The model in Section 2 is written in terms of deviations from the fundamental value, xt. The question is whether this choice is important for the estimation results. Therefore, we also estimate the model in terms of returns (as in, for example, ter Ellen and Zwinkels (2010))11.
To illustrate the difference between the models estimated in deviations from the fun-damental value and in returns, we will also estimate BH98 on returns. Considering this model has slightly different behavioural rules and properties we will now briefly introduce the BH98 model in returns. The price change of the asset is a weighted average of the expectations of fundamentalists and chartists.
∆pt=nF tEF,t−1(∆pt) +nCtEC,t−1(∆pt) (16)
11The underlying reason of this difference lies in the implicit assumption about market clearing. Specif-ically, the model in deviations assumes that markets clear based on a Walrasian auctioneer. The model in returns is disequilibrium model in which a market maker adjusts prices consistent with excess demand.
See Hommes (2006) for an in-depth discussion about the micro-structure of agent-based models
Table 5: The Effect of Memory
Equity Currencies Commodities Macro
S&P500 USDJPY USDUKP Gold Oil CPI House
I = 2
φF 0.942*** 0.932*** 1.078*** 1.026*** 0.904*** 1.075*** 1.344***
(25.72) (37.01) (9.952) (31.49) (16.70) (64.49) (39.74) φC 0.900*** 0.935*** 0.665*** 0.643*** 0.439** 0.907*** 0.735***
(15.03) (13.36) (4.649) (9.461) (2.581) (27.03) (24.32)
β 36.68 0.000 1.931 7.670 51.35 1.671*** 2.166***
(0.146) (0.431) (0.924) (1.329) (0.330) (4.739) (9.236)
PφF=φC 0.612 NA 0.078 0.000 0.012 0.000 0.000
I = 4
φF 1.147** 1.847** 1.267*** 2.043** 1.156 1.135*** 1.275***
(2.256) (2.279) (3.499) (2.484) (1.629) (13.15) (29.98)
φC 0.652 -0.088 0.505 -0.404 0.439 0.772*** 0.815***
(1.268) (-0.111) (1.370) (-0.508) (0.588) (7.653) (24.12) β 0.547 -0.575 0.658 -1.061*** -0.707 -1.394*** -2.400***
(0.461) (-1.472) (1.131) (-4.863) (-0.336) (-5.225) (-6.768)
PφF=φC 0.628 0.226 0.293 0.129 0.620 0.040 0.000
I = 8
φF 1.172 2.064 1.173 1.611 1.133 1.288*** 1.176***
(1.238) (1.386) (0.331) (1.195) (1.231) (3.270) (24.82)
φC 0.642 -0.306 0.656 0.156 0.439 0.625 0.905***
(0.673) (-0.212) (0.186) (0.123) (0.459) (1.597) (25.97)
β -0.371 0.601 -0.129 -0.772* 1.224 1.096*** 3.090***
(-0.278) (1.387) (-0.067) (-1.815) (0.273) (4.257) (3.421)
PφF=φC 0.780 0.419 0.942 0.578 0.711 0.398 0.001
Notes: This table shows the results of estimating Equation (15) on quarterly data using the moving-average fundamental proxy while varying the memory parameter I. *, **, *** represent significance at the 10%, 5%, and 1% level, respectively. PφF=φC denotes the P-value of the Wald test on equality of parametersφF andφC.
