Heterogeneous expectations, forecasting behaviour and policy experiments in a hybrid Agent-based Stock-flow-consistent model

Abstract

This paper presents a hybrid agent-based stock-flow-consistent model featuring heterogeneous banks, purposely built to examine the effects of variations in banks’ expectations formation and forecasting behaviour and to conduct policy experiments with a focus on monetary and prudential policy. The model is initialised to a deterministic stationary state and a subset of its free parameters are calibrated empirically in order to reproduce characteristics of UK macro-time-series data. Experiments carried out on the baseline focus on the expectations formation and forecasting behaviour of banks through allowing banks to switch between forecasting strategies and having them engage in least-squares learning. Overall, simple heuristics are remarkably robust in the model. In the baseline, which represents a relatively stable environment, the use of arguably more sophisticated expectations formation mechanisms makes little difference to simulation results. In a modified version of the baseline representing a less stable environment alternative heuristics may in fact be destabilising. To conclude the paper, a range of policy experiments is conducted, showing that an appropriate mix of monetary and prudential policy can considerably attenuate the macroeconomic volatility produced by the model.

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Notes

  1. 1.

    Although they are often broadly comparable in that SFC models frequently incorporate behavioural assumptions based on the post-Keynesian paradigm (Lavoie 2014).

  2. 2.

    Indeed, even in many otherwise fully-fledged macro-ABMs (e.g. in Assenza et al. (2015) and Seppecher 2012), it is assumed for simplicity that there exists a unique/representative bank.

  3. 3.

    This latter paradigm is arguably closest to the concept of rationality espoused in macroeconomic ABMs.

  4. 4.

    In a stationary state it must be the case that vh = vh,− 1, which implies cd = yd(= yde = c) and hence \(\frac {v_{h}}{yd}=\frac {1-\alpha _{1}}{\frac {\alpha _{2}}{48}}\)

  5. 5.

    In addition to these assets, households are also assumed to privately own the firms and banks in the model. However, as firm and bank equity are assumed to be non-tradeable their rates of return do not enter into the computation of the consumption propensities as households cannot decide to save more or less in order to accumulate more or less firm or bank equity.

  6. 6.

    This implies that monetary policy prima facie has an ambiguous effect on inflation; increases in the central bank rate will tend to decrease aggregate demand and economic activity, which will tend to lead to lower wages and hence prices, but will also increase unit interest cost, which tends to lead to higher prices. The actual effect on the price level will depend on the relative strength of these effects. Such contradictory feedback channels can also be found in some DSGE models (e.g. Christiano et al. 2005).

  7. 7.

    It is assumed that banks always grant loans to firms which are purely aimed at financing replacement investment and only ration loan demand exceeding that needed for replacement investment.

  8. 8.

    In the case of firms, \(\overline {lev_{f}}\) is a one-year moving average of the ratio of loans to capital.

  9. 9.

    The scripts necessary to reproduce the simulations can be downloaded from https://github.com/SReissl/JEEC.

  10. 10.

    Given the absence of competition, the loan and mortgage interest rate setting mechanism used by the single bank is changed such that it no longer compares its rate to the average rate (since these will obviously always be equal) but rather increases the mark-up on loans (mortgages) by a stochastic amount if its revenues on loans (mortgages) have been growing in the recent past and decreases it if the latter have been falling.

  11. 11.

    The numbers in Table 2 as well as all other tables below show the point-estimates and 95% confidence intervals from a Wilcoxon signed-rank test on the simulated statistics across the 100 MC repetitions of the respective simulations. The numbers reported in Table 2 on the other hand represent the unconditional means of the statistics which were used in the empirical calibration, explaining the slight difference between the baseline numbers reported in Table 1 and those shown below.

  12. 12.

    The parametrisation of the mechanism is as follows: ψad = 0.5, ψaa = 0.5, ψtf1 = 0.75, ψtf2 = 1.3, intensity of choice = 5, memonry parameter = 0.7. The functional forms exactly follow those suggested by Anufriev and Hommes (2012).

