Testing models generating time varying asset return expectations and risks

Testing models generating time warying asset return expectations and risks The case of UK private sector pension funds David Blake This paper estimates some models for yeneratin~t time varying asset return expectations and risks. Usin9 the asset returns of U K prit'ate sector pension fimds, we found stron 9 et'idence fi~r dynamic expectations but little evidence for dynamic risks; only for U K property return risks was the ~:ffect significant. Keywords: Characteristic indicator variables; Dynamic expectations; Common and specific dynamic risks

In this paper we estimate it number of asset return modcls which can bc uscd for generating expectations and risks, l-xpcctations and risks arc, of course, two of thc most important asset characteristics assessed when determining portfolio behaviottr.t In many models of portfolio behavior.r, the estimates of the expectations and risks are treated its b a c k w a r d looking, often determined its constants or moving averages from the historical sample. In this paper the estimates of expectations and risks are both forward looking and time vltrying, it is essential to model expectations and risks in a forward looking m a n n e r since the past is often a very poor guide to the future. it is desirable to allow for the possibility that expectations and risks are time varying. Models that generate dynamic expectations are now fairly c o m m o n p l a c e eg Ferson [ 13] and G i b b o n s and Ferson [ 14]. Models generating d y n a m i c risks are much rarer, despite growing evidence for them eg Officer [ 2 3 ] , F a m a [ I ! ], pp 14 -- 17 and Bollerslev et al [4].-" There are, however, few empirical studies of dynamic risks. O n e example

The author is with the Department of Banking and Finance, The City University Business School, Frobisher Crescent, Barbican Centre, London EC2Y 8HB, UK. This research was carried out under ESRC grants Nos B00220011 (at the London School of Economics) and B01250012 (at the London Business School). The author is extremely grateful for the comments of Meghnad Desai, Adrian Pagan. Stephen Pudney. Stephen Schaefer and David Webb on earlier drafts of the paper. Final manuscript received 5 May 1988.

220

is Bollerslev et al [ 4 ] , which allows the risk fitctors to follow a generalized autoregressivc conditional hcteroscedasticity ( G A R C H ) process. This assumes that individuals update their return expectations and risks each period using the prediction errors in the previous period's returns. The model proposed here itllows individuals to incorporate more general information than that simply on market returns. The basic idea underlying our models generating expectations and risks is that asset prices appear to respond to observable information. We will call these pieces of information characteristic indicator variables, because asset characteristics ultimately depend on them. We will assume that the most important indicator variables are m a c r o e c o n o m i c variables. This is because it is generally m a c r o e c o n o m i c information which h:ts the greatest impact on asset price movements as a whole. Microeconomic (ie firm-specific)information

ZExpectations and risks are treated in this paper as portfolio characteristics along the lines proposed in the Gorman-Lancaster characteristics model of the demand for related goods. For a formal treatment of this model in portfolio behaviour, see Blake [ I ]. "A number of studies have considered what factors might determine dynamic risks. For example, Grossman and Shiller [ 16] consider whether they can be attributed to uncertainty concerning discount factors (ie the real rates of interest at which dividend payments are discounted etc) which, in turn. are related to levels of economic activity. Alternatively, Breeden [6] and LeRoy and LaCivita [22] have argued that consumption variability m:O" induce asset price variability, the magnitude of which depends o~n the degree of risk aversion. Tobin [28] considers inflation and exchange rate uncertainty.

0264-9993/'89 020220-2(I 503.00 "~ 1989 Butterworth & Co (Publishers) Ltd

Testing models generating time varyintl asset return expectations and risks: D. Blake

will certainly have an influence on the m o v e m e n t of individual asset prices but in an aggregate model it is impossible to capture all the microeconomic variables that influence price movements at home and abroad. It is also possible that the form in which these variables appear is of importance. We postulate two different forms for the indicators, designated 'exogenous indicators" and "cyclical indicators'. Models with exogenous indicators have been estimated by F a m a [ 12], for example. Models with cyclical indicators are motivated by the observed correlation between asset prices and aggregate economic activity over the business cycle that has been recognized at least since Burns and Mitchell [7]. The structure of this paper is as follows. In the next section we present the theoretical models used for generating dynamic expectations and risks. In the third section we discuss which macroeconomic variables would make suitable characteristic indicator variables. We then estimate the expectations and risks models based on this choice ofcharacteristic indicator variables. We use the asset returns generated by UK private sector pension funds for this purpose. O u r procedure is to estimate very general initial versions of the expectations and risks models and then to sequentially simplify the models to arrive at the tinal versions chosen. These linal versions were selected on the basis of a wide range of diagnostic tests. A s u m m a r y and conclusions arc given in the last section.

Models generating time varying expectations

Time varying expectations are derived from (1) as follows: "~

d.o6 \x~*,/

z,_ l

t = 1, T

(2)

where E is the expectations operator conditioned on information at t - 1. The predictions in (2) therefore have a rational expectations interpretation. Time varying risks, on the other hand, depend on the structure of the error terms in (1). With s i m p l e rational expectations, the error covariance matrix is treated as time invariant:

Lq~ot I et~t J

=r"

(3)

where a~ is the variance of inflation (equivalent to the risk on the liquid asset), ~ is the covariance vector between risky asset excess returns and inflation, and F is the covariance matrix of excess returns. We propose two d y n a m i c versions of FL The simplest allows all the elements of F ~' to be subject to the same c o m m o n dynamic effects: r~=a~

"x

t=l,

T

(4)

and risks Suppose that we have the following linear reduced form between asset returns and characteristics indicator variables : , _ , + ', ,:, /

t = l, T

{n)

where n~, is the real return on the liquid (ie constant nominal price) asset and n* are the total real returns on the risky assets (yield plus capital value change) in excess o f the real return on the liquid asset (ie the risk premiums). The set of characteristic indicator variables is denoted by : and may comprise either exogenous indicators or cyclical indicators. ~ JThe one period lag in z, _ 1 follows becau~ ( 1) is to be used as the basis of one step ahead forecasts and =, will not be known at the forecast date. With a quarterly model this suggests that only information which is twelve or more weeks old is used in the construction of the forecasts, more than adequate to cover the publication lag of any of the variables used in this analysis. This minimal lag length is consistent with some asset pricing models {eg Gibbons and Ferson [ 14]} although others employ two-period lags to minimize the chance of spurious predictive relations {eg Ferson [13]).

ECONOMIC MODELLING

April 1989

where o';~, the c o m m o n d y n a m i c risk component, is a positive scalar and r "~ is a symmetric positive definite matrix (proportional to F ~) with time invariant elements. One possibility for modelling a,-' is a/~ = exp(2b'z,_~)

t = 1, T

(5)

where :,_ ~ are m a c r o e c o n o m i c variables appropriate for explaining the risks in the economy. The intercept of 2 b ' : , _ , is normalized on zero. The second version of dynamic risks exploits the decomposition F[ = A, RXA;

(6)

where R:' is the time invariant matrix of asset return correlations (with typical element q~~) and where A, is "~we will call (2) simple rational expectations; Wills [29] uses the same nomenclature for a similar model. But this should not be confused with Muth's original version of rational expectations in which asset price expectations are determined from the interaction of expected demands and supplies.

221

Testing models generating time varying asset return expectations and risks:. D. Blake Table I. Exogenous and cyclical indicators, UK and US, 1963:i-1978:iV, quarterly.

Exogenous indicators UK Gross Domestic Product. £million (1975 prices), not seasonally adjusted" Gross trading profits. £million. not seasonally adjusted • Money supply, MI. £million. not seasonally adjusted ~ Public Sector Borrowing Requirement (deficit +, surplus - ) . £million. not seasonally adjusted ~ Working days lost (all industries and services). 000. not seasonally adjusted 'j Balance of payments for official financing (deficit - . surplus + ). £million. not seasonally adjusted •

USA Gross National Product, $million. not seasonally adjusted" Implicit price deflator for GNP. 1975 : I = 100. seasonally adjusted f• Profits (before tax). $million, seasonally adjusted "f Money supply. MI. SmiUion. not seasonally adjusted r~ Public Sector (federal government) Borrowing Requirement (deficit +, surplus - k $million. seasonally adjusted "f Working days IosL 000, not seasonally adjusted rh Balance of payments for official financing (deficit - . surplus +). $million. seasonally adjusted •

Cyclical indicators

UK, detrended and smoothed by a .r-year moving average ~ Credit extended by finance houses. £million ( 1975 prices), seasonally adjusted ~ Retail sales (volumeL 1975 = 100. seasonally adjusted j Manufacturing production. 1975 = 100. seasonally adjusted j Level of manufacturing stocks. £million ( 1975 prices), seasonally adjusted k Investment in plant and machinery by manufacturing industry. £million ( 1975 prices), seasonally adjusted k Unemployment level (excluding school leavcrs). 000, seasonally adjusted ( inverted )~

USA, detrended and smoothed by a 17-quarter moving average" Consumer instalment debt (outstanding). $million (1975 prices). seasonally adjusted k Sales at retail stores (value), $million (1975 prices), seasonally adjusted j Industrial production. 1975: ! = 100, seasonally adjusted j Manufacturing inventories of finished goods (book value), $million ( 1975 prices), seasonally adjusted k Business expenditure on new plant and equipment. $miffion ( 1975 prices), seasonally adjusted k Unemployment rate (total), seasonally adjusted (inverted) k

