Real Estate Modelling and Forecasting by Chris Brooks and Sotiris Tsolacos_11 ppt

measured quarterly rises on average in 2004 and 2005 are not allowed forand the system completely misses the acceleration in rent growth in 2006.Part of this has to do with the vacancy f

Table 10.4 Actual and simulated values for the Tokyo office market

Actual Predicted Actual Predicted Actual Predicted Actual Predicted

four-to the actual values in conjunction with the under-prediction in absorption(in relation to the actual values, again) results in a vacancy rate higherthan the actual figure Actual vacancies follow a downward path all theway to 2Q2007, when they turn and rise slightly The actual vacancy ratefalls from 7 per cent in 4Q2003 to 1.8 per cent in 1Q2007 The prediction

of the model is for vacancy falling to 6.1 per cent Similarly, the forecastsfor rent growth are off the mark despite a well-specified rent model The

measured quarterly rises (on average) in 2004 and 2005 are not allowed forand the system completely misses the acceleration in rent growth in 2006.Part of this has to do with the vacancy forecast, which is an input into therent growth model In turn, the vacancy forecast is fed by the misspecifiedmodels for absorption and completions This highlights a major problemwith systems of equations: a badly specified equation will have an impact

on the rest of the system

In table 10.4 we also provide the values for three forecast evaluationstatistics, which are used to compare the forecasts from an alternative modellater in this section That the ME and MAE metrics are similar for the rentgrowth and vacancy simulations owes to the fact that the forecasts of rentgrowth are below the actual values in fourteen of sixteen quarters, whereasthe forecast vacancy is consistently higher than the actual value

What comes out of this analysis is that a particular model may not fit allmarkets As a matter of fact, alternative empirical models can be based on

a plausible theory of the workings of the real estate market, but in practicedifferent data sets across markets are unlikely to support the same model

In these recursive models we can try to improve the individual equations,which are sources of error for other equations in the system In our case, therent equation is well specified, and therefore it can be left as is We focus onthe other two equations and try to improve them After experimentationwith different lags and drivers (we also included GDP as an economic driveralongside employment growth), we estimated the following equations forabsorption and completions

The revised absorption equation for the full-sample period (2Q1995 to4Q2007) is

peri-space/employment ratio, were not significant in the presence of % GDP.

Moreover, past values of absorption did not register an influence on current

absorption In this market, we found % GDP to be a major determinant

of absorption Hence the occupation needs for office space are primarily

reflected in output series Output series are also seen as proxies for revenue.GDP growth provides a signal to investors about better or worse times to

follow Two other observations are interesting The inclusion of %GDP

has eliminated the serial correlation and the DW statistic now falls withinthe non-rejection region for both samples The second observation is that

the impact of GDP weakens when the last four years are added This is a

we add more observations

We rerun the system to obtain the new forecasts The calculations arefound in table 10.5 (table 10.6 makes the comparison with the actual data)

Completions 1Q04: 307.63 + 8.374 × 1.07 − 35.97 × 4.4 = 158

Absorption 1Q04: 107.77 + 95.02 × 1.53 = 253

The new models over-predict both completions and absorption but bybroadly the same amount The over-prediction of supply may reflect thefact that we have both rent growth and vacancy in the same equation Thiscould give excess weight to changing market conditions, or may constitutesome kind of double-counting (as the vacancy was falling constantly andrent growth was on a positive path)

The forecast for vacancy is definitely an improvement on that of the vious model It overestimates the prediction in the vacancy rate but it does

pre-Table 10.5 Simulations from the system of revised equations

turn-Table 10.6 Evaluation of forecasts

Actual Predicted Actual Predicted Actual Predicted Actual Predicted

deceler-in table 10.6 The forecasts for vacancy and rent growth from the secondsystem are more accurate than those from the first For absorption and com-pletions, however, the first system does better, especially for absorption Onesuggestion, therefore, is that, depending on which variable we are interested

in (say rent growth or absorption), we should use the system that better casts that variable If the results resemble those of tables 10.4 and 10.6, it

