tag:blogger.com,1999:blog-5908830827135060852.post6235225925539946515..comments2020-02-26T02:28:51.160-05:00Comments on Bond Economics: Models Are Not Frequency InvariantBrian Romanchukhttp://www.blogger.com/profile/02699198289421951151noreply@blogger.comBlogger41125tag:blogger.com,1999:blog-5908830827135060852.post-49747744685378697142016-03-18T11:41:46.665-04:002016-03-18T11:41:46.665-04:00Hi, Tom. :)
Yes, I did find that equation in the ...Hi, Tom. :)<br /><br />Yes, I did find that equation in the text, along with their explanation of it.<br /><br />I haven't been responding because of illness. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-1793291992110312312016-03-17T13:32:51.289-04:002016-03-17T13:32:51.289-04:00Also, I took another look at Jason's ΔH = Γ*(G...Also, <a href="http://informationtransfereconomics.blogspot.com/2016/03/more-on-stock-flow-models.html?showComment=1458192351122#c1952505938711252475" rel="nofollow">I took another look at Jason's ΔH = Γ*(G - T) expression</a>, and it's pretty clear what Γ represents in a state space description of the model... which explains his <a href="http://informationtransfereconomics.blogspot.com/2016/03/more-like-stock-flow-in-consistent.html" rel="nofollow">Update 4, part 2 and 3 here</a> in another way. Exemplified by his figure <a href="https://1.bp.blogspot.com/-EOE_ag28n0Q/Vtn7-en8v8I/AAAAAAAAJFM/1Mu3YObYe2c/s1600/stock%2Bflow%2B0.5.png" rel="nofollow">here (with Γ=0.5)</a>.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-12785973764372756032016-03-17T13:26:53.262-04:002016-03-17T13:26:53.262-04:00"for example a lag creates an infinite dimens..."for example a lag creates an infinite dimensional state space"<br /><br />Yes, but as I recall from my robust controls class, a Padé approximation can be useful there.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-19859865771437715622016-03-17T13:18:20.390-04:002016-03-17T13:18:20.390-04:00You need to convert to discrete time in order to m...You need to convert to discrete time in order to make the model coherent to observed data. For toy models that you are not fitting to data, that is not an issue, but sooner or later you need to worry about that.<br /><br />You also face a lot of modelling problems in continuos time - for example a lag creates an infinite dimensional state space. The only real analytical tools that work with infinite dimensional systems are frequency domain, which precludes nonlinearities.<br /><br />These are not simple systems; there's a lot of deep mathematical issues that are faced by continuous time models. Does a solution exist, is it unique? Etc. Many of these problems disappear in discrete time. Relying on numerical simulation just hides the discretisation within your software, and you still do not know the properties of the systemBrian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-55540428120024056602016-03-17T12:32:54.029-04:002016-03-17T12:32:54.029-04:00Or you could just keep the continuous time model w...Or you could just keep the continuous time model without any extra terms in b. Then you can do an observability and controllability analysis (easy in this case), do pole and zero placement with a fixed feedback law, add noise to the analysis, design a Kalman filter, design a time varying optimal feedback control law for various objective functions or design a robust control law to achieve a minimum performance level over a whole set of plants that could exist due to our uncertainty about the plant. These things could be done in either discrete or continuous time, but the model is simpler in continuous time. ... and I'm sure you already know all that. :DTom Brownhttp://www.google.comnoreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-84082403363737940462016-03-16T19:02:58.527-04:002016-03-16T19:02:58.527-04:00Using the Taylor Series to encode information is a...Using the Taylor Series to encode information is a way of sneaking higher frequency data into the model. You would not be able to do that if all you have is quarterly data to begin with. A "quarterly" model is one that is applied to quarterly data (only).<br /><br />Think of how you would apply this to real world data. Critical data like GDP components and Flow of Funds data are only available at a quarterly frequency. (They are critical for a SFC model, at least.) Even though you have other monthly data, how would you convert the quarterly data to monthly? The errors created by making up your own monthly GDP (relative to a hypothetical "true" monthly GDP) is going to be larger than the errors created by sampling the dynamics. <br /><br />The fact that a lot of data is only available at relatively low frequencies (quarterly...) is why I have a hard time worrying about this issue in the first place.<br /><br />[OK, Canada has a monthly GDP figure, but it is only a subset of the national accounts; a model builder would need all of the GDP components, like investment, which are only available quarterly.]Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-48537149830700729802016-03-16T16:56:59.937-04:002016-03-16T16:56:59.937-04:00Yes, it's a problem if G changes constant leve...Yes, it's a problem if G changes constant levels on a monthly basis and you're sampling quarterly. As you mention in the post you'd essentially have to keep track of each monthly change in the last quarter... which <i>almost</i> amounts to sampling monthly and then down-sampling (I treat a similar case in my <a href="" rel="nofollow">SIM4</a>).<br /><br />However, if G can be approximated with arbitrary accuracy by a finite Taylor expansion over the last quarter, and you have access to the relevant derivatives... then you can expand my B from a scalar to a row vector and handle that case as well in a sample period invariant way w/o having to do monthly sampling. Contrived? Perhaps... :D<br /><br />Or, like you mention, <a href="https://1.bp.blogspot.com/-Rj8IG9i17QQ/VunHu1WHJ-I/AAAAAAAAAxs/RERW_WGqlgEytze1jhTKtCCPGum8O-qPQ/s1600/h_continuous_SIM_equation.png" rel="nofollow">solve this equation numerically</a> for general g' = dg/dt, where g = integral of all government spending. You'd of course replace 0 with t0 and t with t1 (the start and stop times of your sample period, resp.).Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-25055695334597260952016-03-16T16:14:49.629-04:002016-03-16T16:14:49.629-04:00I have not had a chance to look at this. Yes, you ...I have not had a chance to look at this. Yes, you could match up to a steady G, but what happens when you change G on a monthly basis? The quarterly series only has the sum of the monthly values to work with, which is the effect that I am describing in the article. <br /><br />I am now looking at the equilibrium notion within G&L. They have different notions about expectations, and the dynamics are slightly different for each. I believe that you will definitely see some differences if you drop the "perfect foresight" assumption; if you are adapting to errors, the adaptation is quicker on a monthly basis. Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-17544175168671817242016-03-16T15:22:06.970-04:002016-03-16T15:22:06.970-04:00alpha1_0, alpha2_0, and Ts_0 can be any initial va...alpha1_0, alpha2_0, and Ts_0 can be any initial values actually, not just those specified by G&L, and the change to a new sample period Ts should work with the above. However, if you want to match their results then stick with G&L's values.<br /><br />I did not attempt to make this work with any other functions for G other than a scaled unit step. You can choose G&L's $20/period to match theirs, or any other. More general <a href="http://banking-discussion.blogspot.com/2016/03/sim4.html" rel="nofollow">G functions addressed here</a>. But even with the spread sheet above, you could make it work for G at a constant rate per period, by typing over the G values in the calculation table: so long as you matched the function precisely: for example originally you may have G be $20/period for period 1 and then $40/period for period 2. If you halved the sample rate to 0.5 periods, then you'd leave G at $20/period (=$10/half-period in the spreadsheet) for the 1st 2 sample periods, and then switch it to $40/period (=$20/half-period in the spreadsheet) for the next two, etc.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-71066548372455344002016-03-16T15:10:18.614-04:002016-03-16T15:10:18.614-04:00I did find a closed form way (SIM6) to adjust both...I did find <a href="http://banking-discussion.blogspot.com/2016/03/sim6-updated-sim-to-preserve-time.html" rel="nofollow">a closed form way (SIM6)</a> to adjust both alpha1 and alpha2 for different sampling periods with SIM so as to keep it truly sample period invariant and so that it satisfies all of G&L's accounting expressions at all times. Of course I'm assuming continuous compounding. It's curious to me that the third parameter (theta) did not require adjustment... I wonder if there are a whole set of solutions with different thetas.<br /><br />Basically you make these calculations: defining Ts_0 := the original sample period = 1 "period" (I call this period a year):<br /><br />First calculate an "original" A_0 and B_0, from G&L's original alpha1 and alpha2 (called alpha1_0 and alpha2_0 respectively) and theta:<br /><br />A_0 = 1-theta*alpha2_0/(1-alpha1_0*(1-theta))<br />B_0 = 1-theta/(1-alpha1_0*(1-theta))<br /><br />Then for the new desired sample period Ts, calculate:<br />T_ratio = Ts/Ts_0 = Ts<br /><br />Then:<br />A = A_0^T_ratio<br />B = (B_0*(A-1)/(A_0-1))/T_ratio<br />alpha1 = ((1-B)-theta)/((1-B)*(1-theta))<br />alpha2 = (1-A)*(1-alpha1*(1-theta))/theta<br /><br />You'll find that the time constant doesn't change:<br />Original Tc := Tc_0 = -Ts_orig/LN(A_0)<br />New Tc = -Ts/LN(A) = Tc_0<br /><br />And that all the rest of the parameters (calculated only in terms of alpha1, alpha2 and theta) produce outputs (Y, T, YD and C) satisfying all of G&L's expressions.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-58547216327049984162016-03-15T13:20:54.385-04:002016-03-15T13:20:54.385-04:00BTW, my model above:
h' = a*h + b*g'