• l':c(momic Trendy Annual Supph.ment 1981. Central Statistical Office; h Bank of England. Statistical Ahstract, No 2. 1975; c Financial Statistics, January 198 I, Central Statistical Office; a Monthly Digest o/'Statisth's, Central Statistical Office; ° Survey of Current Busim'ss. US Department of Commerce; f Bu.~im.ssStatisth's ( biennial supplement to the Survey of Current Busim'ss. US Department of Commerce ); • Economic Indicators, Economic Report to the President; t, llandb.ok of Basic Economic Statistics. Bureau of Economic Statistics. Washington, DC; ~shorter leading indicator; tcoincident indicator; ~'lagging indicator; ~Central Statistical Office data bank; "Business Conditions D#lest, US Department of CoIll

nlerCe.

the diagonal matrix of asset return standard errors which are assumed to be time dependent (with typical diagonal element a,). If ai, is modelled as a. =exp(b~:,_t)

t = I, T

(7)

then a typical element of (6) is ltix j , = t l i X~ e x p ( b i 'r- , _ t + b ~ z , _ t )

t=

1, T

(8)

so that the dynamic component differs as i and j vary. We denote this version specific dynamic risks. The intercept of b~:,_ t is normalized on zero. For more details of the theory of dynamic expectations and risks see Blake [2]. The characteristic indicator variables We have said very little about which particular macroeconomic variables would make suitable characteristic indicators, since clearly the suitability of any given set of indicators depends on whose particular expectations and risks are being modelled. In order to test our models we intend to investigate the expectations and risks on the returns of assets held by the UK private 222

sector pension funds (PSPFs). The PSPFs hold more than 30 classes of primary assets, but we reduce this to eight composite assets: bills in the UK (the liquid asset), property in the UK, loans in the UK, shares in the UK, bills in the USA, bonds in the USA and shares in the USA (for more details, see Blake 1"3]). We plan to model the returns on the PSPFs holdings ofcomposite assets between 1963 : I and 1978 : IV. This is the period before the abolition of exchange controls in the UK when most overseas investment activity was confined to the USA. We have assumed, therefore, that in the absence of detailed information on the precise geographical destination of overseas investment flows, all overseas investment was destined for the USA. 5 The set of exogenous and cyclical indicators which we believe to be relevant for the UK PSPFs is therefore composed exclusively of UK and US indicators. These are listed in Table 1.6 Fama [ 12] uses a similar set of exogenous variables in his study. He suggests the following link between

5This assumption is likely to be only about 60% accurate since on a very rough estimate only about 60% of overseas investment flows went to the USA; the remainder went to the Far East and Europe.

E C O N O M I C M O D E L L I N G April 1989

Testin# models generatin# time varyin# asset return expectations and risks:. D. Blake

business activity and asset returns. Impulses to business activity are first revealed in the money supply and industrial production and then in real GDP. The increase in pressure on the existing capital stock raises the rate of return which encourages real investment. Finally, the stock market responds to these real events. Roll and Ross [26] also argue that fundamental economic aggregates such as G D P are important determinants of the systematic components of risk. Hansen and Singleton [18] and Ferson [13] stress the importance of consumption. An even richer model specification might include the effect of factors such as announcement dates, data revisions, or bid-ask spreads on asset prices (see Ferson [ 13] notes 14 and 17, and Schwert [27]). With the cyclical indicators model, the cyclical indicators are intended to capture some of the cyclical variations observed in asset prices. Because asset prices are generally considered to be part of the longer leading indicators of the state of the economy and because our cyclical indicators model is intended to be used for forecasting, we cannot have other longer leading indicators in our set of cyclical indicators since these are observed only at the same time as the asset prices themselves. We shall therefore confine our set of included indicators to the shorter leading, coincident and lagging indicators and hope that changes in these which are designed to signal more definite turnings out of the latest recession, for example, will prove to be successful predictors of asset price changes. The volume of retail sales and the level of manufacturing production are from the set of coincident indicators while the level of manufacturing stocks, investment in plant and machinery by manufacturing industry and the level of unemployment are part of the lagging indicators. These are in the same categories for both the UK and USA. The final indicator is hire purchase credit extended by finance houses (consumer instalment debt outstanding in the USA), which happens to be a shorter leading indicator in the UK but a lagging indicator in the USA, reflecting the differing financial behaviour of UK and US consumers: hire purchase is the first thing that the British give up in hard times but the last thing that the Americans give up. E s t i m a t i n g a n d s i m u l a t i n g the m o d e l s In this section we estimate one set of models of asset 6 For both the cxogenous and cyclical indicators we have attempted to match the US variables with the U K variables as closely as possible. But this was not always possible since definitions can differ between countries and it was not always possible to collect the US data in seasonally unadjusted form. However, the correspondence between the cyclical indicators is probably greater than that between the exogenous indicators. Further. for both the USA and UK, none of the exogenous indicators is used as a cyclical indicator.

E C O N O M I C M O D E L L I N G April 1989

return expectations and risks which depends on exogenous indicators and one set which depends on cyclical indicators. We then undertake some simulation exercises.

The expectations and risks models with exogenous indicators The dynamic expectations model Estimation. The general form of the dynamic expectations model to be estimated with exogenous indicators is given by: 3

4

~q,=ao+ ~. aj~,.,_i+ ~ a3+jln(Y/P,),- j j= l.

./=1

4

+E j=l

a 7 +j In(PROF/Py),_i

4.

+ ~, att+~ln(Mt/P~,),-~ j=l 4.

+ ~ ats.~(PSBR/P~,),-j ./=t 4

+ ~ at.~+~in(WDL),-~ 4

+ ~ az3+j(BP/ey),_ j+e,,

(9)

j=l

where Kit

n*o,for i = 0, the real rate of return on

the liquid asset (bills in the UK) n?, for i = 1, 7, the real excess rate of return on the risky financial assets (note i = 0, 4 refers to UK assets, i = 5, 7 refers to US assets) Y/P, = real income in 1975 market prices PROF/Pr = real gross trading profits in 1975 prices (using the income deflator) MI/Pr = real money supply MI in 1975 prices (using the income deflator) PSBR/Pr = real public sector borrowing requirement in 1975 prices (using the income deflator) WDL = working days lost BP/Pr = real balance of payments for official financing (using the income deflator) The initial specification of (9) was overparametrized. After sequential elimination of 102 insignificant

223

Testinq models yeneratiny time caryiny asset return expectations and risks: D. Blake

variables "r we arrive at a final version of the model which is presented in Table 2 and which contains 122 parameters altogether. The estimates in Table 2 are ordinary least squares estimates from the time series processor ITSP) program (see Hall andHall [ 17]) and the diagnostic statistics are from the generalized instrument variable estimation (GIVE) program (see Hendry. Morgan and Srba [20]). OLS will provide consistent estimates of the coefficients of(9), given the absence of simultaneity in that set of equations. Examining the system of equations as a whole, we observe that every equation passes the joint significance test, although shares in the UK a and bills in the USA only just pass at the 5 % level. The important diagnostic statistics are the five sets listed at the end of Table 2. The first three sets test for autocorrelation in the residuals and the last two test for parameter constancy. The ;(z version of the Lagrange multiplier test for residual autocorrelation indicates significant residual autocorrelation for bills and loans in the UK, but the more powerful F version indicates no significant residual autocorrelation up to six lags in any equation at the 5 % level. The ARCH tests for first order residual autocorrelation Ion the assumption that quadratic heterosccdasticity is present in the residuals)indicate no significant effect. The two parameter constancy or e x post prediction tests concur on every equation. Every equation predicts the four quarters of 1978 well, except the equation exphtining the total return on property in the UK: the estimated equation grossly underpredicts the huge increase in U K property prices in 1978 but it is arguable that these observations should be treated as untypical outliers. We may therefore conclude that the predictions from these equations satisfy the main criterion of rational expectations, namely that they are unbiased and so differ from realized values by unpredictable and therefore serially uncorrelated errors. In terms of individual parameter estimates, little of interest can be inferred from the particular lag structure of the equations in Table 2. For example, it is difficult to justify on theoretical grounds why the three quarters lag in the real balance of payments deficit should influence the real returns on UK bills while no other lag does so. However, this should not cause too much concern

Dymmtic simulations. Figures I - 8 plot the actual and dynamic solution valucs from the exogenous indicators expectations model of Table 2. We interpret the dynamic solutions as time varying rational expectations (see Equation (2)). The expected values track the actual values tolerably well although, with property in the UK, they exaggerate a number of turning points and, with shares in the UK, they are of considerably lower amplitude than the actual values '~(examine, for example, the plot in Figure 5 over the period 1974:1 -1975: II). Table 3 presents some properties of the expectations. [:or no equation are we able to reject the hypothesis of unbiasedness of the expected values for the actual values, even though for some equations the correlation between actual and expected values is quite small, t° For example, for UK loans the squared correlation is nearly 90% ; but for UK shares it is only 26%. The mean expected error is largest in absolute value for UK shares for which there is on average an underprediction (though as we know this is not significant) and smallest in absolute value for UK bills, for which there is an overprediction (again insignificant). The root mean square expectation error (equivalent to the standard error of the expected errors) is largest for UK shares and smallest for UK loans. The Theil decomposition of the expectation errors confirms that the component of the errors due to bias is zero for all

7Our rule was to eliminate all variablesthe absolute valuesof whose estimates were less than their standard errors beginning with those having the lowest t-ratios. nit is arguable for the UK shares equation, given that so many ( 20 out of the 28) variablesoriginally included in the regre~,~ionhave been eliminated, that the significance of the remaining variables is largely spurious. This suggests that the real return on UK shares may be white noise with perhaps some drift, as represented by a significant constant term. This. of course, is the efficient markets hypothesis for UK share returns. The argument appears to be slightly less persuasive for US share returns, however.