  13. 13.

    Note in particular that average forecast errors of both adaptive expectations and heterogeneous expectations with heuristic switching are not significantly different from zero.

  14. 14.

    In the case of stock variables, the chosen initial period is 1995 Q1.

  15. 15.

    There is clearly a danger for this algorithm to get ‘stuck’ at a local maximum of the objective function, but there is little I can do regarding this issue given limited time and computational resources, and the obtained results seem reasonably good.

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Acknowledgements

The author would like to thank Domenico Delli Gatti, Herbert Dawid, Antoine Godin and Alessandro Caiani for their substantial input during the development of this paper. Insightful hints and comments from Alberto Cardaci, Jakob Grazzini, as well as participants at conferences in Berlin, Budapest, Milan and Bamberg are gratefully acknowledged. Comments from two anonymous referees led to a substantial improvement of the paper. The usual disclaimer applies.

Funding

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 721846 (ExSIDE ITN)

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This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 721846 (ExSIDE ITN).

Appendices

Appendix A: Additional tables

The tables below show the traditional balance sheet and transactions flow matrices which provide an overview of the aggregate SFC structure of the model.

Table 11 Balance sheet matrix
Table 12 Transactions flow matrix

Appendix B: Initialisation, calibration and data sources

The model is initialised to a deterministic stationary state using a script in which initial values and as well as a range of parameters can be sequentially calculated based on the imposition of successive restrictions on some characteristics of the stationary state such as the capital stock, the stock of housing, the investment and government spending to income ratios, and so on. Where possible, these values are chosen so as to correspond roughly to those of the economy of the United Kingdom.Footnote 14 To give an example of how this calibration procedure works, once I impose a stationary state level of the capital stock, the capacity utilisation rate, and a parameter value for the capital depreciation rate, then investment demand,the capital to full capacity output ratio and real GDP are implied by these imposed values jointly with the assumed Leontieff production function and the assumption that the simulation begins in a stationary state. In a stationary state it must be the case that

$$ i=i_{d}=\delta_{k}\cdot k, $$
(28)

i.e. capital investment must equal depreciation for the capital stock to be constant. Next, note that the production function implies

$$ \kappa \cdot y = u\cdot k $$
(29)

where κ is the capital to full capacity output ratio. Next, I can substitute for k from Eq. 28 and rearrange to get

$$ \kappa = u\cdot \frac{i}{y\cdot \delta_{k}}. $$
(30)

Having previously imposed a stationary state value for \(\frac {i}{y}\) this gives me the value for κ which in turn I can use to get a value for y from Eq. 29. The rest of the initialisation protocol proceeds similarly. For instance, by imposing a stationary state value of the government expenditure to output ratio, I get a stationary state level of government expenditure and furthermore a value for consumption since

$$ c=y-i-g. $$
(31)

In many cases, the stock-flow consistent accounting structure of the model is useful in this initialisation exercise as accounting conventions dictate the values of certain variables once a sufficient number of others are determined. Despite the imposition of successive restrictions, this procedure leaves a range of parameter values unidentified (in particular those appearing in behavioural equations written in terms of deviations from ‘normal’ or stationary state values). A subset of these are empirically calibrated below, while the rest are set to values which give rise to reasonable results and are subjected to a sensitivity analysis in online appendix C.

For the empirical calibration of free parameters I make use of the simulated minimum distance approach described by Grazzini and Richiardi (2015) by applying the method of simulated moments (see Gilli and Winker 2003; Franke and Westerhoff 2012; Schmitt 2018) in order to empirically calibrate 8 of the model’s free parameters. This is done through maximising an objective function involving a set of 8 moments/statistics calculated from empirical data along with their equivalents generated by model simulations. In particular, the function to be maximised is

$$ O(\theta)=-(m_{d}(\theta_{0})-m(\theta))'\cdot W \cdot (m_{d}(\theta_{0})-m(\theta)) $$
(32)

where 𝜃 is a vector of model parameters (with 𝜃0 being the vector of their ‘true’ values), md is a vector of empirical moments and m(𝜃) is a vector of simulated moments. W is a weighting matrix. Following Franke and Westerhoff (2012), the weighting matrix used here is the inverse of the variance-covariance matrix of the empirical moments/statistics, which is obtained through the use of bootstrapping. This ensures that the variance of the empirical moments is taken into account in the calibration procedure.

