate, the credit spread and the capital quality shock. We then run a quanta regression of current inflation on the credit spread. Okay. And the chart here shows the quanta regression slopes estimated on simulated data from that model as I just described. So the negative slope coefficient for the 10th quanta echoes our empirical findings that higher credit spreads are associated with an increase in downside risk to inflation. Now, in terms of identification, what delivers this result is that during bad times, those featuring a bank run, the conventional channel whereby lower demand results in subdued price pressures is strongest. But the reason why the 90th quanta indicates a positive relationship is that in good times, without the bank run, a capital quality shock reduces capital and thus results in an increase in the rental rate of capital and thus in marginal costs. Now, the median quanta captures the tension between these two effects. And indeed, it is around zero as a normal times these two effects almost offset each other. Okay, so to test whether this relationship also holds in financial markets, we run an OLS regression of options implied inflation probabilities of one year ahead CPI inflation on the credit spread. And in the left panel of the figure, we present the estimated coefficients of that regression. On the right, we report the estimated quanta regression slopes for us core CPI over the last sub sample that you just saw previously. Now the slopes on the left are rescaled so as to facilitate the comparison with those coming from our quantum Phillips curve model. And you can see that despite the vast disparities in the construction of the tails of the inflation distribution, the estimated slopes are very similar to each other. Most importantly, as the inflation probability cutoffs increase their relationship with credit spreads weekends. And this again is reminiscent of our key result from the estimated quantum Phillips curve model over the last 20 years of data that we have here on the right. Okay, so we also analyze how a no linear models such as a regime switching regression compares to our quanta regression estimates. We move to monthly frequency to allow for more observations and better identification of the regimes. This exercise is meant to show that the relationships we established in our main analysis are not an artifact of the quanta regression, but the genuine feature of the data that also alternative nonlinear models would identify. So when comparing realized average inflation over the next year in black against the estimated regime probabilities. I hope it becomes clear to you that the estimated regimes broadly correspond to states where inflation is low, moderate or high. And here you can now see the regime specific fitted values, along with the estimated contests from our quantum regression model, which as you can see are remarkably similar. In particular, the low inflation regime corresponds to the 10th quantum, the moderate inflation regime to the median and the high inflation regime to the 90th quantum. And for those of you who are interested, this result that the mark of switching regression and the quantum regression when put on equal footing deliver the same predictive densities is carefully studied and described in my recent work with Dario Danilo and Pablo that I mentioned before so you can read more about this there. So in the paper we also compare the United States experience with that of the Euro area. And there are some notable differences across the Euro area and the United States and how the determinants considered in our model affect the inflation outlook. For instance, in the Euro area inflation inertia, the unemployment gap and to a small degree also relative prices still play a role in shaping the inflation outlook. And when it comes to the credit spread it still creates downside risk to the inflation outlook but it does so in a symmetric way. And this can be seen in this figure, where we show the densities of one year ahead inflation in the Euro area using core HICP inflation, and in the United States for the periods at the onset of the great recession that is 2007 Q4 and 2008 Q4. This would be the blue densities in the left and right columns respectively. The black densities captured the partial effect of an experiment in which the credit spread is set to zero. And as you can see, while in the Euro area the distribution experiences a symmetric location shift to the right in the counterfactual experiment. The US distribution is not only pushed to a higher inflation value, but also exhibits way smaller tail risks. Right, so risk to the inflation outlook have been front and center at the peak and in the recovery from the COVID-19 crisis and still are nowadays. So we show how our model augmented by credit spreads allows to identify important changes in the United States inflation distribution during this historical episode. And in doing this we consider a monthly version of our model to allow for a more real time assessment of inflation risks. The model is estimated from January 2000 to April 2020 using as dependent variable one year ahead inflation up to April 2021. The chart shows results for average inflation over the next 12 months with core PC inflation on the left and core CPI inflation on the right. Specifically it shows the distribution since January 2020 with markers given at the median and at the 90th quantum. Now both core PC and core CPI inflation distributions, as