"Given that the variance of the prediction errors is non-zero, the variance of the predictions will always be less than the variance of the actual values. to Pagan [25] arguesthat the unbiasednessoftbe dynamicsimulations over the estimation period is not necessarily an adequate test of model specification. Indeed. he argues that it is fairly likely to occur when the sample period is long. However,it is a necessarycondition for rational expectations. Pagan also argues that (one step ahead) dynamic simulations outside the sample peribd add little to the testing of model specificationbeyond that provided by the diagnostic tests within the sample period.

224

since the ultimate purpose of the equation system is forecasting and the ultimate criterion is how well it does this. This argument is also used to justify another aspect of the selection criterion. Beginning with an initial model in levels, zero restrictions were used to derive the final model reported in Table 2. However, the table indicates that a further set of restrictions, namely restrictions between parameters, is also likely to be valid. For example, with U K property and US bonds, the restriction that real income appears as a first difference lagged one period is likely to be accepted; economic theory would also support a relationship between asset returns and growth rates of real income. Nevertheless, we have chosen not to adopt this slightly more parsimonious model representation.

E C O N O M I C M O D E L L I N G April 1989

Testing models generating time varying asset return expectations and risks:. D. Blake

T a b l e 2- O r d i n a r y least s q u a r e s e s t i m a t e s a n d s u m m a r y 1 9 6 6 : 1 - 1 9 7 8 : IV.

Constant

(SE)

Bills in U K

Property in U K

-649.139

- 1 143.080

Bonds in U K

Loans in U K

9437.880

(521.406)

(I 836.280)

(69.337)

0.326 (0.157)

0.317 10.125)

-0.560 10.152)

0.505 10.117)

-0.242

--

--

In(

Y/Py)-t

- 0.224 (0.116)

In( Y / P , ) - 2

--

- 1.275 (0.582)

In( Y/Py)_ ~

--

-0.967 (0.637)

In( Y/P~)_,

0.377 10.160)

.

-0.081 (0.057)

1.965 (0.639)

0.063 (0.019)

--

--

0.927 (0.617)

-0.034 (0.020)

.

-0.638 (0.460)

0.022 (0.021)

-0.588 (0.167)

In(PROF/I'T) _,

-0.154 (0.059)

-0.616 ( 0.208 )

0.337

.

.

.

.

.

.

.

(0.172)

.

0.206

--

- 1.943 11.570)

--

In(MI/Py)_j

0.518 (0.282)

--

-5.173 (2.026)

0.102 (0.063)

In( M ! / P , ) _ ,

- 0.440 12.510)

.

(PSBR/Py)_ t

-0.936 (0.512)

--

(PSBR/Py)_ 2

-2.219 (0.635)

-1.691 {1.656)

(PSBR/P,)_ j

- 1.672 (0.754)

In( WDL)_ t

- I. 117

.

. 15.307 { 5.361 )

-

(BP/P,)_ t

--

(BP/P,)_ z

.

.

.

.

.

.

.

--

.

.

1.306 (0.438)

-3.522

-4.118 ( 1.531 )

.

-4.985

--4.786

(1.857)

(1.461)

12.167)

3.485 (2.919)

--

--

--

--

4.957 (2.437)

2.820 (1.873)

3.886 (1.404)

--

- 10.781 15.261)

0.348 (0.151)

--

1.122 (0.594)

1.297 (0.568)

1.740 (0.725)

1.840 (I.641)

4.696 (4.463)

0.419 (0.145)

9.614 (6.120)

-0.900 (0.619)

- !.475 (0.643)

--

--

--

0.400 (0.161)

.

20.149

25.754

--

(9.608)

18.430)

- 10.969

(6.494) --

-

5.097 (1.759)

.

0.312

---

(0.1961 0.459

--

(0.193)

--

.

2.063 (0.833) .

.

- 1.400 (0.850)

2.033 (0.730)

3.711

-

0.628 (0.459)

--

(2.115)

-

1.609 (0.585)

0.743 (0.559)

-2.544 - 1.805

.

--

. 0.583 (0.445)

.

1.291 (0.660) .

.

.

--

--

(0.832)

In(WDL)_,

.

-1.299 (0.647)

--

(1.920)

(0.763) In( WDL)_ j

.

4.905

(0.8811 In( WDL)_ 2

.

.

--

.

-1.078 (0.664)

(0.045) 0.755 (0.372)

- 1.480 (0.655)

.

- 1.643 (0.561) .

--

--

(0.141)

.

-0.023 (0.017)

In(MI/P,)_ ~

(PSBR/P,)_ 4

-0.235 10.127)

-0.201

.

--

0.157 (0.077)

-0.179 10.133)

.

.

In(PROF/PI)_ j

-0.193 10.145)

.

2.565 (I,638)

--

960.323

(2 I17.570)

--

.

- 1.740 (1.615)

1.961

-0.118 (0.053)

i

.

2 177.510

(I 761.300)

(0.135) .

Shares in U S A

(I 817.160)

-0.206

--

Bonds in U S A

-2 708.960

--

- 0.431 10.128)

-5.176 11.877)

(0.412)

In(PROF/P,)_ 2

In(MI/P,)_

--

1.581 {0.420)

--

In{ PROF/Py)_,

5 229.620

(0.148)

- 0.287 10.142)

Bills in U S A

(I 570.760)

0.438

(0.155) ~f - s

Shares in U K

155.898

(224.859)

k,_,

n,-2

s t a t i s t / c a f o r the final v e r s / o n o f t h e e x o g e n o u s i n d i c a t o r s d y n a m i c e x p e c t a t i o n s m o d a l .

11.872 (5.988)

0.347 (0.209)

I 1.394 (6.371)

-0.269 (0.197)

--

.

0.596 (0.200)

13.027 19.914) -- 11.792 (9.706) 12.174

.

.

.

-10.499

-12.671

(9.067)

(8.498)

--

--

37.179 (13.258) - 16.678 (11.831)

- I 1.546 (9.916)

-21.981 (9.374)

--

- 10.449 (5.320)

-7.442 (4.915)

--

--

--

--

(9.6O8) continued on page 226

ECONOMIC MODELLING April 1989

225

Testing models generating time varying asset return expectations and risks:. D. Blake Table 2 coadnued. Bills in UK

(BP/Py)_ 3 ( BP / V~,)_ ,

- 1.313 (0.680) --

Property in UK

Bonds in UK

Loans in UK

3.864 11.548)

--

--

--

--

Shares in UK

Bills in USA

--

0.460

.

Bonds in USA

10.310 15.150) .

.

Shares in USA

8.606 (4.958)

21.856 (6.459)

.

(2.040) R2

0.853

0.837

0.585

0.924

0.275

0.473

0,534

0.520

/~z

0.772

0.769

0.412

0.880

0.160

0.254

0.358

0.388

Standard error of the regression

2.847

7.317

22.876

0.727

51.388

23.428

23.258

32.344

Joint significance test

F(df t,dfz) (critical

10.604

12.306

3.383

20.616

2.389

2.157

3.031

3.943

(18, 33) (1.95)

(15. 36) (1.97)

(15, 361 (1.97)

(19, 32) (1.92)

(7, 44)

(15, 36) (1.97)

(14, 37) (1.93)

(11, 40) (2.06)

(2.24)

value at 5*/, level) In likelihood DW

- 116.359

- 167.711

-226.989

-44.556

- 274.290

- 228.228

-228.562

- 247.737

1.950

2.027

1.916

2.053

1.780

2.092

2.283

2.152

12.987

10.490

8.978

16.105

8.933

11.585

8.923

6.534

LMAR test"

~ X2(6) (critical v a l u c = 12.59 at 5"/0 level ) LMAR tesd' F(6. df2) (critical value at 5 % level)

1.498 (27)

1.264 (30)

1.044 (30)

1.944 (26)

1.314 (38)

1.433 (30)

1.070 (31)

0.814 (34)

(2.46)

(2.42)

(2.42)

(2.48)

(2.35)

(2.42)

(2.40)

(2.37)

ARCH test c

1.646

1.803

0.113

0.240

0.518

0.029

0.471

0.002

7.846

19,276

2.492

7.447

0.724

6.712

7.479

1.391

3.233 (32) (2.68)

0.319 (32) (2.68)

0.948 (28) (2.71)

0.163 (40) 12.61)

0.853 (32) (2.68)

1.154 (33) {2.67)

