 We go to the contributed papers. The next paper is by Andrea de Polis. All right. So thank you very much to the organizers for including this paper in this conference. So what I'm talking about today is the ever-changing challenges to press stability. This is a joint work with Leonardo Melosi from the Chicago Fed and Ivan Petrella from the University of Warwick. And given that this conference has been focusing mostly on forecasting, let me put my hands in front and say that at the current stage, this paper is pretty much about measurement rather than forecasting, at least at the current stage. But hopefully some of the result can give ideas on how to set new impression forecasting models. So I think yesterday Matteo in his very nice introduction mentioned that recent shifts in the macroeconomic environment or perhaps in inflation dynamics have basically pushed central banks, the ECB and the Fed, to rethink their monetary policy strategy. So the idea behind this paper is really like to try and understand what are the dynamics of risk to inflation that perhaps have contributed to this thinking process. So what we start with is actually just having a look at inflation expectations. And this is just like too proxy for this changing environment that we have in our mind. So here, for example, you can see that here I report the distribution over time of survey data about inflation forecasts from professional forecasters and consumers. This is US data. And what is interesting to notice here, even though it's probably well known by most of the audience, is that in the aftermath of the great financial crisis, like most of the inflation expectation were actually coming from the tail of the overall distribution of forecasters and as well for consumers to some extent here. And this has remained like a predominant tail for the basically 10 years after, whereas now in the last two years, I mean, as we all know, we can see that there is this drastic change in the prevalence of the right tail compared to the left tail of the predictive distributions. So what we were really trying to get from this picture is that even though this is only for the last 15 years or so, there seems to be like sort of start time variation in the perception of the prevailing risk regimes to inflation both from the forecasters and from consumers. However, surveying the literature a little bit, what emerges is that with some few exceptions, most of the available models to fit and forecast inflation seem to be mainly concerned about the long-run mean or time-barring volatility of inflation, perhaps to understand the persistence of the process. And all of these basically rests on the more or less unspoken assumption of conditional gaussianity of these models. And therefore, from the literature, there seems to be like a little bit of a scarcity of understanding about the broad risks to inflation, perhaps their dynamic, and also to some extent how they relate to macroeconomic conditions or variables. So what we really after in this paper is to sort of give you an idea of what this looks like over the post-war period in the US, trying to basically fit time-barring distributions to the inflation process, and also having in mind that risk does not only change, for example, like at these frequencies, but can also change over different frequencies. To make this point, let me show you here. I take core PC data for the US from the mid-60s to 2020. I split the sample into recessions and expansions, which is truly like what is not a recession in my case. And I fit a kernel distribution to it, nothing fancy. However, the results show that this two sort of, this different phase of the business cycle behave like substantially different. In expansions, we have a very nice distribution, which is centered around three in this case, and it has this very long, gray tail. Whereas during recession, what happens is that inflation seems to be a little bit all over the place. So what really dominates this distribution, as you can understand, is the variance. Now, of course, the first thing that should come to mind is that, wait, in this split, what you're putting together is actually like the high volatility, the high inflation of the 80s, and then the sort of deflationary bias period of the 2000s. So here, I redo the same split, but I actually split the sample over the 2000. So I want to understand what changes, for example, how this distribution look like during the high volatility period and in the recent times. And you see that the distribution now look quite different. So the blue one, which is the early part of the sample, 65 to 2000, has a very high mean, consistent with the fact that the inflation was very high. And it has some positive skewness, so longer, right tail compared to the left one. However, what is very interesting, what I believe is very interesting, is that if we then focus on the second part of the sample, most of these moments seem to have changed dramatically. The mean is well below the 2% target that we would like to have inflation. And this is consistent with the idea that I mentioned before, like the strategy review was mainly spurred by, partly spurred by this data here. Standard deviation is low, but especially skewness as flip sign. So the distribution, the sort of unconditional distribution of this data actually now seem to have negative skewness rather than positive skewness. So if we take these two pictures together, what I want you basically to take away from this is that inflation risk seem to vary not only at higher frequencies of business cycle frequencies, but seems to like vary also slowly over longer periods of time. And this is something that we will try to tackle into our estimation strategy. So let me get into what we do and some preview of the results. So as I said, we are trying to measure the evolution over time of inflation, especially of inflation risk. And we do that by employing a flexible time-varying parameter model. This is something I developed in an earlier paper with Ivan and David who's here in the audience. And