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I'd like to estimate the drift of a continuous-paths, non-stationary, stochastic process $X_t$ from a time series of values $\{X_{i\Delta t}\}_{i=1,\dots,N}$ sampled from a single realisation of that process over $t \in [0,T]$.

Although the objective is to calibrate a pricing model (specified under $\Bbb{Q}$) based on historical series (observed under $\Bbb{P})$, we can forget about this and assume we're working under a single measure $\Bbb{P}$.

I've bumped into a similar question, but unfortunately the references provided apply to processes admitting a stationary distribution (= linear drift Ornstein-Uhlenbeck processes used for interest rates modelling) an assumption which I'd like to move away from.

Actually, my assumptions could even be further simplified to the case of a simple Arithmetic Brownian motion with drift $$ dX_t = \mu dt + \sigma dW_t $$

My question is: can someone point me towards a "nice" method to obtain an estimator $\hat{\mu}$ of $\mu$ from the observation of a $N$-sample $\{X_i\}_{i=1,...,N}$, where by "nice" I mean that the finite sample properties of the latter estimator should be better than the usual MLE (= LSE) estimator $$\hat{\mu} = \frac{1}{N-1} \sum_{i=1}^{N-1} \Delta X_i $$ whose relative error is proportional to $1/\sqrt{N\Delta t}$ i.e. to the time horizon $T=N \delta t$ over which the data sample is collected but not to the number of data points $N$ itself making it almost useless in practice.

I'm particularly interested in answers where the answerer has had a successful experience in implementing the method he/she proposes in practice. Because I've already come across different approaches/algorithms myself, but none were satisfying as far as I am concerned.

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  • $\begingroup$ I'm sorry if this is an incredibly dumb suggestion, but why not just take the average difference in the natural log of discrete observations of X at the highest sampling frequency possible? I've heard numerous quant traders warn of the "volatility drag" on returns. This problem is one of perception: it goes away if you use logs to the same base. The mean is also still the most unbiased measure of central tendency which I know of. $\endgroup$ – David Addison Mar 7 '17 at 3:00
  • $\begingroup$ @DavidAddison Why the natural logarithm ? Maybe you're referring to the GBM and not the ABM? Also it is not seem to me as a bias problem, but rather a variance problem in this case. $\endgroup$ – Quantuple Mar 7 '17 at 8:57
  • $\begingroup$ My bad -- logs don't make sense on ABM. Why not just take the arithmetic means of the differences on discrete observations of X? The CLT works in your favor as your sampling size and rate increase. Shouldn't $\mu_s \to \mu_p$ as $t \to \infty$? $\endgroup$ – David Addison Mar 7 '17 at 20:41
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    $\begingroup$ @DavidAddison, yes precisely but it does so with a variance which is inversely proportional to $T$. Hence, although unbiased, the "precision' of the estimator involves $T$ and not $N$. Hence you would need to collect data over say 100 years to get an 'accurate' estimate, regardless of the frequency of the collected data (monthly, daily or even tick data). I guess this is why OP says it makes this estimator "almost useless in practice". $\endgroup$ – Quantuple Mar 7 '17 at 21:05
  • $\begingroup$ You are missing some basic points. There does not exist a sufficient statistic outside Bayesian methods. The Pitman-Koopman-Damois theorem excludes it. The Bayesian solution is always minimally sufficient. There does not exist a point estimator that captures all the information in the data unless you construct it from the posterior predictive distribution. $\endgroup$ – Dave Harris Apr 7 '17 at 21:10
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I see... the problem with too long a time frame is that most time-series are not stationary -- the means and the variances will tend to fluctuate. Given non-stationary in our time-series, and given that no uncertainty is never possible, we are damned if we wait for a larger sample size, and damned if we don't.

But assuming that you know what the underlying moment-generating function looks like (ABM in this case), you can make some simplifying assumptions which let you determine p-values for rejecting a null hypothesis (i.e., that $\mu_s = \mu_p$) and the statistical power of a given $T$. In other words, if you know that it's AGM and you can define acceptable levels of sensitivity (Type II errors rates) and specificity (Type I error rates), you can then choose a $T$ that balances the ideal of certainty with practicality.

