 Perfect. So Manuel, thanks for the great introduction. As you said, I'm basically an outsider to Bitcoin. I think I have some basic knowledge, but I don't really have an agenda. So it just takes us a chance to look at the data and let's see what the data tells us. So Manuel basically paved the way for this presentation. So I don't have to talk about much anymore about the stock to flow models. Instead, I will just dive a bit deeper into the bit conometric aspect. So the whole motivation for this stock to flow model to a large extent is actually comes from this graph. That shows a nice and strong correlation between the stock to flow ratio and the Bitcoin price over time. But the question then really is, is this really just correlation or is there maybe a more fundamental relationship between these two things? So let's start with how this whole analysis began. And suppose we were running just a simple ordinary lease transfer question of the log Bitcoin price on the log of the stock to flow ratio. What we would get would be a highly statistically significant coefficient beta of magnitude about 3.1. So what does it mean? What would it mean? It could mean if the stock to flow ratio goes up by 1%, then the Bitcoin price would be expected to be to go up by roughly 3%. So obviously given that at each half an event, the stock to flow ratio doubles, that would be a substantial implied price increase. But maybe we should be a bit cautious here. Because what we might be running here could just be a so-called spurious regression. The spurious regression problem could arise if we have what non-stationary variables in our model. These are in simple words variables that tend to have a substantial trending behavior over time, which we have just seen in both the price and the stock to flow ratio. Let's give another example. Let's remove the stock to flow ratio from the picture, but instead add the non-EU net migration into the United Kingdom. And again, we see a substantial positive correlation. But I think you all agree with me that this is just a coincidence and no one would probably argue that the UK migration pattern has anything fundamentally to do with the Bitcoin price. So this is really just a spurious correlation. And then the question is, well, with the stock to flow ratio, is it maybe also just spurious or is it more fundamental? Essentially, such a problem that you observe such high correlations could always happen if you have such so-called non-stationary variables. So let's go a bit into the bit chronometrics. What does it actually mean that a variable is non-stationary? A stationary variable, in simple words, is a variable that has a constant mean, a constant variance, and constant autocorverences over time. So if you look at the lower right panel here, that's a typical stationary variable. It just fluctuates over time around this constant mean close to zero. It never really deviates far from this mean and really reverts back to the mean whenever it deviates. So that's what we call the stationary variable. And if you move up to the upper right picture, you see one case of a non-stationary variable because clearly the mean is not constant over time. The mean constantly increases. Essentially, this variable follows a linear time trend. But once we essentially remove this linear time, we essentially get the stationary process down here. That's exactly how these two pictures are related from the top to the below we get by removing this linear time trend. And that's why we call this top-right process a trend-stationary process. Once we remove the trend, it becomes a stationary process. Now let's move to the top-left picture. That process looks actually very similar to the top-right picture. It also fluctuates more or less around some linear time trend. But once we remove this linear time trend, we go down to the bottom-left picture. We see that this process actually takes much, much longer until it reverts back to the zero line here. So it stays away for a substantial period of time from the zero line. And essentially, there's not even a guarantee, even though it happens here by chance, that this process would ever be revered back to the zero line. This is also called a random walk. What's a random walk? A random walk technically is just a process that at any point in time, its value would be determined just by the previous period's value, xt minus one, plus some random shock. So it could just randomly move up or randomly move down. And importantly, any of these shocks that occur essentially has a permanent effect. If you have a positive shock, what does it need to bring us back to the previous level? It needs an equivalent negative shock in the future. Without such a negative shock in the future, we would not be revered back. That's very different to a stationary process. For stationary process, even if you have a positive shock and nothing happens afterwards, this process would gradually sooner or later revered back to the long-run mean. That's not what's happening with the random walk. Each step is really just random, which basically gives this process its name a random walk. And then if you look to the upper left picture once again, if in addition this process tends to drift away into a certain direction, then we would call it a random walk with drift. And what's also very typical about these random walks is that even if you have the same causes in nature, just by drawing a different sequence of random shocks, these processes can look very different compared to the previous ones. While the stationary or advanced stationary processes, even if you draw different shocks, still look very similar