As outlined by Grazzini and Richiardi (2015), building on Grazzini (2012), the use of simulated minimum distance estimators in agent-based models raises the issues of stationarity and ergodicity, in that a simulated minimum distance estimator will only be consistent if the simulated moments/statistics used are stationary and ergodic. Note that Eq. 32 is somewhat misleading in that in an agent-based model, m may be a function not only of 𝜃 but also of the random seed s and the vector of initial conditions y0 and may in particular be non-ergodic w.r.t. the random seed and/or the initial conditions. Having observed the behaviour of the model across a large number of simulations, it appears reasonable to assume that the stationarity assumption is fulfilled for the simulated moments I use, in particular since I apply the HP-filter to the simulated data before calculation of the objective function. The ergodicity assumption w.r.t. the random seed and initial conditions is somewhat more problematic but I can at least partly overcome this issue on the one hand by choosing initial conditions based on empirical information as far as possible and subsequently keeping them fixed across simulations, and on the other hand by defining m as the Monte-Carlo average of moments from a set of simulations with different random seeds, for which in turn the ergodicity assumption appears less heroic.

More broadly, the empirical calibration procedure is used here primarily to arrive at a reasonable baseline simulation without having to fully parametrise the model by hand, rather than to consistently estimate the ‘true’ values of the parameters (all the more so since, as outlined below, I am not able to cover the entire parameter space in my simulations and instead rely on sampling). The time-series I choose for the empirical calibration procedure are quarterly real GDP, real consumption, real investment and the CPI for the UK from 1994 Q2 until 2019 Q1, such that the length of the empirical time series is equal to that of the simulated ones (all simulations shown below, as well as those used for the empirical calibration have a post-transient duration of 25 years). I apply the HP-filter to each empirical time series and then calculate the standard deviation and first order autocorrelation of each series’ percentage-deviation from its trend component. The same procedure is applied to the simulated quarterly time series which are constructed from the weekly model output. The vector of parameters I am aiming to calibrate consists of the parameters shown in Table 13.

Table 13 Empirically calibrated parameters

The empirical calibration proceeds by sampling the parameter space made up of the eight parameters within the ranges shown in Table 13 above using latin hypercube sampling, simulating each parameter configuration 100 times with different (reproducible) seeds and calculating the values of the objective function. Sampling is then repeated around points which appear promising in terms of the value of the objective function until eventually a satisfactory configuration is reached in the sense that further sampling and simulation generates no notable improvements in the value of the objective function.Footnote 15

The sources of the data used to empirically calibrate the model are as follows:

  • Real GDP (quarterly): Office for national statistics; Source dataset: QNA; CDID: ABMI

  • Real consumption (quarterly): Office for national statistics; Source dataset: PN2; CDID: ABJR

  • Investment (quarterly): OECD; Subject P51

  • Price level/CPI (quarterly): Office for national statistics; Source dataset: MM23; CDID: D7BT