you can see have shifted to the right since the beginning of this year. And indeed the selected months show the quick buildup of downside risk to inflation during the harsh initial months of the global pandemic. January to May 2020 in the top panel, which were then followed by the increase in upside risk to inflation in most recent months in the bottom panels. Alright, as a final exercise, we now ask what would have been the predictive distributions over the next 12 months had the financial conditions deterioration of March 2020 persisted in May 2021. The chart presents distributions again of average inflation over the next 12 months but now is of May 2021 for our baseline in the blue solid lines. And in this quote unquote counterfactual scenario in which credit conditions are those prevailing as of March 2020 in the black dashed lines. Now the easing of financial conditions since the onset of the pandemic has moved the distributions of both core PC and core CPI inflation rates to the right, reducing downside rates of low inflation. And indeed, had the financial conditions not eased during the period from their March 2020 state, the probability of inflation running at or above 2.5% would be about 7 percentage points lower for both core PC and core CPI inflation rates as of May 2021. To summarize, in this paper, we show that one needs to look beyond the conditional mean to fully understand inflation dynamics. And indeed, we find ample viability and the tail risk to inflation, even when focusing on the post 2000 period of stable and low mean inflation. Most importantly, we show that financial conditions carry substantial and persistent downside risk to inflation. Finally, our results, we hope, provide empirical guidance and suggest more efforts in modeling the linkages between the entire inflation distribution and financial markets in the context of nonlinear models. Thank you very much for your attention. All right, we will now turn to Paolo. So please feel free to submit questions in the Q&A and then we'll get to those in the chat and we'll get to those at the end for Q&A. Moving on to Paolo. Looks like I can see your slides and I can see you and I can hear you. So you're all set. Please take it away. You have 20 to 22 minutes. Thank you. Thank you, Ed, and thanks to all the organizers for including this paper in the photo. This is joint work with Guido Scari and Cassi Bacche. So the question of the paper is whether the long run philip school is vertical or not. This issue was highly debated in the past in the 60s. And back then, Phelps and Friedman proposed the natural rate hypothesis, saying that in the long run the philip school is vertical at the natural level of output or at the natural rate of unemployment. So this idea played a cornerstone role in macroeconomics both in theory and in practice. I think we can fairly say that this is one of the working assumptions that central bank use when they implement monetary policy. Now, given the importance of this concept, if you look at the literature, it is surprising to note that empirically there's not so much work devoted to understanding empirical validity of this idea. And from a theoretical point of view, if you consider model macroeconomic sticky price models, in general, they do not imply the absence of a long run relation between inflation and output. In particular, if you take the new case of model and you assume that the steady state of inflation is not zero as in the original simple version, but it is a positive number. Then you'll find that the highest is numbered lower will be the GDP in equilibrium. And these are now called generalized new kinds of models, we do and just put on paper on the journal of economic leader to in which they give a very comprehensive treatment of this topic. The question of this paper isn't difficult. We ask what is the long run relation between inflation and output. And the paper is divided into two parts. In the first part, we answer the question using a time series model. We find that in the long run, the school is not vertical, but is negatively slow, meaning that higher inflation is related to lower output in the long run. And what is key to get this result is to model the long run through school as non linear. And here we also have a methodological contribution we propose that's a convenient way to model non linear. So what we find in the in this first part, I would say is a mere statistical relation between the two unobserved values that are potential output and trend inflation. But we cannot really say anything about cars. So in the second part of the paper, we interpret our findings to the lens of a structural model and we choose exactly the generalized decays and model that I was mentioning before. In this model, the causality goes from inflation to GDP. So it's higher than inflation that causes lower GDP in the long run. And the reason why we choose this model is that it has the two key features that we find in the statistical analysis that are non linear and a negatively slow long run Philip school. So we estimate the model and we show that model is also able to capture the quantitative features of the time series analysis when it comes to measuring the cost related to trend inflation. Okay, so this is the overview of the paper with the results. Let me now describe the time series approach to be used. So I like to call it a time value it could be because it is a generalization of the steady state we are