fore-is advfore-isable to monitor the forecasts from both models Another feature ofthe forecasts from the two systems is that, for vacancy and absorption, theforecast bias is opposite (the first system over-predicts vacancy whereas thesecond under-predicts it) Possible benefits from combining the forecastsshould then be investigated These benefits are shown by the numbers inparentheses, which are the values of the respective metrics when the fore-casts are combined A marginal improvement is recorded on the ME andMAE criteria for vacancy and a more notable one for absorption (with amean error of nearly zero and clearly smaller MAE and RMSE values).One may ask how the model produces satisfactory vacancy and real rentgrowth forecasts when the forecasts for absorption and completions are notthat accurate The system over-predicts both the level of absorption and com-pletions The predicted average gap between absorption and completions issixty-six (189 – 123), whereas the same (average) actual gap is fifty-nine (148 –89) In the previous estimates, the system under-predicted absorption andover-predicted completions The gap between absorption and completionlevels was only five (112 – 107), and that is on average each quarter There-fore this was not sufficient to drive vacancy down through time and predictstronger rent growth (see table 10.4) In the second case, the good resultsfor vacancy and rent growth certainly arise from the accurate forecast ofthe relative values of absorption and completion (the gap of sixty-six) If one

is focused on absorption only, however, the forecasts would not have beenthat accurate Further work is therefore required in such cases to improvethe forecasting ability of all equations in the system

Key conceptsThe key terms to be able to define and explain from this chapter are

● endogenous variable ● exogenous variable

● simultaneous equations bias ● identified equation

● structural form ● instrumental variables

● indirect least squares ● two-stage least squares

Vector autoregressive models

Learning outcomes

In this chapter, you will learn how to

● describe the general form of a VAR;

● explain the relative advantages and disadvantages of VARmodelling;

● choose the optimal lag length for a VAR;

● carry out block significance tests;

● conduct Granger causality tests;

● estimate impulse responses and variance decompositions;

● use VARs for forecasting; and

● produce conditional and unconditional forecasts from VARs

11.1 Introduction

Vector autoregressive models were popularised in econometrics by Sims(1980) as a natural generalisation of univariate autoregressive models, dis-cussed in chapter 8 A VAR is a systems regression model – i.e there is morethan one dependent variable – that can be considered a kind of hybridbetween the univariate time series models considered in chapter 8 and thesimultaneous–equation models developed in chapter 10 VARs have oftenbeen advocated as an alternative to large-scale simultaneous equations struc-tural models

The simplest case that can be entertained is a bivariate VAR, in which there

are just two variables, y1t and y2t, each of whose current values depend on different combinations of the previous k values of both variables, and error

337

As should already be evident, an important feature of the VAR model

is its flexibility and the ease of generalisation For example, the modelcould be extended to encompass moving average errors, which would be

a multivariate version of an ARMA model, known as a VARMA Instead of

having only two variables, y1t and y2t, the system could also be expanded to include g variables, y1t , y2t , y3t, , y gt, each of which has an equation.Another useful facet of VAR models is the compactness with which thenotation can be expressed For example, consider the case from above in

which k = 1, so that each variable depends only upon the immediately

previous values of y1t and y2t, plus an error term This could be writtenas

y 1t = β10 + β11 y 1t−1 + α11 y 2t−1 + u1t (11.3)

y 2t = β20 + β21 y 2t−1+ α21 y 1t−1+ u2t (11.4)or

In (11.5), there are g= 2 variables in the system Extending the model to

the case in which there are k lags of each variable in each equation is also

easily accomplished using this notation:

y t = β0 + β1 y t−1 + β2 y t−2 + · · · + β k y t −k + u t

g × 1 g × 1 g × gg × 1 g × g g × 1 g × g g × 1 g × 1

(11.7)The model could be further extended to the case in which the model includesfirst difference terms and cointegrating relationships (a vector error correc-tion model [VECM] – see chapter 12)

11.2 Advantages of VAR modelling

VAR models have several advantages compared with univariate time seriesmodels or simultaneous equations structural models

● The researcher does not need to specify which variables are

endoge-nous or exogeendoge-nous, as all are endogeendoge-nous This is a very important point,

since a requirement for simultaneous equations structural models to beestimable is that all equations in the system are identified Essentially,this requirement boils down to a condition that some variables are treated

as exogenous and that the equations contain different RHS variables ally, this restriction should arise naturally from real estate or economictheory In practice, however, theory will be at best vague in its sugges-tions as to which variables should be treated as exogenous This leaves theresearcher with a great deal of discretion concerning how to classify thevariables Since Hausman-type tests are often not employed in practicewhen they should be, the specification of certain variables as exogenous,required to form identifying restrictions, is likely in many cases to beinvalid Sims terms these identifying restrictions ‘incredible’ VAR esti-mation, on the other hand, requires no such restrictions to be imposed

Ide-● VARs allow the value of a variable to depend on more than just its ownlags or combinations of white noise terms, so VARs are more flexiblethan univariate AR models; the latter can be viewed as a restricted case of