Table 6: Estimation Results for Model in Returns
Equity Currencies Commodities Macro
S&P500 USDJPY USDUKP Gold Oil CPI House
Static
φF -0.053 -0.054 -0.099* 0.056 -0.225** 0.116*** -0.028***
(-0.941) (-1.008) -(1.809) (1.296) (-2.080) (5.562) (-4.433) φC 0.299* 0.204 0.378** 0.258* 0.151 0.689*** 2.026***
(1.962) (1.255) (2.611) (1.916) (1.105) (6.569) (38.93)
c 0.017** -0.007 -0.003 0.008 0.019 0.001 0.002***
(2.618) (-1.446) (-0.858) (1.002) (1.065) (1.035) (4.216)
LL 217.0 273.4 304.5 168.2 30.5 686.7 742.0
Obs 190 190 190 158 150 190 189
Switching
φF -0.053 -0.094* -0.099 0.049 -0.665*** 0.032** -0.014*
(-0.930) (-1.844) (-1.611) (1.573) (-5.581) (2.191) (-1.866) φC 0.299** 0.168 0.378** 0.271** -0.056 0.751*** 1.321***
(1.961) (0.939) (2.609) (2.635) (-0.626) (13.41) (14.46)
β 0.000 7.049 0.000 18.28 76.29 7.807 1.743***
(0.000) (0.430) (0.000) (0.431) (0.393) (1.286) (8.810) c 0.017** -0.007 -0.003 0.009 0.013 0.003*** 0.001***
(2.616) (-1.629) (-0.842) (1.272) (0.796) (3.744) (3.102)
LL 217.0 274.0 304.5 169.8 37.3 694.6 751.1
Obs 190 190 190 158 150 190 189
LR 0.00 1.30 0.00 3.09 13.66 15.83 18.07
Notes: This table shows the results of estimating the model in terms of returns ∆pt rather than deviations from the fundamental valuept−p∗t using quarterly data. *, **, *** represent significance at the 10%, 5%, and 1% level, respectively.
Fundamentalists expect the price to revert to its fundamental value:
EF,t−1(∆pt) =φF(p∗t−1−pt−1) (17)
Chartists extrapolate past returns:
EC,t−1(∆pt) =φC(pt−1−pt−2) (18)
The weights of the two groups are determined in the same way as in Equation (12), and the performance depends on the forecasting abilities of each strategy:
πht =
I
X
i=1
|∆pt−i−Eh,t−i−1∆pt−i| (19)
Table 6 presents the estimation results for the model in returns, again using quarterly data, I = 1, and a fundamental proxy based on a moving-average of the price level.
The estimation results for the model in returns shows a clear presence of both funda-mentalists and chartists: For most assets, we find a negative φF suggesting the presence of fundamentalists, and a positive φC suggesting the presence of chartists. The CPI constitutes the exception, as fundamentalists expect the price level to deviate further from the fundamental value. This finding could be driven by the strong upward trend in the CPI combined with the moving average fundamental that is by construction lagging behind the current price level. The switching results indicate that the dynamic model adds explanatory power to the model for the commodities and the macro-variables, but not for the equity market and the currencies. Within these assets, the ranking of speed of switching is comparable to the other configurations above. The consistency for the housing market is striking: again we find a β roughly equal to 2.
6 Conclusion
We have estimated a generic heterogeneous agent model on various asset classes, rang-ing from macro-economic variables such as CPI and house prices, to fast movrang-ing financial markets such as the foreign exchange market. We find that whereas switching is more intense in financial assets, the macro-economic variables are more unstable in the sense that behavioural bubbles have a higher probability of occurring. Our findings are qual-itatively robust to the choice of fundamental, data frequency, and model configuration.
The results have important implications for policymakers because whereas the focus has typically been on the risk of financial market volatility, we find that slow-moving macro variables pose a bigger threat to financial-economic stability.
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Appendix
One of the reasons why we use the normalized profit measure is ease of estimation.
To illustrate this point, we run a simulation and estimation exercise. Specifically, we run stochastic simulations of the model, adding a noise term t∼(0; 0.1), withφF = 0.8 and φC = 1.1. We setI = 1 and varyβ in the simulation from zero to two with steps of 0.04.
Subsequently, we estimate the model on the simulated data using maximum likelihood.
We do this 1,000 times for each level of β, with 22012 or 1,000 observations per run to study the effect of small samples. The distribution of estimated values ofβbwill allow us to draw inference about the accuracy and efficiency of the estimation procedure. Figure 10 presents the average as well as the standard error of the 1,000 estimated β’s for eachb level of β for t= 220.