by my TSP land 2009. So consider here this equation x t is the vector with the observed variables at time t, and we call x bar t the vector with the long run values of x t. Now the deviation of x bar from the long run is so these deviations are modeled together with the VR with stochastic bulletin. Now we are not the first to use this framework. The neighbor, Janone, Janone, and Dambalotti used it. Recently, Johannes and Martens and there are also other papers around that are using the same framework. This is a trend cycle decomposition in which the vector of observable x t is decomposing into a long run component x bar and the short run component x. So the short run component is described by this VR and this VR is stable and has unconditional expectation equal to zero so that the gap component here is expected to converge to zero. And this is why we interpret bar component as the long run. And for the bar component we, we use the hypothesis that the x bar are a function of a vector data team. This is a latent vector. And this is a mark of process with stochastic dynamics. Now these two functions h and f are in general that can be no linear. There will be no linear in our case. So in particular what we do, we have a model for three observables GDP per capita inflation and the nominal interest rate. For the short run component, the VR has four legs. And for the long run, so we have three elements in our x bar vector output. So we have potential output in x bar inflation we have trained inflation x bar and the long run nominal interest rate. So for potential output we assume that this is the sum of two components, a trend and a function of trend inflation. Let's first concentrate on the trend. So this trend is quite standard. And many people are recognizing this because it has been used quite extensively since long time. So this trend has two shocks. We have a shock, a YT that changes the level of the trend and a shock to the growth rate of the trend that is around the work. This function delta is what we add on top of what is standard say. And in particular the assumption here is that when trend inflation is equal to zero, also this cost, this function is equal to zero. In this way, we can interpret this trend, y star T, as the counterfactual level of potential output under zero trend inflation. Then we assume that trend inflation is around the mall and for the nominal interest rate, we assume that the long run feature equation holds. So this is equal to trend inflation plus a measure of the real interest rates in the long run. Here with Bobo from Labrack and Williams, we assume that this is a function of the growth rate of potential output plus Z that is a random work that should capture all the slow moving trends that are not directly moded in this form. Now, the first equation here is our long run field score. What is delta? Our choice of delta, the function delta is a piecewise linear function. So we assume that delta is going to be equal to a slope K1 times trend inflation. When trend inflation is below a certain threshold tau, but we allow for that slope to change and add a constant when trend inflation is above that threshold tau. Why this choice? First of all, and the most important aspect is that this model is very simple to do. And here we have our methodological contribution that I will explain in the next slide. In general, we think that this model can approximate the kind of non-linearity we have in mind, and it's quite easy to interpret. Let me explain the methodological contribution. The model that I just described can be written in this state space form. Yt here is a vector of observed variables and theta t is our latent process. D, F, M, G and P are matrices, but these matrices are functions of the vector of latent process theta t. In particular, they are function of one element of theta t, which is trend inflation. We assume, in fact, that these matrices might belong to two groups, group one and group two, depending if trend inflation is below the threshold or above the threshold. Our methodological contribution is that we can find the likelihood function and the posterior distribution of theta t analytically. Now, we think that in general this is nice because this model represents a good compromise between efficiency and mis-specification. As an economist, I always find it desirable to estimate, to specify the model as non-linear, because if I think that there are important non-linearities, I would like to reduce the mis-specification of the model. However, for non-linear models, in general, the likelihood function is not a bit analytically. So I need to approximate it, and the question is how good is the approximation? If the approximation is not very good, the efficiency of the filter in a statistical sense is very low. On the other side of the spectrum, there are linear models that are very efficient because the likelihood function is available analytically. However, the risk is that we are missing important non-linearities, so the model is mis-specified. This was linear model, it's a compromise because we are capturing some of the non-linearity, but at the same time, since we have the analytical form of the likelihood, in terms of efficiency, this model, the estimator that we have is very comparable to the one we would get with the linear model. So we think in our case, this is a very good choice. This is not always the best choice. There is a cost with respect to the linear model, and you