VAR models VAR models can therefore offer a very rich structure, implying

that they may be able to capture more features of the data

● Provided that there are no contemporaneous terms on the RHS of the

equations, it is possible simply to use OLS separately on each equation This

arises from the fact that all variables on the RHS are predetermined – that

is, at time t they are known This implies that there is no possibility

for feedback from any of the LHS variables to any of the RHS variables.Predetermined variables include all exogenous variables and laggedvalues of the endogenous variables

● The forecasts generated by VARs are often better than ‘traditional structural’

models It has been argued in a number of articles (see, for example, Sims,

1980) that large-scale structural models perform badly in terms of theirout-of-sample forecast accuracy This could perhaps arise as a result ofthe ad hoc nature of the restrictions placed on the structural models

to ensure the identification discussed above McNees (1986) shows thatforecasts for some variables, such as the US unemployment rate and realGNP, among others, are produced more accurately using VARs than fromseveral different structural specifications

11.3 Problems with VARs

Inevitably, VAR models also have drawbacks and limitations relative to othermodel classes

● VARs are atheoretical (as are ARMA models), since they use little theoretical

information about the relationships between the variables to guide thespecification of the model On the other hand, valid exclusion restric-tions that ensure the identification of equations from a simultaneousstructural system will inform the structure of the model An upshot ofthis is that VARs are less amenable to theoretical analysis and therefore

to policy prescriptions There also exists an increased possibility underthe VAR approach that a hapless researcher could obtain an essentiallyspurious relationship by mining the data Furthermore, it is often notclear how the VAR coefficient estimates should be interpreted

● How should the appropriate lag lengths for the VAR be determined? There

are several approaches available for dealing with this issue, which arediscussed below

● So many parameters! If there are g equations, one for each of g variables and

with k lags of each of the variables in each equation, (g + kg2)parameters

will have to be estimated For example, if g = 3 and k = 3, there will be

thirty parameters to estimate For relatively small sample sizes, degrees

of freedom will rapidly be used up, implying large standard errors andtherefore wide confidence intervals for model coefficients

● Should all the components of the VAR be stationary? Obviously, if one wishes

to use hypothesis tests, either singly or jointly, to examine the statisticalsignificance of the coefficients, then it is essential that all the compo-nents in the VAR are stationary Many proponents of the VAR approachrecommend that differencing to induce stationarity should not be done,however They would argue that the purpose of VAR estimation is purely

to examine the relationships between the variables, and that differencingwill throw information on any long-run relationships between the seriesaway It is also possible to combine levels and first-differenced terms in aVECM; see chapter 12

11.4 Choosing the optimal lag length for a VAR

Real estate theory will often have little to say on what an appropriate laglength is for a VAR and how long changes in the variables should take to workthrough the system In such instances, there are basically two methods that

can be used to arrive at the optimal lag length: cross-equation restrictionsand information criteria.

11.4.1 Cross-equation restrictions for VAR lag length selection

A first (but incorrect) response to the question of how to determine the

appropriate lag length would be to use the block F -tests highlighted in

section 11.7 below These are not appropriate in this case, however, as the

F-test would be used separately for the set of lags in each equation, andwhat is required here is a procedure to test the coefficients on a set of lags

on all variables for all equations in the VAR at the same time

It is worth noting here that, in the spirit of VAR estimation (as Sims, forexample, thought that model specification should be conducted), the mod-els should be as unrestricted as possible A VAR with different lag lengths foreach equation could be viewed as a restricted VAR For example, consider abivariate VAR with three lags of both variables in one equation and four lags

of each variable in the other equation This could be viewed as a restrictedmodel in which the coefficient on the fourth lags of each variable in thefirst equation have been set to zero

An alternative approach would be to specify the same number of lags ineach equation and to determine the model order as follows Suppose that

a VAR estimated using quarterly data has eight lags of the two variables ineach equation, and it is desired to examine a restriction that the coefficients

on lags 5 to 8 are jointly zero This can be done using a likelihood ratio test(see chapter 8 of Brooks, 2008, for more general details concerning suchtests) Denote the variance–covariance matrix of residuals (given by ˆu uˆ
)asˆ

 The likelihood ratio test for this joint hypothesis is given by

where ˆ ris the determinant of the variance–covariance matrix of the uals for the restricted model (with four lags), ˆ uis the determinant of thevariance–covariance matrix of residuals for the unrestricted VAR (with eight

resid-lags) and T is the sample size The test statistic is asymptotically distributed

as a χ2variate with degrees of freedom equal to the total number of tions In the VAR case above, four lags of two variables are being restricted

restric-in each of the two equations – a total of 4 × 2 × 2 = 16 restrictions In the

general case of a VAR with g equations, to impose the restriction that the last q lags have zero coefficients there would be g2qrestrictions altogether.Intuitively, the test is a multivariate equivalent to examining the extent towhich the RSS rises when a restriction is imposed If ˆ r and ˆ u are ‘closetogether’, the restriction is supported by the data