0.179 136) (2.65)

~xa(1) (critical value = 3.84 at 5% level)

BP test a ~ X2(41

(critical value = 9.49 at 5 % level ) C H O W test' ~ F(4, dr,) (critical value at 5 % level)

0.850 (29) (2.70)

• Lagrange multiplier Xz test of autocorrelated residuals ~ Z" (n) for autocorrelations of length n (see Engle [ 9 ]); b Lagrange multiplier F test of autocorrelated residuals ~ F(n, T - k - n) for autocorrelations of length n, with T observations and k regressors (see Godfrey [ 15], Harvey [19], p 173). 'First order autoregressive conditional heteroscedasticity X2 test of autocorrelated residuals assuming quadratic heteroscedasticity (iv squared residuals) ~ xZ(l)0 (see Englc [ I0]). d Box-Pierce parameter constancy ;Cz test ~ X~(n) for n period ahead ex post predictions (see Box and Pierce [5]). ' Chow parameter constancy F test ~ F(n, T - k - n) for n period ahead ex post predictions with T observations and k regressors (see Chow [8]).

equations. It also shows that for each equation most of the errors arise as a result of missing turning points (the covariance proportions are the largest) and this confirms the evidence of Figures 1-8. Finally, the variance proportions, which have a similar ordering to the root mean square expectation errors, confirm that the UK shares equation has the largest amplitude discrepancy in its expectation error structure of all the 226

equations in the model. That the time varying expectations in Figures 1-8 track the actual values reasonably well follows because they are unbiased rational expectations. Further, the prediction errors will be purely random, even though they sometimes have large variances attached to them. This implies that it is perfectly possible to predict, although with sometimes large variances, negative real ECONOMIC MODELLING April 1989

Testing models generating time varying asset return expectations and risks:. D. Blake Table 3. Properties of the dynamic expectations from the exogenous indicators model 1966:1-1978:1V

Regression of actual on expected values: intercept (SE)

B/lls in UK

Property in UK

Bonds /a UK

Loans ia UK

Shares in UK

B/lls in USA

Bonds in USA

Shares ia USA

-0.216 (0.481)

-0.038 (1.056)

-0.151 (3.256)

-0.109 (0.243)

0.112 (6.764)

0.631 (3.064)

-0.019 (2.888)

0.078 (4.160)

0.913 (0.080)

0.998 (0.068)

0.857 (0.150)

1.025 (0.049)

1.031 (0.246)

0.885 (0.155)

0.952 (0.136)

0.976 (0.142)

Joint test of zero slope and unit intercept ~ F(2, 50) (critical value = 3.19 at 5% level)

0.592

0.002

0.452

0.127

0.008

0.274

0.062

0.014

Squared correlation coefficient

0.725

0.813

0.395

0.896

0.260

0.394

0.496

0.485

Mean expectation error

- 0.001

- 0.055

0.019

0.003

O.155

0.035

- 0.051

0.049

31.239

56.656

-0.195

- 3.803

- 73.972

- 47.080

89.712

102.290

3.136

6.512

23.195

0.670

47.776

21.019

20.434

29.400

265.930

I 332.300

225.870

22.609

1055.600

712.630

319.760

468.360

Bias proportion

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

Variance proportion

0.016

0.049

0.116

0.056

0.345

0.138

0.134

O.! 59

Covariance proportion

0.984

0.951

0.884

0.944

0.655

0.862

0.866

0.841

Slope

Mean percentage expectation error

Root mean square expectation

error Root mean square percentage expectation error

Theil decomposition of expectation errors:

r e t u r n s o n t h e riskless a s s e t ( F i g u r e 1) a n d n e g a t i v e excess r e t u r n s ( r i s k p r e m i u m s ) o n t h e r i s k y a s s e t s (Figure 2-8).

The dynamic risksmodels Estimation. In this subsection, we estimate both the common dynamic and the specificdynamic risk models. The general form of the risk model to be estimated with exogenous indicators is given by:

)-i 3 + ~ b6+jln(a(Ml/P,)),_j /=l 3 + '~. bg+ j i n ( a ( P S B R / P , ) ) , _ J j-I 3

3

+ ~. bt2+jln(a(WDL)),-j

in(a,) = ~ bj ln(u(Y/P,)),_j jet

ECONOMIC

3

+ ~" b3+jIn(a(PROF/P,)),_j

MODELLING

j-i

A p r i l 1989

227

Testing models generating time varying asset return expectations and risk.g. D. Blake 15[ I

4 --m--

10~-

I

Acttmi vol~4~l Oynarnsc ~¢dut.~n velu~

T

I

Oynamlc

SOlution

v a ~

4

~.t'~

,

x

:~', ",

,,

I

1

n,

tl~,-

;

s: "

:,,' ~ l

i

"/i

'

-IO~-

"

i

"t

~

:

, ~ ',' ',

~,' " l

" '

'~ :: '~.i

-15 ~

b

•

I ': "~

• '

L 1 I

Ig~5

19~7

, L z L L ~ L~ L L ~ L L L ~ L Z L L L L ~ 1968 1969 1970 1971 1972 1973

~L~X L~X~ 1975 1976 1977

1974

~* I i 1978

Figure I. Graph of actual and dynamic solution values of real rate of return on UK bills. O/o, rational expectations, exogenous indicators, OLS estimates, 1966-78.

_ ~ - -~- -

Actual vmlu~l Ovl~ic lolutJ~ Vllmll

t

l l l l ~ t

1~

1~7

[l

1~

I

~[A,

L i c i t l y [ t i l l e r

19~9

1970

1971

1972

; [ t t l l l t l t

1973 1974

1975

t

197§

t ~ l l ~ t l ~ l l

1977

1978

Figure 3. Graph of actual and dynamic solution values of real rate of return on UK bonds (in excess of UK bills), % rational expectations, exogenous indicators, OLS estimates, 1966-78.

....

r-..,,+, ,

20

_ ~

OvmPw.c m l u t l o n velue~

!"'i

10

0

h - ~:

~:

I

,/"

.

;-,

' -10

-30 -40 1066

1

1967

19~

19~9

1970

1971

1972

1973

1974

1975

1976

1977

1978

Figure 2. Graph of actual and dynamic solution values of real rate of return on UK property (in excess of UK bills), %, rational expectations, exogenous indicators, OLS estimates, 1966 - 78.

1968

1969

1970

1971

1072

1973

1974

1975

1919

1977

1979

Figure 4. G r a p h o f actual and d y n a m i c solution values of

real rate of return on UK loans (in excess of UK bills), %, rational expectations, exogenous indicators, OLS estimates, 1966-78. 3

3

+E

1966 1967

bl 5 +j In( a( BP / P,)),_j

+ ~ b3o +i in(a( WDL))~_ 1

b,8 +j In(a( Y/Py))~_j

+ ~. b33+jln(a(BP/e,))~-j

)-I

+

]

3

y i-I

+ v,

3

+2 )=1

(10)

b2t ÷j In(a(PROF/Py))~_~

3

+ ~ b:4+~ln(a(Ml/P,))~-j ]-1 3

+ ~ bzT+jln(a(PSBR/P,))~-~

228

j-I

where In(a1) =- 0.5 ln(~,) 2 is the estimate of the logarithm of the standard error of the relevant total real rate of return. T h e righthand side variables are respectively the logarithmic standard errors of real income, real gross trading profits, real money supply M l, real public sector borrowing requirement, working days lost and real balance of p a y m e n t s for official financing,

ECONOMIC

MODELLING

April 1989

Testing models oenerating time varying asset return expectations and risk~. D. Blake 3{]I~ :

;f

Actuad vo*~,4.l - - ~ - D y r ~ l l ~ I o I u t ; ~ VOIU4'!

i

i~

¸

'~

,,~ :i~ ~ ~

"

-SO I,v

Ig~

19~7

lg~8

19~

1970

1971

1971

1973

1974

1975

1976

1977

1 78

Figure 5. Graph of actual and dynamic solution values of real rate of return on UK shares (in excess of UK bills), %, rational expectations, exogenous indicators, OLS estimates, 1966-78. Io0

. -

•

.

.

.

.

~

i i i J iii 1~6

1~7

1968

I l l t l ~ l t l l [ 1969

1970

1971

t J i 11 1972

Ig73

i t I i J ~ l 1974

1975

J i 1976

i I i i t [ i i l 1977

1978

Figure 7. Graph of actual and dynamic solution values of real rate of return on US bonds (in excess of UK bills), %, rational expectations, exogenous indicators, OLS estimates, 1966-78. 100

~--

Actu4l yIIUel ~ - ~

Oynemt¢ I~lution vlllue,I

- Actuol v*lluet - Oynlm~ iolutlo~ v i l u e t

"t

!l i!

,/,

I ~

1~7

19~1

I~0

1970

1071

1972

1973

1974

197S 1976

1977

1978

t~

1~7

,~

~

,o7o

,9.

~9.

,o7~

~9~4

1975

,976

,

1o77

,,~

Figure 6. Graph of actual and dynamic solution values of real rate of return on US bills (in excess of UK bills). %. rational expectations, exogenous indicators. OLS estimates. 1966-78.