if ...BTW, my model above:<br /><br />h' = a*h + b*g'<br /><br />if stimulate with a Dirac delta in g' does produce an instantaneous step up, followed by an exponential (rise or decay, depending on if a > 0 or a < 0, resp.) with time constant 1/|a|. Only if a = 0 does it produce a true step, with an infinite time constant. But whatever it does, it's literally the "impulse response" of the system, and it completely characterizes it. b just plays a scaling role.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-21247790765351875312016-03-14T22:02:26.685-04:002016-03-14T22:02:26.685-04:00Sure, some of them got it right. But the question ...Sure, some of them got it right. But the question is whether that was the result of the theories they follow, or common sense overriding theory. That said, Simon Wren-Lewis is probably the best on fiscal policy.<br /><br />From what I saw, a lot of the justifications I saw on mainstream blogs were pretty sketchy("the magical zero bound changes everything!"). (As for J.W. Mason, I like his work, but I am not sure that I would classify him as mainstream....)<br /><br />Some of the mainstream authors are worth reading, but for myself, I find that I read them less and less. I have read enough of their work to know what they are likely to say on a topic, and there is absolutely no chance that they would ever address my concerns about the blind spots in their theory. I would rather advance my own research agenda than attempt to respond to the controversy du jour.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-55362764922966442342016-03-14T19:41:55.810-04:002016-03-14T19:41:55.810-04:00What about people like Simon Wren-Lewis, Paul Krug...What about people like Simon Wren-Lewis, Paul Krugman and Brad DeLong?... and perhaps Miles Kimball and J.W. Mason? What was their record concerning austerity in Greece? Aren't they all fairly mainstream?Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-6065783988254365282016-03-14T17:27:20.906-04:002016-03-14T17:27:20.906-04:00There were a lot of mainstream people talking abou...There were a lot of mainstream people talking about Ricardian Equivalence, and how there would have been no effect from fiscal tightening. Not all agreed, but they were not necessarily actually using what their theories allegedly say.<br /><br />Outside the mainstream, some Austrians also think that government spending cuts are good for growth, but otherwise, they tend to have a similar view to PK economists with regards to fiscal policy.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-74441432030553678562016-03-14T16:56:19.689-04:002016-03-14T16:56:19.689-04:00What you describe about a useful model for Greece ...What you describe about a useful model for Greece sounds like a 0th order: thumbs up or down. PKE was not alone in that evaluation, was it? Tom Brownhttp://www.google.comnoreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-12025652665057257492016-03-14T15:05:31.605-04:002016-03-14T15:05:31.605-04:00re: continuous compounding rate. There's a hug...re: continuous compounding rate. There's a huge number of interest rate conventions within fixed income mathematics; every market has its own compounding convention. The "continuous compounding" rate allows us to convert amongst the different conventions. But under all conventions there is no interest based on "intraday loans"; all interest is based on end-of-day settlement. (And note that the end-of-day might be different for different segments of the fixed income universe...) (Perhaps loan sharks have intraday loans, but I doubt that they worry about their interest rate quote conventions...)<br /><br />My point is that other than as an approximation, you cannot see a unit step function as a legitimate flow in a model. There is no circumstance under which an entity continuously pays another entity infinitesimal amounts of money; the flows are discrete jumps that occur during the end-of-day settlement. Feel free to approximate these flows with continuous time flows, but there is no legitimate complaint that a discrete time model does not align exactly with such an *approximation*.<br /><br />As for falsifiability, it's a big subject. My complaint about the DSGE framework - as it is used in practice -- is non-falsifiable. You could search my posts to see why I say that; it's related to how the natural interest rate is calculated. As for post-Keynesian approaches, I believe that the authors are more honest about what can and cannot be done with models.<br /><br />One can distill some post-Keynesian views as saying that we cannot expect to be able to predict the economic future. Although that's a fairly depressing negative result, the general failure of mathematical forecasting models is consistent with that view. The only way to falsify the PK view is to come up with an accurate forecasting methodology...<br /><br />But we need to be careful about making statements about the quality of the data. The destruction of the Greek economy by austerity policies was predictable; one was not able to to pinpoint how big a disaster it was going to be, but one knew it was going to be bad. Therefore, we have a useful recommendation for policy -- don't be idiots like the European policy makers. I certainly think that we can use economic theory to be able to make statements like that. That's the sort of question that we need to answer; generating a probability distribution for the next quarter's GDP result is not actually that useful, but that is what the "scientific" approaches to economics want to answer.