7 thoughts on “4-3 Anthony Britto (Replacing Joshua Lawson). Towards a Statistical Approximation of Growth Rate Transformations.”
Colm Connaughton
Hi Anthony. Excellent work and excellent talk. It’s very satisfying to see this working out. I’m wondering whether you had any problems with negative values for x_t in the multiplicative-additive model? This seems to be a recurring problem for Box-Cox type transformations if you have non-multiplicative dynamics.
Thanks, Colm! You raise an interesting point. I didn’t have any instances of negative values in x_t in my simulations, but in any case, the “statistical fix” should be rather simple: just add a constant, or move to the Yeo-Johnson transformation. My preference is for the first approach, and my intuition is that you would end up with lambdas that are > 1, so that you have risk-seeking behaviour in situations of ruin. But this is for sure an avenue worth exploring further – thanks for the question!
Note to self: what is the relation of the uncertainty around lambda (which could result from the time frame of the decision problem) to psychological differences between agents (Matthew’s talk)?
Your idea of fitting a max-likelihood dynamic-structure parameter to observed data looks like a major advance on the mapping presented https://arxiv.org/abs/1801.03680. I am very pleased to see this work.
On slide 6, I suggest Dx ~ N(mu*Dt, sigma^2*Dt) to handle arbitrary time steps that may be present in observed data.
Slide 14 on short-time GBM raises some central questions, which run throughout your talk. Let me try to organise my thoughts.
1) For all dynamics, the “true” value of (effective) lambda should be better resolved as sample size increases, i.e. with more observed wealth increments.
2) We can observe more wealth increments either by extending the observation window (increasing H) or increasing the observation frequency (decreasing Dt). Both are trivially possible in a simulation.
3) In a GRT dynamic, there is a “true” value of lambda which is fixed over time. Therefore, you should be able to resolve lambda better either by increasing the window or by reducing the time step. You could test this by reducing the time step in your short-time GBM simulation in slide 14. If this improves the resolution of lambda = 0, then you could conclude that lambda_hat >> 0 was a sample-size effect and not short-time effect.
4) In a non-GRT dynamic, the character of the dynamic and, therefore, the effective lambda change over time. For instance, in your household wealth model, depending on parameter values, either additive or multiplicative terms can dominate in the long-time limit. This temporal evolution of effective lambda happens in model time, which means it should not be affected by changing the time step.
5) If the above is correct, then dependence of fitted lambda on window length would be a key differentiator between GRT and non-GRT dynamics.
6) Finally, I would be inclined to investigate these matters using BM and not GBM as the exemplar GRT dynamic, since there is no non-trivial mapping to confound findings.
Thanks for the excellent notes, Alex!
– Your hint with “dt” is really helpful: I knew that I was missing something. I haven’t tested it yet, but I am pretty confident that this would “fix” the GBM problem on slide 14, in which case you would be right on the money with your point 5.
– Using BM instead of GBM is a great suggestion!
Addendum for posterity: Making Dt small does in fact dramatically reduce the spead in lambdas observed for the GBM in slide 14. Assuming that this is the case for other processes as well (i.e. small Dt implies less uncertainty about lambda) raises the following epistemological question:
Since Dt –> 0 is in fact a mathematical construct, what does the EE decision rule mean for a “real” agent, who typically receives information about her wealth dynamic in “chunks” (Dt >> 0) and therefore has to contend with some uncertainty around her utility function (lambda)?
I feel there is a hierarchy of (at least) three questions here.
1) It is possible to assign a fixed effective lambda parameter to an arbitrary non-GRT dynamic? To answer this, you need to demonstrate that max likelihood fitted parameters are robust, i.e. that they converge to the same value for the same dynamic for large sample size. This is a theoretical question which can be tackled by simulation (and maybe analytically).
2) Armed with knowledge of sample sizes required for good resolution of lambdas, is it feasible to apply your fitting procedure to observed increments in real wealth processes (under an implicit assumption that they are well modelled as generated by stable dynamics)? This is an empirical question.
3) The question you ask above, i.e. whether an agent exposed to a real dynamic at finite time steps can learn enough about the dynamic (e.g. following the fitting procedure in 2) to inform their decisions. This seems a complex question, as the agent’s decisions while they are learning could influence the wealth outcomes at each time step. It feels closely related to some of the questions faced by Ollie Hulme and his team in the Copenhagen and subsequent experiments.