Figure 8. Graph of actual and dynamic solution values of real rate of return on US shares (in excess of UK bills), %, rational expectations, exogenous indicators, OLS estimates, 1966- 78.

for both the UK and the USA, where these are constructed from prediction errors where the expected value is defined as a four quarter moving average. By the relevant total real rate of return, we mean, in the case of common dynamic risks, the total real rate of return (in excess of the liquid asset) for the portfolio as a whole (where this is defined by the portfolio share weighted average) and, in the case of specific dynamic risks, the total real rate of return (in excess of the liquid asset) for each of the risky assets. In addition, in the latter case, we set b t 9 - ba6 equal to zero for U K assets and b t - b t s equal to zero for US assets. Note that the constant term is set to zero. By taking the exponential of twice the fitted values of the righthand side of (I0) we arrive at exp(2b'z,_l), the dynamic

common risk component in (5). By taking the exponential of the fitted values of the righthand side of(10) for each asset separately we arrive at exp(b;z,_ t), the specific dynamic risk component in (7). Table 4 presents the least squares estimates and summary statistics for the final versions of the two risk models, tt It was necessary to impose a large number of zero restrictions in order to derive these final versions. The joint significance test indicates that, with the exception of bills in the USA, the remaining explanatory variables are jointly significantly different

E C O N O M I C M O D E L L I N G April 1989

11The model (I0) involves a generated regressand but no generated regressors, so the caveats of Pagan [24] do not apply.

229

Testing models generating time varying asset return expectations and risky. D. Blake Table 4. Ordinary least squares estimates and summary statistics for the final versions of the exogenous indicators dynamic risks models, 1966: ! - 19"78: IV. Common dynamic risks 0.0

Constant (SE)

In(o( Y/P'))-

Specific dynamic risks Property Bonds in UK in UK 0.0 -0.291

t

(0.149)

0.0

0.0

-t

0.137 (0.099)

0.266 (0.138)

--

In(o(PROF/P,))_

z

--

--

0.203 (0.121)

--

--

0.243 (o.!12)

--

--

-0.281 (0.133)

In(a( WDL))_ t

--

--

--0.149 (0.116)

In(a(WDL))_ z

--

0.128 (0.129)

-0.225 (0.1211

--

--

z

t

In(o(PSBR/Py))_

BP/P,))_

0.0

t

Bills in USA 0.0

Bonds in USA

Shares in USA

0.0

0.0

-0.214

-0.066

0.288

In(o(PROF/P,))_

In(a(

Shares in UK

(0.119)

In(a( Y / P,))- z

In(o(MI/P,)_

Loans in UK

--

-0.297 (0.049)

0.211 (0.099) --

- 0.246 (0.158)

O. 127 (0.097) 0.218 (0.127)

0.106 (0.109)

0.296 (0.164)

In( o( BP / P,) )_ z In(a( y/p,))S_,

(0.095) In(a( y/p,))s_z

0.294 (0.072)

In(o( y/p,))s_ s

-0.233 (0.085)

In(oPROF/P,)) s- t

--

In(o(MI/P,))s_z

-

(0.073)

-0.146 (O.t06)

-0.205

(0.127)

In(o(

PSBR

0.178

0.142

(0.097)

(O.lO4)

0.178 (0.110)

0.441 (0.134)

/ p,))S_ z - 0 . 1 3 5 (0.094)

In(o(WDL))S_z In(o(BP/P,))s_z

-0.401

--

--

-0.247

--

(0.148) In(o(BP/P,))S-s

0.189 (0.152)

--

--

-0.173

(0.158)

(0.166)

0.343 (0.083) --

--

Rz

0.307

O. 157

0.264

O. 123

0.113

0.110

0.233

0.083

Standard error of the regression

0.950

1.454

0.991

1.182

1.050

0.992

1.064

0.926

Joint significance

4.077 (5, 46)

3. I 19 (2, 49)

1.970 (3, 48)

3.569 (4, 47)

4.506 ( 1, 50)

test ~ F(dfl. dr2) (criticalvalue

(2.43)

2.983

3.298

3.423

(3, 48) (2.80)

(5, 46) (2.43)

(2. 49) (2.19)

(2.19)

(2.80)

(2.58)

(4.04)

at 5*/, level) In likelihood DW

-67.904

-91.173

-70.135

-80.925

- 74.797

- 71.265

- 74.402

- 68.795

1.731

1.384

2.139

1.818

2.172

2.305

1.857

1.518

continued

230

ECONOMIC

MODELLING

on page

231

April 1989

Testing models generating time varying asset return expectations and risks: D. Blake Table 4 continued.

Collrllmoll dynamic

risks LMAR test ~ X2(6) (critical value = 12.59 at 5% level)

4,416

LMAR test ~ F(6. df2) (critical value at 5°,/* level)

0.619 (40) (2.34)

ARCH test

Specific dynamic risks Property Bonds is UK in UK

Shares in UK

Bills in USA

Bonds in USA

Shares in USA

8.396

3.218

7.734

3.806

7.197

4.620

2.058 (42) (2.33)

1.284 (40) (2.34)

0.473 (43) (2.32)

1.252 (43) (2.32)

0.553 (42) (2.33)

1.098 141 ) (2.33)

0.715 (44) (2.32)

1.067

2.679

0.101

0.033

0.273

0.025

1.065

0.113

BP test ~Xz(4) (critical value =9.49 at 5% level )

8.164

8.539

6.563

1.424

3.998

2.699

5.086

2,722

Chow test ~ F(4. dr2) (critical value at 5 % level)

1.400 (42) (2.61)

1.734 (44) (2.60)

1.471 (42) (2.61)

0.353 145) (2.59)

0.845 (45) (2.59)

0.517 (44) (2.60)

1.008 (43) (2.60)

0,626 (46) (2.50)

~

11.816

Loam in UK

zZ(l)

(critical value = 3.84 at 5% level )

from zero at the 5% level,tz But the explanatory power of the included variables is not very high: the best equation is the common dynamic risks equation and this has an explanatory power of only 30°/,. The Durbin-Watson statistics, both the X2 and F Lagrange multiplier tests for residual autocorrelation and the ARCH test, indicate the absence of residual autocorrelation in every equation at the 5 % level. The Z2 and F parameter constancy tests indicate that each equation forecasts within the 5 % significance band across four quarters. Taken as a whole, therefore, the two models of Table 4 cannot be rejected, although the rationale underlying the particular lag structure is not obvious. However, a number of counterintuitive aspects of the model remain, the most important of which concerns the signs of the individual parameter estimates. Given that the explained variables and all the explanatory variables are in the form of non-negative standard errors, we might have expected all the parameter estimates in Table 4 to be positive, so that, for example, an increase in the variance of the real public sector borrowing requirement would increase the variance of the return from holding risky assets. Table 4 reveals that this is not the case, 12However, it is arguable, especially with the specific dynamic risks, that in view of the large number of zero restrictions necessary to derive the final specification in Table 4 the significance of the remaining variables is largely spurious. This argument suggests that specific risks, at least, may not actually be very dynamic.

ECONOMIC MODELLING April 1989

The expectations and risks models with cyclical indicators The dynamic expectations model Estimation. The general form of the expectations model to be estimated with cyclical indicators is given by: 3

4

~ a ~ . , _ / + ~ a3÷~In(CFH),_ t

~,=a.+

j-I 4.

j=t 4

+ ~ a7+Iln(RSV),-I+ ~ att+lln(MFP),-i j=l 4

j=l

4.

+ ~ at~+~ln(MFS),-~+ ~ ato+~ln(MFl),-i j=t

j=t

4

+ ~ a23+lln(UNl),_j+e .

(!I)

j=l

where

CFH = credit extended by finance houses in 1975 prices (note i = 0, 4refers to UK credit in £million; i = 5, 7 refers to US credit in $million) RSV = retail sales volume as a 1975 base index MFP= manufacturing production as a 1975 base index MFS = manufacturing stocks in 1975 prices MFI =manufacturing investment in plant and machinery in 1975 prices UNI = unemployment level (inverted)

231

Testing

models

generating

time

varying

asset

return

expectations

and

risks:. D. Blake

Table 5. Ordinal least sqw.res estimates and summary statistics for the final version of the cyclical indicators dynamic expectations model, 1966: I - 1978: IV. Bills in U K

Property

Bonds

Loans

Shares

Bills

Bonds

Shares

in U K

in U K

in U K

in U K

in U S A

in U S A

in U S A

239.643 (214.520)

-721.024 (584.718)

-821.554 (2200.960)

-147.644 (60.383)

-12913.700 ( 4 191.610)

-2078.770 (I 998.050)

-2316.440 (2 3 0 1 . 3 9 0 )

-11397.900 (3 131.9901

5,_ I

0.115 10.1361

0.352 (0.130)

-0.330 10.1231

0.342 (0.134 )

-0.385 10.161 )

-0.209 10.131 )

-0.162 10.1481

-0.233 (0.133)

5, _ z

--

--

-- 0.465 10.1321

- 0.426 10.1421

-- O. 181 (0.133)

nr-3

--

-0.154 (0.136)

-0.284 (0.1421

Constant

(SE)

t

In(CFH)_

InICFH)_

z

In(CFtt)_

~

In( C F t l ) _ .~

--

0.436 (0.143)

-0.214 (0.144)

--

-0.467 (0.139)

0.252

--

0.197 10.1041

--

.