compared to before. That's again a distinguishing feature between a non-stationary and a stationary process. So let's look at these two random walks that we've just seen together in one picture. On the left-hand side, we have the random walk with drift. Both went upwards because they have the same underlying drift. And if we just look at their overall correlation over time, it's substantial, 0.74 in this case, mainly because they have the same underlying time trend. The same underlying drift. But clearly, this correlation is completely spurious because these shocks are drawn completely independent. These processes are completely independent. To illustrate this spurious correlation a bit further, let's just separate the whole 1,000 time periods into four equally spaced periods of 250 time periods each. Then we see in the first period, the blue dots, you have a similar correlation than in the overall time span. But if we move to the second sub-peard, suddenly the correlation becomes zero or even slightly negative. If we then move further to the third or fourth sub-peard, the correlation even becomes strongly negative. And here's a problem. If you would then use the second sub-peard, for example, to make a prediction about what follows afterwards in the third sub-peard, you would get it completely wrong because the correlation suddenly changed. Now, the drift is one reason why we have this strong correlation. But even if you look at the random walks without drift, you still see a significant substantial correlation. In this case, a negative correlation of almost minus 0.5. Even so, again, these processes are completely independent from each other. And this just happens because just by chance, these processes exhibit some extensive trending behavior. And it happens that they just diverge in different directions. This can create a strong negative correlation. So this is something you would not observe that strongly, at least the stationary processes. Okay, doesn't mean we can't do anything with non-stationary processes. The general answer is it depends. If the processes are independent, there's really not much we can do. But let's generate one process, the blue one here, as a function, essentially, of the other one. The red process is x. It's, again, just a random walk with drift. The blue process follows from the red process. It inherits the properties from the red process. And we see a nicely essentially co-move with each other. And in fact, it turns out that the blue process is essentially just about one half of the red process. And this ratio remains constant over time. That is what constitutes a co-integrating relationship between these two non-stationary processes. So essentially at any point in time, our best guess for the blue process might be half of the red process if we know the value of the red process. In other words, a linear combination between the two processes would yield a stationary process. That's what we see in the lower right picture. That's the so-called error correction term, the deviation from this long-run equilibrium. And the deviation from this long-run equilibrium is just a nice stationary process. And also, if we look again at the scatter plot, the correlation pattern, we see that now all of these dots nicely align along this regression line. So just please some fundamental underlying relationship between these two processes. Okay, let's go back to the actual Bitcoin data. First of all, we see both variables are clearly non-stationary. They are upward trending. As a trend stationary or really just following random walks, a defining pattern of the stock-to-flow ratio are really these jumps. So maybe let's first separate the jumps, which are completely deterministic. They are just deterministically determined by the halving event, the halving of the rewards for the miners. Let's separate these jumps from the effects of the remaining fluctuations. And you have just mechanically taken out the jumps from the halving effects events from the stock-to-flow ratio. And instead, I will look at the effect as well of the halving of the miners' rewards. Well, the stock-to-flow ratio now, after accounting for maybe a halving period-specific time trend, looks very stationary. It just tends to fluctuate randomly around without deviating much into one or the other direction, which is kind of different for the Bitcoin price. We could apply statistical tests to see whether these variables are actually stationary, trans-stationary or not. And typically these statistical tests, I don't want to talk too much about them, would confirm or at least not reject the hypothesis that the Bitcoin price tends to be a non-stationary variable. And the stock-to-flow ratio, at least after accounting for the halving effects and the potentially different time trends within these halving periods, is likely a stationary process. But eventually, for what I'm going to do, we actually don't necessarily need to do these separate tests for stationary or non-stationary. We could just estimate a model that allows for the combination of both stationary and non-stationary variables. This model explains the log Bitcoin price, not just by the current stock-to-flow ratio as since the very beginning, but it also adds the time lags of the Bitcoin price and the time lags of the stock-to-flow ratio. In addition, I would also like to throw in the rewards to capture the deterministic effects of the halving breaks. Now, adding these time lags in particular of the log price implies that this model can allow for the Bitcoin price to be a pure random work as a special case, which the initial model that we just were running, the