Table 14 Parameters & Exogenous variables

Table 14 below shows the values of all parameters and exogenous variables used in the baseline simulation. In addition it shows whether a given value is empirically calibrated (“emp”), imposed to produce the initial stationary state (“pre-SS”), implied by the stationary state (“SS-given”), or free. Where applicable, the range of values used for the sensitivity analysis is also shown. For parameters and initial values which need to be set “pre-SS” (i.e. they are needed to identify the initial stationary state rather than being implied by the latter or being calibrated empirically), I try where possible to use rough empirical values. Thus for instance, the fixed housing stock and the initial capital stock are set so as to roughly correspond to their empirical counterparts in the UK in 1995 Q1 according to the national balance sheet. Similarly, conditions such as the ratios of government consumption and capital investment to GDP, the labour share in GDP, depreciation and labour productivity are set to values close to their empirical counterparts. The conditions thus imposed are kept fixed across all simulations. Once a sufficient number of such conditions have been imposed, a large part of the remaining free parameters and initial values is implied by those already set together with the SFC structure and the assumption of a stationary state. Of the rest (category “free”), a subset is calibrated empirically as discussed above while most others are subjected to a sensitivity analysis which is discussed in online appendix C.

Table 15 below shows the aggregate initial values which are needed to initialise the model for the simulations shown in the paper. Variables pertaining to banks (e.g. stocks such as deposits, loans, mortgages etc. but also flows such as interest payments or profits) are set by imposing an initial market share for each bank (assumed equal in all markets) and then distributing each stock and flow according to these shares. The shares assumed here for the twelve banks are 0.13, 0.11, 0.11, 0.1, 0.09, 0.08, 0.07, 0.07, 0.06, 0.06, 0.06 and 0.06. Due to the way the model is set up, all banks offer equal rates on loans and deposits in the initial, deterministic stationary state. Initial values for flows refer to weekly values in all cases.

Table 15 Initial values

Appendix C: Sensitivity analysis

Recall that in the baseline, the central bank follows a pure inflation-targeting policy rule. Here I generalise the policy rule to

$$ r_{cb, d}=r_{0}+\pi^{e}+\phi_{\pi}\cdot (\pi^{e}-\pi^{t}) + \phi_{u}\cdot (u^{e}_{cb}-u_{n}) $$
(33)

meaning that the central bank can also react to gaps between expected capacity utilisation and its normal or conventional value. In the baseline, ϕπ = 0.25 so that the Taylor principle holds (recall that πt = 0). I then simulate the model for a range of values for both parameters, the range being − 1 to 1 for ϕπ and 0 to 1.5 for ϕu with step-size 0.25 in both cases. All parameter combinations are simulated for 100 MC-repetitions as in the baseline. Note that if ϕπ < 0, the Taylor principle does not hold and when ϕπ = − 1 monetary policy does not react to inflation dynamics at all. Figures 16 and 17 show the response of the standard deviations of (filtered) real output and the (filtered) price-level to variations in ϕπ (axis label π) and ϕu (axis label u) using heatmaps.

Fig. 16
figure16

Response of output volatility to changes in interest rate rule parameters

Fig. 17
figure17

Response of price level volatility to changes in interest rate rule parameters

It can be seen that simulation results are fairly sensitive to changes in the parametrisation of the monetary policy rule. A look at the results concerning ϕπ suggests that price level volatility is minimised around the value of ϕπ in the baseline (0.25), with ϕu being close to 0. Output volatility, on the other hand, is minimised then phiπ is close to zero while ϕu reacts moderately to utilisation gaps, suggesting a weak trade-off between price and output stabilisation. Overly strong reactions of monetary policy to output gaps, on the other hand, tend to lead to greater volatility in both output and inflation (indeed for high values of ϕu the model gives rise to extreme volatility or breaks down completely, which explains the missing observations in the plots). Similarly, very strong (but also very weak) reactions of monetary policy to inflation appear disadvantageous for macroeconomic stability.

In addition to the parameter sweep of the monetary policy rule, I conduct a basic sensitivity analysis on those 12 parameters for which a sensitivity range is shown in Table 14. This is done by varying the value of each parameter, one by one, along the range and according to the step sizes shown in the table. I simulate each parameter configuration for 100 Monte Carlo repetitions and compare the results to the baseline by inspecting time-series plots as well as the volatility of the time series which were used in calibrating the model. The results for variations in each parameter are discussed below in turn. Results indicate that most of the non-empirically calibrated parameters analysed here have little influence on model dynamics if varied along the ranges considered, suggesting that the choice of parameters for the empirical calibration procedure was broadly appropriate.