the first key result that we find is that inflation asymmetry varies substantially over time. Or if you prefer, the inflation process seems to feature time-varying skewness. We also try to understand what drives these processes, especially the risk process, so the skewness process. And so we try to add some exogenous predictors to the law of motion. I will show you in detail how we set everything up. And what comes out is that risk that we think of as the combination of variants and skewness seem to be mainly related to policy regimes, to changes in policy regimes. And then lastly, through the lenses of our model, we basically can understand how some of these predictors that we use sort of feed into the forecasting process of expected inflation. And this is really a byproduct of our modeling strategies. And especially, I will show you something about the Phillips curve. And what we find is that these relations seem not to be stable over time. And really, again, seems to be related to the level of the perceived risk in the inflation outlook. So before I move on, so I just want to give three main takeaways that we were really thinking hard with Leonardo in terms of how to frame all of this to be relevant for policy. First one is that we find that monetary and fiscal regimes are what really matters to understand the long-run dynamic of inflation risk. And I will show you later that this picture shows up quite nicely in one of the decomposition that I will have. So because there is this dependence on the prevailing policy regime, what we argue is that there is no one-size-fit-all policy framework to stabilize inflation. And why is that? Well, this is because, as I will hopefully get to show you in a very simple example, higher-order moments in the inflation distribution should actually be taken into account when thinking about make-up strategies based, for example, on the average inflation targeting. So these are the three sort of takeaways that we want to highlight in terms of how do we relate our findings to the policy process. Let me now dive into the model specification. So the model in its first part is quite simple. We have a measurement equation for inflation where mu t is a time-varying location. So you can think of this as the central tendency of a distribution. And then epsilon t is an innovation which is distributed according to a QT distribution. And then we have sigma t and rho t. And these are like a time-varying scale and a time-varying shape parameter. So the shape parameter really sort of captures the asymmetry of the distribution. And so you can think of these three parameters to map more or less linearly into the mean variant, mean standard deviation, actually, and skewness of the distribution. A nice feature of this distribution that we use is that we have a very actually well-behaved and nice likelihood function from which I just want to highlight that basically almost all the QT that we are now used to since the last few years, it then compasses like symmetric distribution, like the t, but especially the Gaussian. And so this is to say that even if we will have time-varying parameter, the model is free to actually reject completely the presence of time variation in any of the moment, especially skewness if that's what you might be worried about and just collapse to a conditional Gaussian model. How do we allow for time-varying in the parameter? Well, we place ourselves basically within the score-driven setting. We have Simeon here, which is one of the main, the first author that proposes this framework. And the idea is that we have these three parameters. Actually, we are going to, for like easiness of the estimation, we're going to model like the log scale and a transformation of the asymmetry parameter. And we're going to allow this score-driven law of motion, which is really like a random workflow of motion for the parameter, plus some additional predictors. And I will talk about this in a second, from which you can see I split between this x-bar and x tilde. And this is because we will have some predictors for the long-run process and some predictors which seem to be more related to shortly fluctuation in the inflation process. And then this ST, which is what is generally called the scale score, to give you an idea sort of captures the unexplained time variation in this process. And this is nothing more than a rescaled version of the first derivative of the log likelihood function with respect to the parameter of interest. I mean, this might be like not intuitive at first, but I mean, these models seem to work very nicely. They are very nice applications that are using this model. And this is the way I always rationalize the role of the score to give a sort of an easy interpretation. That is, the scale score simply maps the prediction error into an appropriate update for the parameter of interest. Why do we believe that this framework is actually like a good one within our framework? This is because our update mainly has two interesting features. The first one is the outlier discounting mechanism. This really comes from the t-distribution. And the idea is that our model, sort of whenever it faces an outlier, decides how much it wants to learn from it. So if something, if an observation is quite unusual, the model is able to downplay it. And so it's not going to incorporate a lot of noise. But whenever the model understands that there is perhaps like a change in the level of the parameter, actually the model is going to go in that direction. It's not going to treat it anymore as an outlier. So this seems to be like a first nice feature. The second one concerns mainly the updating of the symmetry parameter. And the updating of the symmetry parameter means that here we have on the x-axis like the standard dice prediction error, so 0, 1 variable. Here we have this score variable. And what you see is that in the case in which we have positive skewness, that is this green light, so rho is positive. We are mainly expecting positive observations. However, if the model then sees