11.4.2 Information criteria for VAR lag length selection

The likelihood ratio (LR) test explained above is intuitive and fairly easy

to estimate, but it does have its limitations Principally, one of the twoVARs must be a special case of the other and, more seriously, only pairwisecomparisons can be made In the above example, if the most appropriatelag length had been seven or even ten, there is no way that this informationcould be gleaned from the LR test conducted One could achieve this only

by starting with a VAR(10), and successively testing one set of lags at a time

A further disadvantage of the LR test approach is that the χ2 test will,strictly, be valid asymptotically only under the assumption that the errorsfrom each equation are normally distributed This assumption may not beupheld for real estate data An alternative approach to selecting the appro-priate VAR lag length would be to use an information criterion, as defined

in chapter 8 in the context of ARMA model selection Information criteriarequire no such normality assumptions concerning the distributions of theerrors Instead, the criteria trade off a fall in the RSS of each equation asmore lags are added, with an increase in the value of the penalty term.The univariate criteria could be applied separately to each equation but,again, it is usually deemed preferable to require the number of lags to bethe same for each equation This requires the use of multivariate versions

of the information criteria, which can be defined as

num-which will be equal to p2k + p for p equations in the VAR system, each with

k lags of the p variables, plus a constant term in each equation As ously, the values of the information criteria are constructed for 0, 1, , ¯k lags (up to some pre-specified maximum ¯k), and the chosen number of lags

previ-is that number minimprevi-ising the value of the given information criterion

11.5 Does the VAR include contemporaneous terms?

So far, it has been assumed that the VAR specified is of the form

y 1t = β10 + β11 y 1t−1 + α11 y 2t−1 + u1t (11.12)

y 2t = β20 + β21 y 2t−1 + α21 y 1t−1 + u2t (11.13)

so that there are no contemporaneous terms on the RHS of (11.12) or (11.13) –

i.e there is no term in y2t on the RHS of the equation for y1t and no term

in y1t on the RHS of the equation for y2t What if the equations had acontemporaneous feedback term, however, as in the following case?

y 1t = β10 + β11 y 1t−1 + α11 y 2t−1 + α12 y 2t + u1t (11.14)

y 2t = β20 + β21 y 2t−1 + α21 y 1t−1 + α22 y 1t + u2t (11.15)Equations (11.14) and (11.15) can also be written by stacking up the termsinto matrices and vectors:

This would be known as a VAR in primitive form, similar to the structural form

for a simultaneous equation model Some researchers have argued that theatheoretical nature of reduced-form VARs leaves them unstructured andtheir results difficult to interpret theoretically They argue that the forms ofVAR given previously are merely reduced forms of a more general structuralVAR (such as (11.16)), with the latter being of more interest

The contemporaneous terms from (11.16) can be taken over to the LHSand written as

u 1t

u 2t

(11.17)or

This is known as a standard-form VAR, which is akin to the reduced form from a

set of simultaneous equations This VAR contains only predetermined values

on the RHS (i.e variables whose values are known at time t), and so there is

no contemporaneous feedback term This VAR can therefore be estimatedequation by equation using OLS

Equation (11.16), the structural or primitive-form VAR, is not identified,since identical predetermined (lagged) variables appear on the RHS of bothequations In order to circumvent this problem, a restriction that one of

the coefficients on the contemporaneous terms is zero must be imposed In

(11.16), either α12 or α22must be set to zero to obtain a triangular set of VARequations that can be validly estimated The choice of which of these tworestrictions to impose is, ideally, made on theoretical grounds For example,

if real estate theory suggests that the current value of y1tshould affect the

current value of y2t but not the other way around, set α12= 0, and so on.Another possibility would be to run separate estimations, first imposing

α12= 0 and then α22= 0, to determine whether the general features ofthe results are much changed It is also very common to estimate only areduced-form VAR, which is, of course, perfectly valid provided that such

a formulation is not at odds with the relationships between variables thatreal estate theory says should hold

One fundamental weakness of the VAR approach to modelling is that itsatheoretical nature and the large number of parameters involved make theestimated models difficult to interpret In particular, some lagged variablesmay have coefficients that change sign across the lags, and this, togetherwith the interconnectivity of the equations, could render it difficult tosee what effect a given change in a variable would have upon the futurevalues of the variables in the system In order to alleviate this problem par-tially, three sets of statistics are usually constructed for an estimated VARmodel: block significance tests, impulse responses and variance decompo-sitions How important an intuitively interpretable model is will of coursedepend on the purpose of constructing the model Interpretability maynot be an issue at all if the purpose of producing the VAR is to makeforecasts