Also, if we know that

$$\sqrt{N}(\frac{1}{N}\sum_{i=1}^{N}(X_{i}-X_{i-1}) - \mu) \to ^d N(0,\sigma^2)$$

we can also deduce that the LSS technique for the estimating $\mu_p$ of ABM converges to the arithmetic mean. Download this discrete model for proof of that assertion: http://the-world-is.com/blog/wp-content/uploads/2017/03/AGM-Estimators-with-LSS.xlsm

An alternative to the arithmetic mean includes a class of estimators known as the maximum likelihood estimators (MLEs). MLEs tend to be biased... eliminating that bias is somewhat involved. Still, they provide a general purpose way of fitting parameters to observations. Personally, I use these only when trying to develop an intuition regarding whether a given set of parameters is reasonable.

Exponentially weighted functions and regressions are another option. Exponential weighting reflects the intuition that recent observation carry more weight, but that time-series do in practice tend to have long-term memories. If you know you're dealing with ABM, but you don't know if the moments are non-stationary, time-weighted regressions will produce better estimates of the present underlying moments than other methods. MLEs can be used to calibrate the weights -- I usually use just figure that I want to build in a 10% annual decline in weightings, so the actual discrete factor depends on sampling frequency and sample size. Personally, I've had a lot of success "setting and forgetting" these in the real world. Moreover, it is relatively simple to come up with a class of functions such that discrete weights sum to one.

G-ARCH models combine exponential weighting with a long-term mean reversion process. MLEs are typically used to estimate the parameters. If you assume you're dealing with ABM without mean reversion, there is no reason to use any of the ARCH family of models. Even if you're testing for stationarity, unit root tests are far simpler.

Hope that response is apropos!

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There is a simple Bayesian model, that is only slightly less simple if there is a stochastic budget constraint. The likelihood would be $$\mathcal{L}(x_1\dots{x_{n}}|\mu;\sigma)=\frac{1}{\pi}\frac{\sigma}{\sigma^2+(x_{t+1}-\beta{x_t}-\alpha)^2}.$$

I have successfully implemented this. If you want to test this, you should use the Bayesian predictive distribution as your test using a scoring method.

The Bayesian posterior density, which is $$p(\mu;\sigma|\mathbf{X})\frac{\mathcal{L}(\mathbf{X}|\mu;\sigma)p(\mu;\sigma)}{\int_{-\infty}^\infty\int_0^\infty\mathcal{L}(\mathbf{X}|\mu;\sigma)p(\mu;\sigma)\mathrm{d}\sigma\mathrm{d}\mu},\forall\mu;\sigma\in\Theta$$ is the basis for the prediction,where $\Theta$ is the parameter space. This is converted to a predictive distribution by integration as $$p(\tilde{x}_{t+1}|\mathbf{X})=\int_{-\infty}^\infty\int_0^\infty\mathcal{L}(\tilde{x}_{t+1}|\mu;\sigma)p(\mu;\sigma)\mathrm{d}\sigma\mathrm{d}\mu,\forall{x_{t+1}\in\chi},$$ where $\chi$ is the future sample space. Notice that the predictions, the distribution on the left, no longer depend upon the estimates for the parameter that are on the right and now depend only on your observed data. You will want to apply a scoring function that is less than quadratic.

There is an existing proof by White in 1958 that there is no non-Bayesian solution to this problem. FYI this was for your arithmetic time series.

You will want to think through your prior, for example, if $\beta>1$ then it must be the case that the prior distribution is zero at values less than or equal to one. As a Bayesian method, it is impossible to find a more efficient estimator anyway as Bayesian methods with a prior that reasonably addresses your beliefs are intrinsically an admissible estimator.

Also note that $\sigma$ is not a standard deviation and there is no expectation in this model. $\beta$ is not like the OLS estimator. It is a weaker construction. It does satisfy the linear absolute loss function. This model appears to have heteroskedasticity, but that is incorrect. It is an askedastic model, not a heteroskedastic model. Sample heteroskedasticity is what you get, with volatility clustering. Ignore it. That is predicted by the math for a fixed value solution. In fact, in this case, $\sigma$ is actually a direct measure of price heteroskedasticity and simultaneously a scale parameter for the model.

To test it out, download the Dow and make adjustments for the number of days that the market is closed. Compare it to any fancy method you can think of. You will probably want to construct your own Metropolis-Hastings algorithm. Start with a hill climbing program and allow tens of millions of iterations and build a method to get out of local maxima. If you get near enough to the global maximum a posteriori estimate then there is no MCMC burn in when you run the M-H algorithm.

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  • $\begingroup$ Could you please derive your first equation and explain what $(\alpha, \beta)$ is? Would you also care to compare your method with those referenced in quant.stackexchange.com/a/2957/6686? Thank you, Dave. $\endgroup$ – Hans Sep 2 '18 at 9:23

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