something that is negative or even deeply negative, the model will want to update the symmetry quite strongly. The data is telling you that there is evidence perhaps for a change in design of skewness. So the model wants to pick this up, incorporate it into the parameter dynamic, and then 1. So the idea that what I want to highlight from this part is that the model is really able to capture turning points in the underlying series quite quickly. And this, I believe, is a strong feature of our model. We have a very nice closed form expression for the expectation of this distribution, so for expected inflation. And this is basically like a linear combination of the location, so again the central tendency of the distribution, and then this nonlinear function of the scale and shape. So in this term, basically, you will have volatility and skewness, so to speak, interacting a nonlinear way. And so what we are doing in reality is that the model starts from a linear dependence between predictors and parameters and transforms it into nonlinear relation between predictors and the moments of the predictive distribution, which is what we are really after here. And this is going to come later when I will show you something about the Phillips curve. OK, let me talk about predictors very briefly. Nine minutes, OK, predictors very briefly. So we want to investigate these nonlinear relations between inflation moments and some well-known predictors of inflation. We consider some short-term parameters, like monetary policy stance, the unemployment gap, measures of the acyclical measure of unit labor cost, and then we have commodity price, which includes oil, so we put all together, and the real exchange rate. And then we consider some variable for the long run. And this variable are like money growth, the measure of the fiscal stance, the trend of unit labor cost, and the long run real rate. And I will show you in a second what we apply a filter to actually do to this raw data to try and extract like smooth trends to relate to the dynamic of inflation in our model. And we use like the Mueller and Watson filter, which basically what it does, it decomposes like the long run dynamic, in our case at frequencies like of longer than 10 years, by basically creating a linear combination of sign and cosine functions. So we will have something very smooth. Let me show you what I mean when I talk about this decomposition. So here I give you an example. This is just one of the variables that we use is the long run real rate, which we find quite interesting. This is the long run real rate is the black line in this picture. And then I take, for example, like a model-free measure of skewness of time-varying skewness from the data. In this case, I use like a five-year rolling window quantized skewness. We reproduce this for like with sample skewness. You can use different rolling windows. You can use like different quantized skewness. You will still get like a pretty much the same picture. And what you can see here is that at the beginning of the sample, there seemed not to be like much of a correlation between the two variables. But things seem to like move together like much more in the second part of the sample. So because we believe that the long run real rate can create some long run, can actually give some information about inflation risk over the long run, we want to understand what this is trend. This is what we do. We fit like a series of sine and cosine function to the series. We do the same for the red line. And again, we find that here the dynamic does not seem too much very well. But here, there seem to be like some sort of predictive relation between the black line and the red line. And what we are trying to do next is basically to exploit this evidence in our model. If you're curious about the other variables, I mean, we also plot like an in-sample long run covariability, as the authors call it. And we see that basically among the four variables that are left significant in our panel, actually like three of them seem to be like strongly correlated with the skewness, which is really what we're after. I will actually skip this in the interest of time. Results. So I'll skip this. Anyway, we go Bayesian so that we fit with most of the papers so far. Here, I think our first model check that we always need to have a look at is do our first and second moment that we all know makes sense. And I think it pretty much, they all look like nice. So the mean process seems to be very in line with what you will get with a standard stock and Watson model, where you see that basically most of the time variation is picked up by this red line, which in our case is what we call like the trend. That is basically the time-varying mean if we exclude all the short-run predictors. And we see that basically there are meaningful deviations only here in the 70s and 80s and perhaps something here before the 2000. Inflation also seems to behave like as expected. We have strong peaks in the 70s and 80s. But then we capture this dynamic consistent with the great moderation where basically in the mid-80s inflation starts to go down and remains quite slow until basically at least the great financial crisis. I think this is the most interesting plot that I would like to discuss. And this is a measure of time-varying skewness of the US core PC from the 60s. I mentioned before in the introduction that we talk about. I use the word regimes a lot. And this is because what we find is that skewness seemed to move in sort of regimes or a regime-like pattern, where basically we have high skewness, as one would expect in the 70s and 80s, this starts to go down from the mid-80s to basically in turns negative here in the early 90s and stays negative ever since until the last period. The last period here, we see that there is this uptick of this longer skewness even though total skewness seemed to be negative. This perhaps going to be puzzling. How can we estimate negative skewness in these last two years? Well, this is because you should think of skewness about