can see it from Equation 4. The piecewise linear specification requires the estimation of a higher number of parameters, including the threshold down. So going a bit more into the specific, our sample is, so we use US data for 1960 Q1 to 2008 Q2, we use a Bayesian approach. So remember, there are two sources of non-linearity. We have stochastic volatility, and we have a piecewise linear long-run Phillips group. So we find it convenient to estimate it to a particle filter. There are two aspects that I want to stress. The first is that, thanks to the analytical results on the piecewise linear model, we can improve the efficiency of the filter through the so-called Raoul black liquidization. So we use the Raoul black liquid theorem. And the second aspect is that we use the particle filter also to approximate the posterior distribution of the parameters. In particular, we use, we combine two techniques. The particle learning by Carvalho-Ionis, Lopez and Paulson, and the mixture of normal distributions, as in U.N. West, 2001. Okay, let me show you the results. The first thing that we do, we estimate the model under the assumption that the long-run Phillips group is linear. So we only have one slope at this cap. Here you see the posterior and the prior distribution. When we specify a linear long-run Phillips group, we estimate the slope equal to zero, as you can see. When we, instead, we estimate the non-linear version, the piecewise linear version, so we have two slopes. Again, here I'm comparing the prior in blue and posterior distribution of the two parameters. When trend inflation is below the threshold tau, then the slope is zero. But when inflation is above the threshold, the estimated slope is negative. It is important to stress that the linear model is a particular case of the piecewise linear. But it's rejected by the data, because the data, when you give the opportunity to the data to the model to have a non-linearity, then the model wants the non-linearity. And estimates the second slope in the negative table. So this is, instead, the estimated threshold tau. So the slope changes when trend inflation becomes above 4%, basically a bit more than 4%. When we combine this information together, then we get our estimated long-run Phillips group. Here you see on the x-axis values for trend inflation and on the y-axis the difference between the potential output and the counterfactual potential output under zero trend inflation. So these are the costs from trend inflation. You can see that the estimated costs are basically zero until we get to the threshold of 4%. After that threshold, this relation becomes non-linear. So this is an estimate of trend inflation plotted together with inflation. And if you combine the information in this chart with the long-run Phillips group of this chart, then we can get what we call the long-run output gap. This is the cost of trend inflation estimated over time. So here on the x-axis you have time, and here you have the distribution of, again, the difference between potential output and the counterfactual potential output under zero trend inflation over time. So before the 70s and during the great moderation when trend inflation was below the threshold, then this cost estimated to be equal to zero. While during the great inflation period, the median cost is roughly is around 3%. So potential output during the great inflation period is estimated to be 3% lower with respect to the case of zero trend inflation. So the point here is that the costs from trend inflation are quite substantial. Now we want to understand if we can interpret the findings in this model through the lens of the generalized Newtonian model. So we consider generalized Newtonian model with three equations. An inter-temporal Euler equation with external evidence assumption. We have a generalized Newtonian Phillips group and the Taylor type monetary policy rule. Importantly, we allow for time variation in trend inflation. And this gives us, through the mechanism of the model, a non-linear and negative result long-run Phillips group. Another important assumption is that when taking decisions, the agent considers trend inflation as a constant parameter. So we are under what is called the anticipated utility model by Krebs. And here we're following the problem's border in 2008. Finally, we put stochastic volatility into the four shots we have. The discount factor of technology, monetary policy and trend inflation. Now, before showing you the results, I just want to stress what is the mechanism behind the model. So why do we have a long-run Phillips group, which is negative? In the Newtonian model, the friction is price taken. So the assumption, we have calval pricing. So the assumption here is that firms have a positive probability of not changing the price or changing the price at time t. So this creates price inspection, which leads to an inefficiency in the quantity problem. Now, the idea is the following, intuition. If there is high trend inflation, the high is a trend inflation. The high will be the price inspection because firms, when they get the opportunity to change the price, they will take into account that the cost may be quite high. So they have the incentive to increase the price even more when they get the opportunity to change it. And this increases price dispersion even more, increasing the inefficiency in output. Formally, you can see that considering the aggregate employment and the definition of