Testing for persistence in US mutual funds’ performance: a Bayesian dynamic panel model

Abstract

We provide a Bayesian panel model to consider persistence in US funds’ performance while we tackle the important problem of errors in variables. Our modelling departs from prior strong assumptions such as error terms across funds being independent. In fact, we provide a novel, general Bayesian model for (dynamic) panel data that is stable across different priors as reported from the mapping of the prior to the posterior of the Bayesian baseline model with the adoption of different priors. We demonstrate that our model detects previously undocumented striking variability in terms of performance and persistence across funds categories and over time, and in particular through the financial crisis. The reported stochastic volatility exhibits a rising trend as early as 2003–2004 and could act as an early warning of future crisis.

Appendices

Appendix 1: Sequential Monte Carlo/particle-filtering (SMC/PF) techniques

Chopin (2002) proposed a sequential PF for static models. Given a target posterior $p(\theta |Y): = p(\theta |Y_{1:T} )$ a particle system is a sequence $\{ \theta_{j} ,w_{j} \}$ such that $E(h(\theta )|Y): = \smallint h(\theta )p(\theta |Y)d\theta \cong lim_{J \to \infty } \tfrac{{\sum\nolimits_{j = 1}^{J} \,w_{j} h(\theta_{j} )}}{{\sum\nolimits_{j = 1}^{J} \,w_{j} }}$, almost surely, for any measurable function $h$, provided the expectation exists. We consider the sequence of posterior distributions $p_{t} : = p(\theta |Y_{t} )$. The PF algorithm is as follows.

Step 1: Reweight: update the weights $w_{j} \leftarrow w_{j} \tfrac{{p_{t + 1} (\theta_{j} )}}{{p_{t} (\theta_{j} )}},j = 1, \ldots ,J$.

Step 2: Resampling: resample $\{ \theta_{j} ,w_{j} J_{j = 1}^{H} \to \{ \theta_{j}^{r} ,1\}_{j = 1}^{J}$.

Step 3: Move: draw $\theta_{j}^{m} \sim \,K_{t + 1} (\theta_{j}^{r} ),j = 1, \ldots ,J$, where $K_{t + 1}$ is any transition kernel whose stationary distribution is $p_{t + 1} .$

Step 4: Loop: $t \leftarrow t + 1,\{ \theta_{j} ,w_{j} \}_{j = 1}^{J} \leftarrow \{ \theta_{j}^{m} ,1\}_{j = 1}^{J}$ and return to Step 1.

Chopin (2002) recommends the independence Metropolis algorithm to select the kernel, which requires a source distribution. A possible choice, if we sampled from $p_{n}$ (n < T), with respect to $p_{n + s}$ is $N(\mathop {\hat{E}}\nolimits_{n + s} ,\mathop {\hat{V}}\nolimits_{n + s} )$ where

$$ \mathop {\hat{E}}\nolimits_{n + s} = \frac{{\sum\nolimits_{j = 1}^{J} \,w_{j} \theta_{j} }}{{\sum\nolimits_{j = 1}^{J} \,w_{j} }},\:\mathop {\hat{V}}\nolimits_{n + s} = \frac{{\sum\nolimits_{j = 1}^{J} \,w_{j} \left( {\theta_{j} - E_{n + p} } \right)\left( {\theta_{j} - E_{n + p} } \right)}}{{\sum\nolimits_{j = 1}^{J} \,w_{j} }}. $$

The strategy can be parallelized easily. If K processors are available, we can partition the particle system into K subsets, say $S_{k} ,k = 1, \ldots ,K)$, and implement computations for particles of $S_{k}$ in processor k. The algorithm can deal with new data at a nearly geometric rate and therefore the frequency of exchanging information between processors (after reweighting) decreases at a rate exponential to n, which is highly efficient.