<br /><br />Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-69789239762293466512016-03-14T14:34:27.564-04:002016-03-14T14:34:27.564-04:00"Whatever log-linear models are, they are not..."Whatever log-linear models are, they are not models of economic systems that obey accounting identities."<br /><br />Well, OK. It seems like they might still be something useful though (I'm far from being able to judge!), especially if you can demonstrate they have some empirical validity (I'm not saying you can show that!).<br /><br />"continuous flows are never, ever, observed in the real world"<br /><br />Well, banks do offer continuous compounding though, don't they? Isn't that a kind of continuous flow?<br /><br />"other than by forcing all of the flows to be a comibination of Dirac delta functions. In which case, you cannot have nice clean exponential response functions in your dynamics"<br /><br />Well, OK, but a Dirac delta in g' in this model:<br /><br />h' = a*h + b*g'<br /><br />produces a step in h. I'd say the dynamics of h are still described by an exponential with time constant 1/|a|, regardless of the forcing function g'. The time history of h may not always look exponential, e.g. if g' is a sinusoid with frequency f, then h is too of course. Maybe it's just semantics, but I'd say the dynamics of that system are determined exclusively by a.<br /><br />I'm not going to argue with you about what's better: continuous vs discrete, because I have no idea.<br /><br />However, if the empirical data is so crappy (Noah Smith has stated repeatedly that "macro data is uninformative") that all that's warranted is a 0th or 1st order model, it seems to me you'd be justified in cutting the parameters to their bare minimum. That is if you're interested in falsifiability. In other words, perhaps macro data CAN be informative if your model is simple enough. <br /><br />One of the economists I asked "What would convince you that you're wrong?" (Nick Rowe) also seemed to indicate that the data was too crappy to tell (<a href="http://worthwhile.typepad.com/worthwhile_canadian_initi/2015/06/back-propagation-induction-does-not-work-under-inflation-targeting.html?cid=6a00d83451688169e201b8d1227795970c#comment-6a00d83451688169e201b8d1227795970c" rel="nofollow">in the particular case I asked him about</a>).<br /><br />Regarding DSGE, Jason asks this in a post today:<br /><br /><a href="http://informationtransfereconomics.blogspot.com/2016/03/is-dsge-framework.html" rel="nofollow">"Is DSGE a framework?"</a><br /><br />Spoiler alert: his first line: "TL;DR = No" Lol.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-50947518866567801072016-03-14T13:46:55.531-04:002016-03-14T13:46:55.531-04:00"And yes, the Lucas Critique does mean someth..."And yes, the Lucas Critique does mean something to me, but I'm far from an expert on it. Thanks. I'll look up "single period rational expectations" in G&L."<br />They definitely do not call it that, that is how I would phrase it. The discussion in section 3.7.2 (page 80-81 on my copy) covers the notion of how the household needs to know the "equilibrium" value of income for the period in order to decide upon its spending plans. It's been awhile since I looked at this, but that equilibrium is what makes the G&L SFC model solutions not immediately obvious.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-67756396789963302922016-03-14T13:33:30.302-04:002016-03-14T13:33:30.302-04:00I notice that Bill's expression:
G/(1 − α1 · ...I notice that Bill's expression:<br /><br />G/(1 − α1 · (1 − θ))<br /><br />is the "G" component (DY*G) of "measurement" Y in my expressions:<br /><br />Y[n+1] = CY*H[n] + DY*G[n+1]<br /><br />All those are listed at the <a href="http://banking-discussion.blogspot.com/2016/03/sim.html" rel="nofollow">bottom of this post (SIM)</a>. I don't know if that means anything. I suppose Bill found that in G&L's text.<br /><br />And yes, the Lucas Critique does mean something to me, but I'm far from an expert on it. Thanks. I'll look up "single period rational expectations" in G&L.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-91702933427488826922016-03-14T13:30:12.893-04:002016-03-14T13:30:12.893-04:00"Also what about Jason's point (in that c..."Also what about Jason's point (in that comment, and the ones immediately below) expressed by this interchange between he and I:<br /><br />Me:<br />"Jason, wouldn't these problems affect other discrete time models, for instance the New Keynesian model[s]...?"<br /><br />Jason:<br />"It wouldn't affect all models because not all finite difference models have this same invariance. NK DSGE models are also written in log linear variables which don't have the same issues.""<br /><br />If one insist on creating a continuous time model to replicate a discrete time model, like he does, you get a dependence upon the sample time. No ifs, ands, or buts about it.