Hi Anthony. Excellent work and excellent talk. It’s very satisfying to see this working out. I’m wondering whether you had any problems with negative values for x_t in the multiplicative-additive model? This seems to be a recurring problem for Box-Cox type transformations if you have non-multiplicative dynamics.
Thanks, Colm! You raise an interesting point. I didn’t have any instances of negative values in x_t in my simulations, but in any case, the “statistical fix” should be rather simple: just add a constant, or move to the Yeo-Johnson transformation. My preference is for the first approach, and my intuition is that you would end up with lambdas that are > 1, so that you have risk-seeking behaviour in situations of ruin. But this is for sure an avenue worth exploring further – thanks for the question!
Note to self: what is the relation of the uncertainty around lambda (which could result from the time frame of the decision problem) to psychological differences between agents (Matthew’s talk)?
Anthony
Your idea of fitting a max-likelihood dynamic-structure parameter to observed data looks like a major advance on the mapping presented https://arxiv.org/abs/1801.03680. I am very pleased to see this work.
On slide 6, I suggest Dx ~ N(mu*Dt, sigma^2*Dt) to handle arbitrary time steps that may be present in observed data.
Slide 14 on short-time GBM raises some central questions, which run throughout your talk. Let me try to organise my thoughts.
1) For all dynamics, the “true” value of (effective) lambda should be better resolved as sample size increases, i.e. with more observed wealth increments.
2) We can observe more wealth increments either by extending the observation window (increasing H) or increasing the observation frequency (decreasing Dt). Both are trivially possible in a simulation.
3) In a GRT dynamic, there is a “true” value of lambda which is fixed over time. Therefore, you should be able to resolve lambda better either by increasing the window or by reducing the time step. You could test this by reducing the time step in your short-time GBM simulation in slide 14. If this improves the resolution of lambda = 0, then you could conclude that lambda_hat >> 0 was a sample-size effect and not short-time effect.
4) In a non-GRT dynamic, the character of the dynamic and, therefore, the effective lambda change over time. For instance, in your household wealth model, depending on parameter values, either additive or multiplicative terms can dominate in the long-time limit. This temporal evolution of effective lambda happens in model time, which means it should not be affected by changing the time step.
5) If the above is correct, then dependence of fitted lambda on window length would be a key differentiator between GRT and non-GRT dynamics.
6) Finally, I would be inclined to investigate these matters using BM and not GBM as the exemplar GRT dynamic, since there is no non-trivial mapping to confound findings.
Best wishes
Alex
Thanks for the excellent notes, Alex!
– Your hint with “dt” is really helpful: I knew that I was missing something. I haven’t tested it yet, but I am pretty confident that this would “fix” the GBM problem on slide 14, in which case you would be right on the money with your point 5.
– Using BM instead of GBM is a great suggestion!
Addendum for posterity: Making Dt small does in fact dramatically reduce the spead in lambdas observed for the GBM in slide 14. Assuming that this is the case for other processes as well (i.e. small Dt implies less uncertainty about lambda) raises the following epistemological question:
Since Dt –> 0 is in fact a mathematical construct, what does the EE decision rule mean for a “real” agent, who typically receives information about her wealth dynamic in “chunks” (Dt >> 0) and therefore has to contend with some uncertainty around her utility function (lambda)?
Hi Anthony
I feel there is a hierarchy of (at least) three questions here.
1) It is possible to assign a fixed effective lambda parameter to an arbitrary non-GRT dynamic? To answer this, you need to demonstrate that max likelihood fitted parameters are robust, i.e. that they converge to the same value for the same dynamic for large sample size. This is a theoretical question which can be tackled by simulation (and maybe analytically).
2) Armed with knowledge of sample sizes required for good resolution of lambdas, is it feasible to apply your fitting procedure to observed increments in real wealth processes (under an implicit assumption that they are well modelled as generated by stable dynamics)? This is an empirical question.
3) The question you ask above, i.e. whether an agent exposed to a real dynamic at finite time steps can learn enough about the dynamic (e.g. following the fitting procedure in 2) to inform their decisions. This seems a complex question, as the agent’s decisions while they are learning could influence the wealth outcomes at each time step. It feels closely related to some of the questions faced by Ollie Hulme and his team in the Copenhagen and subsequent experiments.
Best wishes
Alex