.

.

.

.

.

.

.

In(RSV)

1.497 (0.910}

j

.

.

.

--

.

z

-4.104 (2.41 l)

- 18.279 (5.121)

-8.025 (4.738)

9.462 (2.282)

7.155 (5.366)

--

--

3.506 (3.463)

--

5.025 12.126)

3.621 (2.522)

5.551 (3.264)

15.125

2.508

(6.040)

(2.126)

2.755 (2.270)

(3.013)

--

--

--

--

1.578 (0.569)

.

.

-0.272 (0.098)

.

.

.

0.190

8.469

10.102)

(4.9271

.

"

.

.

- 18.592 15.6551

.

-2.311 (0.778)

.

In(MFI') _ i

In(M F I ' ) _

.

-0.604 (0.273)

In(RSV).,

10.428 (2.793)

.

- 0.621

--

~

In(RSV) _ z

8.029 (5.022)

--

.

10.329) In( R S V ) _

5.254 {3.860)

--

10.1391

-0.208 (0. I I I )

3.192

0.138

(2.591)

(0.104)

.

.

--

--

-

7.054

--

-9.313

(2.319)

.

.

3.932

(3.0581 In ( M F P ) _ ~

-

In( M F I ' ) _ ~

5.943 (3.091)

-

.

.

.

- 0.137

-- 22.436 (5.651)

(3.1351 In( M F S ) _ t In( M F S 1 , 2

1.861

1.360

(0.6011

(1.330)

- 3.063

--

In(MFS) _ j

--

2,198 ( 1.4261

In(MFS)_,L

1.230 (0.410)

--

In( M F I ) _

t

.

In( M F I ) _

z

--

In ( M F I ) _ j

.

In(MFI)..~

--

In(UNI) _ s

--

In( U N I ) _ z

.

.

. --

.

-0.156 (0.073)

.

.

- 23.664

--

--

9.371 (4.4481

0.433 (0.152)

-7.211 (3.280)

-0.242 10.145)

- 14.050 (4.3341

0.060 (0.0301

1.795 (1.365)

. - 1.566 (0.727)

.

56.300 (11.282) --

.

--

.

2.732

4.989

(1.668)

11.539 (2.762)

--

--

--

-- 6.545 ( 1.7501

-- 2,166 (1.594) .

.

--

--

.

.

.

.

.

0.337

0.727

.

.

.

.

.

(0.1991

10.5131 -0.991 (0.485)

0,082 (0.030)

2.301 (0.825)

--

--

--

--

-4.507 13.0121

--

4.595 (2.336)

--

--

continued

232

ECONOMIC

-9.120 12.6141

.

2.418 (0.779)

-0.372 (0.236)

--

(1.568)

- 5.064 (1.426)

.

0.055 (0.028)

.

0.587 (0.290)

--

--

( 8.9631

- 3.174 12.001 )

10.701 )

- 4.600

(2.263)

7.937

.

--

(0.113)

MODELLING

on page

April

233

1989

Testing models generating time varying asset return expectations and risks:. D. Blake Table 5 continued.

Bills in UK

In(UNI)_~

Property in UK --

Bonds in UK

Loans in UK

--

Shares in UK

- 0.139

Bills in U S A --

Bonds in U S A

--

1.038

10.0441 ln l UXI)_,t

0.177 (0.068)

0.334 (0.138)

Rz

0.613

0.762

/~z

0.530

0.689

Standard error of regression

4.087

Joint significanoe test

7.398 19. 42) (2.12)

~ F(df t, df z)

--

Shares in U S A --

t0.653)

0.004 (0.0261

- 1.369 (0.724)

-0.826 10.3371

- 1.335 10.726)

-0.889 (0.498)

0.551

0.871

0.590

0.465

0.525

0.567

0.413

0.822

0.463

0.318

0.362

0.403

8.491

22.853

0.884

41.071

22.393

23.183

31.942

10.400 112. 39) (2.00)

3.993 (12, 39) (2.00)

17.817 (14.37) (1.93)

4.671 (12,39) 12.00)

3.165 111,40) (2.04)

3.227 113.38) (2.00)

3.463 114.37) (1.93)

(critical value at 5% level) In likelihood

DW L M A R test ~ z:(bl (critical value

- 141.442

- 177.531

-229.017

1.932

2.030

2.231

1.597

2.288

2.022

1.990

2.026

13.022

20.547

7.856

13. I 13

14.436

9.481

8.144

9.283

0.979

1.742

2. I 14

1.264

0.990

1.123

-58.510

-259.501

-228.618

-229.088

-245.061

= 12.59 at 5% level ) L M A R test

~ F(6, dr,) (critical value at 5*/, level)

2.004 (36) ( 2.38 )

3.593 (33) (2.411)

(33)

131)

133)

(341

(32)

(31)

12.40)

(2.42)

12.41))

12.39)

(2.41)

12.42)

11.012

0. I 16

2.386

0.373

0.323

1.655

11.470

8.429

12.237

ARCII test xZ(l) (critical wdue = 3.84 at 5% level )

0.597

I 1.584

0.(X)7

BP test ~ Zz(4) (critical value

7.737

23.637

5.404

20.817

= 9.49 at 5°/, level I

C H O W test F(4, df 2 ) (critical value at

1,250 138) (2.62)

5.065 (35) (2.65)

0.680 (35) (2.65)

2.1)80 133) (2.67)

0.348 (35) (2,65)

2.143 (36) {2.63)

1.714 (34) (2.66)

1.776 (33) (2.67)

5 % level )

The initial version of (!1) was overparametrized. After sequential elimination of ! 19 insignificant variables, we arrive at a final version of the model which is presented in Table 5 and which contains 105 parameters; again the estimates are least squares from TSP and GIVE. The variables which remain are jointly significantly different from zero at the 5"/0 level. The UK property equation shows significant signs of residual autocorrelation, although this is not picked up by the Durbin-Watson statistic. According to the LMAR F and ARCH tests, none of the other equations exhibits residual autocorrelation, although from the lower powered LMAR X2 test the UK bills, loans and shares equations appear to indicate some residual autocorrelation. The UK property equation also fails the parameter constancy test over the four quarters of 1978 but, as in the case with exogenous indicators, this

E C O N O M I C M O D E L L I N G April 1989

could be as a result of dramatic but largely untypical changes in the UK property market in that year. According to the C H O W test none of the other equations fails the parameter constancy test, although from the lower powered BP test, the UK loans and the US bills and shares equations are suspect. How does this fit in with the original purpose of cyclical indicators as timing indicators of the current state of the business cycle? CFH, for example, is a shorter leading indicator in the UK and a lagging indicator in the USA and yet it has precisely the same effect on comparable UK and US assets. It foreshadows an increase in the total return on shares and property as the recovery gets under way, but a fall in the total return on bills and bonds as inflationary expectations raise nominal interest rates, which leads to a collapse of the capital values of fixed interest stock. RSV and

233

Testin# models ~leneratiny time varyin~l asset return expectations and risks:. D. Blake 15. Z~O - -~ - DV~

~ u a l volu~

-

lolullon voluql4

IQ-

i

jI t, e? •

'~':'

i

,,,

,'~

SO

¢

,k

1966

lg~7

I~

I~

1970

1971

I972

197~]

1974

1975

1976

1977

1978

i~

Figure 9. Graph of actual and dynamic solution values of real rate of return on UK bills, %, rational expectations, cyclical indicators, OLS estimates, 1966-78.

,~0 [ ~ I + •

-

,~7

1~

,~9

1~7~

i~71

19,~

~97~ ,~74

197~ 1979 ,~77

,~7~

Figure I I. Graph of actual and dynamic solution values of real rate of return on UK bonds (in excess of UK bills), %, rational expectations, cyclical indicators, OLS estimates, 1966-78. g .

ACtUll VlI~4~II OVl~l~r.c Ioltllio~ vglu¢~

.

.

.

.

.

.

q

-- ~ - Actu "a v 4 d ~ Ovn4mlc

~luhO~

vqlu~'l

6 ii ¢,

•

t~

7 J ¢ ,-

6

(

'

1o i

, ,

/'/;

,,'.,,,~

°i:I ?-"." b . . . .

:

', b:.:/

.0l ~.

}

',[

,,

"I

,

,i~ t "

'".'.*/

,, ,

¢

/(

v

"

':m.'

:,.

.~

7oi./ 30,

o

• ;'~/

, ,,

'~¢' II1",, ,I

,I

V,

il;

i i

i

!

~t

ill~~

a

~" ~ ~i

,'

*

19q~

lCJ~l

1~

1~9

1970

1971

1977

1913

1974

1975

1976

1977

1978

19~[I

1~7

1968

1969

1970

1971

19/2

197 "1 1974

1975

197e

1977

1918

Figure 10. Graph of actual and dynamic solution values of real rate of ret urn on U K property (in excess of U K bills), %, rational expectations, cyclical indicators, OLS estimates, 1966-78.

Figure 12. Graph of actual and dynamic solution values of real rate of return on UK loans (in excess of UK bills), %, rational expectations, cyclical indicators, OLS estimates, 1966-78.