11.6 A VAR model for real estate investment trusts

The VAR application we examine draws upon the body of literature ing the factors that determine the predictability of securitised real estatereturns Nominal and real interest rates, the term structure of interest rates,expected and unexpected inflation, industrial production, unemploymentand consumption are among the variables that have received empirical sup-port Brooks and Tsolacos (2003) and Ling and Naranjo (1997), among otherauthors, provide a review of the studies in this subject area A common char-acteristic in the findings of extant work, as Brooks and Tsolacos (2003) note,

regard-is that there regard-is no universal agreement as to the variables that best predictreal estate investment trust returns In addition, diverse results arise fromthe different methodologies that are used to study securitised real estate

returns VAR models constitute one such estimation methodology Clearly,this subject area will attract further research, which will be reinforced bythe introduction of REIT legislation in more and more countries.

In this example, our reference series is the index of REIT returns inthe United States These trusts were established there in the 1960s, andresearchers have long historical time series to carry out research on the pre-dictability of REIT prices In this study, we focus on the impact of dividendyields, long-term interest rates and the corporate bond yield on US REITreturns These three variables have been found to have predictive powerfor securitised real estate The predictive power of the dividend yield isemphasised in several studies (see Keim and Stambaugh, 1986, and Famaand French, 1988) Indeed, as the study by Kothari and Shanken (1997)reminds us, anything that increases or decreases the rate at which futurecash flows are discounted has an impact on value Changes in the dividendyield transmit the influence of the discount rate The long-term interest rate

is sometimes viewed as a proxy for the risk-free rate of return Movements

in the risk-free rate are expected to influence required returns and yieldsacross asset classes The corporate bond yield guides investors about thereturns that can be achieved in other asset classes This in turn affects theirrequired return from investing in securitised real estate, and hence pricing

We therefore examine the contention that movements across a spectrum ofyields are relevant for predicting REIT returns

The data we use in this example are as follows

● All REIT price returns (ARPRET): the return series is defined as the

differ-ence in the logs of the monthly price return index in successive months.The source is the National Association of Real Estate Investment Trusts(NAREIT)

● Changes in the S&P500 dividend yield (SPY): this is the monthly

abso-lute change in the Standard and Poor’s dividend yield series Source:S&P

● Long-term interest rate (10Y ): the annual change in the ten-year

Trea-sury bond yield Source: Federal Reserve

● Corporate bond yield change (CBY): the annual change in the AAA

cor-porate bond yield Source: Federal Reserve

We begin our analysis by testing these variables for unit roots in order

to ensure that we are working with stationary data The details are notpresented here as the tests are not described until chapter 12, but suffice

to say that we are able to conclude that the variables are indeed stationaryand we can now proceed to construct the VAR model Determining that the

Table 11.1 VAR lag length selection

AIC value: AIC value:

Lag ARPRET equation system

Note: Bold entries denote optimal lag lengths.

variables are stationary (or dealing appropriately with the non-stationarity

if that is the case) is an essential first step in building any model with timeseries data

We select the VAR lag length on the basis of Akaike’s information

cri-terion Since our interest is the ARPRET equation in the system, we could

minimise the AIC value of this equation on its own and assume that thislag length is also relevant for other equations in the system An alternativeapproach, however, would be to choose the lag length that minimises theAIC for the system as a whole (see equation (11.9) above) The latter approach

is more in the spirit of VAR modelling, and the AIC values for both the whole

system and for the ARPRET equation alone are given in table 11.1.

The AIC value for the whole system is minimised at lag 2 whereas the AIC

value for the ARPRET equation alone is minimised with a single lag If we

select one lag, this may be insufficient to capture the effects of the variables

on each other Therefore we run the VAR with two lags, as suggested by theAIC value for the system The results for the VAR estimation with two lagsare given in table 11.2

As we noted earlier, on account of the possible existence of ity and other factors, some of the coefficients in the VAR equations may not

multicollinear-be statistically significant and take the expected signs We observe these

in the results reported in table 11.2, but this is not necessarily a problem

if the model as a whole has the correct ‘shape’ To determine this wouldrequire the use of joint tests on a number of coefficients together, or anexamination of the impulse responses or variance decompositions Thesewill be considered in subsequent sections, but, for now, let us focus on the

ARPRET equation, which is our reference equation We expect a negative

impact from all yield terms on ARPRET This negative sign is taken only

by the second lag of the Treasury bond yield and the corporate bond yieldterms Of the two corporate bond yields – i.e lag 1 and lag 2 – it is the second