Panel (a) of Figure 10 displays the results for the model with absolute profit differ-ences. The upper left figure shows the average estimatedβ. Whereas the actualb β ranges from zero to two, the estimatedβbhovers around 1,000 with a high level of variation. The rough coefficient estimate, therefore, is not informative about the underlying switching mechanism. We therefore winsorize the coefficient estimates at the 10% level13. The resulting figure, in the lower left panel, shows an upward sloping line from a little below zero to two, reflecting the underlying β values.
Panel (b) of Figure 10 displays results for the model with relative profit differences.
The upper left figure shows a noisy, but clearly upward sloping line. Whereas the actual values range from zero to two, the estimates appear to range from zero to approximately 100. The standard error of the estimatedβs is substantially lower (approx. 20 times) thanb for the model with absolute profit differences. After winsorizing the coefficient estimates at the 10% level, we observe a very smooth line exactly reflecting the actual underlying β’s in the lower left panel. The standard errors are also very low, and appear to decrease as the underlyingβincreases. Overall, it is clear that the estimation produces much more accurate and efficient estimates for the model with relative profit differences.
To formalize the findings in Figure 10, we estimate the following equation βbi =c1+ c2βi+i. An unbiased βbi should give c1 = 0 andc2 = 1. Table 7 presents the results.
The upper half of Panel (a) in Table 7 shows results for the absolute profit differences.
12This is roughly equal to the number of observations we have in the empirical part of the paper.
13Specifically, we remove the 10% highest values of
Figure 10: Estimation
Notes: This figure shows estimated β coefficients for the absolute versus relative profit difference
39
Table 7: Estimation Bias
Coeff. St.Err. Win. Coeff Win. St.Err.
Panel (a): t = 220 Absolute profit difference
c1 682.285*** 20.101*** -0.743*** 0.009***
(4.778) (8.067) (-7.082) (-27.816) c2 230.504 2.621 1.350*** -0.001***
(1.492) (1.054) (18.433) (-2.072)
R2 0.01 -0.01 0.89 0.06
Relative profit difference
c1 -41.915*** 0.792*** 0.151*** 0.001***
(-3.313) (2.872) (5.318) (14.520) c2 87.813*** 0.544*** 0.825*** -0.000***
(7.102) (2.217) (37.912) (-5.352)
R2 0.48 0.07 0.99 0.73
Panel (b): t = 1,000 Absolute profit difference
c1 -18.451 5.807* -0.020* 0.002***
(-0.181) (1.746) (-1.692) (47.62) c2 399.84*** 6.913* 1.009*** 0.000**
(3.025) (1.878) (54.72) (-2.601)
R2 0.11 0.03 0.98 0.11
Relative profit difference
c1 -0.019 0.006 0.097*** 0.000***
(-0.117) (1.389) (8.897) (15.21) c2 1.318*** 0.003 0.906*** 0.000***
(5.864) (0.503) (108.23) (-3.557)
R2 0.50 -0.02 1.00 0.54
Notes: This table shows the results of estimatingβbi=c1+c2βi+iusing βb estimates from simulated data. An unbiased βbi should give c1 = 0 and c2 = 1. *, **, *** represent significance at the 10%, 5%, and 1%
level, respectively.
Consistent with Figure 10, we observe that the estimated βbis not informative regarding the actual β. After winsorization, we do observe a positive and significant c2, although it is significantly larger than unity. The winsorized standard error decreases somewhat as β increases. The lower half of Panel (a) in Table 7 shows the results for the relative profit differences. The raw βbestimates show a positive correlation with β, although the c2 coefficient is far too high. After winsorization, the estimates are very well behaved, with c1 = 0.151 and c2 = 0.825.
Panel (b) in Table 7 presents the results for the simulation of t = 1,000 periods.
Overall, the results are very similar to those with t= 220 indicating that the differences between the models with absolute and relative profit differences are not driven by the small sample issue. We do observe that the non-winsorized results are somewhat stronger for the longer simulation paths. This implies that the longer simulation paths eliminate some of the extreme values in the β estimates.