Resampling according to $\theta_{j}^{m} \sim \,K_{t} (\theta_{j}^{r} ,.)$ reduces particle degeneracy (Gilks and Berzuini 2001) since identical replicates of a single particle are replaced by new ones without altering the stationary distribution. For this application using $J = 2^{12}$ particles gave a mean squared error in posterior means of 10⁻⁵ over 100 runs.

Chopin (2004) introduces a variation of MSC in which the observation dates at which each cycle terminates (say $t_{1} , \ldots ,t_{L} )$ and the parameters involved in specifying the Metropolis updates (say $\lambda_{1} , \ldots ,\lambda_{L} )$ are specified. Therefore, $0 = t_{0} < t_{1} < \cdots < t_{L} = T$ and we have the following scheme (we rely heavily on Durham and Geweke 2013).

Step 1: Initialize $l = 0$ and $\theta_{jn}^{(l)} \sim p(\theta )$, $j \in J,n \in N$.

Step 2: For $l = 1, \ldots ,L$:

(a) Correction phase:

(i) $w_{jn} (t_{l - 1} ) = 1,\:j \in J,n \in N$

(ii) For $s = t_{l - 1} + 1, \ldots ,t_{l}$

$$ w_{jn} (s) = w_{jn} (s - 1)p(y_{s} |y_{1:s - 1} ,\theta_{jn}^{(l - 1)} ),\:j \in J,n \in N. $$

(iii) $w_{jn}^{(l - 1)} : = w_{jn} (t_{l} ),\:j \in J,n \in N.$

(b) Selection phase, applied independently to each group $j \in J$ : Using multinomial or residual sampling based on $\left\{ {w_{jn}^{(l)} ,n \in N} \right\}$, select

$$ \{ \theta_{jn}^{(l,0)} ,n \in N\} $$

from $\{ \theta_{jn}^{(l - 1)} ,n \in N\}$.

(c) Mutation phase, applied independently across $j \in J,n \in N$:

$$ \theta_{jn}^{(l)} \sim \,p(\theta |y_{1:t} ,\theta_{jn}^{(0)} ,\lambda_{l} ) $$

(A1)

where the drawings are independent and the pdf above satisfies the invariance condition:

$$ \int_{\varTheta } \,p(\theta |y_{{1:t_{l} }} ,\theta^{ * } ,\lambda_{l} )p(\theta^{ * } |y_{{1:t_{l} }} )d\nu (\theta^{ * } ) = p(\theta |y_{{1:t_{l} }} ). $$

(A2)

Step 3. $\theta_{jn} : = \theta_{jn}^{(l)} ,\:j \in J,n \in N.$

At the end of every cycle, the particles $\theta_{jn}^{(l)}$ have the same distribution $p(\theta |y_{{1:t_{l} }} ).$ The amount of dependence within each group depends upon the success of the Mutation phase which avoids degeneracy.

Appendix 2

See Table 11 and Figs. 7, 8, 9, 10, 11, 12, 13 and 14.

Table 11 Yearly 10 average performance indicators over time

Fig. 10

Posterior distributions of $\gamma_{it}$ s (performance): a 10 best funds, b 10 worst funds. c 10 average funds

Fig. 11

Posterior distributions of $\gamma_{it}$ for the best-performing fund for years 2000, 2008, 2012 and 2014. Note The posterior distributions are obtained with Sequential Monte Carlo/Particle Filtering draws. We have used 15,000 draws the first 5000 of which are discarded to mitigate possible start up effects

Fig. 12

The posterior distribution of optimal portfolio. Note The diagram presents results for 20 different priors from the last column in Table 1

Fig. 13

Marginal posterior distribution of average persistence ($\alpha_{i}$) of its component returns. Note 50 different priors from the last column in Table 1

Fig. 14

Marginal posterior distribution of persistence when average persistence is ‘significant’. Note 50 different priors from the last column in Table 1

Fig. 15

Marginal posterior distributions of α_i for each fund only when its average turns out ‘significant’. Note 50 different priors from the last column in Table 1