M F P are coincident indicators but they tend to have the opposite long-run effect on asset returns. So, for instance, an increase in retail sales (corresponding to an increase in aggregate demand) is good for share prices, while an increase in manufacturing production (corresponding to an increase in aggregate supply) is bad for share prices (both in the UK and USA). The lagging indicators ( M F S , M F I and U NI), on the other hand, tend to influence asset returns in the same direction and usually (the major exceptions being US bills and bonds) the positive direction, so that they therefore provide the strongest evidence that the worst of the recession is over and that genuine recovery is on the way.

Dynamic simulations. Figures 9 - 1 6 plot the actual and dynamic solution (time varying expectation) values from the cyclical indicators equations of Table 5. The tracking of actual values by expected values is not too bad on the whole, although some equations track better than others. The tracking of both UK and US shares is very good (Figures 13 and 16) but the UK bills equation has a tendency to miss turning points by one quarter and the residual autocorrelation that Table 5 warns us is present in the U K property equation is clearly apparent in Figure 10. Table 6 presents some properties of the expectations. They are unbiased estimates of the actual values. The correlations are on balance higher for this model than

234

E C O N O M I C M O D E L L I N G April 1989

Testing models generating time ~'arying asset return expectations and risks:. D. Blake 300 r

• ¢lua¢ ~ u m Oynem~ IohJtllm'~ vllu,n

i 290~

,o I

180-

[ 100~

.

!

,

~,~

,

•

'~''' '~ '' s ~i i ~

'

~;";'~; '. 'i

~

~:*

I

-I00~

i

-6oi-

'

lO~l

1~7

19~

19~9 1970

1971 1972 197:) 1974

1975 1976

1977 1 9 7 8

Figure 13. Graph of actual and dynamic solution values of real rate of return on UK shares (in excess of UK bills), %, rational expectations, cyclical indicators, OLS estimates, 1966- 78. O0 F . . . . . . . . i ~ Aclt~ll velu,i,l ~!~. --J- O V U l e I~tula~¢l VIIU4111

.

.

.

.

.

.

19645 1967 19~48 19~9 1970

1971 1972

1973

1974 1975 1976 1977

1978

Figure 15. Graph of actual and dynamic solution values of real rate of return on US bonds (in excess of UK bills), %, rational expectations, cyclical indicators, OLS estimates, !966- 78.

. [

~

Actual vllU41~

°.- O y n ~ ' ~ solul,on voluel 150

,;,

20~ 0

,

.

;

'..,

\

-20

-440 -60

100

-80

tbO

- 100

1960 19457 19ea

1969 1970 1971

1972

191:)

1974 197S

1978 1977 1978

,966

,~7

i~

1~9

,070

,97,

1972 ,9,~

1974 197~ ,978

,977

1978

Figure 14. Graph of actual and dynamic solution values of real rate of return on US bills (in excess of UK bills), %, rational expectations, cyclical indicators, OLS estimates, 1966- 78.

Figure 16. Graph of actual and dynamic solution values of real rate of return on US shares (in excess of UK bills), %, rational expectations, cyclical indicators, OLS estimates, ! 966- 78.

for the exogenous indicators model, especially for UK shares (compare with Table 3). in terms ofexpectation errors, every equation, except that for UK loans, tends on average to overestimate the actual values, although we know that this is not significant. The root mean square expectation error is largest for UK shares and smallest for UK loans. The Theil decomposition of the expectation errors confirms the absence of bias in every equation. It also shows that for each equation most of the errors arise from missed turning points (ie the covariance proportions dominate) and this confirms the evidence of Figures 9-16. Finally, the variance proportions indicate that the US bills equation has

the largest amplitude discrepancy between actual and simulated values ofall equations in the model. But they also show that now the UK shares equation has one of the smallest amplitude discrepancies. Compared with Table 3 the variance propdrtion for UK shares has fallen from 0.345 to 0.070, which again shows the superiority of the cyclical indicators version of the UK shares equation. Overall, then, we do not find the need to reject the cyclical indicators model. Indeed, as far as the shares equations are concerned, it has some rather appealing properties. However, it is less successful with the property equation.

E C O N O M I C M O D E L L I N G April 1989

235

7ksriny

modrls yrtwrariny rime raryiny ~SSSYI mum

r.xprctations

and risks: D. Blake

Table 6. Properties of the dynamic expectations from the cyclical indicators model. %%:I-1978: Bills in UK

Property in L’K

Loans in UK

EkdS

in CK

IV.

Bonds

Shares in L’K

in CSA

in USA

Shares in USA

Repression of actual on expected values:

-0.009

intercept

(0.59’)

-0.027 (I.3241

-0.072 (3.090)

0.0’6 (0.367)

- 0.023 (5.647)

-0.829 (3.032)

- 0.494

(SE)

(3.090)

-0.253 (3.899)

dope

1.004 (0.114)

0.984 (0.088)

0.967 (0.150)

0.995 (0.075)

0.862 (0.1’6)

I.071 (0.174)

0.929 (0.153)

I.034 (0.133)

Joint test of zero slope and unit intercept _ F(2. 501 (critical value = 3.19 at 5% level)

0.001

0.027

0.024

0.027

0.604

0.095

0.124

0.033

Squared correlation coetticien t

0.610

0.716

0.45s

0.780

0.484

0.430

0.424

0.548

- 0.425

-O.SRO

-0.204

16.569

4 13.980

2 I .xb4

27.551

3 19.190

2 669.5ot)

-0.018

Mean enpcctntion error

48.510

Mcm pcrccnt;tgc c’cpcctatwn cfror

-0.154

-0.035

305.730

- 60.3X9

- 7.556

2 I .X26

0.973

Root mean squaw cnpectation error

3.h’)O

x.042

Root mean square pcrccnt;rgc erpcctxtion error

Ix?.7tw~

2 53B.500

OSM~I

372.240

37.x37

-0.271

- 2%. I40

-24.133

4037x

20.3 I5

17X7.4(30

505.690

Thcil dccomposittor ol capcctation error: Bias proportion

O.(W)0

0.000

0.000

o.ooo

0.000

tJ.000

o.ooo

0.000

V;tri;mcr proportion

Ct.126

0.069

0.168

0.057

0.070

0.263

0.155

0.17x

Covariance proportton

0.874

0.93 I

0.832

0.943

0.930

0.737

o.n45

0.822

Conipurisott

willt the exoyowus

the cyclical indicators

itdictrlors

rtiodd.

dynamic expectations

perform mom or less favourably

Does model

than the exogenous

the two sets of equations in Tables 2 and 5 make the F test statistics non-comparable) we see that, in terms of residual autocorrelation,

the exogenous indicators

model dominates in the UK

bills, shares (both UK and

Tables 2 and 5, WC observe that, in terms of goodness

US) and especially the UK

property equation (indeed

of fit. the exogenous indicators

significant

indicators

dynamic cxpcctations model? Comparing model error variance

residual autocorrclation

dominates in the UK bills, property and loansequations

the exogenous

and the cyclical indicators

equation),

model

error

variance

indicators

whereas

the

version cyclical

is not present in of the property indicators

model

dominates in all the remaining equations and especially

dominates in all the remaining equations; in terms of

in the UK

parameter stability, we see that the exogenous indicators

shares equations where there is a reduction

in residual error variance of 36 % (see standard errors

model dominates in all equafiions except for UK

of regression in Tables 2 and 5). Comparing the x2 test statistics (the dilfcring dcgrces of freedom between

and there the difference is not likely to be significant.

236

bills

On the basis of the last point, we should, on balance,

ECONOMIC

MODELLNG

April

1989

Testiny models qeneratiny time varyiny asset return expectations and risks: D. Blake prefer the exogenous indicators version (certainly for UK property), even though the shares equations are better with the cyclical indicators version. A comparison of the tracking record of the two sets of equations also supports this conclusion.

The dynamic risks models Estimation. The general form of the risk model in the case of cyclical indicators is given by:

In(o,)= ~ bsln(a(CFH)),_ j j=l

+ ~. b;+sln(o(RSV)),_j j=l

+ ~ b~+sln(o(MFP)),-s I=1

+ ~.. h,, ~jln(a(lVll:S)),_j i~1 2

+ ~. l, s,Jln(a(Ml''l)),_j i

+

" I

2

E- I

hto, j ln(o(UNl)),_s

Comparison with the exoyemms indicators model. Comparing the dynamic risk components from cyclical indicators (Table 7) with those from exogenous indicators (Table 4), we see that, with common dynamic risks, the exogenous indicators model variance dominates the cyclical indicators model by about 1 2 " . With specific dynamic risks, the exogenous indicators model variance dominates the cyclical indicators model for UK shares and for all US assets, while for UK property, bonds and loans the dominance is reversed: this is not very surprising given the extremely poor fit of the UK shares and US assets equations discussed above. What may be a little surprising about this result, however, is the fact that the cyclical indicators dynamic expectations model for shares (both in the UK and the USA) outperformed the exogenous indicators dynamic expectations model. This suggests that, unlike property, time invariant risk factors may not be inappropriate for shares.