<br /><br />Since the underlying data is discrete time anyway, he is completely and utterly wrong in his critique. The accounting is determined by a sum of discrete cash flows over a time interval -- continuous flows are never, ever, observed in the real world. Since we are summing a sequence of non-zero flows, there is no way that you can fit that to a clean continuous time model, other than by forcing all of the flows to be a comibination of Dirac delta functions. In which case, you cannot have nice clean exponential response functions in your dynamics.<br /><br />He's right that DSGE models do not have this problem -- they have worse problems. The act of log-linearisation breaks accounting identities. Whatever log-linear models are, they are not models of economic systems that obey accounting identities.Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-732521776332979312016-03-14T13:24:01.228-04:002016-03-14T13:24:01.228-04:00This is what Bill wrote:
"Tom, there are two...<a href="http://informationtransfereconomics.blogspot.com/2016/03/more-on-stock-flow-models.html?showComment=1457713979719#c5378746408969842272" rel="nofollow">This is what Bill wrote:</a><br /><br /><i>"Tom, there are two equilibrations in their SIM model. One occurs across time periods, one occurs within them. The within period equilbrium value of GDP is given by this equation:<br /><br />Y∗ = G/(1 − α1 · (1 − θ))<br /><br />What happens after within period equilibrium is reached? Apparently nothing, until the next time period.<br /><br />The model does not work if the time period is too short for within period equilibration, so there is a minimum time period. It doesn't make much sense to have an extended period within which nothing happens after equilibration, either. So I think that there is an implied time period in the model, we just don't know what it is. ;)"</i>Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-34950243418057517312016-03-14T13:17:43.500-04:002016-03-14T13:17:43.500-04:00"For the particular case of SIM, I think it&#..."For the particular case of SIM, I think it's a closed form solution for theta, alpha1 and alpha2 as a function of (Ts2/Ts1), assuming continuous compounding. Extending this to a broad class of exogenous inputs (government spending functions) is also closed form. I can try it out and see"<br /><br />SIM may be too simple for this effect to show up (I have not had a chance to look too carefully at it). It definitely shows up when fiscal policy is introduced.<br /><br />When I created my SFC models, I simplified them by eliminating that "in-period" equilibrium, so that I could add complexity in other directions. I think G&L like it because it is similar to other Keynesian models with multipliers. Additionally, it embeds a certain amount of expectations effects into the model; the models allow for "single period rational expectations"; all that is lost is forward-looking expectations. It's a partial answer to the Lucas Critique (if that means anything to you).Brian Romanchukhttps://www.blogger.com/profile/02699198289421951151noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-12786272694431695912016-03-14T12:49:24.641-04:002016-03-14T12:49:24.641-04:00I don't know much about this "in-period&q...I don't know much about this "in-period" equilibrium you mention (a commenter on Jason's blog "Bill" mentioned the same thing, producing an expression for the case of SIM, but didn't tell me how he obtained it). I'm just looking at this from a general sampled system point of view... so I'll have to read G&L.<br /><br />Also what about Jason's point (in that comment, and the ones immediately below) expressed by this interchange between he and I:<br /><br />Me:<br /><i>"Jason, wouldn't these problems affect other discrete time models, for instance the New Keynesian model[s]...?"</i><br /><br />Jason:<br /><i>"It wouldn't affect all models because not all finite difference models have this same invariance. NK DSGE models are also written in log linear variables which don't have the same issues."</i>Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-43758216041205166622016-03-14T11:54:54.243-04:002016-03-14T11:54:54.243-04:00"I think you need to solve a particular case ..."I think you need to solve a particular case numerically, and you could then try to back out parameter values for the new system."<br /><br />For the particular case of SIM, I think it's a closed form solution for theta, alpha1 and alpha2 as a function of (Ts2/Ts1), assuming continuous compounding. Extending this to a broad class of exogenous inputs (government spending functions) is also closed form. I can try it out and see.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-5908830827135060852.post-85063788706719407092016-03-14T09:43:38.478-04:002016-03-14T09:43:38.478-04:00But it would be better to say:
Or rainfall could ...But it would be better to say:<br /><br />Or rainfall could be measured in weight. Or volume. The unit of time we select is the size of the <i> <b> time </b></i> container.Roger Sparkshttps://www.blogger.com/profile/01734503500078064208noreply@blogger.com