the mean. Our mean is already very high. And so the model is telling us there is not much historical evidence for inflation to go even higher, but perhaps it's more plausible that it's going to come down at a certain point. I'm going to also show you why we think that that is the case. So here, I decompose the time variation of the asymmetry parameter. So this is not the skewness, but it pretty much relates to that into the contribution of the long-run parameters of the long-run predictors. And for example, here you can see that there is a prevalence of unit labor cost pushing inflation up during the 70s and 80s. This is pretty much consistent with actually historical observations. And then we see that most of the reason why skewness is coming down is because of reducing long-run real rate and negative fiscal surpluses in this period of time. And in recent time, we see that, and I would be more specific about this later, we see that there is a negative drug coming from, for example, the long-run real rate that, if you will, is a proxy for the zero lower bound. So what explains skewness to be negative in this time? Especially in the very last part of the sample, this is pretty much like the zoom in that you saw before. You see that monetary policy in our setup, which is this purple, which is like in purple, seem to be actually exerting some negative drug to skewness. So to some extent, what we see from the data is that monetary policy seems to actually go in the right direction of trying to reduce the upside risk to inflation. All right, so I have a minute. I want to show you this. So in line also with some of the results that we saw before about the Phillips curve, we cannot really devise a full-blown Phillips curve as we have seen before, but we can understand the elasticity of the expected inflation to some of the predictors that we include. And this is easy because, as I show you, the equation for the expectation is quite simple. We simply differentiate it with respect to XT. XT, in our case, is going to be the unemployment gap. And here, we basically estimate a measure of time-barring Phillips curve. And what we see is that this relation seems to be strongly negative in the 80s and 90s, and it flattens right after during the early 90s and all around the 2000. And there seems to be some evidence of the Phillips curve resurrecting in this last time. Why this happens in our model? Well, let me try to explain this as fast as I can. In here, we have the Phillips curve coefficients that are highlighted in these colored sort of scatters. The magnitudes are these iso-quants. And here, we have the asymmetric coefficient and the variance, the volatility. And basically, what comes up is that the Phillips curve relation seems to be relevant only when risk is very high. Whenever there is not much risk about inflation, the Phillips curve relation seems to be much more muted. And this is something that we find interesting, and we believe that can be informative about how policy makers think about the Phillips curve. We can also deal about the balance of risk. I'm going to skip that. It's in the paper. And so I will skip also this. And we basically model time variation in skewness. We find that it relates to many historical facts about US inflation. And I look forward to the discussion from Julia. Thank you very much. The discussion is Julia Sjärnbult from Bilti University. Hi, I'm Sjärnbult. Right. Yeah, thank you very much to the organizers for inviting me to discuss this very interesting paper today. And thank you, Andrea, for this nice presentation. So this is a cool paper. I'm going to start by giving an overview. So broadly, the topic and the goal of the paper is to model and analyze the dynamics and also the drivers of US inflation since the World War II, so basically across a long time period with many shifts in market conditions but also shifts in monetary and fiscal policy regimes. The methodology could be described as a score-driven, non-linear trend cycle model for three of the parameters of the conditional distribution. So the idea is to take the location, the scale, and the asymmetry parameters of the conditional distribution and have them and give them a trend cycle specification. And this is also then enriched by observed regressors that then help to predict and to disentangle long-run and short-run dynamics of the inflation, the conditional inflation moments. This model can be used for policy. So I think this is actually really relevant for monetary policy because the findings can also be interpreted in a quite rich way. So one of the findings that Andrea didn't have time to present was these shifts in the balance of risks, what they call balance of risks over time, implying that there is not one particular monetary policy or also that exists for all the different regimes, as you called it in the sample. Also, they find that fiscal policy, the fiscal policy stands, plays an important role, especially for the long-run dynamics. And the example of the time-varying slope of the Phillips curve, which is also in line with what Simian presented earlier today, is also one of the features of the model that makes a lot of sense and can be interpreted and used. OK, so the model has been presented, of course. It just always helps me to look at the equations to really understand what's going on. And I have to say I had to go to the earlier paper, the methodology one with Davide. So we have the observation equation here, where we then have these three time-varying parameters, mu, sigma, and rho. And they are, again, decomposed into a long-term and a short-term component, each that are then driven, that have a score-driven specification, dynamics. And these matrices A, B, and C here are restricted coefficients that allow to identify this decomposition into long and short-run dynamics. And S, again, is a scale score that serves as an innovation to those time-varying parameters here. So here, what is really intuitive and interesting, I think, about this approach, is that you can basically obtain conditional