compression of the log price and stock-to-flow ratio, couldn't. By allowing this special case, we could essentially distinguish between the situations where the price is really just a random work or where it may be driven by the stock-to-flow ratio. Let's maybe look at a slightly different formulation of the model since it's really just a reformulation of the initial log-to-flow ratio and we've got order-requisite distributed lag model, and this reformulation might be easier to interpret. It says that the change now in the log Bitcoin price is essentially a function of the deviations of the price from its long-run equilibrium, assuming that the long-run equilibrium might exist. And if it exists, this long-run equilibrium would be determined by the stock-to-flow ratio and maybe a linear time trend. So this coefficient beta here is essentially just the same coefficient from an interpretation point of view as we had in the very beginning. In addition, because we had these lags initially in the model for certain short-run effects, it would just tell us whether we could potentially predict the Bitcoin price over the next couple of periods, while the long-run equilibrium really aims to make a prediction more into the future. What happens with our data without showing you the details is essentially that these short-run effects don't really matter. So let's just remove them from the model. The model becomes even simpler. And another key parameter in this model is the coefficient alpha that we didn't get to talk about. So suppose this long-run equilibrium exists, what does the coefficient alpha tell us? It tells us how strongly does the price react to a deviation from the long-run equilibrium. Suppose the price is too high relative to its equilibrium, then alpha will tell us by how much the price then would be expected to readjust downwards towards the equilibrium. So clearly there are two key parameters, alpha and beta, that we need to look at. And for the existence of such a long-run equilibrium, of course we need that beta as non-zero. But we also need that alpha is non-zero because if alpha is zero, then essentially this whole term here drops out. If there's no reaction to a deviation from the long-run equilibrium, then essentially that cannot exist any long-run equilibrium. So both alpha and beta need to be equal to zero. Fortunately, there is a nice statistical test that we can apply and that test allows for stop-to-flow ratio in particular to both stationary or non-stationary. And this test we could use to check whether alpha and beta are equal to zero. What happens with the data here is actually that we cannot reject the null hypothesis that alpha and beta are equal to zero. This would be evidence that such a long-run equilibrium does not exist. So if it doesn't exist, we could just remove this term. What's left is just a simple model, very similar to what Manuel just showed us with his two dummy variables. Instead of the two dummy variables, I essentially just have the log-rewards. The nice thing about the log-rewards is, instead of the dummy variables, because we know how the rewards look like in the future, that's just deterministically given, we could still use this model to make some predictions, some forecasts. Let's first estimate this model. Again, I don't show you the numbers. What would happen or what we would find is, again, similar to Manuel's results, that essentially this coefficient gamma is not statistically significantly different from zero. We cannot establish any effect statistically of these log-rewards on the Bitcoin price. And if you then remove the log-rewards from the model-rewards left, it's really just that the log-price depends on its price in the previous period, plus possibly some drift. And this drift term would then be statistically positive, so we would have a random walk with drift. Now let's assume for a moment, and that will be a last picture that I want to show you. Let's assume for a moment this gamma is not zero. And in fact, whenever you estimate the model gamma is usually not exactly equal to zero, we can just not statistically reject the hypothesis that it is equal to zero. So let's suppose we forget about the statistical significance, take the point estimate at face value, and make some predictions. And I warn you ahead of the next picture, what I show you now is not to be taken too serious, basically because this effect is not statistically significantly different from the end. What we get is something like this. The gray area is essentially the drift over time, adjusted for the halving period, given by the halving in the log-rewards, but the halving in the rewards. So the boundaries are really just arbitrary, just picked them based on the peak and the drafts. But the direction essentially indicates what this model would predict for the future, the drift of the Bitcoin. And we see it would essentially reverse and go down again. If you like, you could actually add rainbow colors, then you would get a rainbow that touches it twice, as many rainbows do in reality. But once again, as I said, don't take this too seriously. This coefficient is essentially not statistically significantly different from zero. And what we have is just a random walk with drift, and the prediction for this random walk with drift would just be given by this blue triangle. And by the very nature of a random walk, it's essentially unbounded above and below. It can easily go through the roof, can easily go down through the floor, or it could just walk around somewhere in the middle. And that's what I want to finish, and I'm happy to discuss further now and in the panel session later.