ψ a d

: In contrast to varying only the expectations mechanism of banks, as was done in the experiments above, jointly varying the adaptation parameter in adaptive expectations for all sectors (including banks) at once has a slight effect on macroeconomic volatility. A larger (smaller) value of ψad leads tends to increase (decrease) fluctuations as expectations which feed into the determination of various decision-variables become more (less) sensitive to forecast errors.

ε d2

: A higher value of εd2 than in the baseline implies a greater sensitivity of the deposit rates offered by banks to their clearing position. Overall this increases the range of variation in deposit interest rates and also leads to greater short-term fluctuations in deposit rates. This in turn translates into a slight increase in macroeconomic volatility. In the case of a lower value for εd2 than in the baseline, the opposite applies

σ I B

: An increase (decrease) in σIB, the sensitivity of the interbank interest rate to excess demand or supply on the interbank market obviously increases (decreases) the volatility of the interbank rate. Beyond this, however, there is no noticeable effect on model dynamics, which is in line with the passive role played by the interbank market in the model.

ι 1

: ι1 determines the sensitivity of banks’ market shares to interest rate differentials. Consequently, a higher value of ι1 leads to larger variations in market shares but for the range of values considered does not give rise to persistent monopolisation tendencies. The effects of varying ι1 on macroeconomic dynamics are slight, with higher (lower) values somewhat increasing (decreasing) the volatility of the price level due to larger variations in bank interest rates as a result of stronger (weaker) price competition.

ι 2

: At the level of individual banks, the effects of variations in ι2, which determines the sensitivity of banks’ market shares in loans and mortgages to their history of credit rationing, are similar to those caused by varying ι1. However, there is no significant effect on macroeconomic volatility for the range of values used here.

A R d i s

: An increase or decrease in the persistence of shocks to the distribution of deposits and loan demand between banks does not appear to have any systematic impact on simulation outcomes for the range of values of the parameter which are considered here.

σ d i s

: σdis denotes the standard deviation of shocks to the market shares of banks. Similarly to the effect of varying the persistence of these shocks, varying σdis along the range of values considered here has no significant impact on simulation outcomes.

C C d e f

: As one might suspect, an increase (decrease) in the cross-correlation of default shocks among banks significantly increases (decreases) macroeconomic volatility. More systemic fluctuations in defaults produce an increased volatility of interest rates as well as greater correlation in the fluctuations of individual banks’ capital adequacy ratios, both of which feed back on the aggregate sectors and ultimately lead all macro time-series to become more volatile.

s t e p

: step gives the mean value of the normal distribution which banks use to draw mark-up revisions when changing their interest rates on loans and mortgages. Decreasing or increasing this mean value along the range indicated above has no significant impact on simulation outcomes.

σ s t e p

: σstep is the standard deviation of the normal distribution which banks use to draw mark-up revisions when changing their lending rates. Varying the value of this parameter, similarly to what was found for step, does not significantly alter simulation results.

ξ 1

: ξ1 gives the upper bound of the rationing indicators on loans and mortgages calculated in Eq. 16, which feed into the distribution of loan and mortgage demand between banks. Varying this parameter has no effect on simulation results, suggesting that the indicators never reach their upper bound in the simulations considered.

ξ 2

: ξ2 measures the sensitivity of the credit rationing indicators to the intensity with which a bank rationed credit in the past. Varying this parameter along the range indicated above has no significant impact on model dynamics.

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Reissl, S. Heterogeneous expectations, forecasting behaviour and policy experiments in a hybrid Agent-based Stock-flow-consistent model. J Evol Econ 31, 251–299 (2021). https://doi.org/10.1007/s00191-020-00683-7

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Keywords

  • Stock-flow consistent models
  • Agent-based models
  • Expectations formation
  • Monetary policy
  • Prudential policy

JEL Classification

  • E12
  • E52
  • E58
  • E61
  • G28