expectations for inflation that incorporate the other features of the distribution apart from the location parameter as well. And so time-varying scale and time-varying shape parameter enter the conditional distribution. And this also then gives rise to these time-varying elasticity. So the impacts of changes in the regressors will then also, potentially, depend on time-varying skewness, on time-varying asymmetry, and time-varying scale. And this also just comes out of the model specification. So that's why I think it's really nice and gives rise to this rich, has a potential to give rise to very rich interpretations. So my first set of comments is about the model specification. So it's just striking. And I suppose this is not unexpected that I would bring this up, right? So you have four parameters in your skew T distribution, and three of them have dynamics. But the degrees of freedom parameter that one would maybe associate with the cortosis of the distribution is kept constant. And I was, so this is not discussed in the paper. I was just wondering if this is something that is just not supported by the data. Maybe time-varying asymmetry is enough. But it would be definitely interesting to learn what was the motivation to keep the degrees of freedom constant. So what I presented earlier was this conditional expectation. Also, under other conditional moments, they are features of the model of this model that is fully parametric. But there are also some choices. So it seems very flexible, but there are also some choices, not only leaving the degrees of freedom constant, but also, for example, leaving beta constant, which is the vector of coefficients related to the regressors. And especially since you were talking a lot about shifts in regimes, one could also, I just wondered what happens if you would merely split the sample, for example, and whether the time variation, this massive time variation that you pick up in the moment, is maybe also related to time-varying impacts of these different groups of regressors. And also, again, a somewhat related comment. So I'm not entirely sure in what the relationship between the model in this paper is and the model in the earlier paper. But in your equation here, you basically so in equation 7 of the paper, and I think you presented another version of this earlier as well. So there is basically the short-run dynamics or the short-run fluctuations are only driven by those short-run regressors. So the xt tilde, as you call them. So that means that if there are some xts that you may be missing in the model, they all go into the long-term component. So I think in the other version of the model, there was a score in the short-term dynamics as well. And I was just wondering what happened to that. And maybe also related to the earlier presentation, maybe there are some short-term dynamics or some short-term regressors that, especially in this recent period, have become more relevant, such as supply side shocks in particular that are not. Yeah, I mean, you have the commodity prices, but maybe there is also some more, maybe, from that that could be included. Yeah, all in all, I would like to see some diagnostics. So I mean, this is a really nice model. And it has a great interpretation if it's the right model. So if we look at the time variation, if we look at those dynamics, and yeah, do we want to believe them or not? I think it depends on what model diagnostics would look like, especially. So for example, we could look at the scores. So if there's some patterns in the scores under correct specification, we would have martingale differences for the scores of the model. Another way would be to do an out-of-sample forecasting exercise. So some more minor comments about the terminology. So I was a bit confused at times what exactly is the difference between inflation risk versus volatility, skewness, shape, asymmetry, things like that. So it would be nice to fix the terminology clearly from the start. I was also, yeah, so this is also a notational thing, and I'm out of time. You condition on the past of inflation here and your conditional expectations. And I was just wondering, so you should condition on the regressors as well, I think, right? So that, yeah. And then you provide the analytic closed-form expressions for conditional mean and variance, but not the skewness. And I was also wondering if that, why that is? Is there a question? I was wondering if that, why that is? Because that would be especially interesting since we're all looking at the plot. And then all in all also clarify the relationship to the earlier model. And those were my comments. So super interesting, very useful, and some model diagnostics would be interesting to see. Thank you. Questions from the floor? Thank you. It's, again, Blažek Mazur from National Bank of Poland. I congratulate you on the excellent paper you're on co-authors. I like it very much. However, I would like to ask a couple of questions. Well, first, it seemed to me, I'm not sure, but that you missed that autoregressive part in this score updating. So I would like to ask, what if you include this autoregressive part and why you prefer not to have it, right? And second, what if you, instead of core, if you look at total inflation? Because could you also specify what kind of? Is it just quarter on quarter blown up to annual? Or is it year on year inflation, right? And the last one is the answer to the question about degrees of freedom. I mean, degrees of freedom reflect rare events. And it's extremely difficult to figure out short run dynamics for something that rules rare events. Because then if you have a count period, they will mean revert. And then you see some rare event that kicks in, right? So I totally agree with keeping constant degrees of freedom in this distribution. Thank you. Any questions? This is Davide. And there are just a few questions about the specification. I'd like to see what the contribution of the X variable on driving short run and long run, which is a sort of like refiltered data. So I think a sort