Pension Funds: Financial Econometrics on the Herding Phenomenon in Spain and the United Kingdom

Abstract

This work reflects the impact of the Spanish and UK pension funds investment on the market efficiency; specifically, we analyze if manager’s behavior enhances the existence of herding phenomena.

To implement this study, we apply a less common methodology: the estimated cross-sectional standard deviations of betas. We also estimate the betas with an econometric technique less applied in the financial literature: state-space models and the Kalman filter. Additionally, in order to obtain a robust estimation, we apply the Huber estimator.

Finally, we apply several models and study the existence of herding towards the market, size, book-to-market, and momentum factors.

The results are similar for the two countries and style factors, revealing the existence of herding. Nonetheless, this is smaller on size, book-to-market, and momentum factors.

Appendices

Appendix 1: The State-Space Models

We remark that the herding parameter is not observed, so in order to extract herding parameter we apply state-space models.

A state-space model is defined by two equations:

$$ {Y}_t=c+S{X}_t+{e}_t $$

(66.17)

$$ {X}_t=d+H{X}_{t-1}+{z}_t $$

(66.18)

where:

• X_t is the hidden vector at time t.

• Y_t is the observation vector at time t.

• c and d are vectors with constants.

• e is the error.

• z is the state error.

e and z are both multivariate normally distributed, with mean zero and covariance matrices of R and Q, respectively.

Those models can be estimated by using the Kalman filter, which is an algorithm to perform filtering on the state-space model.

The estimate of the state equation by the Kalman filter algorithm also offers a smoothing time series, by performing fixed interval smoothing, i.e., computing Y _t|t = P[Y _t ∣Y ₁, … ,Y _{t − 1}] for t ≤ T.

The objective is, in the formula (66.17), to minimize the difference between the observation Y_t, and the prediction based on the previous observations, (Y _t|t = P[Y _t ∣Y ₁, … ,Y _t−1]) by recursive maximum likelihood estimation.

The Kalman filter can be considered as an online estimation procedure, which is used to estimate the parameters online when new observations are entered after they have already been estimated. On the other hand, the smoothed Kalman filter is a method only used when the total series are observed.

The Kalman filter results are close to the maximum likelihood estimates, while the smoother results are exact to the maximum likelihood estimates.

Appendix 2: Robust Estimate of the Betas

In order to calculate the different betas from the CAPM model and from the four-factor Carhart model, the ordinary least squares (OLS) estimation is the most common technique for estimating betas; however, this has some drawbacks.

Firstly, they behave badly when the errors are not from a normal i.i.d. (independent and identically distributed) distribution, particularly when the data is heavily tailed, which are very frequent in return data.

Furthermore, the existence of outliers may also influence on the OLS beta, thus leading to a distorted perspective on the relationship between asset returns and index returns.

In order to overcome these disadvantages and provide a better fit, Martín and Simin (2002) indicate that a robust estimation of beta should be implemented. One of the most commonly applied methods of robust regression is the M-estimation method, a generalization of maximum likelihood estimation.

In order to explain this estimate method, we considered a linear model as a starting point:

$$ {y}_i={X}_i\beta +{\varepsilon}_i $$

(66.19)

where i = 1,.., n

Thus, the fitted model is

$$ {y}_t={X}_tb+{e}_t $$

(66.20)

The M-estimate principle is to minimize the objective function:

$$ {\displaystyle \sum_{i=1}^n\rho \left({e}_i\right)=}{\displaystyle \sum_{i=1}^n\rho \left({y}_i-{X}_ib\right)} $$

(66.21)

where the function ρ(.) gives the contribution of each residual to the objective function.

If we define ψ = ρ′, as the first order derivative of ρ(.), by differentiating the objective function with respect to b and setting the partial derivatives to zero, we obtain a system of estimating equations:

$$ {\displaystyle \sum_{i=1}^n\psi \left({y}_i-{X}_ib\right){X}_i^{\mathit{\prime}}=}0 $$

(66.22)

If the weight function is w(e) = ψ(e)/e and w _i = w(e _i ), the estimating equations become

$$ {\displaystyle \sum_{i=1}^n{w}_i{e}_i{X}_i^{\prime }=}0 $$

(66.23)

These equations can be solved as a weighted least squares problem, with the objective of minimizing:

$$ {\displaystyle \sum_{i=1}^n{w}_i^2{e_i}^2} $$

(66.24)

The weights depend on the residuals, the residuals depend on the estimated coefficients, and the estimated coefficients depend on the weights, so an iteration procedure is needed in order to solve the problem.

To solve this iterative procedure, we apply the Huber estimation, given that this allows us to determine the weighted, the residuals, and the estimated coefficients.

In order to compare the OLS estimator with the robust Huber estimator, Table 66.7 distinguishes the objective functions and weighted functions for each one of the methods.

Table 66.7 Objective functions and weight functions for the ordinary least squares estimation and the Huber estimation

In Table 66.7 we observe that both functions increase without bound, as the residuals departs from zero; nonetheless, the Huber objective function increases more slowly.

In fact, the least squares assigns equal weight to each observation, but the weights of the Huber estimator decline for ∣e∣ > k, where e is the residual term and k is called a tuning constant for the Huber estimation.

In the OLS estimation, a smaller k parameter provides more resistance to outliers; however, it offers a lower efficiency when the errors are normally distributed. In contrast, with Huber estimation, k has a general value of k = 1.345 σ (where σ is the conventional standard deviation), producing 95 % efficiency when the errors are normal, and it also offers protection against outliers; therefore, this estimation is better than the OLS estimation.