aggregate employment is just the integral of the employment of the firm over the firm's eye. If you substitute in first the production function of the firm and then the demand for the good eye that the firms face, then you get this equation. And this integral here is what we call price dispersion because this gives you a measure of how prices differ from the general level of prices. So you can write all of these, making the output wide explicit, and then you get something that looks very familiar. So you say that output depends on technology and on employment, the usual fashion, but now you see that it also depends negatively on price dispersion. Importantly, price dispersion in the long run is a positive function, non-linear function of trends. For the reasons I tried to explain before. So we estimate the generalized organism model and then we can compare what are the costs that this model implies and compare it with what we found in the statistical analysis. In this graph, you see the same picture that I showed you before. So it's the long run Phillips curve from the time series model in blue compared with the long run Phillips curve implied by the estimated generalized organism model in black. As you can see, the cost implied by this model are very much in line with what we find from the statistical analysis. So I can conclude saying that, okay, the question of the papers, what is the long run relation between inflation and output? So we use a time series model and the time series model suggests that the long run Phillips curve is non-linear and negatively slowed. We then interpret these findings to the lens of a generalized organism model and we find that this model is able to measure the cost implied by the long run Phillips curve in a very consistent way with the time series model. Thank you for staying on time. All right, so at this point we have some time for questions. So I see we have one from Christian in the chat or Francesca. So maybe we can answer that one first. So Francesca, this one's to you. The credit channel can explain downside risk to inflation. What about upside risk? What kinds of variables would you use to proxy supply bottlenecks or other types of supply side shocks? Of course, this is extremely relevant for our present day circumstances and I think on many people's minds so we'd be happy to hear your thoughts on it. Yeah, as you anticipated, Christian hit the nail on its head with this question. So as you can see in the results that I've presented you, the model in its current setting already has the ability to capture some of those upside risk that we have seen in recent months. However, if I had to make this model fully operational for policy purposes, I would indeed have to think about supply bottlenecks and also pent up demand. Thinking about pent up demand, I would probably need to think about some measures of disposable income that would better correlate with inflation than say the unemployment gap in current times. As to supply bottlenecks, any variables relating to shipping costs or the cost of used goods on top of the measure of import and oil prices that is already considered in the model would more appropriately describe the inflation process as well. But at the end of the day, I think that including these variables probably won't be enough to make the model fully operational for this current times. And probably what is needed is also to look at this model in a panel dimension that also considers some sectoral measures of inflation rates, and also, you know, different inflation components in separation of each other. Since, as we know, the pandemic has had very asymmetric effects across, you know, different sectors of the economy, notably services versus manufacturing, and also across inflation components, some of which, as we know, now more prominently feature in their past through to realize the inflation. Alright, thank you. Maybe we'll move to Ali's next. Is there any empirical, this is for Paolo. Is there any empirical reason to think causation runs from inflation to output and principle output can worsen budget balances and increase the incentives of policymakers. I've lost the question to raise inflation to finance those deficits leading to a negative steady state correlation with opposite causation. How can we distinguish between these two. Please Paolo. Yeah, so it's perfectly possible, I think, and so our starting point is the time series model doesn't give it doesn't give any answer in that so what we do we interpret this finding with the with the generalizing case and model with that causation but we can't really say if the story that that all years in mind is perfectly, perfectly fine. So it's important to stress that the point of the second part of the paper is just to say that the workers model and this and the cover pricing way. It does a good job in, in measuring the cost from time to inflation, but we can't really say, you know, written install that causation is that what we just saying that is a good model when it comes to to measuring those costs, but I think he has a point. So the answer is no we can't say for sure that the causation is that. Thanks Paolo so Andrea Tomilotti had a question for each of you so Andrea I will send a request to unmute you and you can ask your questions at the same time if you wouldn't mind. Awesome. Thanks so much. So Francesca I was curious to know if you guys looked at the connection between financial conditions and inflation