of robustness analysis could be that you first fit the model, like our original paper without putting the X variables. And so sort of like it's completely data driven. And then check if there is somehow a square with your predictors. Otherwise, it seems that it's too much driven by your pre-filtering operation. And then it's also to show how the various driver that you put in your model specification are contributing to drive the asymmetry versus the volatility, right? So then final question is more applied, since probably you use US data with Leonardo. Seems that there's much role for energy prices. For us, it's much more important. Thank you, Andrea, for the very nice presentation. I guess I have a couple of questions. One, I guess, is related to what Davide and the other question you received. So correct me if I'm wrong, but you have an underwall plus something else. And then there is the score. And then you try to separate it long and short run effect. But in principle, if you have very outlier, you update the parameter, it's around the walk, so it's already a super persistent shock. So I was wondering, can you really avoid the out-of-aggressive part if you want to separate long and short run effects? That's the first question. The second one is about if you use different measures of inflation, I guess would be interesting for instance, if you exclude like you look at the core CPI, you exclude food and energy, does this curious really matters that much or is mostly volatility, if you can elaborate on that. Thanks. That's it. All right, so thank you very much for the nice question. Thank you, Julia, for the nice discussion. So I'll try to address all the points. So time, but in degrees of freedom, I think you already had like a sort of good answer. I mean, we did try to feed it. We did try also before in the other paper. It's not always nice what comes out. It doesn't really help like for the distribution. Like the other moment does not change as much, but it makes the estimation like much more difficult. So we did not really see like much scope for that, but that is clearly like a sensible point to address in the paper. And then like time varying betas or like the sort of the static parameters of the model, especially the betas, that makes absolute sense. We were actually thinking in that direction already to understand like breaks, for examples, in this Philip Kerr relation, not only due to the parameters of the model. We were like reluctant at first because we would like an extra layer of sort of complexity to the model that is already like not very straightforward. But this is absolutely like a well taken point. We were thinking about this. And the same goes for like forecasting. We are working on that. We have an issue that has been like sort of flagged by David as well with this prefiltering variable. We want to sort of try to still say something about like long-gram predictors, long-gram predictions trying to use like long-gram predictors and that creates like a little bit of an issue. So this is also something that we are working on. And same with the analytics. I mean, we are putting things together like pretty much as we did in the other paper. So like absolutely like you made like some very good points about sort of robustifying the methodological contributed by sort of part of this paper. And last thing, like why we did not show like an analytical formula for students. We have it. It's just that we were in the paper, we are interested in understanding like the interplay between like the symmetry parameter, so skewness in a sense and the moments that we are most mainly used to think about. So mean and variance skewness would only depend on the degrees of freedom and the symmetry parameter. So you would like lose all the sort of location and scale effect. And so it seems for now that it's like less relevant, but you're right. Like the skewness that I presented is just like fitting the time bar in grow and degrees of freedom into the skewness equation that I have. So the other question is why not what's the other aggressive law of motion. So there are two main reasons that we found like while we were experimenting. First one is that if we add like another aggressive law of motion, even if we try to shrink it to like no sort of persistence so to speak, the parameter would like basically always go to the 9999 or whatever is the upward bound that you would put. So we were just like sort of easing a little bit the estimation because we have a lot of bidas and we were just like putting it to one. Second, because we have the split between like short and long, the short run parameters actually like they already embed a lot of persistence. So in that case, we would like sort of build up like much more persistent that we would like. But yeah, this is something that we've tried. I mean, we would also like revert to the other aggressive law of motion. I know that there is this issue with like sort of the filter like the random work filter, but it doesn't seem to be an issue per se. And we did try, we don't have it in the paper because like this paper originally was motivated by a policy sort of question that we had with Leonardo. So core PC is the relevant inflation measure in the US. We have estimates are reported in the current version for PC headline and CPI core and headline. The image, the picture that you will get is pretty much the same. Actually for CPI, you would get even like nicer sort of skewness dynamics. So I've always been a fun to use CPI, but there is an interest for PC. And then I mean, we also had, we also, I also experimented with Euro area data and then of course the set of predictor must change. I mean, I did not really play a lot with energy data yet. This is again in the pipeline. I mean, the original paper actually was a comparison between Euro area US and Japan. Now we move to the US for this policy reasons, but we are actually planning to go back to that. So hopefully we will have an answer for that. And I don't know if I missed anything. Otherwise thank you very much.