2

+ ~. hi, ~jln(cr(Cl"ll))7_j 2

+ 7. hla~ln(~(RSV))7-i 2

+ ~ hl,,+jln(a(Ml"P))7-j 2

+ ~. fitu+jln((a(3,tFS))~-j 2

+ ~ h,_,,,jln{a(MFl))~_j 2

+E

h22 +j In(a( UNI))~_j

+ v,

risks sets bt3 - b z . t to zero for UK assets and bt - b t 2 to zero for US assets. Again the intercept is set to zero. Table 7 presents the estimates and summary statistics for the final versions of the two risk models after sequential elimination of insignificant variables. The variables which remain have estimates equal to or larger in absolute value than their standard errors. So while the remaining individual coefficients may have some claim to being significant, their joint significance is rejected in the UK shares and all the US equations, implying that dynamic risks are not important for these equations. All the other equations are jointly significant at the 5% level, but the overall explanatory power of these equations is still fairly low. Nevertheless, no equation fails the residual autocorrelation or parameter constancy tests and so on these grounds we have accepted the equations of Table 7, as constituting the 'best' dynamic risk models with cyclical indicators. The best dynamic risks equation is that for UK property: it has by far the highest explanatory power at nearly 35%. Given how poorly fitting the dynamic expectations equation was for UK property and given how this latter equation assumed a constant (ie time invariant) risk fitctor, it is clear how important dynamic risk htctors can be in adding to the expllmatory power of equations which sock to capture the variability of certain asset returns.

112)

Summary and conclusions where the righthand side variables are respectively the logarithmic standard errors of C F tf. RS V, M F P, M FS, MF! and UNI for both the UK and USA. The model with common dynamic risks allows all the bj to be estimated freely, but the model with specific dynamic ECONOMIC M O D E L L I N G April 1989

The objective of this paper has been to test models generating time varying asset return expectations and risks, using the returns on assets held by UK private sector pension funds between 1963 and 1978. A key feature of the test was the different sets of macro237

Testing models generating time varyin# asset return expectations and risks: D. Blake Table 7. Ordinary least squares estimates and summary stabistics for the llnal version of the cyclical indicators dynamic risks models, 1966:1-1978: IV. Common dynamic risks Constant

Specific dynamic risks Property Bonds in UK in UK

0.0

0.0

Loans in UK

0.0

Shares in UK

0.0

Bills in USA

0.0

Bonds in USA

0.0

0.0

Shares in USA 0.0

(SE) In(a(CFH))_ t In(oICFH))_

2

In(o(RSV))_

t

In(o( RSV))_

z

In(o(

MFP))_ t

In(o(MFP)L2

In( M F S ) ) _

t

In(a(AIFS))_

2

- 0.244 (0.148)

--

-0.271

-0.168

--

(0.128)

0,237 (0.150)

- 0.295 (0.161)

0.211 (0.1441

-0.195 10.1371

--

--

--0.453 (0.192)

0.578 (0.2691

-0.313 10.1791

0.514 (0.239)

--

--

.

.

.

0.284 (0.2421

.

--

- 0.382 (0.136)

--

0.385 10.132)

--

0.532 10.152)

--

--

--

0.394 (0.096)

In(at MFI)) _ i

"

--0,109 (0.095)

- 0.278

.

.

.

.

.

-0.223 10.2121

--

0.277 (0.112)

--

--

--

--

--

0.301 10.104)

--

--

--

-0.265 (0.180) 0.192 (0.172)

-0.176

10.1621 lnlo(UNI)) _ t

--

(0.148)

-0.228

(0.121)

(0.1391 0.196

.

(0.138) In(o(UNI)) . ~

-0.344 {0,163)

.

In( ct( C l " l l )IS l In(a(Cl.'ll))S

In(a(RSV))

2

s- ,

ln(a(RSV))S_,

In(a{ M I - ' I ' ) ) s_ ,

InIo(MFI'))s_;

.

.

.

.

-~ .

.

.

0.172 (0.1031

.

-0.139 (0.105)

.

.

.

0.300

.

.

.

.

.

.

.

.

.

.

.

0,290 (0,024)

.

0.278 (0,167)

.

.

-0.271

.

(0.1671 In(o{ M F S ) ) s_ i

0.141 (0.09O)

---

. .

.

0.199 (0.020)

--

.

.

.

.

.

.

.

.

.

0.228 (0.025)

10.1641

.

.

.

.

.

.

.

.

R"

0.241

0.348

0.270

0.276

0.079

0.04 1

0.024

0.043

Standard error of the regression

1.005

1.321

0.987

1.I 20

1.082

1.009

1.152

0.988

J o i n t significance test

F(df t, dfz)

2.379

3.996

3.401

2.865

1.369

2.116

0.000

0.000

(6, 45) (2.32)

(6, 45) (2.32)

(5. 46) (2.43)

(6, 45) (2.32)

(3, 48) (2.80)

11, 501 (4,041

10,511

11,511

( -

(4.04)

(critical v a l u e at 5"/0 level) -69.920

- 75.914

- 75.778

-73.207

-80.664

DW

2.458

1.509

2.103

1.922

2.327

2.408

2.146

1.623

L M A R test ~ Z2(6) (critical

4.396

6.073

10.403

1.517

8.620

4.501

7.589

5.874

In l i k e l i h o o d

-70.277

-84.515

-72.142

value = 12.59 at 5 0

level) continued

238

ECONOMIC

MODELLING

on parle 239

April

1989

Testing models generating time ~'arying asset return expectations and risks:. D. Blake Table 7 continued.

Common dynamic risks

Specific dynamic risks Property Bonds in UK in UK

LMAR test ~ F(6, df2) (critical value at 5% level)

0.600 (39) (2.35)

0.860

ARCH test ~ ,t2( ! ) {critical value = 3.84 at 5% level) BP test ~ 3C'(4) {critical value =9.49 at 5*/, level) CHOW test F(4. df~) (critical value at 5*/, level)

1.667

Loans in UK

Shares in UK

0.196

1.391

Bills

Bonds

in USA

in USA

0.695

! .282

Shares in USA

0.934

(39)

(40)

(39)

(42)

(44)

(45)

(44)

1Z35)

(2.34)

(2.35)

(2.33)

(2.32)

(2.32)

(2.32)

0.164

4.108

0.389

1.209

0.872

0.144

0.088

0.900

5.427

1.578

5.816

1.686

0,953

2.971

2.377

1.046

1.339

0.333

1.377

0.343

0.210

0.703

0.579

0.238

(41) (2.61)

(41 I

(42)

(41)

(44)

146)

(47)

(46)

(2.61)

(2.60)

(2.61)

(2.59)

(2.58)

(2.58)

(2.58)

economic variables selected as indicators of the expectations and risks characteristics. When asset returns were modelled in terms of exogenous indicators we found, from both misspecification and forecast statistics, that the strongest equation overall was that explaining the return on UK loans and the weakest equation was the one explaining the return on UK shares. Every equation passed the test of joint significance of parameters and showed no sign of signiticant residual autocorrelation. In terms of the parameter constancy or e x post prediction tests, every equation was able to predict reasonably well the four quarters of 1978 except for the one explaining the total return on property in the UK which underpredicted the large and untypical increase in UK property prices in 1978. The standard error of the expectation errors was largest for UK shares and smallest for UK loans. With the exogenous indicators there appeared little evidence for the existence of dynamic risks: the explanatory power of the common dynamic risks equation was only 30%, for example; with specific dynamic risks, only the two bonds equations possessed explanatory power above 20%. When asset returns were modelled in terms of cyclical indicators, we found that the misspecification and forecast statistics suggested that there was a marked improvement in the performance of the UK shares equation which was matched by a deterioration in that of the UK property equation. The property equation showed signs of significant residual autocorrelation and also underpredicted the increase in property prices in 1978. Again the standard error of the prediction errors was largest for UK shares and smallest for UK loans; but the squared correlation

E C O N O M I C M O D E L L I N G April 1989

between the actual and expected returns for the UK shares equation had risen from 26% with exogenous indicators to 48'/0 with cyclical indicators. With the cyclical indicators there was even less evidence of the existence of common dynamic risks than there had been with the exogenous indicators. The only equation which showed any real sign of specific dynamic risk was that for UK property with explanatory power of 35%. Given the poor performance of this equation with constant risks, it may well be the case that for some assets we can improve the explanatory power by taking into account time varying risks. On balance do we prefer the model with exogenous indicators or the one with cyclical indicators? What appears to be fairly unambiguous and what is perhaps the most interesting result is that for UK shares the best equation is that with cyclical indicators and constant risks while for UK property the best equation is that with exogenous indicators and dynamic risks. With the remaining equations the evidence is less clear cut.

References I D. Blake, 'Portfolio behaviour and asset pricing in a characteristics framework', Department of Banking and Finance, City University Business School, Discussion Paper No 44, August 1986. 2 D. Blake, 'A non-linear model of portfolio behaviour with time varying expectations and risks', Department of Banking and Finance, City University Business School, Discussion Paper No 45, August 1986. 3 D. Blake, 'The investment behaviour and performance of UK private sector pension funds 1963-1978", Department of Banking and Finance, City University Business School, Discussion Paper No 46, September 1986.

239

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ECONOMIC MODELLING April 1989