say three years on the road, as opposed to sort of more contemporaneously the way the way you're doing it and and you know that's the same flip there I'm thinking of the work that David has with the Stein and the ego and I think about again sort of compress the spreads sort of leading to problems may say three years down the road. And then the question for for Paolo is very interesting paper what is it possible that what you guys are picking up, which I mean again is probably there in the data is essentially the fact that over those 15 or so years in the 70s when inflation was high, output was low. And this goes a little bit to all this question, you know, maybe it's a little bit hard to say what the channels through which that correlation, you know, does it go from up to inflation or the other way around this is probably a little bit hard to tell. I mean, I think it's a reasonable way of characterizing the data but is there more that your model gives you then just as people for the long run correlation. Thank you so much. All right, I'm going to start. Good. Thank you. Go ahead. Thank you for your question there. So actually, no, we have not looked at that. I can see where that question is coming from. I was already a bit skeptical or yet skeptical in using our model to explain such, you know, long horizon movements in inflation. Because, you know, we have both a cyclical and the trend component in there and I always thought that the three years ahead with asking too much from the data in terms of the forecasting ability of credit spreads. The reason why when I wanted to dig further into the role of inflation expectations. Also, at the horizons of three or five years in some separate work that we have done, I reverted to that model that I mentioned in my presentation of Elmar Mertens, where he he looks at such long horizons and rather at that more slow moving trend component of inflation. This being said, I will just take the suggestion and perhaps also look at measures of leverage that I know have good ability in forecasting GDP growth. You know, it's such long horizons and see whether there's something more than I can say there I can see where the question comes from and obviously nowadays that's a really very relevant question. Okay, so thank you Andrea for the question and so it is to so what we are capturing is exactly what you were described so the great inflation period is a period in which inflation is high GDP is low. And we give the opportunity in the time series model to interpret this or as a cyclical temporary phenomenon, right, because we have the VR that describes the short term dynamics, you know, as rich dynamics, four laks and stochastic volatility, you know, or as something that pertains to slow moving trend. And the model says, well, it's something that is better described by the slow moving part, the trend part because it's persistent phenomenon. And just to clarify, you know, in the statistical model that this is exactly a correlation. So when you look at the Phillips there, you know, the slow parameters appear in the correlation matrix. So we can't really say that we can deduce where the causality goes, you know, and so we are very clear on that. So the causality can go both ways. So in some sense, what we find is very much in line with the traditional concept of Phillips, right, is a statistical relation, what we find between two unobserved variables that are trend inflation and potential. So, if you have, if you have in mind models that can interpret this in another way, so there's a possibility to see as we did for the new kinds of model, the quantitative implications are similar or not. I don't know if I answered the question. Thanks, Paulo. It looks like the next two are for you. All right, maybe even the next three. So, the 4% from the statistical exercise of possible indication we may consider a higher target. Are there other possible costs not considered a higher volatility of output, for example. Also, just as the result of your vertical Phillips curve from 0 to 4%, an assumption or result from your method. And then one from Todd that another check is it possible to estimate the model for a couple of other economies that had less like Germany or more of a run up in inflation during the great inflation to see if the estimated output costs change as expected, given the model. Okay, yeah. So let me start then. Okay, so this is me right is 4%. So, okay, the, the, the point here so when we interpret is the going back to the causality issue. And if the causality goes from inflation to GDP, then we say 4% is when there is a probability distribution so 4% is already in a, in a zone where it's kind of risky to stay right you already might have substantial losses that are statistically right so, but we don't you know we don't want to go into these, these are more policy issues that we need to policy makers, we just want to document this, this variation. So the, the, the, the, the, this, this first result so we're applying to Augustino. So, from zero to four, this is estimated. Okay, so the model wants so when we saw the model estimates the first look. Equal to zero case or is not is a result from the estimation we're not imposing it we were thinking about maybe a model which we can even impose that in the first part but we haven't done it so what you see is the result of an estimation. And while you're planning to, to talk about this is a very good, very good suggestion. I think we should probably try to do that. Yes. Yes, and I think it would be also interesting to see how the costs are different because the cost front and inflation might really depend also on the structure of the economy so it's a very good suggestion. Thank you. So Francesca I had a question for you if we can shift gears for a moment. So, and this relates to your direct approach in essence the question that I had was, to what extent are these financial conditions affecting inflation directly. And can you use your framework to assess that they're affecting inflation directly, or to what extent are the financial conditions affecting inflation indirectly potentially through affecting the forecast for the real economy. Inflation expectations, commodity prices. So in other words, you know, can you can you leverage some of the growth at risk literature because you're kind of building on that so can you use your framework and basically separate those two things. Yes, so a very cheap answer to your question would be that well you know we have a model where we are also controlling for a measure of inflation expectations and of the unemployment gap on top of these financial conditions. But the I perhaps I think more appropriate answer to your question would be that in an ideal world one would precisely use that methodology that I developed with Dario Danilo and Pablo and use Markov switching because they are featuring you know multiple variables that can interact with each other to be able to answer to that question and in an ideal world also you know identify a very exogenous structural measure of financial shocks and see how financial shocks affect the entire inflation distribution. So this is just work for us down the road and you know, so your suggestion is a spot on I think. I'll follow one more question for you in the chat there. So models, featuring downward now I'm going to wait for Judy going back to Akerlof Dickens and Perry also imply that the long run Phillips curve is nonlinear in that lower inflation implies an increase in longer run unemployment would it be feasible to assess the relative importance of that mechanism from that of relative price distortion. So Manu is suggesting I think another paper is very interesting question I think. I think in general it is possible so if you estimate model, a model like that, you can check you know the empirical performance of this model and and but that's another, this is another paper I don't think we're going into this direction in this paper. Thanks for the thanks for for suggesting. Right, looks like we still have a few more minutes so we'll just keep on coming so okay next one for you Paolo. So linking back to the question asked that Kira asked Hassan yesterday about high inflation and attention. If high inflation would put inflation back on the radar. If you have an effort and change prices more often, then why would you have higher price dispersion thinking also mixed evidence on higher price dispersion higher inflation. So how central is that to your model, and you know I would voice the same question you're using a calval framework to think about changes to the inflation steady state you know I think that that always raises some concerns there so any thoughts that you had along those lines. Yeah, so now it's clear that if you have a model like that in mind, you know the cost from price dispersion will be lower. If you have a state dependent model for example right so the cost will be lower yes that that's perfectly true. So when we when we choose that model the question that we have in mind is really about if the calval pricing you know is a is a good way to not not in general you know but to the specific dimension which is capturing the cost from an inflation is really exposed it does. Then, if it does because maybe you know it's capturing something that is not in the model that I don't know. In general one might think about when you when I think about the cost from an inflation I think about for example labor market is not present in our, in our simple model you know, for example that this is another one that we should consider as well right so there are many aspects that we should consider, but the point of the exercise is just to say okay this class of models in general they do imply costs from an inflation that tie in line with what we find in the statistical this is the key message then if you want to enhance the model one direction or the other that's a good these are good points but you know this is not the purpose of our exercise. So I think you know we missed a question from Mikhail. I can answer very, so Mikhail is asking if it could be that inflation and real activity transfer common factor. I think it can definitely be the case and this is something for example we can check so we have the model and we can put this common factor in the model and check so we can actually check for this perfectly possible thank you. Thank you for catching that Paulo sorry about that McKellie. Alright so with that word 11 o'clock Eastern time, which means that we will conclude this session. So big thank you to our presenters. You know, thank you for this work and thank you for presenting it at our conference we really appreciate and thanks to our audience for all of the great questions. So we have a 15 minute break for now, we will resume at 1115 Eastern time, which I assume is 515 Frankfurt time with structural Phillips curves so we'll take a short break. See you in a little bit.